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Student difficulties with volumetric analysis Anamuah-Mensah, Jophus 1981

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STUDENT DIFFICULTIES WITH VOLUMETRIC ANALYSIS by JOPHUS ANAMUAH-MENSAH B.Sc.(Ed.)/ U n i v e r s i t y of Cape Coast, Ghana, 1971 B.Sc.(Chem.), U n i v e r s i t y of Cape Coast, Ghana, 1972 M.Sc.(Chem.), U n i v e r s i t y of Cape Coast, Ghana, 1974 M.A., The U n i v e r s i t y of B r i t i s h Columbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS AND SCIENCE EDUCATION We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1981 ( c ^ Jophus Anamuah-Mensah, 1981 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t 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 , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f M/V-rKE MAT\C<L A^ £> J O S U C E L v3 C f y r \ ^ 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 A M G r M J T 1 \ A b s t r a c t The study was designed to i n v e s t i g a t e the d i f f i c u l t i e s t h a t grade 12 chemistry students have wi t h v o l u m e t r i c a n a l y s i s . In the f i r s t p a r t of the study, a model of performance on vo 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 developed from the i n -t e g r a t i o n of two t h e o r e t i c a l p e r s p e c t i v e s of i n t e l l e c t u a l performance, namely, the p r o p o r t i o n a l r e a s o n i n g schema i n P i a g e t ' s theory and t h e cummulative l e a r n i n g theory o f Gagne. T h i s i n t e g r a t e d model hypothesized some r e l a t i o n s h i p s among the v a r i a b l e s : d i r e c t p r o p o r t i o n a l r e a s o ning, i n v e r s e p r o p o r t i o n a l r e a s o ning, 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 . The second p a r t examined s p e c i f i c c o n c e p t u a l and manual d i f f i c u l t i e s t h a t the students have with v o l u m e t r i c a n a l y s i s . T e s t s developed i n a p i l o t study were used to measure d i r e c t p r o p o r t i o n a l r e a s o ning, i n v e r s e p r o p o r t i o n a l reasoning, knowledge of subsumed 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 . The f i r s t t h ree measures were admi n i s t e r e d to s u b j e c t s e n r o l l e d i n the grade 12 chemistry course p r i o r to the te a c h i n g of the v o l u m e t r i c a n a l y s i s u n i t . A f t e r the teachers had taught the v o l u m e t r i c a n a l y s i s u n i t , a t e s t measuring 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 was admi n i s t e r e d to the students. The f i n a l sample s i z e was 328. A f t e r admin-i s t r a t i o n of the t e s t s , a subsample of 47 s u b j e c t s was i n t e r v i e w e d on a t i t r a t i o n t a s k . The t e s t data were analyzed u s i n g path a n a l y s i s techniques. The psychometric p r o p e r t i e s of the t e s t s were assessed. The i n t e r n a l c o n s i s t e n c y estimate of r e l i a b i l i t y f o r each of the t e s t s was above 0.75. The i n t e r v i e w data and the w r i t t e n work of the s u b j e c t s on 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 were analyzed u s i n g q u a l i t a t i v e procedures (e.g. c a t e g o r i z i n g the frequency of response p a t t e r n s or conceptual e r r o r s ) . The 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 performed and performance or 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 analyzed u s i n g c o r r e l a t i o n a l a n a l y s i s . The e v a l u a t i o n of the proposed i n t e g r a t e d model r e v e a l e d t h a t the performance of s u b j e c t s i d e n t i f i e d as u s i n g a l g o r i t h m s without understanding c o u l d be adequately e x p l a i n e d by a trimmed i n t e g r a t e d model i n which d i r e c t p r o p o r t i o n a l reasoning and 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 were assumed to have n e g l i g i b l e d i r e c t i n f l u e n c e on p r e r e q u i s i t e concepts and 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 , r e s p e c t i v e l y . The e v a l u a t i o n of the proposed 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 as u s i n g a l g o r i t h m s w i t h understanding r e v e a l e d t h a t a trimmed i n t e g r a t e d model i n which 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 was assumed to have n e g l i g i b l e d i r e c t e f f e c t on p r e r e q u i s i t e concepts and 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 p r o v i d e d a reasonable e x p l a n a t i o n of t h e i r performance. The a n a l y s i s of the s u b j e c t s ' c a l c u l a t i o n s on the items i n the v o l u m e t r i c a n a l y s i s instrument r e v e a l e d a number of conceptual e r r o r s (e.g. i n d i s c r i m i n a t e assumption of 1:1 mole r a t i o s ) made by the s u b j e c t s . i v The a n a l y s i s of the manipulatory s k i l l s of the s u b j e c t s d u r i n g the i n t e r v i e w r e v e a l e d t h a t while some s k i l l s seemed to be adequately developed, other important s k i l l s seemed to be l a c k i n g . The a n a l y s i s of the s t u d e n t s 1 understanding of concepts i n v o l v e d i n an acid-base t i t r a t i o n r e v e a l e d t h a t the concepts of pH and i n d i c a t o r behaviour as w e l l as the use of c e r t a i n s c i e n t i f i c t e r m i n o l o g i e s such as endpoint were not w e l l understood. The a n a l y s i s of the approaches used by the s u b j e c t s to c a l c u l a t e the c o n c e n t r a t i o n of the a c i d s o l u t i o n from t h e i r own data, d u r i n g the i n t e r v i e w , r e v e a l e d t h a t two main approaches -the P r o p o r t i o n a l Approach and the Formula Approach - were used by the s u b j e c t s i n t h e i r s o l u t i o n . However, the Formula Approach was used by a g r e a t e r p r o p o r t i o n o f the s u b j e c t s . S e v e r a l i m p l i c a t i o n s f o r chemical education i n the secondary school were i n f e r r e d from the i n t e r p r e t a t i o n of the r e s u l t s . Such i m p l i c a t i o n s i n c l u d e the development of d i f f e r e n t i n s t r u c -t i o n a l s t r a t e g i e s r e f l e c t i n g the d i f f e r e n t path models f o r students who use al g o r i t h m s w i t h and without understanding. V TABLE OF CONTENTS Page CHAPTER I INTRODUCTION 1 Background of the Problem 1 Analytic Orientations 4 De f i n i t i o n of Terms 7 Statement of the Problem 9 Research Questions 10 Overview of the Study 14 II REVIEW OF LITERATURE 16 Introduction 16 The Proportionality Schema i n Piaget's Model of Adolescent Reasoning 16 Gagne's Theory of Learning 3 3 Comparison of Gagne's and Piaget's Theories 38 Related Studies: Proportional Reasoning and Content 41 Proportional Reasoning Studies 41 Proportional Reasoning as an Explanatory Construct 42 Content as an Explanatory Construct 48 Influence of Content on Logical Reasoning 51 Path-Analytic Model of Performance 54 Studies Related to the Secondary Questions i n t h i s Study 60 III PILOT TESTING OF INSTRUMENTS AND PROCEDURES 6 6 Introduction 66 Instruments 66 Classroom Proportionality Test 67 Subconcepts Test , 69 vi CHAPTER Page III ..Volumetrie Analysis Test 73 Laboratory S k i l l s i n T i t r a t i o n ...... 74 The Interview Task . . . . . .•„•-.•. .... 76 P i l o t Study ..... ................. ..... 78 Pu.3rpo S S • • •>••«.- •>••••« ,• *•>••••-••«>-*'•-• * • • • • • • -"78 The Sample 7 8 Data C o l l e c t i o n Procedure 79 Scoring 83 Data Analysis 83 Results and Discussion 85 Revisions of the Instrument 97 Volumetric Analysis Test 97 Classroom Proportionality Test 98 Subconcepts Test 98 Laboratory S k i l l s i n T i t r a t i o n 99 The Interview Task 99 IV METHODS OF THE STUDY 101 Introduction 101 Instruments 101 Volumetric Analysis Test 10 2 Classroom Proportionality Test 102 Subconcepts Test 102 Population 102 Sample 103 Data C o l l e c t i o n Procedures 104 Test Administration 104 Interview Sample 105 Interview Data C o l l e c t i o n Procedure ... 106 Data Analysis 108 Data Preparation 108 Scoring 109 Preliminary Analyses of Data for Model Testing 109 R e l i a b i l i t y of Instruments 109 Differences of Classes I l l Path Analysis of Data 114 Analysis of Data for the Sp e c i f i c Student D i f f i c u l t i e s 117 Analysis of Conceptual Errors 117 Analysis of Interview Data 120 Manual S k i l l s i n T i t r a t i o n 120 Ideas About the Concepts Involved i n T i t r a t i o n 120 Problem Solving Approach and Predictive Behaviour 122 Performance on VAT and Number of T i t r a -tion Experiments 124 v i i CHAPTER Page V RESULTS A.ND DISCUSSION OF MODEL TESTING ..... 126 Introduction 126 Evaluation of the Integrated Model 127 S t a t i s t i c a l Hypotheses 127 Model Evaluation for Subjects Using Algorithms without Understanding 128 Model Evaluation for Subjects Using Algorithms with Understanding 141 V I R E S U L T S A N D D I S C U S S I O N O F D A T A O N S P E C I F I C S T U D E N T D I F F I C U L T I E S 152 Introduction 152 Conceptual Errors on Volumetric Analysis Test 153 Manual S k i l l s i n T i t r a t i o n 162 Concepts i n T i t r a t i o n 17 2 Student D i f f i c u l t i e s with pH 172 Student D i f f i c u l t i e s with Concentration and i t s Relationship to Moles and pH.. 180 Student D i f f i c u l t i e s with Indicator Behaviour 18 6 Student D i f f i c u l t i e s with S c i e n t i f i c Terminology 200 Approaches used i n the Computation of Solution Concentration 204 Relationship Between Prior Number of T i t r a t i o n s and Performance on VAT 231 Relation of Student D i f f i c u l t y Data to the Validated Model 233 VII SUMMARY AND IMPLICATIONS 237 Summary 237 Purpose 237 Procedure 238 Analysis 239 Results 241 Delimitations of the Study 248 Implications 249 Future Research 253 REFERENCES 255 APPENDIX A VOLUMETRIC ANALYSIS TEST 266 B CLASSROOM PROPORTIONALITY TEST 276 B.l Answer Booklet 277 B.2 Testing Procedures for each Item 285 v i i i APPENDIX Page C SUBCONCEPTS TEST ...... . , . . . ...... . . .... 290 D LABORATORY S K I L L S I N TITRATION CHECK-L I S T 300 E INTERVIEW PROCEDURE 303 F SUMMARY ITEM S T A T I S T I C S FOR DIRECT PRO-PORTIONALITY SUBTEST 306 G SUMMARY ITEM S T A T I S T I C S FOR INVERSE PRO-PORTIONALITY SUBTEST 308 H SUMMARY ITEM S T A T I S T I C S FOR THE TOTAL TEST AND SUBTESTS OF THE SUBCONCEPTS TEST 310 I SUMMARY ITEM S T A T I S T I C S FOR THE VOLUMETRIC ANALYSIS TEST 315 J MEAN AND STANDARD DEVIATION FOR THE DIRECT PROPORTIONALITY, INVERSE PRO-PORTIONALITY, SUBCONCEPTS AND VOLUMETRIC ANALYSIS TEST SCORES BY CLASS, SEPARATELY AND COMBINED 317 K DATA FOR THE LABORATORY S K I L L S I N TITRATION INSTRUMENT 320 L INTERVIEW DATA FOR EACH INTERVIEW SUBJECT ON CONCEPTS INVOLVED I N A TITRATION 322 M INTERVIEW DATA FOR EACH SUBJECT ON THE COMPUTATION PROBLEM 408 ix LIST OF TABLES TABLE Page 1 The Combinatorial of the Propositional Elements of Piaget's Model of Formal Thought , 23 2 Relationship between Propositions : The I,N,R, and C Operations 25 3 Test S t a t i s t i c s of the Classroom Pro-p o r t i o n a l i t y Test 86 4 Test S t a t i s t i c s of the Subtests i n the Subconcepts Test 8 8 5 Test S t a t i s t i c s of the Volumetric Analysis Test 91 6 Test Administration and Inter-test Breaks 108 7 Summary Test S t a t i s t i c s for a l l 4 Tests I l l 8 Test of Differences i n Means and Homo-geneity of Dispersion among Direct Proportionality, Inverse Proportion-a l i t y , Subconcepts and Volumetric Analysis Test Scores of Classes 112 9 Test of Differences i n Means and Homo-geneity of Dispersion among Direct Proportionality, Inverse Proportion-a l i t y , Subconcepts and Volumetric Analysis Test Scores of Classes 113 10 Covariance Matrix for Subjects Using Algorithms Without Under-standing 128 11 Parameter Estimates, Standard Errors and C r i t i c a l Ratios for Subjects Using Algorithms Without Understanding 129 12 Residual Matrix Resulting from the Difference Between the Sample Covariance Matrix and the Reproduced Covariance Matrix for Subjects Using Algorithms Without Understanding 132 X TABLE. Page 13 Parameter Estimates, Standard Errors and C r i t i c a l Ratios of Trimmed Inte-grated Model for Subjects Using Algorithms Without Understanding 133 14 E f f e c t s Analysis i n the Integrated Model for Subjects Using Algorithms Without Understanding 135 15 Parameter Estimates, Standard Errors and C r i t i c a l Ratios for Subjects Using Algorithms With Understanding 14 2 16 Covariance Matrix for Subjects Using Algorithms With Understanding 14 3 17 Residual Matrix Resulting from the Difference Between the Sample Covariance Matrix and the Reproduced Covariance Matrix for Subjects Using Algorithms With Understanding 144 18 Parameter Estimates, Standard Errors and C r i t i c a l Ratios of Trimmed Inte-grated Model for Subjects Using Algorithms With Understanding 147 19 E f f e c t s Analysis i n the Integrated Model for Subjects Using Algorithms With Understanding 147 20 Parameter Estimates, Standard Errors and C r i t i c a l Ratios for Subjects Using Algorithms with Understanding 148 21 Residual Matrix for the Next Trimmed Model 149 22 Conceptual Errors on the Volumetric Analysis Test 154 23 Manual S k i l l s i n T i t r a t i o n 163 24 Manual S k i l l s Displayed 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 16 6 xi TABLE Page 25 Questions and Response Patterns Derived from Interview Data i n the area of pH 173 26 Questions and Response Patterns Derived from Interview Data i n the Area of Concentration 181 27 Questions and Response Patterns Derived from Interview Data i n the Area of Indicator Behaviour 187 28 Questions and Response Patterns 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 Solving Em-ployed by Subjects 206 3 0 Approaches to Problem Solving Em-ployed by Subjects 2 07 31 D i s t r i b u t i o n of Students According to Approach 214 32 D i s t r i b u t i o n of Interview Sample by Strategy and Achievement 215 33 D i s t r i b u t i o n of Subjects According to Approach and Number of Correct Responses on the 3 Prediction Questions 219 34 Student Predictions of the Concen-t r a t i o n of Acid from T i t r a t i o n Results 223 x i i LIST OF FIGURES FIGURE Page 1 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 C a l c u l a t i o n s 11 2 H i e r a r c h i c a l Arrangement o f L e a r n i n g Types 35 3 H i e r a r c h i c a l A n a l y s i s o f T i t r a t i o n C a l c u l a t i o n s 50 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 A n a l y s i s o f T i t r a t i o n C a l c u l a t i o n s (Modi f ied) 70 6 O v e r a l l Des ign o f E x p e r i m e n t a l P r o -cedure 107 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 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 130 8 Trimmed 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 Wi thout U n d e r s t a n d i n g . . . 133 9 Proposed 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 U n d e r s t a n d i n g 143 10 Trimmed 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 U n d e r s t a n d i n g 146 11 The F u r t h e r Trimmed 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 U n d e r s t a n d i n g 14 9 12 The Formula Approach to Problem S o l u t i o n - the b a s i c 208 13 The Formula Approach t o Problem S o l u t i o n - V a r i a n t 209 14 The P r o p o r t i o n a l Approach to Problem S o l u t i o n - B a s i c 211 15 The p r o p o r t i o n a l Approach to Problem S o l u t i o n - V a r i a n t 212 16 D i a g r a m a t i c R e p r e s e n t a t i o n o f the P r e -d i c t i o n s o f S u b j e c t s U s i n g the D i f f e r e n t Approaches 220 x i i i ACKNOWLEDGEMENTS The author would l i k e to express his sincere gratitude to Dr. Gaalen Erickson and Dr. P. James Gaskell, his thesis advisors, for t h e i r invaluable guidance, encouragement and generous contribution of t h e i r time and expertise throughout the period of research and i n the preparation of thi s manuscript. The author would also l i k e to thank Dr. Todd Rogers for the generous contribution of his expertise throughout the period of research and his comments to the i n i t i a l d r a f t of this manuscript. The other members of my Committee, Drs. Robert Thompson, Pat A r l i n and Reginald Wild are g r a t e f u l l y acknowledged for t h e i r valuable comments and suggestions since the inception of t h i s research. The author would also l i k e to thank Dr. Stanley J. Taylor for his invaluable help throughout the period of the author's study at U.B.C. F i n a l l y , I would l i k e to express my sincere gratitude to my wife, Beatrice Anamuah-Mensah and my children, Ebow, Ekua, Maame and Staylor for t h e i r patience and emotional support throughout my four year sojourn at U.B.C. 1 CHAPTER I INTRODUCTION Background to the Problem In recent years science education researchers have been c a l l e d upon to increase the amount of research on the d i f f i c u l t i e s students have with s p e c i f i c science concepts (Shulman and Tamir, 1973; Driver and Easely, 1978; Levine and Linn, 1977). Kuhn (1979) has suggested that some of the problems science educators encounter i n applying psychological theories to t h e i r s p e c i f i c areas r e f l e c t the ambiguities inherent i n the theories themselves. I t has been noted that the simultaneous application of more than one psychological theory may remove some of the problems encountered when they are applied separately (Sticht, 19.71; G r i f f i t h s , 1979) i n investigating children's d i f f i c u l t i e s with school concepts. As such, there exists a need for research that i s science concept s p e c i f i c and that attempts to integrate more than one psychological theory. The present study, i n keeping with the positions stated above, examined a persistent problem i n high school chemistry, namely, the d i f f i c u l t i e s that students experience 2 w i t h v o l u m e t r i c a n a l y s i s . The c h o i c e o f v o l u m e t r i c a n a l y s i s grew from the w r i t e r ' s e x p e r i e n c e i n t e a c h i n g t h i s a spec t o f the i n t r o d u c t o r y c h e m i s t r y cour se i n h i g h s c h o o l and by the c o n c e r n expres sed by o t h e r c h e m i s t r y educa tor s (Herron , 1975; Wheeler and Kas s , 1977) . Both c h e m i s t r y t e a c h e r s and r e s e a r c h e r s have been aware o f the apprehens ions s tudent s have and the d i f f i c u l t i e s they encounter when d o i n g 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 (Duncan and Johns tone , 1973; J ohns tone , M o r r i s o n and Sharp , 1976) . A l t h o u g h no r i g o r o u s s tudy has been done t o i d e n t i f y f a c t o r s c o n t r i b u t i n g t o s t u d e n t s 1 d i f f i c u l t i e s i n p e r f o r m i n g computat ion problems i n v o l u m e t r i c a n a l y s i s , these d i f f i c u l t i e s have been a t t r i b u t e d t o the ta sk components ( i . e . content ) or s t r u c t u r e i n h e r e n t i n the ta sk (Duncan and Johns tone , 1973; Her ron 1975; Wheeler and Kas s , 1977). I f those t a sk components o r s t r u c t u r e which c o n t r i b u t e t o the d i f f i c u l t i e s t h a t s tudent s encounter i n s o l v i n g such t a s k s c o u l d be i d e n t i f i e d , then i t may be p o s s i b l e to d e s i g n t e a c h -i n g s t r a t e g i e s which would h e l p t o a l l e v i a t e these d i f f i c u l -t i e s . A l s o , i s o l a t i o n o f the sources o f d i f f i c u l t y may h e l p i n u n d e r s t a n d i n g how these sources i n d e p e n d e n t l y and '. '. c o l l e c t i v e l y a f f e c t p r o c e s s i n g a b i l i t i e s o f s tudent s and how s tudent s come t o s o l v e problems i n v o l v i n g v o l u m e t r i c a n a l y s i s . C o n s i d e r the f o l l o w i n g t y p i c a l v o l u m e t r i c a n a l y s i s p rob lem: 25 mL o f 1 M s o l u t i o n o f h y d r o c h l o r i c a c i d n e u t r a l i z e s 20 mL o f sodium h y d r o x i d e s o l u t i o n . What i s the c o n c e n t r a t i o n o f the sodium h y d r o x i d e s o l u t i o n ? 3 In s o l v i n g such a problem a s tudent may r e a s o n : " 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 , the number o f moles o f HC1 i n the r e a c t i o n would be e q u a l t o the number o f moles o f NaOH i n the r e a c t i o n . S i n c e I can c a l c u l a t e the number o f moles o f HC1 from i t s c o n c e n t r a t i o n and volume, I can determine the c o n c e n t r a t i o n o f the NaOH s o l u t i o n from the e q u a l i t y . " Such r e a s o n i n g i n v o l v e s : a) the r e c o g n i t i o n o f d i r e c t p r o p o r t i o n a l i t y 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 number o f moles o f the r e a c t a n t s ; b) the r e c o g n i t i o n o f i n v e r s e p r o p o r t i o n a l i t y between c o n c e n t r a t i o n and volume of the r e a c t a n t s ; and c) the u n d e r s t a n d i n g o f s u b o r d i n a t e c o n t e n t ( i . e . concept s and r u l e s ) such as the mole , b a l a n c i n g o f c h e m i c a l e q u a t i o n s , and c a l c u l a t i o n o f amount o f r e a c t a n t s from s t o i c h i o m e t r i c e q u a t i o n s . The 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 c o n s t i t u t e the s t r u c -t u r a l components o f the problem w h i l e the s u b o r d i n a t e concept s and r u l e s c o n s t i t u t e the c o n t e n t o f the p rob lem. A l t h o u g h a l l t h r e e r e q u i r e m e n t s , t aken t o g e t h e r , may be nece s s a ry f o r a s u b j e c t ' s u n d e r s t a n d i n g o f the problem they may not a l l be r e q u i r e d f o r a s u b j e c t t o s u c c e s s f u l l y s o l v e the p rob lem. T h i s i s because s u b j e c t s may per form the ta sk w i t h o u t showing the u n d e r s t a n d i n g l a i d out above by r e s o r t i n g t o the use o f formulas such as CHC1 VHC1 _ CNaOH V NaOH, a b 4 where C and V r e p r e s e n t 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 , and a and b are 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 (or mole r a t i o s ) f o r the a c i d and the base , r e s p e c t i v e l y . T h i s a l t e r n a t e s o l u t i o n o f u s i n g a memorized a l g o r i t h m has been noted i n s t u d i e s o f o t h e r areas i n c h e m i s t r y which have r e p o r t e d the predominant use o f formulas w i t h o u t adequate u n d e r s t a n d i n g (Chappet ta , 1974; S m i t h , 1978) . Such l a c k o f u n d e r s t a n d i n g may be seen i n the c h o i c 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 s . To f u r t h e r c l a r i f y t h i s i s s u e , s u b j e c t s i n the p r e s e n t s tudy were i d e n t i f i e d who performed 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 the c o r r e c t c o e f f i c i e n t s and those who assumed unit v a l u e s f o r the c o e f f i c i e n t s r e g a r d l e s s o f the a c t u a l r e a c t i n g c o e f f i c i e n t s . A n a l y t i c O r i e n t a t i o n s : Chemica l problems o f the type o u t l i n e d above have g e n e r a l l y been a n a l y z e d o n l y i n terms o f the o p e r a t i o n a l s t r u c t u r e s demanded by the problem (Herron , 1975; I n g l e and Shayer , 1971; M e a l i n g s , 1969) u s i n g P i a g e t ' s t h e o r y , w i t h o u t g i v i n g adequate c o n s i d e r a t i o n t o c o n t e x t u a l v a r i a b l e s such as the c o n t e n t o f the p rob lem. C u r r e n t l y , t h e r e seems t o be a growing c o n c e r n over the s o l e r e l i a n c e on o p e r a t i o n a l s t r u c t u r e s i n a n a l y z i n g s t u d e n t ' s performance i n problem s o l v i n g s i t u a t i o n s . Some o f these concerns were expres sed as a r e a c t i o n t o the d i f f e r e n t i a l performance o f e x p e r i m e n t a l s u b j e c t s on t a s k s r e q u i r i n g e q u i v a l e n t l o g i c a l s t r u c t u r e s (Brown and D e f o r g e s , 1977; Smedslund, 1977) . As a r e s u l t 5 many r e s e a r c h e r s have c a l l e d f o r an a n a l y s i s o f the c o n t e n t o f the ta sk i n a d d i t i o n to the o p e r a t i o n a l s t r u c t u r e s i n v o l v e d (Abramowitz, 1975; D r i v e r and E a s e l y , 1978; L u n z e r , 1965; 1975) . Other s have suggested t h a t more a t t e n t i o n be p a i d t o the s u b j e c t ' s f a m i l i a r i t y w i t h the ta sk ( L o v e l l , 1971a; 1973; Berzonsky , 1971). But as J o h n s o n - L a i r d , L e g r e n z i and L e g r e n z i ' s (1972) study, i n d i c a t e d , f a m i l i a r i t y may be n o t h i n g more than knowledge o f the c o n t e n t . To t h i s e n d , s t u d i e s have been done which attempted to examine the e f f e c t o f the c o n t e n t on performance ( e . g . G r i f f i t h s , 1979) u s i n g Gagne ' s cummulative. t h e o r y . In the p r e s e n t s t u d y , two t h e o r e t i c a l frameworks — P i a g e t ' s t h e o r y o f i n t e l l e c t u a l development ( Inhe lder and P i a g e t , 1958) and Gagne ' s (1977) cummulative l e a r n i n g t h e o r y — were taken as the b a s i s f o r d e v e l o p i n g an i n t e g r a t e d model i n an at tempt t o i d e n t i f y the i n f l u e n c e s o f s t r u c t u r e and c o n t e n t upon performance on v o l u m e t r i c a n a l y s i s prob lems . The s t r u c t u r e — 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 — and the c o n t e n t — the subsumed p r e r e q u i s i t e concept s — c o n s t i t u t e d the p r e d i c t o r v a r i a b l e s i n the s t u d y . The dependent v a r i a b l e i n the s tudy was the s t u d e n t ' s performance on a s e t o f v o l u m e t r i c a n a l y s i s p rob lems . A s u b j e c t who posses sed a l l t h r e e r e q u i r e m e n t s , namely 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 , and subsumed c o n c e p t s , was expected to s o l v e the v o l u m e t r i c a n a l y s i s problems s u c c e s s f u l l y . In c o n t r a s t , s u b j e c t s m i s s i n g any o f these s k i l l s were expected t o have some d i f f i c u l t i e s w i t h the prob lems . In examining the i n t e r r e l a t i o n s h i p s among these v a r i a b l e s , pa th a n a l y s i s was employed. 6 A p a r t from the development and s t a t i s t i c a l v a l i d a -t i o n o f the above r e l a t i o n s h i p s , the s tudy a l s o examined t h r e e o t h e r areas r e l e v a n t t o g a i n i n g f u r t h e r u n d e r s t a n d i n g o f s t u d e n t s ' d i f f i c u l t i e s w i t h v o l u m e t r i c a n a l y s i s . One o f these i n v o l v e d the 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 o f the concep-t u a l e r r o r s made by s u b j e c t s when 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 c a l c u l a t i o n prob lems . The second area was an examina t ion o f the l a b o r a t o r y s k i l l s and c o n c e p t u a l u n d e r s t a n d i n g which s tudent s d i s p l a y e d when d o i n g a p r a c t i c a l t i t r a t i o n t a s k . F u r t h e r m o r e , an a n a l y s i s was made o f the types o f s t r a t e g i e s used by these s tudent s when they a p p l i e d the r e s u l t s o f t h e i r t i t r a t i o n t o a s e t o f v o l u m e t r i c a n a l y s i s p rob lems . The t h i r d a rea i n v o l v e d an i n v e s t i g a t i o n o f the r e l a t i o n s h i p between 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 and the number o f l a b o r a t o r y t i t r a t i o n s per formed . Three d i f f e r e n t a n a l y t i c t e c h n i q u e s were used i n examining these c o n c e r n s . These were r e s p e c t i v e l y , ( 1 ) the examina t ion o f s t u d e n t s ' w r i t t e n work f o r the type and f requency o f e r r o r s , made, ( 2 ) the use o f c l i n i c a l i n t e r v i e w s i n a t i t r a t i o n exper iment and ( 3 ) the use o f c o r r e l a t i o n a l a n a l y s i s . These i n v e s t i g a t i o n s p r o v i d e d : ( 1 ) a d d i t i o n a l i n s i g h t i n t o the d i f f i c u l t i e s s tudent s have i n v o l u m e t r i c a n a l y s i s , and ( 2 ) da ta t o e n r i c h 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 from the pa th a n a l y s i s . 7 D e f i n i t i o n of Terms Volumetric Analysis: the process whereby the amount of a chemical substance i n a given sample involved i n a quantitative chemical reaction i s determined with the aid of a chemical indicator i n a process c a l l e d t i t r a t i o n . In volumetric analy-.' s i s , the amount of one of the reactants i s unknown. The objective therefore i s to determine the mass or concentration of the desired substance i n a sample whose quantitative composition i s unknown (Toon and E l l i s , 1973). Volumetric Analysis Calculation or Problem: the determination of the concentration or mass of one chemical substance from the known concentration of the other substance i n an acid-base reaction. In the present study, performance on t h i s variable was measured by the 'Volumetric Analysis Test' (see Appendix A). Direct Proportionality Reasoning: two variables so related that t h e i r r a t i o i s constant, are d i r e c t l y proportional to each other (Vance, 1962). In t h i s study, d i r e c t proportionality was measured with the f i r s t subtest of the 'Classroom Propor-t i o n a l i t y Test' (see Appendix B). Inverse Proportionality Reasoning: two variables so related that t h e i r product i s a constant, are inversely proportional to each other. In the present study, inverse proportionality was defined i n terms of scores on the second subtest of the 'Classroom Proportionality Test' (see Appendix B). Subsumed or Prerequisite Concepts: concepts hypothesized to 8 be neces sa ry f o r u n d e r s t a n d i n g v o l u m e t r i c a n a l y s i s p rob lems . These subsumed concept s a r e , t h e r e f o r e , i n t e g r a l l y r e l a t e d t o v o l u m e t r i c a n a l y s i s problems i n t h a t i t i s h y p o t h e s i z e d t h a t such problems cannot be under s tood i f the subsumed concept s are not a v a i l a b l e t o the s u b j e c t . In t h i s s t u d y , the subsumed p r e r e q u i s i t e concept s i n c l u d e d e s s e n t i a l l y the concept s i d e n t i f i e d by Gower, D a n i e l s and L l o y d (1977a) t o be nece s s a ry f o r 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 c a l c u l a t i o n s . These concept s are the mole , gram/mole c o n v e r s i o n s , b a l a n c i n g and i n t e r p r e t -i n g e q u a t i o n s , c a l c u l a t i n g r e l a t i v e mass o f r e a c t a n t s and p r o d u c t s from e q u a t i o n s , c o n v e r t i n g a s o l u t i o n from one concen-t r a t i o n t o a n o t h e r , and 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 s o l u t i o n s from the mass o f the s u b s t a n c e s . U s i n g Gagne ' s cummulative mode l , these concept s were a r ranged i n a h i e r a r c h i c a l o r d e r even though, as w i l l be shown i n Chapter I I ; the 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 i n t h i s s t u d y . F o r the purposes o f t h i s s tudy, s t u d e n t s ' knowledge about the subsumed p r e r e q u i s i t e concept s was d e r i v e d through c o n t e n t a n a l y s i s o f 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 the Sub-concept s T e s t ( c o n t a i n e d i n Appendix C) completed by the s t u d e n t s . Reasoning S t r a t e g y : a p l a n , o r s t r a t e g y f o r g e n e r a t i n g o r d e r l y r e l a t i o n s h i p s out o f e x p e r i e n c e , i n o t h e r words , f o r s o l v i n g problems and g e n e r a t i n g new knowledge. Reasoning s t r a t e g i e s opera te i n problem s o l v i n g as p a r t o f one b a s i c p r o c e s s which has , as i t s a im, the o r g a n i z a t i o n o f e x p e r i e n c e i n t o meaning-9 f u l systems o f o b j e c t s , e v e n t s , and s i t u a t i o n s . They are i n a sense e x p l i c i t gu ides t o problem s o l v i n g (Lawson, 1979) . The r e a s o n i n g s t r a t e g i e s o f i n t e r e s t i n the p r e s e n t s tudy were the procedures o r approaches used by s u b j e c t s as they v e r b a l i z e d t h e i r s o l u t i o n s t o a v o l u m e t r i c a n a l y s i s problem i n an i n t e r -view s i t u a t i o n . C o n c e p t u a l E r r o r : an e r r o r i n r e a s o n i n g o r a m i s c o n c e p t i o n d i s p l a y e d by a s u b j e c t i n h i s / h e r c a l c u l a t i o n as he or she s o l v e d a v o l u m e t r i c a n a l y s i s p rob lem. A l g o r i t h m : r u l e or formula employed by i n d i v i d u a l s as a i d s i n s o l v i n g problems or making a s e t o f da ta m e a n i n g f u l . In t h i s s t u d y , formulas and r u l e s i n v o l v e d i n s t o i c h i o m e t r i c c a l c u l a -t i o n s were c l a s s i f i e d as a l g o r i t h m s . Statement o f the Problem The p r e s e n t s tudy i n t e g r a t e d a spec t s 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 o f i n t e l l e c t u a l per formance , namely, the p r o p o r t i o n a l schema i n P i a g e t ' s t h e o r y and the cummulative l e a r n i n g t h e o r y o f G a g n e , i n t o a model t o account f o r an i n d i v i d u a l ' s performance o f 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 . As w i l l be argued i n Chapter I I , P i a g e t ' s t h e o r y suggested t h a t 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 precedes 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 and t h a t b o t h i n v e r s e and 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 i n f l u e n c e achievement on c e r t a i n p r e r e q u i s i t e c h e m i c a l c o n c e p t s . I t f u r t h e r suggested t h a t 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 i n f l u e n c e s performance on v o l u m e t r i c 1 0 a n a l y s i s c a l c u l a t i o n s . The cummulative l e a r n i n g t h e o r y o f Gagne suggested t h a t the p r e r e q u i s i t e concept s o r c o n t e n t i n f l u e n c e .' 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 . Taken t o g e t h e r , these i n t e r r e l a t i o n s h i p s suggested the i n t e g r a t e d model r e p r e s e n t e d i n F i g u r e 1 . The p r imary purpose o f the s tudy was, t h e r e f o r e , t o t e s t the v a l i d i t y o f the i n t e g r a t e d model proposed i n F i g u r e 1 , thereby g a i n i n g g r e a t e r u n d e r s t a n d i n g o f the r e l a t i v e i n f l u e n c e o f 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 , and sub-sumed concept s on v o l u m e t r i c a n a l y s i s per formance . A secondary purpose o f the s tudy was t o examine the f o l l o w i n g i s s u e s r e l a t e d t o s t u d e n t s ' d i f f i c u l t i e s w i t h v o l u m e t r i c a n a l y s i s : a) the c o n c e p t u a l e r r o r s made by s tudent s i n t h e i r c a l c u l a t i o n s , b) the s u b j e c t s ' l a b o r a t o r y t e c h n i q u e s and u n d e r s t a n d i n g o f the concept s i n v o l v e d i n a t i t r a t i o n exper iment , and c) the r e l a t i o n s h i p between the number o f p r i o r l a b o r a t o r y t i t r a t i o n performed by the s u b j e c t s and t h e i r 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 . E x p l o r a t o r y i n n a t u r e , the main aim o f these i n v e s t i g a t i o n s was to p r o v i d e a d e t a i l e d view o f the s u b j e c t s ' d i f f i c u l t i e s w i t h v o l u m e t r i c a n a l y s i s . Research Quest ions The g e n e r a l concerns o u t l i n e d above were grouped i n t o two d i s t i n c t but r e l a t e d groups o f q u e s t i o n s a c c o r d i n g 1 1 Subsumed Concepts 3 F i g u r e 1 : 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 * * 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 o f t h i s model 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 used i s g i v e n i n Chapter Two. 12 t o whether they f e l l under the pr imary purpose o f the s tudy ( i . e . model t e s t i n g ) o r the secondary and e x p l o r a t o r y s e c t i o n o f the s t u d y . The f i r s t q u e s t i o n which r e l a t e d t o the t e s t i n g o f the model was: Ques t ion 1: Is the arrangement o f v a r i a b l e s r e p r e s e n t e d i n the i n t e g r a t e d model v a l i d a c c o r d i n g t o the a p p l i c a t i o n o f pa th a n a l y s i s ? S p e c i f i c a l l y , does the i n t e g r a t e d model c l o s e l y m i r r o r the s u b j e c t s ' performance on the t e s t s ? T h i s q u e s t i o n r e l a t e d t o the d e t e r m i n a t i o n o f the v a l i d i t y o f the model as a whole t h r o u g h : ( i) the use o f model compar i son t e c h n i q u e s and ( i i ) examina t ion o f the s i g n i f i c a n c e o f each o f the paths c o n n e c t i n g p a i r s o f v a r i a b l e s ( that i s , the h y p o t h e s i z e d i n f l u e n c e o f one v a r i a b l e on a n o t h e r ) . As suggested i n the l i t e r a t u r e , s tudent s who use c e r t a i n a l g o r i t h m s i n t h e i r s o l u t i o n s to c h e m i c a l problems 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 the a l g o -r i t h m ( C h i a p p e t t a , 1974; S m i t h , 1978) . Thus , an attempt was made i n t h i s s tudy t o i d e n t i f y these two groups o f s tudent s through the form o f t h e i r responses t o the V o l u m e t r i c A n a l y s i s T e s t . S tudents who used a l g o r i t h m s i n d i s c r i m i n a t e l y , ( e . g . they used a 1:1 mole r a t i o when i t was not a p p l i c a b l e ) formed one group w h i l e those who used a l g o r i t h m s d i s c r i m i n a t e l y ( e . g . they used a 1:1 mole r a t i o w i t h an u n d e r s t a n d i n g o f when to a p p l y i t ) compr i sed the second group . I t was 13 h y p o t h e s i z e d t h a t s i n c e these groups o f s u b j e c t s might respond d i f f e r e n t l y t o the t e s t s used i n the s t u d y , the proposed model might f i t the da ta f o r one group more c l o s e l y than f o r the o t h e r . Hence, the second major q u e s t i o n addres sed was: Ques t ion 2: Does the i n t e g r a t e d model account f o r the observed da ta f o r the two groups o f s u b j e c t s t o the same ex tent ? The s t a t i s t i c a l hypotheses f o r the above r e s e a r c h q u e s t i o n s w i l l be s t a t e d i n Chapter I V . The r e s e a r c h q u e s t i o n s addressed i n the secondary p a r t o f the s tudy were: Ques t ion 3: What c o n c e p t u a l e r r o r s are made by s tudent s on the V o l u m e t r i c A n a l y s i s T e s t ? Ques t ion 4: What p r a c t i c a l l a b o r a t o r y s k i l l s do s u b j e c t s d i s p l a y i n a t i t r a t i o n exper iment? Quest ion 5: What d i f f i c u l t i e s d o ; s u b j e c t s have w i t h concept s i n v o l v e d i n an a c t u a l t i t r a t i o n exper iment? Quest ion 6: What are the d i f f e r e n t approaches used by the s u b j e c t s i n s o l v i n g a v o l u m e t r i c a n a l y s i s problem? Quest ion 7: I s the number o f l a b o r a t o r y t i t r a t i o n s per formed r e l a t e d t o s t u d e n t s 1 performance on the V o l u m e t r i c A n a l y s i s T e s t ? 1 4 The t h i r d question sought to i d e n t i f y the errors made by the subjects as they solved the problems on the Volumetric Analysis Test. The next three questions explored the type of understanding that subjects bring into actual t i t r a t i o n s i t u a t i o n s . The manipulatory behaviour ( i . e . prac-t i c a l s k i l l s ) of the subjects as well as t h e i r understanding of the concepts involved i n t i t r a t i o n (e.g. pH, indicators, terminology) were the subjects of the fourth and f i f t h questions respectively. The sixth question examined the approaches used by the subjects while doing a Volumetric Analysis problem. In the seventh question, the relationship between pr i o r involvement i n performing t i t r a t i o n s and performance on volumetric analysis calculations was examined. The degree of involvement was defined as the number of laboratory t i t r a t i o n s done since grade eleven. Overview of the Study In t h i s chapter, the background, d e f i n i t i o n of terms and the research questions posed i n the study were described. In Chapter II, relevant l i t e r a t u r e related to the present study are reviewed. In p a r t i c u l a r , l i t e r a t u r e related to the content of a task and the i n t e l l e c t u a l structure ( i . e . propor-t i o n a l reasoning) demand by a task are reviewed. Also the empirical and t h e o r e t i c a l underpinnings of the hypothesized model are discussed. In Chapter I I I , the data c o l l e c t i o n instruments and the p i l o t study are described. Following t h i s , the 1 5 procedures employed in the main study for 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 reported and discussed for the research questions related to the testing of the model. The results for the questions posed i n the secondary part of the study are then presented and discussed i n Chapter VI. F i n a l l y , i n the l a s t chapter, Chapter VII, a summary of the study together with the th e o r e t i c a l and educational implications are presented. 1 6 CHAPTER II REVIEW OF THE LITERATURE I n t r o d u c t i o n The focus of t h i s study - the e f f e c t of u n d e r l y i n g s t r u c t u r e and content on performance - suggests the review of l i t e r a t u r e r e l a t e d to these two areas. In t h i s chapter, the p r o p o r t i o n a l r e a s o n i n g schema i n P i a g e t ' s formal o p e r a t i o n a l theory are d e s c r i b e d and r e l a t e d to the problem a t hand. F o l l o w i n g t h i s , Gagne's cummulative model are summarized. St u d i e s employing the s t r u c t u r e and/or the content as the e x p l a n a t o r y c o n s t r u c t are d i s c u s s e d . The s t u d i e s r e p o r t e d i n the l i t e r a t u r e are then used to c o n s t r u c t an i n t e g r a t e d model of performance on 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 . F i n a l l y , s t u d i e s r e l a t e d to the secondary p a r t of the study are d i s c u s s e d . The P r o p o r t i o n a l i t y Schema In P i a g e t ' s Model of  Adolescent Reasoning P i a g e t p o s t u l a t e d two g e n e r a l conceptual a c q u i s i -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 t h i n k i n g (or formal o p e r a t i o n s i n P i a g e t ' s t e r m i n o l o g y ) . These are: 1 7 (a) p r o p o s i t i o n a l l o g i c , and (b) fo rmal o p e r a t i o n a l schemata. T h i s i s c l e a r l y s t a t e d by I n h e l d e r and P i a g e t ( 1 9 5 8 ) as f o l l o w s : fo rmal t h i n k i n g ( i . e . a d o l e s c e n t reasoning) makes i t s presence known not o n l y by the c o n s t a n t u t i l i z a t i o n o f 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 combina-t i o n s which d e r i v e from them but a l s o by the s p o r a d i c e l a b o r a t i o n o f some concept s or schemata which are i n a c c e s s i b l e a t the c o n c r e t e l e v e l (p. 3 0 8 ) S i n c e a comprehensive r ev i ew of the formal s tage i s not i n t e n d e d h e r e , a t t e n t i o n w i l l be focus sed on the fo rmal o p e r a t i o n a l schemata w h i l e the p r o p o s i t i o n a l l o g i c w i l l o n l y be r e f e r r e d t o where i t s e rves t o c l a r i f y P i a g e t ' s e x p l a n a -t i o n o f the former ( i . e . the schemata) . A c c o r d i n g t o I n h e l d e r and P i a g e t ( 1 9 5 8 , pp . 3 0 8 - 3 0 9 ) , the o p e r a t i o n a l schemata c o n s t i t u t e g e n e r a l concept s which m a n i f e s t themselves a t the fo rmal l e v e l when the s u b j e c t i s f a ced w i t h c e r t a i n k i n d s o f d a t a , but which do not m a n i f e s t themselves o therwi se ( p . 3 0 8 ) . They are g e n e r a l i n the sense t h a t they reappear i n many d i f f e r e n t problems o r are a p p l i c a b l e i n the most d i v e r s e s i t u a t i o n s . In the view o f I n h e l d e r and P i a g e t , these g e n e r a l concept s are not d i s c o v e r e d i n the o b j e c t s themselves but are a b s t r a c t e d o r deduced by the s u b j e c t from h i s own ' o p e r a t i o n a l s t r u c t u r e s ' , and t h a t a l l o f these concept s "show some r e l a t i o n s h i p t o the l a t t i c e o r group s t r u c t u r e s , and s e v e r a l o f them t o the INRC 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 " ' (pp. 3 0 9 - 3 1 0 ) . These schemata or g e n e r a l concept s i n c l u d e c o m b i n a t o r i a l :reasoning., . p r o p o r t i o n s , double systems o f 1 8 reference, mechanical equilibrium, p r o b a b i l i t i e s , correlations, m u l t i p l i c a t i v e compensations, and forms of conservation which go beyond d i r e c t empirical v e r i f i c a t i o n . The following section w i l l present a summary of Piaget's explanation of the proportionality schema. Proportion i s a r e l a t i o n of one variable to another; i t i s the equivalence of two rel a t i o n s or r a t i o s . Thus i n solution chemistry, the r e l a t i o n : concentration = moles/ volume, i s a proportional r e l a t i o n s h i p . Proportional reasoning therefore involves the a b i l i t y to use proportions and r a t i o s . Inhelder and Piaget ( 1 9 5 8 , p. 3 1 7 ) view the propor-t i o n a l schema as having two aspects - l o g i c a l proportions and metrical proportions. They point out that l o g i c a l proportion precedes metrical proportion. That i s , l o g i c a l proportion, which expresses the compensation between two heterogeneous factors such that an increase i n the value of one gives the same r e s u l t as an increase or decrease i n the value of the other (in q u a l i t a t i v e terms), i s achieved e a r l i e r than metrical (or numerical) proportion which expresses these same compensa-tions i n quantitative terms. They noted that whereas with l o g i c a l proportion, compensation may be additive or m u l t i p l i c a -t i v e , metric proportionality involves only m u l t i p l i c a t i v e compensations. Inhelder and Piaget point out that t h i s accounts for the " i n i t i a l tendency of the c h i l d to look for proportionality i n the equality of additive differences" ^ ( 1 9 5 8 , p. 2 2 3 ) . They also stated that t h i s compensation, which forms an important aspect i n the acq u i s i t i o n of proportion, derives 1 9 d i r e c t l y from the idea of r e c i p r o c i t y . The position that proportion i s a second-order operation i s c l e a r l y expressed by Inhelder and Piaget, "In t h i s sense proportions presuppose second degree operations and the same may be said of propositional l o g i c i t s e l f , since interpropositional operations are performed on statements whose intrapropositional content consists of class and r e l a t i o n a l operations" ( 1 9 5 8 , p . . . 2 5 4 ) . Inhelder and Piaget state further that the a c q u i s i t i o n of the operational schema of inverse proportion presupposes an understanding of both r e c i p r o c i t y (that i s , compensation by equivalence) and proportions, a feat which i s achieved, only during late adolescence. However, Piaget points out that problems of d i r e c t proportionality as i n the experiments on the notion of similar 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 vessels (Inhelder and Piaget, 1 9 5 8 , pp. 1 3 3 - 1 4 7 ) tend to be solved during early adolescence (that i s , around ages 1 1 and 1 2 ) . This means that relations of inverse proportionality come l a t e r than those of d i r e c t proportionality (Rogers, 1 9 7 7 ) . Piaget's (Inhelder and Piaget, 1 9 5 8 ) interpretation of the ess e n t i a l mechanism underlying the understanding of problems of r e c i p r o c i t y and proportions can be presented i n two stages. F i r s t , Piaget showed that wherever two physical systems of actions are r e c i p r o c a l l y related to one another, any action may be cancelled either by executing the opposite action, i . e . performing the inverse action within the same 2 0 a c t i o n system - a p roce s s c a l l e d I n v e r s i o n o r Nega t ion - o r e x e c u t i n g an a c t i o n w i t h i n the r e l a t e d a c t i o n system - a p roce s s c a l l e d R e c i p r o c i t y - t o o b t a i n s i m i l a r r e s u l t . Thus u s i n g the e q u i l i b r i u m between communicat ing v e s s e l s as an example, they showed t h a t the a c t i o n o f l o w e r -i n g the l e v e l i n one v e s s e l by r a i s i n g the v e s s e l w h i l e l e a v i n g the o t h e r v e s s e l 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 by l o w e r i n g the same v e s s e l , i . e . Nega t ion - p e r f o r m i n g the . i n v e r s e a c t i o n w i t h i n the same a c t i o n system - or by r a i s i n g the o t h e r v e s s e l , i . e . R e c i p r o c i t y - e x e c u t i n g an a c t i o n r e c i p r o c a l to t h a t o f the o r i g i n a l a c t i o n but i n r e l a t e d v e s s e l . S ince each v e s s e l can be r a i s e d and lowered t h i s produces four p o s s i b l e a c t i o n s f o r the two v e s s e l s as a whole . P i a g e t f u r t h e r p o i n t s out t h a t these two forms o f r e v e r s i b i -l i t i e s i . e . Nega t ion (or I n v e r s i o n ) and R e c i p r o c i t y , which r e p r e s e n t separa te t r a n s f o r m a t i o n s d u r i n g the p r e a d o l e s c e n t stage become i n t e g r a t e d i n t o a s i n g l e system a t the ' f o r m a l s t a g e ' . Thus , i r r e s p e c t i v e o f the a c t i o n taken as the s t a r t i n g p o i n t f o r these t r a n s f o r m a t i o n s , the in te rdependence o f i n v e r s i o n and r e c i p r o c i t y c o n t i n u e t o h o l d . Thus , i n a l o g i c a l sense , the four a c t i o n s (or t r a n s f o r m a t i o n s ) i n the system o f communicat ing v e s s e l s , c o n s t i t u t e a g roup . The group na ture o f the four a c t i o n s can be shown ( a c c o r d i n g t o L u n z e r , 1965) by assuming any o f the a c t i o n s to be a p o i n t on a v e c t o r where the v e c t o r r e p r e s e n t s the p a r t i c u l a r type o f a c t i o n ( d i s r e g a r d i n g the d i s t a n c e , i . e . the magni tude , on i t ) . I f one o f these four v e c t o r s i s chosen as 2 1 the s t a r t i n g point and c a l l e d I ( i d e n t i t y operator), then the vector representing i t s exact opposite can be c a l l e d N(negation, r a i s i n g or lowering the vessel), R(reciprocity) which has similar e f f e c t to N can stand for the t h i r d vector and the fourth vector which i s the exact opposite of R can be c a l l e d C ( c o r r e l a t i v e ) . The I,N,R, and C operations therefore constitute a group of transformations which describe the rela t i o n s between the four vectors and has the following m u l t i p l i c a t i v e table. 2 2 2 2 I = N = R = C = I IR = RI = R; IN = NI = N; IC = CI = C NR = RN = C CR = RC = N CN = NC = R IN = RC NRC = . I This group - INRC - i s isomorphic to a well-known mathematical group c a l l e d the four group. Although, the INRC group can be seen to be connected to the physical system, i n the second stage of his interpreta-t i o n , Piaget argued that t h i s occurs as a r e s u l t of the adolescent's use of propositional l o g i c . Thus, he t r i e d to show the interconnection between the INRC connected with the physical system and the isomorphic group connected with the propositional l o g i c . I t i s t h i s aspect of his theory which has raised a number of c r i t i c i s m s . 22 According to Piaget, for two elementary propositions, P and q, propositional reasoning constitutes an integrated system of the combinatorial of the two propositions - thus producing 16 binary propositions (Table 1) - and a group of four transformations which transform the propositions into one another (Table 2). Table 1 gives alternate representations of the 16 l o g i c a l propositions which can be obtained from the set of four base elements, p.q, P.q, P.q and p.q (obtained from the i n i t i a l two propositions) by considering them one-by-one, two-by-two, three-by-three, a l l four and including the n u l l set. As i n the physical system, Piaget argued that the 16 binary propositions can be related to one another by the group of inversions and r e c i p r o c i t i e s - the INRC (Table 2). Thus,if we l e t P stand for "he i s hungry" and q for "he i s t i r e d " , then t h e i r cancellations P and q represent "he i s not hungry" and "he i s not t i r e d " respectively. Now, i f we choose t h e i r conjunction, i . e . p.q as the i d e n t i c a l operation, then the following four r e l a t i o n s r e s u l t . I (P.q) =che i s hungry and t i r e d N(P.q) = P v q = he i s not hungry or he i s not t i r e d . This statement i s a strong denial of the o r i g i n a l and hence i t s inversion. R(P.q) = P.q = he i s not hungry and he i s not t i r e d . This statement i s a weak denial of the o r i g i n a l pro-position and hence i t s r e c i p r o c a l T h e C o m b i n a t o r i a l o f t h e P r o p o s i t i o n a l E l e m e n t s o f P i a g e t ' s M o d e l o f F o r m a l T h o u g h t V e r b a l L a b e l C l a s s N o t a t i o n L o g i c N o t a t i o n P i a g e t N o t a t i o n 1. C o m p l e t e A f f i r m a t i o n A 1 A 2 + A 1 A 2 + !- I t A j A j A ^ ' ( P . q ) v ( P . q - ) v ( P . q ) v ( P . q ~ ) P * q 2 . C o m p l e t e N e g a t i o n 0 0 0 3 . C o n j u n c t i o n A 1 A 2 P . q P . q k. 1ncompa t i b i 1 i t y A 1 A 2 + A i A 2 + A ! A 2 ( P . q X F . q ) v ( P \ q } P/q 5 . D i s j u n c t i o n A 1 A 2 + A 1 A 2 + A 1 A 2 ( P . q ) v ( P . q ) v ( F . q ) P v q 6. C o n j u n c t i v e N e g a t i o n A ' A ' A 1 A 2 P .q " F.q 7 . I m p l i c a t i o n A 1 A 2 + A | A 2 + A ' A ' A 1 A 2 ( P . q ) v ( F . q ) v ( P \ q ) P D q 8 . N o n - l m p l i c a t i o n A 1 A 2 P .q " P . q 9 . R e c i p r o c a l I m p l i c a t i o n A 1 A 2 + A A' + A ' A 1 A 1 A 2 (P.q)v(P.q^v(F.'ql q D P 10 . 1 1 . N e g a t i o n o f R e c i p r o c a l I m p l i c a t i o n Equ i v a 1 e n c e A ! A 2 A 1 A 2 + A ' A ' F.q ( P . q ) v(F . q ) F.q P = q ; o r P ^ . q Pv v q 12 . R e c i p r o c a l E x c l u s i o n A 1 A 2 + A 1 A 2 ( P . q l v ( F . q ) 13 . A f f i r m a t i o n o f P A 1 A 2 + A 1 A 2 ( P . q ) v ( P . " q l P [ q ] ]k. N e g a t i o n o f P A ; A 2 + A 1 A 2 (F .q)v(F .q) F [ q ] 1 5 . A f f i r m a t i o n o f q A 1 A 2 + A 1 A 2 ( P . q ) v(F . q ) q [ P ] 16 . N e g a t i o n o f q A 1 A 2 + A ' A ' 1 2 ( P.i ) v(F . q ) ? [ P ] Key on n e x t p a g e . to 00 2 4 KEY TO T A B L E ' J VERBAL L A B E L S : T h e s e a r e P i a g e t ' s l a b e l s . Some s y n o n y m o u s l a b e l s a r e : T a u t o l o g y = C o m p l e t e A f f i r m a t i o n ; C o n d i t i o n a l = I m p l i c a t i o n ; B i c o n d i t i o n a l = E q u i v a l e n c e ; " t h e i n c l u s i v e o r " = D i s j u n c t i o n ; " t h e e x c l u s i v e o r " = R e c i p r o c a l E x c l u s i o n . CLASS NOTAT ION : A j = o c c u r r e n c e o f C l a s s 1 A ^ = o c c u r r e n c e o f C l a s s 2 T h e r e a r e k p o s s i b l e p a i r s : E a c h o f t h e s e p a i r s i n t u r n r e p r e s e n t s a n e l e m e n t o f a new c a t e g o r y ; t h e s e p a i r s a r e t h e p r o d u c t o f l o g i c a l m u l t i p l i c a t i o n o f c l a s s e s . LOGIC NOTAT ION : P i s t h e g e n e r a l i z e d a b s t r a c t i o n c o r r e s p o n d i n g t o A ^ ; a s s e r t i o n P i s t r u e q i s t h e g e n e r a l i z e d a b s t r a c t i o n c o r r e s p o n d i n g t o A ^ ; a s s e r t i o n q i s t r u e P i s t h e g e n e r a l i z e d a b s t r a c t i o n c o r r e s p o n d i n g t o A J ; a s s e r t i o n P i s f a l s e q i s t h e g e n e r a l i z e d a b s t r a c t i o n c o r r e s p o n d i n g t o A ^ ; a s s e r t i o n q i s f a l s e . C o m b i n a t i o n s a r e r e p r e s e n t e d b y t h e l o g i c a l c o n j u n c t i o n • a n d v • = a n d / b o t h ( e . g . , P . q - b o t h P a n d q a r e t r u e ) ; i t r e p r e s e n t s l o g i c a l m u l t i p l i c a t i o n . 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 t o + i n C l a s s N o t a t i o n ; i t s t a n d s f o r l o g i c a l a d d i t i o n ( e . g . P v q = e i t h e r P o r q o r b o t h a r e t r u e ) . P a r e n t h e s e s ( ) , Bound o r s e t o f f c o m p o n e n t s . i A^ = n o n - o c c u r r e n c e o f C l a s s 1 A ^ = n o n - o c c u r r e n c e o f C l a s s 2 A ^ 2 > ^ ] ^ 2 ' ^ 1 ^ 2 ' ^ 1 ^ 2 ' 2 5 T a b l e 2 R e l a t i o n s h i p B e t w e e n P r o p o s i t i o n s : T h e 1, N , R, and C O p e r a t i o n s . I DENT ITY : A p p l i c a t i o n o f t h e I d e n t i t y O p e r a t o r t o a l o g i c a l r e p r e s e n t a t i o n o f a p r o p o s i t i o n y i e l d s an e q u i v a l e n t r e p r e s e n t a t i o n o f t h e same p r o p o s i t i o n . E q u i v a l e n c e i s d e t e r m i n e d b y c o n s t r u c t i o n o f t r u t h t a b l e s a c c o r d i n g t o t h e c o n v e n t i o n s o f s t a n d a r d l o g i c . E v e r y p r o p o s i t i o n h a s a u n i q u e i d e n t i f y i n g t r u t h t a b l e . E x a m p l e : l { ( P . q ) v ( P . q ) v (F.q)'} = P v q I ( D i s j u n c t i o n ) = D i s j u n c t i o n C O R R E L A T I O N : T h e 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 c o n j u n c t i o n s i n t h e l o g i c a l r e p r e s e n t a t i o n o f a p r o p o s i t i o n b y s u b s t i t u t i n g " v " f o r " . " a nd v i c e v e r s a . E x a m p l e : C ( P . q ) = P v q C ( C o n j u n c t i o n ) = D i s j u n c t i o n . R E C I P R O C I T Y : T h e R e c i p r o c a l O p e r a t o r c h a n g e s t h e s i g n s i n t h e l o g i c a l r e p r e s e n t a t i o n o f a p r o p o s i t i o n b y a d d i t i o n o f t h e n e g a t i o n s y m b o l ( - ) , w h e r e i t i s a b s e n t a n d i t s d e l e t i o n w h e r e p r e s e n t . E x a m p l e : R ( P . q ) = ~P.q R ( C o n j u n c t i o n ) = C o n j u n c t i v e - N e g a t i o n NEGATION : T h e N e g a t i o n O p e r a t o r c h a n g e s b o t h s i g n s a n d c o n j u n c t i o n s i n t h e l o g i c a l r e p r e s e n t a t i o n o f a p r o p o s i t i o n : E x a m p l e : N ( P . q ) = P v q N ( C o n j u n c t i o n ) = I n c o m p a t i b i l i t y C(P.q) = P v q = he i s hungry or t i r e d This statement i s nearer to the o r i g i n a l than the others 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 four interpropositions, P.q, P v q, P.q and P v q, form a group of four transformations isomorphic to the INRC group just as problems of r e c i p r o c i t y i n physical systems form a similar group. That i s , the propositional r e c i p r o c i t i e s and inversions express the r e c i p r o c i t i e s and inversions operating in the equilibrated system under study. Inhelder and Piaget (1958) explained t h i s psycho-l o g i c a l unity between propositional l o g i c and r e c i p r o c i t y i n physical systems by saying that the INRC group functions at two l e v e l s - i n i t i a l l y , " . . . . i t governs the propositional operations which the subject uses to describe and explain r e a l i t y ; as such i t constitutes an integrated structure at the i n t e r i o r of his thought, ... (and then) as a re s u l t of t h i s f i r s t function i t i s projected out-side into the phenomena under study (since i n the given data, these consist of a physical system whose equilibrium represents the very problem to be resolved)." (Inhelder and Piaget, 1958, p. 32) According to Inhelder and Piaget, the INRC group i s therefore not r e s t r i c t e d to mechanical systems alone but i s applicable i n a l l situations involving the coordination of two d i s t i n c t reference variables l i k e concentration and solution volume i n t i t r a t i o n i n chemistry. The relationship of the INRC group to the propor-t i o n a l schema, i . e . to concepts involving both r e c i p r o c i t y and proportion, i s expressed by Inhelder and Piaget as follows: 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 terms o f a group s t r u c t u r e - INRC - i n d i c a t e s our u n d e r s t a n d i n g o f the e q u a l i t i e s NR = I C , RC = IN , NC = IR, e t c . , the 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 the INRC group i s i t s e l f e q u i v a l e n t t o a system of l o g i c a l p r o p o r t i o n s : Ix _ Rx Rx _ Cx Cx Nx Ix Nx S i n c e IN = RC (where x = the o p e r a t i o n t r ans formed by I , N , R, or C ) . ( Inhe lder and P i a g e t , 1958, p . 177) . That i s , p r o p o r t i o n a l i t y i s a consequence o f the d i s c o v e r y o f l o g i c a l r e c i p r o c i t y . U s i n g the E q u i l i b r i u m i n the Ba lance Task , as an example, they argued t h a t the above p r o p o r t i o n cor re sponds t o the n u m e r i c a l 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 the weight and d i s t a n c e from the f u l c r u m , r e s p e c t i v e l y ; and n = c o e f f i c i e n t g i v e n t o an i n c r e a s e i n x o r y . S i n c e i n t h e i r v i e w , the INRC group i s fundamental t o p r o p o s i t i o n a l l o g i c , they argue t h a t s u b j e c t s from 12 year s t o 15 year s 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 i n terms o f p r o p o s i t i o n a l l o g i c and t h e r e f o r e are a b l e t o under-s t and and t r a n s f o r m the e q u a l i t y o f two p r o d u c t s . One o f the exper iments de s igned by I n h e l d e r and P i a g e t (1958) t o i l l u s t r a t e p r o p o r t i o n a l r e a s o n i n g i s the E q u i l i b r i u m i n the B a l a n c e . T h i s w i l l be used t o show the a p p l i c a t i o n o f h i s a n a l y s i s o f t h i s schema. T h i s problem i s chosen because i t i n v o l v e s r e c i p r o c a l p r o p o r t i o n ( i . e . 2 8 r e c i p r o c i t y and proportion) in contrast to the problem of the communicating vessels which involves only r e c i p r o c i t y ; also, i t i s isomorphic to some of the calculations i n volumetric analysis as w i l l be shown l a t e r . The balance task concerns the discovery and explana-ti o n of the r e l a t i o n between the weights placed on the balance and t h e i r distances from the fulcrum. That i s , W/W = L'/L where W and W are two unequal weights L and L 1 are t h e i r corresponding unequal distances, The balance scale used i s a simple seesaw type balance with varying weights which can be hung at d i f f e r e n t points along i t s two arms. The successful solution of t h i s problem i s assumed to depend on two conceptual systems: the r e c i p r o c i t y that obtains between weight and distance and which according to Piaget i s r e f l e c t e d i n the propositional reasoning of the subject, and the notion of proportion i t s e l f as opposed to additive differences or s e r i a t i o n and correspondences. That i s , instead of subtracting from the distance along the arm, L, whatever i s added to the weight W, or adding to L whatever i s subtracted from W, the successful application of the pro-p o r t i o n a l i t y schema involves the abstraction of a second order r e l a t i o n from the elementary r e l a t i o n s . Thus given: W L W L' the discovery of proportion l i e s i n the a b i l i t y to abstract the second order r e l a t i o n : 1 W:W* = L:L' instead of r e l y i n g on the additive differences, W-W1 = L'-L. In short, successful solution of the balance problem involves the use of true proportions or r a t i o s . Piaget analyzes the proportionality involved i n t h i s task as follows: If the increase i n weight i s represented by P i t s decrease or negation can be denoted by P. However, the increase in weight can be compensated by a corresponding decrease i n distance from the fulcrum and t h i s can be denoted by q while the negation of t h i s , q, i . e . increasing the distance from the fulcrum, compensates for the decrease i n weight. The following formulation then r e s u l t s : I(P) ; N(P) ; R(q) ; C(q) This i s equivalent to the proportionality: P q ,, Ix Cx , _ — — = — f — thus — = — where x = P q P Rx Nx P P - -Also, = R—— , and as such (P.q) = R(P.q) q q 3 0 That i s , understanding the system of inversions and recipro-c i t i e s follows from an understanding of the above proportional r e l a t i o n . In other words, increasing the weight and reducing the distance on one arm of the balance i s similar to reducing the weight and increasing the distance on the other arm. A similar analysis could be applied to a t y p i c a l problem i n volumetric analysis such as: If 20 mL of 2M H2S0& neutralizes 100 mL of NaOH solution, what w i l l be the molarity of the NaOH solution? Successful solution of t h i s problem may involve the recognition of the two types of proportions - inverse proportion and di r e c t proportion i n the problem. In the f i r s t case, as i n the problem of Equilibrium in the Balance, the subject must be able to recognize the re c i p r o c i t y that obtains between the concentration and the volume of the solutions and at the same time be able to use ra t i o s and proportions instead of additive differences. That i s , the f i r s t part of the problem i s solved i f the subject i s able to i d e n t i f y the inverse proportionality that obtains between the concentration and the volume of the solutions. The solution to the problem i s f i n a l l y completed by recognizing the d i r e c t proportion that obtains between the product of the concentration and volume of the solutions ( i . e . the number of moles of the reactants) and the c o e f f i c i e n t of the reactants (H 2S0 4 and NaOH) i n the st o i c h i o -metric equation. Symbolically, i t can be stated that: 31 or r e c i p r o c a l proportion and, 1/2 d i r e c t proportion where C . J stands for the concentration of acid or base V. J stands for the volume of acid or base 1 and 2 stand for the c o e f f i c i e n t i n the s t o i c h i o -metric equation. Considering the r e l a t i o n between volume and concentration, i f the increase i n concentration i s set forth as the i d e n t i c a l operator and represented by P, then i t s decrease or negation i s the inverse transformation P. But the decrease i n the volume of the solution compensates for the increase i n concen-t r a t i o n without cancelling i t . Hence i t plays the part of r e c i p r o c a l transformation, q. The inverse of the re c i p r o c a l produces a similar e f f e c t as the decrease i n concentration and can be regarded as playing the role of the correlate, q. Thus, the following transformations: I(P); N(P); R(q); C(q) r e s u l t . This y i e l d s the following proportion, P P or (P.q) = R (P.q) 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 of moles of reactants used in the t i t r a t i o n , the following s i t u a t i o n e x i s t s : If an increase i n the stoichiometric 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, P, then i t s decrease i s the inverse operation, P. The decrease i n the number of moles of reactants compensates for the decrease i n stoichiometric c o e f f i c i e n t , thus acting as the reci p r o c a l transformation, q. The inverse of the r e c i p r o c a l , q, i . e . increase i n the number of moles, produces an e f f e c t similar to the increase i n the stoichiometric 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 proportional r e l a t i o n r e s u l t i n g from the above transformation i s : I R P q x x = — 2 — or — = — q P C N M x x In other words, the increase of stoichiometric c o e f f i c i e n t i s to the increase of number of moles as the decrease of number of moles i s to the decrease of stoichiometric c o e f f i c i e n t . I t should be noted that the above r e l a t i o n requires knowledge or a b i l i t y to write and interpret a balanced equation of the reaction between the reactants, v i z . , H 2 S 0 4 + 2 NaOH = Na 2 S 0 4 + 2 H 2 0 That i s , knowledge of content may play an important part i n obtaining a successful solution to the problem. Piaget (1972) has acknowledged that the content i n a p a r t i c u l a r area may help 3 3 a s u b j e c t to use fo rmal r e a s o n i n g which i n c l u d e s p r o p o r t i o n a l r e a s o n i n g i n s i t u a t i o n s r e q u i r i n g i t . I t has been argued t h a t Gagne ' s cummulative l e a r n i n g t h e o r y o f f e r s the o p p o r t u n i t y t o examine the i n f l u e n c e o f the c o n t e n t i n l e a r n i n g or problem s o l v i n g ( G r i f f i t h , 1979) . The next s e c t i o n w i l l t h e r e f o r e b r i e f l y summarize Gagne 1 s t h e o r y and attempt t o i d e n t i f y d i f f e r e n c e s between Gagne 1 s t h e o r y and P i a g e t ' s t h e o r y . Gagne ' s Theory o f L e a r n i n g In c o n t r a s t to P i a g e t , Gagne views development as s u b o r d i n a t e t o l e a r n i n g . A c c o r d i n g t o h im, l e a r n i n g i s a cummulative p r o c e s s i n v o l v i n g the t r a n s f e r o f c e r t a i n i n t e l -l e c t u a l s k i l l s such as d i s c r i m i n a t i o n s , concept s and r u l e s t o a v a r i e t y o f problem s o l v i n g s i t u a t i o n s and i n l e a r n i n g a number o f h i g h e r - o r d e r s k i l l s . As s u c h , development i s seen as the l o n g term e f f e c t o f l e a r n i n g . From t h i s b e l i e f i n the cummulative e f f e c t o f l e a r n -i n g , Gagne' argues t h a t i n d i v i d u a l s a c q u i r e c a p a b i l i t i e s i n an o r d e r e d and a d d i t i v e manner. That i s , the l e a r n i n g o f a more complex s k i l l i s preceded by the l e a r n i n g o f l e s s complex one which i s a l s o preceded by a much l e s s complex s k i l l and so o n , u n t i l the most b a s i c component o f the s k i l l i s e n c o u n t e r e d . Gagne' (1970, p . 20) d i s t i n g u i s h e s h i m s e l f from the t r a d i t i o n a l p o s i t i o n t h a t a l l l e a r n i n g can be accounted f o r by any p a r t i c u l a r p r o t o t y p e o f l e a r n i n g such as a s s o c i a t i o n o r i n s i g h t , by b e l i e v i n g t h a t t h e r e might be as many d i f f e r e n t types of learning as there are conditions under which learning can occur. He distinguishes eight d i f f e r e n t types of learning which i n his view form a h i e r a r c h i c a l structure i n which the subordinate types are prerequisite to the learning of the superordinate ones. The developmental nature of t h i s cummula-ti v e model i s represented i n Figure 2 (Gagne, 1970, p. 70). According to Gagne, the hierarchy can be generated by asking the question "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 ? " This i s repeated for any new c a p a b i l i t y generated u n t i l simple and less demanding c a p a b i l i t i e s are arrived at. Applied to the Ideal Gas Law, the complex superordinate rule to be learned can be mathemati-c a l l y expressed as follows: PV = nRT where P = pressure, V = volume, n = moles, R = universal gas constant, and T = temperature. I t may be hypothesized that t h i s complex rule requires the learning of rules governing- temperature and volume ( i . e . Charles Law) and pressure and volume when temperature i s held constant ( i . e . Boyle's Law). The learning of these two rules can be further hypothesized to require the following concepts as prerequisites - temperature, volume^pressure, mass and mole. It i s noteworthy that Gagne has changed his t h e o r e t i c a l stance to some extent since the publication of his book, Conditions of Learning (Gagne', 1970) . In his e a r l i e r writings Gagne (1970) believed that the h i e r a r c h i c a l structure could be applied to any content area. He stated, "learning hierarchies are the best way to describe the structure of any Problem Solving (Type 8) which requires as p re requ i s i t e s : Rules (Type 7) I which requires as p re requ i s i t e s : Concepts j lype 6) which requires as p re requ i s i t e s : D iscr iminat ions (Type 5) which requires as prerequ i s i tes Verbal Assoc iat ions (Type h) or chains (Type 3) which require as prerequ i s i tes o r Stimulus-Response Connections (Type 2) Signal Learning (Type 1) Figure 2 Hierarch ica l Arrangement of Learning Types (Gagne', 1970, p .66) 3 6 t o p i c , c o u r s e , o r d i s c i p l i n e " (Gagne',- 1970, p . 245) . However, ' i n h i s more r e c e n t works he has moved from t h i s p o s i t i o n by sugge s t ing 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 can be a p p l i e d o n l y t o c e r t a i n k i n d s o f l e a r n i n g (Gagne, 1972; 1977; Gagne' and B r i g g s , 1974) . In these r e c e n t w r i t i n g s , he d i s t i n g u i s h e s f i v e domains o f l e a r n i n g which he c a l l s motor s k i l l s , v e r b a l i n f o r m a t i o n , i n t e l l e c t u a l s k i l l s , c o g n i t i v e s t r a t e g i e s , and a t t i t u d e s . Gagne (1972) suggests t h a t o f the f i v e domains, i t i s o n l y the i n t e l l e c t u a l s k i l l s l e a r n i n g which r e q u i r e p r i o r l e a r n i n g o f s u b o r d i n a t e p r e r e q u i s i t e s k i l l s . That i s , the h i e r a r c h i c a l model can be a p p l i e d o n l y t o the i n t e l l e c t u a l s k i l l s . The i n t e l l e c t u a l s k i l l s domain c o n s i s t s o f the e i g h t types o f l e a r n i n g o r i g i n a l l y p o s t u l a t e d by Gagne' (1970) and shown i n F i g u r e 2. A l t h o u g h l e a r n i n g i n any p a r t i c u l a r domain may be a f f e c t e d i n some way by l e a r n i n g i n the o t h e r domains, o n l y the l e a r n i n g o f i n t e l l e c t u a l s k i l l s and i t s r e l a t i o n t o c o g n i -t i v e s t r a t e g i e s w i l l be c o n s i d e r e d . T h i s i s because i t i s the o n l y domain which has been a n a l y z e d i n d e t a i l i n Gagne's t h e o r y and a l s o the domain which r e l a t e d t o the c o n t e n t o f a t a s k ; thus making i t a p p l i c a b l e t o the proposed s t u d y . Gagne makes a d i s t i n c t i o n between i n t e l l e c t u a l s k i l l s and c o g n i t i v e s t r a t e g i e s i n the f o l l o w i n g way. Whereas the a p p l i c a t i o n o f r u l e s o r concept s i n a meaning fu l way i n any s u b j e c t a rea r e p r e s e n t s i n t e l l e c t u a l s k i l l s ; the way an i n d i v i d u a l a t t a c k s a n o v e l problem c o n s t i t u t e s h i s c o g n i t i v e s t r a t e g y (Gagne and B r i g g s , 1974, p . 49 ) . Gagne' and B r i g g s 37 (1974) c o n s i d e r c o g n i t i v e s t r a t e g y as an i n t e r n a l l y o r g a n i z e d s k i l l . They r e l a t e i t t o i n t e l l e c t u a l s k i l l s by r e f e r r i n g t o c o g n i t i v e s t r a t e g y as a v e r y s p e c i a l k i n d o f i n t e l l e c t u a l s k i l l i n t h a t i t deve lops out o f " s p e c i a l l y l e a r n e d i n t e l l e c -t u a l s k i l l s by a p roce s s o f g e n e r a l i z a t i o n " (Gagne' and B r i g g s , 1974, p . 48) and are t h e r e f o r e impor tant i n n o v e l or problem s o l v i n g s i t u a t i o n s . They suggest t h a t a major d i f f e r e n c e between c o g n i t i v e s t r a t e g i e s and i n t e l l e c t u a l s k i l l s i s the o b j e c t o f the s k i l l . Whereas i n t e l l e c t u a l s k i l l s a re o r i e n t e d toward the o b j e c t s i n the i n d i v i d u a l ' s environment such as g raphs , f o r m u l a s , and s tatement o f l aws , c o g n i t i v e s t r a t e g i e s have as t h e i r o b j e c t s the i n d i v i d u a l ' s own thought p r o c e s s e s such as how c r i t i c a l l y he t h i n k s and how f l u e n t l y he t h i n k s . That i s , i n t e l l e c t u a l s k i l l s a re s u b j e c t mat ter o r i e n t e d whereas c o g n i t i v e s t r a t e g i e s p e r t a i n more t o the b e h a v i o u r o f the 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 s t u d y i n g o r the problem he i s s o l v i n g , and are as such ' c o n t e n t f r e e ' . A l t h o u g h Gagne suggests t h a t c o g n i t i v e s t r a t e g i e s are o f p a r t i c u l a r importance i n problem s o l v i n g , he m a i n t a i n s t h a t i f an i n d i v i d u a l i s t o a r r i v e a t a s p e c i f i c s o l u t i o n to a p a r t i c u l a r p rob lem, the i n t e l l e c t u a l s k i l l s or more s p e c i f i c a l l y , the concept s and r u l e s r e q u i r e d f o r i t s s o l u t i o n must be a v a i l a b l e t o the l e a r n e r . In the next s e c t i o n , a compari son o f the two .. . t h e o r i e s , namely Gagne ' s t h e o r y and P i a g e t ' s t h e o r y and t h e i r p o s s i b l e combina t ion from a t h e o r e t i c a l v i e w p o i n t w i l l be c o n s i d e r e d . 38 Comparison o f Gagne ' s and P i a g e t ' s T h e o r i e s I t i s i n t e r e s t i n g t h a t Gagne (Gagne and B r i g g s , 1974, p . 48) a t tempts t o r e l a t e the domain o f c o g n i t i v e s t r a t e g i e s i n h i s t h e o r y t o the i n t e l l e c t u a l s t r u c t u r e o f the i n d i v i d u a l as p o s i t e d by P i a g e t . T h i s i s impor tant g i v e n the attempt i n the p r e s e n t s tudy t o examine s tudent s performance on a c h e m i c a l problem i n terms o f b o t h the p r o p o r t i o n a l r e a s o n i n g schema i n P i a g e t ' s t h e o r y and the c o n t e n t requ i rement s as i d e n t i f i e d u s i n g Gagne ' s t h e o r y . A c c o r d i n g t o P i a g e t (1964), the i n d i v i d u a l ' s i n t e l l e c -t u a l s t r u c t u r e s e t s the l i m i t s to the k i n d s o f problem s o l v i n g an i n d i v i d u a l can s u c c e s s f u l l y per form w h i l e a t the same time accommodating t o the e n v i r o n m e n t a l s t i m u l u s . In c o n t r a s t , the Gagnean t h e o r y suggests t h a t the s u b o r d i n a t e p r e r e q u i s i t e c a p a b i l i t i e s se t the l i m i t s f o r what can be l e a r n e d . Thus Gagne (1970, p . 290) argues t h a t " d i f f e r e n c e s i n deve lopmenta l r e a d i n e s s are p r i m a r i l y a t t r i b u t a b l e t o d i f f e r e n c e s i n the number and k i n d o f p r e v i o u s l y l e a r n e d i n t e l l e c t u a l s k i l l s " . S t r a u s s (1972) suggests t h a t on the s u r f ace,, the two t h e o r i e s appear t o agree on the f o l l o w i n g : (1) t h a t an i n d i v i -d u a l deve lops an i n c r e a s i n g number o f i n t e l l e c t u a l c a p a b i l i -t i e s as he grows o l d e r , (2) t h a t such c a p a b i l i t i e s are the p r o d u c t o f an i n d i v i d u a l ' s i n t e r a c t i o n w i t h h i s env i ronment , and (3) t h a t the a c q u i s i t i o n o f these c a p a b i l i t i e s i s s e q u e n t i a l . He, however, notes t h a t these apparent g e n e r a l i z a -t i o n s are m i s l e a d i n g s i n c e the two t h e o r i e s e n t e r t a i n e n t i r e l y d i f f e r e n t p h i l o s o p h i c a l p o s i t i o n s . 39 S t r a u s s suggested t h a t a l t h o u g h both P i a g e t and Gagne v iew i n t e l l e c t u a l c a p a b i l i t i e s as forms o f thought which can be a p p l i e d t o d i f f e r e n t c o n t e n t s , t h e r e are t h e o r e t i c a l d i f f e r e n c e s i n the na ture o f the forms . He argues t h a t i n Gagne ' s t h e o r y the forms o f thought are n a r r o w l y d e f i n e d s t r u c t u r e s such as the s p e c i f i c r e s p o n s e s , d i s c r i m i n a t i o n s , c o n c e p t s , r u l e s and problem s o l v i n g which c o n s t i t u t e a hier-v. a r c h y ; i n the P i a g e t i a n t h e o r y , they are p o t e n t i a l o r g a n i z a -t i o n s o f menta l s t r u c t u r e s and are t h e r e f o r e much more comprehens ive . A second d i f f e r e n c e i d e n t i f i e d by S t r a u s s i s t h a t i n the Gagnean t h e o r y , the l e a r n e r c o p i e s r e a l i t y r a t h e r than c o n s t r u c t s r e a l i t y . Hence the l e a r n e r i s a r e l a t i v e l y p a s s i v e r e c i p i e n t o f the e n v i r o n m e n t a l s t i m u l i , w h i l e i n ....... P i a g e t ' s t h e o r y , the l e a r n e r i s v e r y a c t i v e i n c o n s t r u c t i n g h i s i n t e l l e c t u a l s t r u c t u r e s . T h i s i n t e r p r e t a t i o n o f Gagne ' s t h e o r y may be j u s t i f i a b l e o n l y t o the i n t e l l e c t u a l s k i l l s s i n c e Gagne c l a i m s t h a t i t i s i n t h i s domain t h a t l e a r n i n g can be c a r e f u l l y c o n t r o l l e d t o a i d the a c q u i s i t i o n o f h i g h e r -o r d e r s k i l l s . As G r i f f i t h s (1979) sugges t s , the d i f f e r e n c e between P i a g e t and Gagne i s more l i k e l y t o c e n t r e around the e x t e n t t o which the l e a r n e r i s c o n t r o l l e d . An impor tant a spec t o f t h i s c o n t r o l i s whether the m a j o r i t y o f i n d i v i d u a l s need t o f o l l o w a s i n g l e h i e r a r c h i c o r g a n i z a t i o n o f a p a r t i c u l a r concept as Gagne 1 s t h e o r y sugges t , or whether a v a r i e t y o f p o t e n t i a l r o u t e s s h o u l d be a v a i l a b l e as P i a g e t ' s t h e o r y sugges t s . Rowel l and Dawson (1979) suggests t h a t 40 Gagne does not c o n s i d e r a ' m u l t i - p a t h ' t o development because he approaches 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 i n s t r u c t i o n r a t h e r than a c o n s i d e r a t i o n o f n a t u r a l o r ' u n f o r c e d ' development . Some ev idence suggests t h a t a s i n g l e h i e r a r c h y o f s u b s k i l l s i s u n l i k e l y t o c h a r a c t e r i z e the development o f any-p a r t i c u l a r concept s i n c e c h i l d r e n o f d i f f e r e n t ages use d i f f e r e n t methods and d i f f e r e n t p r o c e s s e s , o r the same proce s se s i n d i f f e r e n t o r d e r s , i n r e a c h i n g a deve lopmenta l g o a l ( S c h a e f f e r , E g g l e s t o n and S c o t t , 1974) . The d i f f e r e n c e s i n the t h e o r e t i c a l p o s i t i o n s as noted by S t r a u s s (1972) and Rowel l and Dawson (1979) need not p r e c l u d e the a p p l i c a t i o n o f bo th t h e o r i e s i n s i t u a t i o n s i n which t h e i r use may be advantageous . In f a c t , these d i f f e r e n c e s when combined may h e l p i n b e t t e r u n d e r s t a n d i n g a s u b j e c t ' s performance on a t a s k . As noted by L o v e l l (1971a), P i a g e t ' s t h e o r y does not say much about the c o n t e n t o f a t a s k , a s i t u a t i o n which can be c o n t r a s t e d w i t h h i s d e t a i l e d a n a l y s i s o f the i n t e l l e c t u a l s t r u c t u r e s needed f o r i t s s o l u t i o n . On the o t h e r hand, Gagne ' s t h e o r y g i v e s a d e t a i l e d h i e r a r c h i c a l d e s c r i p t i o n o f the c o n t e n t o f the p rob lem. The combina t ion o f the two t h e o r i e s may remove some o f the problems encountered when e i t h e r t h e o r y i s employed a lone i n r e s e a r c h . T h i s need i s e x e m p l i f i e d by W h i t e ' s f i n d i n g which i n d i c a t e d the " s u b j e c t ' s f a i l u r e t o l e a r n even when he posses ses a l l the a p p a r e n t l y nece s s a ry s u b o r d i n a t e e lements " (White , 1972) . The examinat ion o f the i n t e l l e c t u a l r e a s o n i n g o f the s u b j e c t may e x p l a i n t h i s f i n d i n g . On the other hand Sti c h t (1971) suggests that the problem of hori-... zontal decalege i n Piagetian research may be removed by provid-ing the appropriate prerequisite content. Thus i t can be seen that the combination of the two theories may be f r u i t f u l i n affording a better explanation of students' d i f f i c u l t i e s with certain tasks i n the school curriculum. The next section w i l l be concerned with the discus-sion of studies related to proportional reasoning structures and the content of science tasks, e s p e c i a l l y those concerned with chemical problems. Related Study; Proportional Reasoning and Content The following section w i l l attempt to show that Piaget's theory and Gagne's cummulative theory have been used independently i n analysing a subject's performance on a task. A possible combination of both theories i n a single study i s suggested af t e r reviewing studies which demonstrate the influence of content on l o g i c a l reasoning as proposed i n Piaget's theory. Proportional Reasoning Studies The importance of studying the proportional reason-ing of students cannot be underestimated since i t forms a v i t a l aspect of the understanding of quantitative r e l a t i o n s i n science and also because i t forms an e s s e n t i a l part of Piaget's formal operations. 42 During the l a s t two decades, a substantial portion of the science education l i t e r a t u r e has been geared to applying Piaget's theory to science curriculunr.and i n s t r u c t i o n (e.g. Karplus, 196 5). A f a i r proportion of t h i s l i t e r a t u r e has been concerned with the proportional reasoning of students i n both the elementary and high schools (e.g. Karplus and Peterson, 1970; Chiappetta, 1974). It should be mentioned that a number of studies have not dealt with proportional reasoning per se but only as part of the t o t a l a b i l i t i e s which characterizes formal operations. Examples of such studies include those by L o v e l l (1961) and Lawson and Renner (1975) where tasks requiring propor-t i o n a l reasoning were used with tasks requiring control of v a r i -ables or other schema in deciding on the cognitive developmental l e v e l of the subject. Since.the' aim of such studies was not to i d e n t i f y proportional reasoning or i t s r e l a t i o n to some other construct per se, these studies w i l l not be discussed here. The studies discussed here deal s p e c i f i c a l l y with proportional rea-soning. The order of presentation include references to the analysis of chemical concepts i n terms of Piaget's theory, and the presence of proportional reasoning i n adolescents. Proportional Reasoning as an Explanatory Construct From Piaget's theory i t appears that the i n t e l l e c t u a l structure of the subject should dictate his performance on a task requiring t h i s structure. To t h i s end a number of studies have appeared in the l i t e r a t u r e which attempt to explain students d i f f i c u l t i e s i n terms of the i n t e l l e c t u a l s t r u c t u r e . M e a l i n g s (196 3) and a number o f r e s e a r c h e r s ( Ing le and Shayer , 1971; Shayer , 1972; H e r r o n , 1975; H a r f o r d and Good, 1976; K a r p l u s , 1977; Chappeta 1978) have a n a l y z e d concept s i n s c h o o l s c i e n c e c u r r i c u l a r m a t e r i a l s s o l e l y i n terms o f t h e i r u n d e r l y i n g i n t e l l e c t u a l s t r u c t u r e . M e a l i n g s (196 3) conducted a s tudy i n which he i n v e s t i g a t e d the r e a s o n i n g used by s tudent s i n s o l v i n g c e r t a i n s c i e n c e prob lems . From the r e s u l t s , he suggested t h a t s tudent s under 16 y e a r s o f age w i l l f i n d i t d i f f i c u l t t o under take t o p i c s 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 m e t r i c p r o p o r t i o n such as 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 Law. He a l s o suggested t h a t under 14 y e a r s o f age, a s tudent w i l l not be a b l e t o d i s t i n g u i s h between pota s s ium s u l p h a t e , po ta s s ium c a r b o n a t e , magnesium s u l p h a t e and magnesium carbonate by means o f p r a c t i c a l t e s t s . From t h i s l e a d , I n g l e and Shayer (1971) a n a l y z e d the i n t e l l e c t u a l demands o f the t o p i c s i n the N u f f i e l d ' 0 ' L e v e l c h e m i s t r y cour se i n terms o f P i a g e t ' s s t age s . A c c o r d i n g t o t h e i r a n a l y s e s , most secondary s c h o o l s tudent s w i l l encounter d i f f i c u l t i e s w i t h t o p i c s such as 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 s i n c e p r o p o r t i o n s and r a t i o s are i n v o l v e d . In a s i m i l a r a n a l y s i s r e l a t e d t o the t o p i c s i n the N u f f i e l d ' 0 ' L e v e l p h y s i c s c o u r s e , i t was c o n c l u d e d t h a t t o p i c s such as k i n e t i c t h e o r y and energy w i l l p r e s e n t a c o n s t a n t source o f d i f f i c u l t y f o r s tudent s (Shayer , 1972) . H a r t f o r d and Good (1976) a n a l y z e d the c o n c e p t u a l demands o f c e r t a i n t o p i c s i n the CHEM Study t e x t s i n terms o f P i a g e t ' s fo rma l o p e r a t i o n t h e o r y and noted t h a t c h e m i c a l bond-4 4 ing and equilibrium require a high l e v e l of cognitive develop-ment. Herron ( 1 9 7 5 ) presented a l i s t of 1 6 commonly expected competencies which chemistry students who are concrete opera-t i o n a l could be expected to exhibit. Each of these competen-cies were compared with the i n t e l l e c t u a l demands judged to be required by the science curriculum. He argued that a number of areas required formal operational a b i l i t i e s to successfully understand the.concepts and problems involved. For example, given the volume of base needed to neutralize 1 . 0 g of acid, a concrete operational student can calculate the volume of base needed to neutralize any amount of acid. However, according to Herron, the same student cannot calculate the concentration of an acid i f he i s given the concentration and volume of base needed to neutralize a certain volume of the acid. Herron 1s analysis suggests that students d i f f i c u l t i e s i n volumetric analysis calculations i s attributable to lack of understanding of the inverse p r o p o r t i o n a l i t i e s involved. Karplus ( 1 9 7 7 ) has also suggested that certain concepts such as chemical bonding, periodic system and i d e a l gas which are usually included i n secondary school science courses may be more d i f f i c u l t for students i n that they demand formal reasoning. There i s much\recent evidence that suggests that"as many as 5 0 % of some samples of high school and college students have f a i l e d to acquire a working understanding of the proportionality schema (e.g. Karplus and Peterson, 1 9 7 0 ; Lunzer and Pumfrey, 1 9 6 6 ; L o v e l l , 1 9 6 1 ; Lunzer, 1 9 6 5 ; Wollman and Karplus, 1 9 7 4 ) . In Karplus and P e t e r s o n ' s (1970) s tudy a Paper C l i p s Task (which measures d i r e c t p r o p o r t i o n ) was a d m i n i s t e r e d t o s tudent s from Grade 4 t o 12. T h e i r f i n d i n g i n d i c a t e d t h a t a l a r g e p r o p o r t i o n o f h i g h s c h o o l s tudent s use a d d i t i v e s t r a t e g i e s i n s i t u a t i o n s r e q u i r i n g the use o f p r o p o r t i o n . A s i m i l a r c o n c l u s i o n was a r r i v e d a t by L o v e l l and B u t t e r w o r t h (1966). A s tudy by C h i a p p e t t a (1974) u s i n g a d u l t s as s u b j e c t s showed t h a t about 50% o f the a d u l t s used a d d i t i v e s t r a t e g i e s on a s o l u b i l i t y problem which r e q u i r e d d i r e c t p r o p o r t i o n . C h i a p p e t t a (1976) a r r i v e d a t a s i m i l a r c o n c l u s i o n a f t e r r e v i e w i n g P i a g e t i a n s t u d i e s r e l a t e d t o h i g h s c h o o l and c o l l e g e science- e d u c a t i o n . T h i s apparent l a c k o f congruence between the c u r r i c u l u m m a t e r i a l s used i n the s c h o o l s and the i n t e l l e c t u a l development o f h i g h s c h o o l s tudent s l e a d s t o the q u e s t i o n o f whether an e m p i r i c a l r e l a t i o n s h i p e x i s t s between the i n t e l l e c t u a l r e a s o n i n g o f s tudent s and t h e i r achievement . G r i f f i t h s (1979) rev iewed s t u d i e s r e l a t i n g c o g n i t i v e d e v e l o p -menta l l e v e l t o s c i e n c e achievement and suggested t h a t such ev idence i s s c a r c e . Of p a r t i c u l a r i n t e r e s t t o the p r e s e n t s tudy and even more s c a r c e i s the ev idence needed t o answer the q u e s t i o n : i s t h e r e an e m p i r i c a l r e l a t i o n s h i p between p r o p o r t i o n a l r e a s o n i n g o f h i g h s c h o o l s tudent s and t h e i r achievement i n s c i e n c e ? The o n l y s t u d i e s r e p o r t e d on t h i s q u e s t i o n are those conducted by C h i a p p e t t a (1974) and Wheeler and Kass (1977) . C h i a p e t t a (1974) i n v e s t i g a t e d the r e l a t i o n s h i p between female i n - s e r v i c e t e a c h e r s ' a b i l i t y t o reason i n terms of p r o p o r t i o n and t h e i r achievement i n p h y s i c a l s c i e n c e . The s u b j e c t s were f i r s t g iven the E q u i l i b r i u m i n the Balance task to assess t h e i r p r o p o r t i o n a l thought development. They then p a r t i c i p a t e d i n a s e l f - p a c e d 12-unit p h y s i c a l s c i e n c e l a b o r a -t o r y course. One of the u n i t s was chemical s o l u b i l i t y whose o b j e c t i v e s were concerned w i t h c a l c u l a t i o n s i n v o l v i n g propor-t i o n s and r a t i o s . The p h y s i c a l s c i e n c e achievement of the s u b j e c t s was assessed a t two p o i n t s u s i n g a paper and p e n c i l t e s t f o l l o w e d by an i n t e r v i e w . One achievement t e s t was given at the completion of the u n i t on s o l u b i l i t y and the other a t the end of the course. A c o r r e l a t i o n of 0.60 was found . between performance on the E q u i l i b r i u m i n the Balance Task and achievement i n s o l u b i l i t y w h i le a c o r r e l a t i o n of 0.80 was found between the E q u i l i b r i u m i n the Balance Task and a c h i e v e -ment on a l l the p h y s i c a l s c i e n c e l a b o r a t o r y u n i t s . C h i a p p e t t a concluded t h a t the p r o p o r t i o n a l r e a s o n i n g of the s u b j e c t s appeared t o be d i r e c t l y r e l a t e d t o t h e i r p h y s i c a l s c i e n c e achievement. However, no attempt was made to r e l a t e t h e i r achievement t o t h e i r understanding of the content area. Only the l o g i c a l s t r u c t u r e was c o n s i d e r e d as r e l e v a n t i n e x p l a i n i n g achievement. Wheeler and. Kass (1977) s t u d i e d the r e l a t i o n s h i p of the p r o p o r t i o n a l r e a s o n i n g schema to achievement i n f o u r areas of chemistry - chemical nomenclature and w r i t i n g of formulae, chemical r e a c t i o n s , the mole concept and g r a v i m e t r i c s t o i c h i o -metry. The s u b j e c t s i n v o l v e d were 168 high s c h o o l chemistry students. The s u b j e c t s were i n i t i a l l y a d m i n i s t e r e d t h r e e proportional reasoning tasks and a general proportional reasoning t e s t . The tasks used were Equilibrium i n the Balance, the Ratio Task which i s a modification of the Paper Clip s Task (Wollman and Karplus, 1974) and the Metric Puzzle which involved conversion from inches to centimeters. They also employed the Island Puzzle (Karplus and Karplus, 1970) which was employed to assess deductive reasoning. At the conclusion of each of the four introductory chemistry units which were taught by the regular teachers, a subtest measuring content i n that unit was administered. At the end of the course a chemistry achievement te s t which had items similar to the items i n the chemistry subtests were administered. The s i m i l a r i t y of the chemistry achievement te s t and the subtests i s shown by a high positive 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 reported 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 proportional reasoning and achieve-ment i n chemistry. However, i n t h i s case the authors also examined the e f f e c t of content but t h i s i s discussed i n the next two sections. Even though the tasks on proportional reasoning included both inverse and d i r e c t proportionality tasks, no attempt was made to examine how these two aspects of proportionality separately and i n t e r a c t i v e l y a f f e c t subjects achievement i n chemistry. Thus with the exception of the study by Wheeler and Kass, the above studies have mainly examined the s t r u c t u r a l demands of chemical concepts i n c u r r i c u l a r materials or have examined only the influence of proportional reasoning on 4 8 achievement . G e n e r a l l y , the c o n t e n t has not been c o n s i d e r e d as a r e l e v a n t v a r i a b l e i n the s t u d i e s . Content as E x p l a n a t o r y C o n s t r u c t A p a r t from the use o f s u b j e c t ' s p r o p o r t i o n a l r e a s o n -i n g i n the above s t u d i e s t o e x p l a i n t h e i r d i f f i c u l t i e s , o t h e r s t u d i e s have focus sed on the c o n t e n t o f the t a s k s i n e x p l a i n i n g these same d i f f i c u l t i e s . Duncan and Johnstone (1973) conducted a s tudy i n an at tempt t o i d e n t i f y the d i f f i c u l t i e s encountered by h i g h s c h o o l s tudent s i n u n d e r s t a n d i n g the mole concept and some of i t s a p p l i c a t i o n s - 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 work. A paper 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 sed . They found t h a t the s tudent s d i f f i c u l t i e s were r e l a t e d t o t h e i r u n d e r s t a n d -i n g o f the concept s subsumed by the mole . S p e c i f i c a l l y , they r e p o r t e d t h a t the d i f f i c u l t i e s seem t o r e l a t e t o (i) overcoming the mi sapprehens ion t h a t one mole o f a compound w i l l a lways r e a c t w i t h one mole of a n o t h e r , r e g a r d l e s s o f the s t o i c h i o -metry ,of the r e a c t i o n , ( i i ) b a l a n c i n g e q u a t i o n s , and ( i i i ) m a n i p u l a t i o n o f m o l a r i t y o f s o l u t i o n s . A s tudy by Kass (1977) on the mole concept r e p o r t e d d i f f i c u l t i e s s tudent s have i n c o r r e c t l y b a l a n c i n g e q u a t i o n s p r i o r to t h e i r use i n s t o i c h i o -m e t r i c c a l c u l a t i o n s . N o v i c k and Menis (1976) conducted a s tudy o f h i g h s c h o o l s t u d e n t s ' p e r c e p t i o n o f the mole concept u s i n g a s t r u c t u r e d i n t e r v i e w p r o c e d u r e . They d i s c o v e r e d t h a t the s t u d e n t s ' d i f f i c u l t i e s i n v o l v e d (i) t h i n k i n g o f the mole i n terms o f mass and not i n terms o f number; t h i s , they b e l i e v e d 49 originates from quantitative operations based on mass measure-ments, ( i i ) r e s t r i c t i n g the mole concept to a cert a i n number of p a r t i c l e s of gas, and ( i i i ) thinking of the mole as a property of a molecule. The above studies demonstrate the importance of prerequisite content i n chemistry achievement. To anticipate students' d i f f i c u l t i e s i n terms of the subsumed prerequisite concepts demanded by a p a r t i c u l a r task, a content analysis can be performed on the task. Gagne's h i e r a r c h i c a l theory provides such a means for getting at the subsumed concepts. A number of studies have analyzed c e r t a i n chemical concepts using t h i s approach.' Gower, Daniels and Lloyd (1977b) analyzed the mole concept using Gagne'-type of h i e r a r c h i c a l analysis and found that i t subsumes other lower l e v e l concepts which students have to master before understanding the mole. Their study involved the written responses of 24 high school students to a set of written items representing the elements of the hypothesized hierarchy. They found that the connections established cor-responded c l o s e l y to those predicted i n the hypothesized hierarchy. However, the authors noted that the small number of subjects involved i n the study precludes generalization of the findings. In another work^Gower, Daniels and Lloyd, (1977a), analyzed volumetric analysis calculations using Gagne's theory and found i t to subsume a number of important lower-order concepts (Figure 3). A si m i l a r l i n e of approach was followed by G r i f f i t h s and Kass (1979) i n t h e i r attempt to i d e n t i f y a hierarchy 50 T i t r a t ion C a l c u l a t i o n s S o l u t ion Concent r a t ion C a l c u l a t ions C a l c u l a t i o n s from E q u a t i o n s C o n c e n t r a t i o n M o l a r i t y C o n v e n t i o n s g/mole Convers ion I n t e r p r e t Equat i o n s W r i t e E q u a t i o n s Mole F i g u r e 3 H i e r a r c h i c a l A n a l y s i s o f T i t r a t i o n C a l c u l a t i o n s (Gower, D a n i e l s and L l o y d , 1977a) 5 1 r e p r e s e n t i n g the mole . U s i n g a l a r g e sample o,f h i g h s c h o o l c h e m i s t r y s t u d e n t s , they were a b l e t o o b t a i n a r e l a t i o n s h i p between the e m p i r i c a l c o n n e c t i o n s and the h y p o t h e s i z e d one . That i s , achievement i n the mole concept was found t o be dependent on the subsumed c o n c e p t s . A s i m i l a r r e s u l t was o b t a i n e d by Okey and Gagne ( 1 9 7 0 ) i n a s tudy i n which they used s o l u b i l i t y p r o d u c t problems as the concept o f i n t e r e s t . A g e n e r a l c o n c l u s i o n from the above s t u d i e s i s t h a t the subsumed concept s may be c r i t i c a l t o a s u b j e c t ' s performance on a t a s k . However, a g a i n o n l y one v a r i a b l e , the c o n t e n t was used i n e x p l a i n i n g the performance o f the s u b j e c t s . The s t r u c t u r a l component o f the t a s k s was not c o n s i d e r e d . I n f l u e n c e o f Content on L o g i c a l Reasoning The above d i s c u s s i o n s d e a l t w i t h s t u d i e s which employed e i t h e r p r o p o r t i o n a l r e a s o n i n g o r c o n t e n t i n e x p l a i n i n g s t u d e n t s ' d i f f i c u l t i e s . The s t u d i e s t o be d i s c u s s e d below demonstrate the p r o b a b l e i n f l u e n c e o f the na ture o f the c o n t e n t oh l o g i c a l r e a s o n i n g and hence demonstrates the need t o examine b o t h i n o r d e r t o a c h i e v e a b e t t e r u n d e r s t a n d i n g o f s t u d e n t s ' d i f f i c u l t i e s . A number o f s t u d i e s have shown t h a t the c o n t e n t o f a t a sk may have a s i g n i f i c a n t e f f e c t on the r e c o g n i t i o n o f u n d e r l y i n g l o g i c a l s t r u c t u r e o f the t a s k . A c l a s s i c example o f t h i s was demonstrated by W i l k i n s ( 1 9 2 8 ) i n her at tempt t o f i n d the e f f e c t o f f a m i l i a r everyday c o n t e n t , u n f a m i l i a r c o n t e n t and symbol i c c o n t e n t on the s o l u t i o n o f s y l l o g i s m s . 52 She d i s c o v e r e d t h a t her s u b j e c t s found the problems w i t h f a m i l i a r c o n t e n t e a s i e r t o handle than problems w i t h symbol i c o r u n f a m i l i a r c o n t e n t . In a s i m i l a r s tudy conducted by Roberge and Paulus (1971) a s u b s t a n t i a l and h i g h l y s i g n i f i c a n t c o n t e n t e f f e c t was found . Another c l e a r demons t ra t ion o f the e f f e c t o f c o n t e n t i s g i v e n by a s e r i e s o f s t u d i e s based on Wason's (1966) Four Card Problem ( e . g . J o h n s o n - L a i r d , L e g r e n z i and L e g r e n z i , 1972; Wason and S h a p i r o , 1971; L u n z e r , H a r r i s o n and Davey, 1972) . A l t h o u g h o n l y d e d u c t i v e r e a s o n i n g was c o n s i d e r e d i n the above s t u d i e s , one i m p l i c a t i o n i s t h a t the c o n t e n t o f a t a sk may be more impor tan t than the p r o p o r t i o n a l r e a s o n i n g u n d e r l y i n g i t . Another i m p l i c a t i o n i s t h a t the c o n t e n t and p r o p o r t i o n a l r e a s o n i n g may i n t e r a c t t o a f f e c t a s u b j e c t ' s performance on a t a s k . There e x i s t the need t h e r e f o r e , f o r s t u d i e s which examine the r e l a t i o n s h i p among c o n t e n t , p r o p o r -t i o n a l r e a s o n i n g and achievement . The importance o f the c o n t e n t i n problem s o l v i n g s i t u a t i o n s has been r e i t e r a t e d by a number o f w r i t e r s (Levine and L i n n , 1977; L o v e l l , 1971a; 1973; L u n z e r , 1965; Nagy and G r i f f i t h s , 1979) . L e v i n e and L i n n (1977) i n a r ev i ew o f r e -s e a r c h on a d o l e s c e n t r e a s o n i n g noted the need f o r r e s e a r c h d i r e c t e d t o the c o n t e n t o f s c h o o l s u b j e c t s , the l o g i c a l s t r u c t u r e and s tudent s p r e v i o u s e x p e r i e n c e w i t h the v a r i a b l e s i n the s tudy i n e x p l a i n i n g t h e i r per formance . Other rer-s e a r c h e r s ( K e a t i n g , 1979; L o v e l l , 1971a; 1973; L u n z e r , 1965; 1975) have made s i m i l a r s u g g e s t i o n s . I t i s noteworthy t h a t i n 53 his more recent writings Piaget ( 1 9 7 2 ) has come to acknowledge th i s p o s i t i o n . The recognition of the need to consider not only the structure but also 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 i s relevant to the present study. The study conducted by Wheeler and Kass ( 1 9 7 7 ) and discussed e a r l i e r seems to f a l l in t h i s domain. The authors considered the proportional reasoning structures and the content of cert a i n introductory chemistry concepts on the performance of high school students on a chemistry achievement t e s t . Their analysis showed that both content and proportional reasoning were s i g n i f i c a n t l y related to achievement. However, content accounted for a higher percentage (63.4%) of the t o t a l variance of the chemistry achievement te s t scores. Further, a stepwise regression analysis for the prediction of chemistry achievement also revealed that the proportional reasoning tasks did not add s i g n i f i c a n t l y to the regression equation once the scores from the chemistry content subtests had been entered. A similar finding was reported by G r i f f i t h s ( 1 9 7 9 ) when develop-mental l e v e l scores were entered into a regression equation i n which concepts subsumed by the mole had f i r s t been entered. These studies considered only the influence of structure and content on achievement but did not examine the i n t e r r e l a t i o n -ships among them. To examine the relationships among a l l the relevant variables 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 relationships among2 the variables are s p e c i f i e d . Such a model i s developed i n the next section. 54 P a t h - A n a l y t i c Model o f Performance So f a r i n d i v i d u a l s t u d i e s have been c i t e d t o j u s t i f y the need t o s tudy the e f f e c t o f c o n t e n t and p r o p o r t i o n a l r e a -son ing on achievement i n c h e m i s t r y . In the p r e s e n t s e c t i o n , r e s u l t s o f s t u d i e s r e p o r t e d i n the l i t e r a t u r e and the t h e o r e t i -c a l assumptions o f the s t r u c t u r a l t h e o r y o f P i a g e t and the cummulat ive l e a r n i n g t h e o r y o f Gagne w i l l be used t o c o n s t r u c t a c a u s a l model i n v o l v i n g the four v a r i a b l e s - d i r e c t p r o p o r t i o n , i n v e r s e p r o p o r t i o n , subsumed p r e r e q u i s i t e concept s and 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 . S t u d i e s conducted by P i a g e t and h i s a s s o c i a t e s i n d i c a t e t h a t problems o f 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 exper iments on the n o t i o n o f s i m i l a r t r i a n g l e s ( P i a g e t , 1957) t end t o be s o l v e d d u r i n g e a r l y ado le scence ( i . e . around ages 11 and 12) w h i l e problems o f i n v e r s e p r o p o r t i o n a l i t y such as the E q u i l i b r i u m i n the Ba lance Task ( Inhe lder and P i a g e t , 1958) are s o l v e d a t a l a t e r age ( L o v e l l , 1971b). That i s , t h e r e i s a t empora l o r d e r o f development between d i r e c t p r o -p o r t i o n and i n v e r s e p r o p o r t i o n . S t r o n g e r ev idence for' t h i s was g i v e n by Rogers (1977) i n an e x p e r i m e n t a l s tudy aimed a t t e s t i n g f o r a deve lopmenta l sequence o f a c q u i s i t i o n o f p r o p o r -t i o n a l r e a s o n i n g i n 16 yea r o l d h i g h s c h o o l p h y s i c s s t u d e n t s . The Paper C l i p s Task (Wollman and K a r p l u s , 1974) (a f i r s t - o r d e r d i r e c t 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 i n the Ba lance Task (a f i r s t - o r d e r i n v e r s e p r o p o r t i o n a l i t y task) , and two o t h e r t a s k s measur ing second-order d i r e c t and second-order i n d i r e c t p r o p o r t i o n a l i t y were used i n the s t u d y . U s i n g an o r d e r i n g - t h e o r e t i c approach , Rogers found t h a t a p r e r e q u i s i t e sequence e x i s t between the f i r s t o r d e r d i r e c t p r o p o r t i o n a l i t y and the f i r s t o r d e r i n v e r s e p r o p o r t i o n a l i t y . A s tudy by Lunzer and Pumfrey (1966) g i v e s f u r t h e r support t o the above f i n d i n g s t h a t a c q u i s i t i o n o f d i r e c t p r o p o r t i o n a l i t y precedes t h a t o f i n v e r s e p r o p o r t i o n a l i t y . The c o n n e c t i o n between d i r e c t p r o p o r t i o n a l i t y and the subsumed p r e r e q u i s i t e concept s (or content ) i s i n d i c a t e d by the work o f I n g l e and Shayer (1971), H e r r o n . (1975), Wheeler and Kass (1977), and C h i a p p e t t a (1979). Both I n g l e and Shayer (1971) and Herron (1975) p r o v i d e d a t h e o r e t i c a l a n a l y s e s o f the mole c o n c e p t , s to i ch iomet ry^ and o t h e r c h e m i c a l c o n c e p t s . T h e i r a n a l y s e s , based on P i a g e t ' s t h e o r y , i n d i c a t e d t h a t the mole , s t o i c h i o m e t r y , gram/mole c o n v e r s i o n s ^ a n d i n t e r p r e t a t i o n o f e q u a t i o n s r e q u i r e d i r e c t p r o p o r t i o n a l i t y f o r u n d e r s t a n d i n g . C h i a p p e t t a (1979) has a l s o i n d i c a t e d t h a t s t o i c h i o m e t r i c c a l c u l a t i o n problems r e q u i r e ( 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 b e f o r e an adequate u n d e r s t a n d i n g can be a c h i e v e d . The s tudy conducted by Wheeler and Kass (1977) and a l r e a d y d i s c u s s e d , showed 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 p r o p o r t i o n a l r e a s o n i n g and achievement i n a c r i t e r i o n t e s t t e s t i n g the mole , g r a v i m e t r i c s t o i c h i o m e t r y and c h e m i c a l r e a c t i o n s which r e q u i r e d i r e c t p r o p o r t i o n ( Ingle and Shayer , 1971) . The p r o p o r t i o n a l r e a s o n i n g i n the s tudy was measured u s i n g both 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 t a s k s . But as the f i n a l concept s t e s t e d i n v o l v e d o n l y d i r e c t p r o p o r t i o n one c o u l d conc lude t h a t d i r e c t p r o p o r t i o n a l i t y was nece s s a ry f o r achievement i n the f i n a l t e s t . 56 By l o g i c a l extension of the preceding discussion, a relationship between d i r e c t proportionality and volumetric analysis calculations can be established. The application of Piaget's theory suggests that d i r e c t and inverse proportion-a l i t y constitute the formal structures underlying volumetric analysis calculations (Herron, 1975; Wheeler and Kass, 1977). The assumptions involved i n the theory suggests that the solution to such a problem ( i . e . volumetric analysis c a l c u l a -tion) w i l l require the p r i o r presence of d i r e c t and inverse proportional reasoning. Indirect evidence i n favour of the connection between inverse proportionality and volumetric analysis calculations i s provided i n the study by Chiappetta (1974) that looked at the inverse proportional reasoning of subjects (measured using the Equilibrium i n the Balance Task) and performance on a chemical s o l u b i l i t y t e s t . A moderate co r r e l a t i o n (0.60) was found which may indicate the dependence of achievement i n chemical concepts on inverse proportionality. One of the subsumed prerequisite concepts i n the present study i s the calculations involved i n the d i l u t i o n of concentrated solutions. Analysis of such problems using Piaget's theory suggests that inverse proportionality underlies such problems (Herron, 1975). The implication i s that solutions to these problems depend on the understanding of inverse proportionality. No empirical study was found i n the l i t e r a t u r e to .„!••... show the influence of subsumed prerequisite concepts on volumetric analysis calculations per se. However, a number 57 o f s t u d i e s have r e p o r t e d on the i n f l u e n c e o f subsumed concept s on o t h e r c h e m i c a l c o n c e p t s . The study by G r i f f i t h s (1979) on the e f f e c t o f concept s subsumed by the mole concept showed t h a t the subsumed c o n t e n t were r e q u i r e d f o r optimum performance on the problems r e l a t e d t o the mole . The subsumed p r e r e q u i s i t e concept s i n the s tudy were genera ted by u s i n g Gagne 1 s h i e r a r c h i c a l t h e o r y . A s i m i l a r s tudy conducted by Gower, D a n i e l s and L l o y d (1977b) gave i d e n t i c a l r e s u l t s . Okey and Gagne (1970) conducted a s tudy i n which the c r i t e r i o n v a r i a b l e was s o l u b i l i t y p r o d u c t c a l c u l a t i o n s . U s i n g the h i e r a r c h i c a l t h e o r y p o s t u l a t e d by Gagne, they found t h a t the d i f f i c u l t i e s encountered by the h i g h s c h o o l c h e m i s t r y s u b j e c t s used i n the s tudy were due to l a c k o f the subsumed p r e r e q u i s i t e concept s h y p o t h e s i z e d i n the s t u d y . The r e s u l t s o f o t h e r s t u d i e s (Duncan and Johns tone , 1971; N o v i c k and M e n i s , 1976; Ka s s , 1977) i n d i c a t e t h a t succes s i n a c h e m i c a l t a sk depends on the p r i o r mastery o f p r e r e q u i s i t e c o n c e p t s . From these s t u d i e s , i t can be h y p o t h e s i z e d t h a t knowledge o f subsumed p r e r e q u i s i t e concept s i s n e c e s s a r y a l t h o u g h not s u f f i c i e n t f o r s u c c e s s f u l p e r -formance 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 . Also , the b a s i c as-sumption o f the Gagnean t h e o r y t h a t the l e a r n i n g o f a h i g h e r o r d e r concept ( e . g . v o l u m e t r i c a n a l y s i s ) depends on the l e a r n i n g o f the lower o r d e r concept s subsumed by i t , l ends suppor t t o t h i s h y p o t h e s i s . A d i a g r a m a t i c r e p r e s e n t a t i o n o f the i n t e g r a t e d model-as p o s t u l a t e d above i s g i v e n i n F i g u r e 4. T h i s model w i l l be r e f e r r e d t o as the proposed i n t e g r a t e d mode l . The o m i s s i o n 58 Subsumed Concepts 3 F i g u r e 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 59 o f the 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 and p e r f o r -mance 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 n the i n t e g r a t e d model i s e s s e n t i a l l y a t h e o r e t i c a l a s s e r t i o n ; s i n c e i t i s subsumed by a l l the o t h e r v a r i a b l e s i n the i n t e g r a t e d mode l , these v a r i a b l e s s h o u l d mediate the i n f l u e n c e o f 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 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 . As shown i n t h i s r e c u r s i v e mode l , d i r e c t p r o p o r t i o n a l i t y i s an exogeneous v a r i a b l e ( i . e . i t i s not i n f l u e n c e d by o t h e r v a r i a b l e s i n the model) w h i l e i n v e r s e p r o p o r t i o n a l i t y , sub-sumed concept s and 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 are endogeneous v a r i a b l e s . S i n c e i t i s never p o s s i b l e t o account f o r the t o t a l v a r i a n c e o f a v a r i a b l e , r e s i d u a l v a r i a b l e s R^ R3 and R 2 are i n t r o d u c e d i n t o the i n t e g r a t e d model t o i n d i c a t e the e f f e c t o f v a r i a b l e s not i n c l u d e d i n the i n t e g r a t e d model such as p r i o r e x p e r i e n c e o f s u b j e c t s . The f i g u r e uses a s t a n d a r d c o n v e n t i o n i n the m e t h o d o l o g i c a l l i t e r a t u r e o f r e p r e s e n t i n g observed o r measured v a r i a b l e s by square s . A l s o by c o n v e n t i o n s i n c e a r e c u r s i v e model i s p roposed , paths i n the form o f u n i d i r e c t i o n a l arrows are drawn t o i n d i c a t e the d i r e c t i o n o f the c a u s a l i n f l u e n c e w i t h the c a u s a l f low o r i g i n a t i n g i n the v a r i a b l e s taken as " cause s " o r e x p l a n a t o r y and y i e l d i n g the v a r i a b l e s t aken as e f f e c t s ( K e r l i n g e r and Pedhazur , 1973, P .308). The use o f the word "cause" i n p a t h a n a l y s i s i s o n l y c o l l o q u i a l ; i t " i s meant t o p r o v i d e no p h i l o s o p h i c a l meaning beyond a shor thand d e s i g n a t i o n f o r a h y p o t h e s i z e d unobserved p r o c e s s " ' . (Bent le r , 1980) . Thus o t h e r words l i k e e x p l a n a t o r y o r p roce s s c o u l d be used f o r the same end . 6 0 Studies Related to  the Secondary Questions i n t h i s Study Questions posed i n the secondary part of the study re l a t e to the i d e n t i f i c a t i o n of s p e c i f i c student d i f f i c u l t i e s . For convenience, these d i f f i c u l t i e s can be grouped into those which re l a t e to a misunderstanding of chemical concepts ( i . e . conceptual d i f f i c u l t i e s ) and those related to the place of laboratory work. Studies w i l l be reviewed which:;relate to these broad areas. A number of studies have been reported i n the l i t e r a -ture which examined the conceptual d i f f i c u l t i e s that students have on chemical concepts. These studies can be grouped into those which assess the conceptual d i f f i c u l t i e s of the students from t h e i r written work and those which employ c l i n i c a l interviews. The former include the studies by Duncan and Johnstone ( 1 9 7 3 ) , Doran ( 1 9 7 2 ) , Wheeler and Kass ( 1 9 7 8 ) , Johnstone, MacDonald and Webb ( 1 9 7 7 ) and Rowell and Dawson ( 1 9 8 0 ) . The study by Duncan and Johnstone ( 1 9 7 2 ) which was reviewed e a r l i e r employed multiple choice techniques to assess secondary school students 1 d i f f i c u l t i e s with concepts including the mole, stoichiometry and volumetric analysis c a l c u l a t i o n s . The studies by Johnstone, MacDonald and Webb ( 1 9 7 7 ) and Wheeler and Kass ( 1 9 7 8 ) also examined the d i f f i c u l t i e s of secondary school students on chemical equilibrium problems using multiple choice t e s t s . Doran ( 1 9 7 2 ) also applied a multiple response format to i d e n t i f y the misconceptions of his subjects on the part i c u l a t e nature of matter. 61 The above s t u d i e s have t h e r e f o r e a s se s sed the c o n c e p t u a l d i f f i c u l t i e s o f s u b j e c t s by d e v e l o p i n g i t em d i s t r a c -t o r s a c c o r d i n g t o p rede te rmined m i s c o n c e p t i o n c a t e g o r i e s . As was noted by Wheeler and Kass ( 1 9 7 8 ) , the i n t e r p r e t a t i o n o f the responses t o 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 t e s t s p r e s e n t s c e r t a i n problems i n t h a t a s u b j e c t may o b t a i n a c o r r e c t answer by gues s ing or by a r g u i n g i n c o r r e c t l y . The s u b j e c t can a l s o o b t a i n a p a r t i c u l a r i n c o r r e c t response through a v a r i e t y o f d i f f e r e n t pa th ways.. Thus , the use o f m u l t i p l e -c h o i c e t e s t s t o i d e n t i f y the c o n c e p t u a l e r r o r s o f s tudent s may not r e v e a l much about the s u b j e c t s ' d i f f i c u l t i e s . S t u d i e s a r e , t h e r e f o r e , needed which examine the c o n c e p t u a l d i f f i c u l -t i e s o f s tudent s u s i n g o t h e r approaches . The s tudy by Rowel l and Dawson (1980) f a l l s i n t h i s domain. These au thor s i d e n t i f i e d the c o n c e p t u a l d i f f i c u l t i e s t h a t secondary s c h o o l s tudent s have w i t h s t o i c h i o m e t r y problems i n v o l v i n g the mole by examining the w r i t t e n work o f the s t u d e n t s . They d i d not employ m u l t i p l e - c h o i c e i t e m s . The s tudent s were g i v e n the q u e s t i o n and were r e q u i r e d to show t h e i r c a l c u l a t i o n s . The s teps used by the s tudent s i n t h e i r w r i t t e n c a l c u l a t i o n s were examined to i d e n t i f y the e r r o r s made by them. The approach used by Rowel l and Dawson a l l o w e d them t o i d e n t i f y , the c o n c e p t u a l e r r o r s made by the s t u d e n t s . T h i s approach shou ld be a p p l i e d t o o t h e r c h e m i c a l problems such as 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 s i n c e Rowel l and Dawson (1980) used i t o n l y f o r s t o i c h i o m e t r i c problems which are p r e r e q u i s i t e t o 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 . 62 The s t u d i e s which employed c l i n i c a l i n t e r v i e w s t o a s ses s the c o n c e p t u a l d i f f i c u l t i e s o f s u b j e c t s i n c h e m i s t r y are e x a m p l i f i e d by the s t u d i e s done by Nov ick and h i s c o l l e a g u e s (Novick and M e n i s , 1976; Nov ick and Nussbaum, 1978) . C l i n i c a l i n t e r v i e w s have a l s o been used t o i d e n t i f y the c o n c e p t i o n s o f c h i l d r e n about phenomena i n o t h e r f i e l d s ( e . g . E r i c k s o n , 1975; A n d e r s o n , 1965; Kargbo, Hobbs and E r i c k s o n , 1980) . The s tudy by Nov ick and Menis (1976) which was p r e -v i o u s l y d i s c u s s e d employed s t r u c t u r e d i n t e r v i e w s t o i d e n t i f y the c o n c e p t u a l d i f f i c u l t i e s t h a t the secondary s c h o o l s u b j e c t s i n the s tudy had w i t h the mole c o n c e p t . A s i m i l a r procedure was used by N o v i c k and Nussbaum (1978) to examine the d i f f i -c u l t i e s t h a t secondary s tudent s had i n u n d e r s t a n d i n g the p a r t i c u l a t e na ture o f m a t t e r . The au thor s noted t h a t t h i s approach was u s e f u l , i n r e v e a l i n g the c o n c e p t u a l d i f f i c u l t i e s o f the s u b j e c t s s i n c e i t a l l o w e d the p r o b i n g o f the i n i t i a l re sponses o f t h e ^ s u b j e c t s . A l t h o u g h , S u l l i v a n (1967) has emphasized the need f o r the a p p l i c a t i o n o f the c l i n i c a l i n t e r v i e w t o i n v e s t i g a t e s p e c i f i c s c i e n c e c o n c e p t s , the p a u c i t y o f r e s e a r c h s t u d i e s u t i l i z i n g c h e m i c a l concept s i n i n t e r v i e w s i t u a t i o n s i s n o t a b l e . The need , t h e r e f o r e , e x i s t s f o r s t u d i e s which employ t h i s approach t o i n v e s t i g a t e the concep-t u a l d i f f i c u l t i e s w i t h c h e m i c a l concept such as v o l u m e t r i c a n a l y s i s . The above s t u d i e s by N o v i c k and h i s c o l l e a g u e s employ ing a ques t ion-and-answer t e c h n i q u e d u r i n g the c l i n i c a l i n t e r v i e w can be d i f f e r e n t i a t e d from c l i n i c a l i n t e r v i e w s which employ the ' t a l k i n g a l o u d 1 o r ("concurrent v e r b a l i z a t i o n " ( E r i c s s o n and Simon, 1 9 8 0 ) t e c h n i q u e . However, a c c o r d i n g t o E r i c s s o n and Simon ( 1 9 8 0 ) b o t h t e c h n i q u e s can produce r e l i a b l e i n f o r m a t i o n about a s u b j e c t ' s c o g n i t i v e p r o c e s s e s . They a l s o s t a t e d t h a t the demand f o r t h i n k i n g a l o u d d i d not a f f e c t the p r o c e s s i n g a b i l i t i e s o f s u b j e c t s . The t a l k i n g a l o u d t e c h n i q u e s has been used s u c c e s s -f u l l y t o i d e n t i f y the s t r a t e g i e s or c o g n i t i v e p roce s se s o f s u b j e c t s on c e r t a i n mathemat ica l problems ( e . g . Pa ige and Simon, 1 9 6 6 ; Newel l and Simon, 1 9 7 2 ; F l a h e r t y , 1 9 7 4 ) . More r e c e n t l y , Gorodetsky and Hoz ( 1 9 8 0 ) a p p l i e d i t t o c e r t a i n s c i e n c e prob lems . However, the au thor s were more i n t e r e s t e d i n the type and f requency o f concept s used by t h e i r s u b j e c t s . They were not i n t e r e s t e d i n the s t r a t e g i e s used by the s u b j e c t s t o a r r i v e a t s o l u t i o n s . However, t h e i r r e s u l t s i n d i c a t e t h a t t h i s approach c o u l d be used t o a s ses s the s t r a t e g i e s used by s tudent 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 p rob lems . S t u d i e s i n v e s t i g a t i n g s t u d e n t s ' d i f f i c u l t i e s w i t h v o l u m e t r i c a n a l y s i s l a b o r a t o r y work and the use o f the l a b o r a t o r y work t o a s s i s t s tudent s i n overcoming t h e i r concep-t u a l d i f f i c u l t i e s i n v o l u m e t r i c a n a l y s i s are s i m i l a r l y l a c k i n g . I n f a c t , a r e v i e w o f the l i t e r a t u r e by Doran ( 1 9 7 8 ) i n d i c a t e s t h a t s t u d i e s examining the l a b o r a t o r y t e c h n i q u e s o f c h e m i s t r y s tudent s i n g e n e r a l are l a c k i n g . The o n l y s t u d i e s r e p o r t e d i n the l i t e r a t u r e which d e a l t w i t h the l a b o r a t o r y s k i l l s o f c..^ c h e m i s t r y s u b j e c t s were those by E g l e n and Kempa ( 1 9 7 4 ) and Johnstone and McCallum (1972; c i t e d i n Johnstone and Sharp, 1979). The study by Eglen and Kempa (1974) sought to examine the degree of concordance among teachers 1,judgments of a student's manipulative laboratory s k i l l s , as presented on a video-tape. Different assessment techniques such as chec k l i s t s andiopen-ended schedules were used by the teachers to compare the r e l a t i v e effectiveness of each of these techniques. This study did not seek to assess or investigate the laboratory s k i l l s per se of the students. However, studies have been conducted i n other science areas to assess the ac q u i s i t i o n of laboratory techniques (e.g. Kruglak, 1954; Tamir and Glassman, 1970) . In some of these studies (e.g. Tyler, 1942) chec k l i s t s were used to assess the s k i l l s of the students while i n others (e.g. Kruglak, 1954) laboratory p r a c t i c a l examinations were employed i n a group s i t u a t i o n . The r e s u l t s of these studies indicate that while certain laboratory s k i l l s are learned quite well, others are poorly learned (Johnstone and Sharp, 1979; Johnstone and Wham, 1979; Doran, 1978). Johnstone and Wham (1979) attribute t h i s to the i n s u f f i c i e n t emphasis on the mastery of s k i l l s . Doran (1978) suggests that t h i s might be due to the emphasis each teacher gives to students' equipment manipulation and laboratory techniques. Although the above r e s u l t may be generalized to include chemistry students, the paucity of studies dealing with laboratory s k i l l s i n chemistry suggest the need for studies which attempt to assess laboratory s k i l l s such as those found i n t i t r a t i o n . 6 5 L a c k i n g i n the l i t e r a t u r e are s t u d i e s r e l a t e d t o the use o f 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 knowledge. However, s i n c e t h i s i s r egarded by c h e m i s t r y t e a c h e r s as an impor tant o b j e c t i v e f o r l a b o r a t o r y work i t seems impor tant f o r s t u d i e s t o be conducted t o a s ses s the c l a i m . In the next c h a p t e r , the da ta c o l l e c t i o n i n s t r u m e n t s and the p i l o t s tudy w i l l be d i s c u s s e d . 66 CHAPTER III PILOT TESTING OF INSTRUMENTS AND PROCEDURES Introduction The previous two chapters dealt with the s p e c i f i c a -t i o n of the questions posed i n t h i s study and the rationale drawn from the l i t e r a t u r e to support the study of these ques-tions. In t h i s chapter, the instruments and the p i l o t study are discussed. The instrumentation and p i l o t t esting involved several stages: preparation of the instruments, pretesting the instruments and other a n a l y t i c a l techniques i n the p i l o t study, analyzing the re s u l t s of the study, and refinement of the instrument and techniques for use i n the main study. Instruments To obtain data for the variables i n the proposed integrated model, three separate t e s t instruments were developed. Two of these instruments - Subconcepts Test and Volumetric Analysis Test were used to measure prerequisite concepts and achievement in.volumetric analysis calculations respectively. The t h i r d t e s t , the Classroom Proportionality Test was used to measure d i r e c t and inverse proportionality. 6 7 On each test, the following information was c o l l e c t e d : ( 1 ) gender, ( 2 ) date of b i r t h , ( 3 ) name, and ( 4 ) school. In addition to the above tests, a c h e c k l i s t and protocols were developed to assess the q u a l i t a t i v e questions posed i n the study. A b r i e f description of each of the above instruments follows. Classroom Proportionality Test This i s a 14-item group administered test used i n assessing the proportional reasoning of subjects. I t consists of two parts - a d i r e c t p roportionality subtest and an inverse proportionality subtest. Since the subtests assess two. variables of intere s t in the study they are described separately. The items i n the inverse proportionality subtest were derived from a classroom t e s t of formal operations developed by Lawson ( 1 9 7 7 , 1 9 7 8 ) . Lawson 1s test was chosen as a source of items because i t i s "at present the most appropriately validated t e s t " (Nagy and G r i f f i t h s , 1 9 7 9 ) with f a i r l y high r e l i a b i l i t y (KR-20 i s . 7 8 ) . In Lawson 1s test only two items were used to investigate inverse proportionality since his objective was to assess a wider range of formal operational thought. However, i n the present study, i n addition to the 1 : 2 and 2 : 3 situations investigated by Lawson, the inverse proportionality items were extended to include the following r a t i o situations - 1 : 3 , 3 : 4 , 3 : 5 , 1:4 and 1 : 5 . That i s , seven items were constructed for the inverse proportions 68 a l i t y subtest. Following the procedure used by Lawson ( 1 9 7 8 ) and Shayer and Wharry ( 1 9 7 5 ) , each item involved a demonstra-t i o n using a balance beam and hanging weights (Inhelder and Piaget, 1 9 5 8 ) . For each item, the demonstration was used to pose a question to the classras a whole or c a l l for a prediction. Each student responded to the questions i n writing in his own test booklet. The te s t booklets contained only the questions followed by a number of possible answers. Students were instructed to respond by checking the box next to the best answer and then to explain why they chose the answer they had for each question. As noted by Lawson ( 1 9 7 7 ) , t h i s demonstration procedure retains some important aspects of the c l i n i c a l interview employed by Inhelder and Piaget ( 1 9 5 8 ) (e.g. asking the-.students to explain t h e i r responses)_ while allowing for a large number of students to be tested i n a r e l a t i v e l y short time using a substantial number of problems. The seven items i n the d i r e c t proportionality subtest involved seven demonstrations using four p l a s t i c graduated cylinders of d i f f e r e n t diameters and a jar of coloured water. These items were also obtained from Lawson's test (Lawson, 1 9 7 7 , 1 9 7 8 ) . Apart from the 2 : 3 s i t u a t i o n investigated by the two items i n Lawson 1s te s t , the following r a t i o situations were investigated — 2 : 5 , 1 : 3 and 1 : 2 . With the exception of the l a s t s i t u a t i o n , each s i t u a t i o n involved two demonstrations and hence two test items. For each s i t u a t i o n , subjects were allowed enough time to complete the item before the next demonstration was performed. 69 The v a l i d i t y of the items was established by Lawson (1977). He compared subject's performance on the written task with t h e i r performance on an interview task, the Equilibrium i n the Balance Task. Performance on the Classroom Proportionality Test was scored as follows: Those who selected the correct answer and reasoned using r a t i o or proportion were awarded three points. Those who used the correct reasoning i . e . , r a t i o or proportion but selected the wrong answer were awarded two points. Subjects who selected the correct answer but used additive or any other reasoning or provided no reason were given one point while subjects who used wrong reasoning and selected a wrong alterna-t i v e were scored zero. This was the same scoring procedure used by Lawson (Lawson, Nordland and DeVito, 1974). Subconcepts Test This was a test designed to test the knowledge of the content represented by the subsumed prerequisite concepts and hypothesized to be required for doing calculations i n volu-metric analysis. I t consisted of a series of subtests representing each of the seven subordinate prerequisite concepts represented i n Figure 5. The subordinate concepts i n the figure were derived from the one proposed by Gower, Daniels and Lloyd (1977a). The only important difference between t h i s figure and the one proposed by Gower, Daniels and Lloyd (see Figure 3), i s that subordinate concepts requiring v e r b a l i z a -t i o n of d e f i n i t i o n s such as 'concentration' and 'molarity conventions' were not included i n Figure 5. This was done for 7 0 DETERMINE AMOUNT OF R E -ACTANT FROM CONCENTRATION OF ANOTHER C A L C U L A T E CONCENTRATION OF SOLUT IONS FROM MASSj OF SUBSTANCES CONVERT SOLUT IONS FROM ONE CONCENTRA-T ION TO ANOTHER C A L C U L A T I O N S FROM EQUATIONS g/mo 1e CONVERS ION WRITE EQUATIONS INTERPRET EQUATIONS F i g u r e 5-H i e r a r c h i c a l A n a l y s i s o f T i t r a t i o n C a l c u l a t i o n s ( M o d i f i e d ) ( G o w e r , D a n i e l s , a n d L l o y d , 1977a) 71 two r e a s o n s : (1) t o decrease the l e n g t h o f the t e s t c o n s t r u c t e d because o f t ime l i m i t a t i o n s i n the s c h o o l s , and (2) t o ensure t h a t o n l y concept s o f h i g h e r o r d e r are i n c l u d e d s i n c e v e r b a l i z a -t i o n o f d e f i n i t i o n s does not ensure u n d e r s t a n d i n g . A d e s c r i p t i o n and i l l u s t r a t i v e example o f each o f the s u b o r d i n a t e concept s l a b e l l e d A t o H i n F i g u r e 6 i s p r e s e n t e d below: A . Determine the c o n c e n t r a t i o n o r mass o f one r e a c t a n t i n v o l v e d i n a t i t r a t i o n from the c o n c e n t r a t i o n o f another r e a c t a n t and the r e a c t i o n s t o i c h i o m e t r y . Example: "What i s the m o l a r i t y o f a h y d r o c h l o r i c a c i d s o l u t i o n 30.0 mL of which i s n e u t r a l i z e d by 48.0 mL o f 0.10 M sodium h y d r o x i d e ? ' B . 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 s o l u t i o n s from the mass o f the s u b s t a n c e s . Example: ' C a l c u l a t e the m o l a r i t y o f a s o l u t i o n c o n t a i n -i n g 5.0 mg o f NaCl per mL o f s o l u t i o n . ' C . Conver t a s o l u t i o n from one c o n c e n t r a t i o n t o a n o t h e r . Example: ' C a l c u l a t e the volume o f 11.70 M h y d r o c h o l o r i c a c i d s o l u t i o n t h a t must be measured out t o p repare 250 mL o f a 0.20 M s o l u t i o n ? 1 ' D. C a l c u l a t e the r e l a t i v e mass o r moles o f r e a c t a n t s and p r o d u c t s from b a l a n c e d e q u a t i o n s . Example: (a) 'How many grams o f CC^ are produced when 50.0 g o f CaCC>3 i s decomposed?' 72 (b) ' C a l c u l a t e the grams o f A g C l formed when 0.500 l i t e r o f 0.10 M AgN0 3 r e a c t s w i t h an excess o f N a C l s o l u t i o n . 1 E . C o n v e r t the mass o f an element or compound t o i t s number o f mole s , and v i c e - v e r s a . Example: 'How many moles o f NaCl are p r e s e n t i n 29.Og o f NaCl?' F . C a l c u l a t e the masses o f d i f f e r e n t c h e m i c a l e lements o r compounds c o n t a i n i n g the same or p r o p o r t i o n a t e numbers o f atoms or m o l e c u l e s . Example: ' C a l c u l a t e the mass o f copper r e p r e s e n t i n g the same number o f moles as 24.0 grams o f c a r b o n . ' G. W r i t e 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 u s i n g c h e m i c a l symbols . Example: 'Ba l ance the f o l l o w i n g c h e m i c a l e q u a t i o n : 7 CO + o 2 -» c o 2 ' H . I n t e r p r e t 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 i n terms o f number o f mole s , m o l e c u l e s o r atoms o f c h e m i c a l s p e c i e s i n v o l v e d . Example: 'What does the f o l l o w i n g e q u a t i o n mean: C2H5OH + 30 2 —> 2 C0 2 + 3 H 20' Each s u b o r d i n a t e concept was r e p r e s e n t e d by f o u r m u l t i p l e c h o i c e ( f i v e - r e s p o n s e ) q u e s t i o n s , thus y i e l d i n g 28 i tems f o r the e n t i r e t e s t . The i tems i n the t e s t were o b t a i n e d from two s o u r c e s . The f i r s t source was a t e s t used by Wheeler and Kass (1977) t o a s ses s knowledge o f i n t r o d u c t o r y 7 3 chemical concepts such as the mole and stoichiometry. The other source was a test used by Johnstone and Duncan ( 1 9 7 3 ) to i d e n t i f y students d i f f i c u l t i e s with certain chemical calculations including volumetric analysis, stoichiometry and gram mole conversions, The score of each student was indicated by the number of correct answers given for the items i n the t e s t . Volumetric Analysis Test This was a written test designed to test subject's a b i l i t y to do volumetric analysis c a l c u l a t i o n s . Eight items i n the test were selected from Duncan and Johnston's ( 1 9 7 3 ) t e s t . Similar items were b u i l t to increase the t o t a l number of items to f i f t e e n . The t e s t items required subjects to calculate the volume, concentration or mass of one of the reactants involved i n a t i t r a t i o n . The subjects were asked to respond to the items by showing t h e i r answers as well as the steps used i n t h e i r c a l c u l a t i o n s . Achievement i n t h i s test was assessed by the number of correct answers obtained i n the test independent of the procedures used. However, the calculations were examined for any conceptual errors that the students may have made. It was also used to i d e n t i f y those who used algorithms with understanding and those who used algorithms without understanding. In the study, using algorithms with understanding was defined as employing the correct stoichiometric 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 referred to the use of 1 : 1 and 1 : 2 s t o i c h i o -74 m e t r i c r a t i o s i n t e s t i tems r e q u i r i n g them. Those who c o r r e c t l y employed t h i s a l g o r i t h m were r e f e r r e d t o as 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 . U s i n g a l g o r i t h m s w i t h o u t under-s t a n d i n g was d e f i n e d as employing i n c o r r e c t s t o i c h i o m e t r i c o r mole r a t i o i n c o m p u t a t i o n s . 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 o f 1:1 mole r a t i o s i n more than one computa t ion r e q u i r i n g o t h e r mole r a t i o s . S u b j e c t s u s i n g such s t r a t e g i e s were c o n s i d e r e d t o be 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 . In a d d i t i o n t o g a t h e r i n g p s y c h o m e t r i c da ta on the above i n s t r u m e n t s , the p i l o t s tudy was a l s o used t o a s ses s the e f f e c t i v e n e s s o f a number o f o t h e r a n a l y t i c a l t e c h n i q u e s used i n the s t u d y . 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 Two major approaches have been u t i l i z e d i n a s s e s s i n g s tudent l a b s k i l l s . These are c h e c k l i s t s o r r a t i n g s c a l e s ( e . g . Ty ler . . , 19 .42 ; l A l l e h y ;197.2) and l a b o r a t o r y p r a c t i c a l exami-n a t i o n s ( e . g . K r u g l a k , 1954) . A c c o r d i n g to Doran (1978), the c h e c k l i s t approach i s a p p r o p r i a t e f o r i d e n t i f y i n g the m a n i p u l a -t i v e t e c h n i q u e s used by s tudent s i n an i n t e r v i e w s i t u a t i o n . The c h e c k l i s t approach was t h e r e f o r e employed i n t h i s s t u d y . The t a sk t o which the c h e c k l i s t was a p p l i e d i s d e s c r i b e d i n the next s e c t i o n . The i tems i n the c h e c k l i s t were based on the Manual S k i l l s c a t e g o r y r e f e r r e d t o by K l o p f e r (1971) i n h i s d i s c u s s i o n o f the e v a l u a t i o n o f l e a r n i n g i n s c i e n c e . K l o p f e r i d e n t i f i e d two s u b c a t e g o r i e s under the Manual S k i l l s . These a r e : (a) development o f s k i l l s i n u s i n g common l a b o r a t o r y equipment and (b) performance o f common l a b o r a t o r y t e c h n i q u e s w i t h c a r e and s a f e t y . W h i l e the former i s concerned w i t h how the s tudent m a n i p u l a t e s v a r i o u s equipment , the l a t t e r i s concerned w i t h the c a r r y i n g out o f a s e r i e s o f m a n i p u l a t i o n s t o r e a c h a d e s i r e d g o a l . Examples under the f i r s t subcategory i n c l u d e m a n i p u l a t i n g g l a s sware , such as p i p e t t e s and b u r e t t e s . An example under the second subcategory would i n c l u d e the sub-s k i l l s r e q u i r e d by a s tudent when d e t e r m i n i n g the volume o f base r e q u i r e d to r e a c t 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 a c i d i n t i t r a t i o n . Some of these s u b s k i l l s a re s e t t i n g up the exper iment , add ing an i n d i c a t o r , and u s i n g a whi te background t o d e t e c t c o l o u r changes . A c c o r d i n g t o K l o p f e r (1971) " s u c c e s s f u l performance o f these and o t h e r t e c h n i q u e s c a l l s  f o r them t o be done c a r e f u l l y , so t h a t good r e s u l t s a re : •, o b t a i n e d , and t o be c a r r i e d out w i t h s u f f i c i e n t a t t e n t i o n to s a f e t y t o p r e v e n t i n j u r i n g e i t h e r the equipment or the exper imenter " ( i t a l i c s added) . T h i s s tatement about good r e s u l t s and s a f e t y r e q u i r e d the s t u d e n t , f o r example, t o e x e r c i s e c a u t i o n near the end p o i n t o f the r e a c t i o n and a l s o t o use f u n n e l s i n t r a n s f e r r i n g s o l u t i o n s . D u r i n g the t i t r a t i o n , t h i s c h e c k l i s t was used by the i n v e s t i g a t o r t o i d e n t i f y the s t u d e n t s ' manual s k i l l s by c h e c k i n g o f f the i t e m s . In s e c t i o n A o f the c h e c k l i s t , the o v e r a l l s k i l l used i n h a n d l i n g the b u r e t t e or p i p e t t e was summarized as i n a d e q u a t e , adequate or s u p e r i o r . I f each o f the t h r e e b e h a v i o u r s under:: a g i v e n s e c t i o n , say the b u r e t t e , were 7 6 checked "yes", the subject was regarded as displaying superior s k i l l s . If only two of the included behaviours were present the subject was regarded as having adequate s k i l l s i n handling the burette. However, i f only one or none of the behaviours was checked "yes" the subject was regarded as having inadequate s k i l l s i n handling the burette. The Interview Task To assess the subjects understanding of the concepts involved i n an acid-base t i t r a t i o n an ind i v i d u a l interview approach was favoured since i t allows one to probe further into a subject's i n i t i a l responses. Following Piaget's pro-cedure, a concrete task — the t i t r a t i o n of HC1 against NaOH — was used as the basis for e l i c i t i n g the subject's responses. The task required the subject to determine the concentration of the HC1 given "the"concentration of NaOH. This task was broken down into two parts. The f i r s t part was to f i n d the volume of base required to neutralize 25 ml of a given acid while the second part required the computation of the concentration of the acid from the data obtained from the f i r s t part. The subjects were supplied with a l l the solutions needed for the t i t r a t i o n including one indicator and a white sheet of paper. The solutions were HC1 (approximately 0.1 M), and a d i l u t e NaOH solution. The indicator used was bromthymol blue. The apparatus consisted of a 50 ml burette, a 25 ml pipette, a funnel, a burette stand, four 250 ml beakers and four 250 ml conical f l a s k s . The concepts involved i n an acid-base t i t r a t i o n that were of in t e r e s t i n the present study were: pH, concentration, mole concept and i n d i c a t o r behaviour. Also of i n t e r e s t was the subjects' understanding of some of the chemical termi-ologies used i n acid-base t i t r a t i o n s . These included endpoint, stoichiometric point, equivalence point, 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 interview s i t u a t i o n , a number of questions were developed. These questions were based upon the procedures used i n doing t i t r a t i o n and the observations made by the subjects. Questions such as: "Why did you choose t h i s indicator" and "Why does the indicator change colour" were asked, for example. Two chemistry teachers who expressed inte 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 the questions. Their major suggestion was to include a question on the pH at the stoichiometric point for the t i t r a t i o n of a weak acid against a strong base. To i d e n t i f y the strategies used by the students i n t h e i r solution to the volumetric analysis calculations during the interview, the following question was asked: "How can you obtain the concentration of the HCl from the obtained data?" Since the interview task involved the t i t r a t i o n of NaOH against HCl, the reacting mole r a t i o i s 1:1. 78 F u r t h e r m o r e , i n o r d e r t o examine how the s u b j e c t s used such t i t r a t i o n da ta t o 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 a problem s i t u a t i o n r e q u i r i n g a mole r a t i o o t h e r than 1 :1 , t h r e e q u e s t i o n s were u s e d . These a r e : 1. I f the a c i d r e a c t e d w i t h the base i n a mole r a t i o o f 2 :1 r e s p e c t i v e l y , what w i l l be the c o n c e n t r a t i o n o f the a c i d ? 2. I f you used 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 i n s t e a d o f h y d r o c h l o r i c a c i d , what would be the c o n c e n t r a t i o n o f the s u l p h u r i c a c i d ? 3. I f i n s t e a d o f sodium h y d r o x i d e , you had used sodium carbonate t o t i t r a t e h y d r o c h l o r i c a c i d , what would be the c o n c e n t r a t i o n o f the h y d r o c h l o r i c a c i d ? P i l o t Study Purpose The purposes o f the p i l o t s tudy were : (1) to d e t e r -mine the p s y c h o m e t r i c p r o p e r t i e s o f the C la s s room P r o p o r t i o n -a l i t y T e s t (CPT) , Subconcepts T e s t (SCT) and the V o l u m e t r i c A n a l y s i s T e s t (VAT) ; (2) t o determine the a p p r o p r i a t e n e s s o f the procedure used t o i d e n t i f y the c o n c e p t u a l e r r o r s made on the VAT; and (3) t o determine the a p p r o p r i a t e n e s s o f the format o f the i n t e r v i e w on the t i t r a t i o n t a s k . The Sample The p i l o t sample c o n s i s t e d o f 158 s tudent s i n two secondary s c h o o l s i n the Vancouver m e t r o p o l i t a n a rea and 5 f i r s t yea r u n i v e r s i t y c h e m i s t r y s t u d e n t s . The secondary s c h o o l s u b j e c t s be longed t o s i x i n t a c t c l a s s e s i n the two s c h o o l s . F o r t y s u b j e c t s i n two c l a s s e s were e n r o l l e d i n the Grade 11 c h e m i s t r y cour se w h i l e the r e m a i n i n g s u b j e c t s i n four c l a s s e s were e n r o l l e d i n the Grade 12 c h e m i s t r y c o u r s e . The Grade 11 s u b j e c t s were used because o f the d i f f i c u l t y i n o b t a i n i n g Grade 12 c l a s s e s i n o t h e r s c h o o l s . The u n i v e r s i t y s u b j e c t s were asked t o v o l u n t e e r f o r the s t u d y . They were i n c l u d e d i n the s tudy because a t the t ime o f the i n t e r v i e w most Grade 12 c h e m i s t r y s tudent s had not y e t been f o r m a l l y i n t r o d u c e d t o l a b o r a t o r y t e c h n i q u e s i n v o l u m e t r i c a n a l y s i s a l t h o u g h they had done 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 . The u n i v e r s i t y s tudent s were thought t o be s i m i l a r t o Grade 12 c h e m i s t r y s tudent s i n t h a t p r i o r t o t h e i r p a r t i c i p a -t i o n i n the p i l o t s t u d y , they had not done any t i t r a t i o n s a t the u n i v e r s i t y . Data C o l l e c t i o n Procedure In May 1980, the V o l u m e t r i c A n a l y s i s T e s t and the Subconcepts T e s t were a d m i n i s t e r e d t o the Grade 12 c h e m i s t r y s t u d e n t s . The four p a r t i c i p a t i n g t e a c h e r s were in formed o f the purposes o f the p i l o t . They were encouraged t o be c r i t i c a l o f the t e s t s and t o i n d i c a t e any sources o f c o n f u s i o n i n the format o f the t e s t s . In September 1980, the C la s s room P r o p o r t i o n a l i t y T e s t was a d m i n i s t e r e d t o the two Grade 11 c l a s s e s . The two t e a c h e r s f o r these c l a s s e s were asked t o be p r e s e n t d u r i n g the a d m i n i s t r a t i o n o f the t e s t so t h a t they c o u l d p r o v i d e t h e i r 8 0 comments on the inst r u c t i o n s . They were also asked to comment on the format. Immediately aft e r the administration of t h i s t e s t , 8 students were randomly selected from the 4 0 students who took t h i s t e s t and interviewed on the Equilibrium i n the )• Balance Task (Inhelder and Piaget, 1 9 5 8 ) . This was done to assess the v a l i d i t y of the Classroom Proportionality Test. The interview task involved the use of a balance beam and hang-ing weights to assess the proportional reasoning of subjects. The task required the subjects to simply restore the e q u i l i -brium of the balance by putting a sp e c i f i e d weight at the correct point where these would counter-balance those placed by the interviewer. After placing the weights, the subjects were asked to explain t h e i r answers. Three d i f f e r e n t r a t i o situations were used. Performance on the Balance Task was assessed on the basis of the quality of the explanations and the a b i l i t y to hang weights i n the correct locations on the beam of the balance. The v a l i d i t y and r e l i a b i l i t y of the Equilibrium i n the Balance Task have been established by numerous investigators (e.g. Bart, 1 9 7 1 ; DeVries, 1 9 7 4 ; Pumfrey, 1 9 6 8 ; Lawson, Nordland and Devito, 1 9 7 4 ) . Following the interview on the Equilibrium i n the Balance Task, f i v e university chemistry students were i n t e r -viewed on the t i t r a t i o n task. The interview was conducted i n one of the lecture rooms i n the Faculty of Education of the University of B r i t i s h Columbia. The interview was divided into two major sect-ions. 81 In the f i r s t part of the interview a type of Piagetian c l i n i c a l interview technique (Piaget, 1929) was used to e l i c i t the student's understanding of the concepts involved i n t i t r a t i o n . The subject was asked questions as he was performing the t i t r a t i o n . He was also given a sheet of paper on which he could write. Also during t h i s part of the interview a check-l i s t , 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 the manipulatory s k i l l s used by the subject as he performed the t i t r a t i o n . In the second part of the interview, the subject was asked to verbalize his thinking as he performed c a l c u l a ^ tions and to predict the concentration of the acid i n three problem sit u a t i o n s . The t a l k i n g aloud technique or concurrent v e r b a l i z a -t i o n (Ericsson and Simon, 1980) employed i n t h i s study, has been used successfully by a number of investigators (e.g. Flaherty, 1974; Gorodetsky and Hoz, 1980) as an alternative approach to the question-and-answer technique i n i d e n t i f y i n g the problem solving processes of subjects. I t must be noted that although a review a r t i c l e by Nisbett and Wilson (1977) sought to c r i t i z e the use of t h i s technique, a recent reanalysis of t h e i r paper by Ericsson and Simon (1980), using a model of v e r b a l i z a t i o n which predicts the kind of content that can r e l i a b l y be reported, showed that the experimental conditions and procedures used i n most of the studies reviewed by Nisbett and Wilson were such that "verdical reports could hardly be expected"(Ericsson and Simon, 1980) . Using t h e i r 82 model, Ericsson and Simon concluded that verbal reports e l i c i t e d by asking subjects to think aloud produce a valuable and thoroughly r e l i a b l e source of data about cognitive processes. In the present study, after the subjects had completed the t i t r a t i o n they were asked to think aloud as they used the obtained data to calculate the concentration of the acid. The subjects were given sheets of paper with t h e i r names on so that they could show t h e i r calculations on the sheets as they verbalized t h e i r thinking. After c a l c u l a t i n g the concentration of the acid used i n the t i t r a t i o n , the subject was asked to use the data obtained in the t i t r a t i o n to predict the concentration of the acid in three situations requiring a mole r a t i o of 2:1. This was done in order to f i n d out i f problem situations requiring the use of 2:1 reaction mole r a t i o s would present any d i f f i c u l t i e s to the subjects — that i s , whether the subject r e a l i z e d the need to use the correct stoichiometric r a t i o s i n situations requiring t h e i r use. I t was also done to f i n d out i f the same approach used i n the i n i t i a l s i t u a t i o n requiring a 1:1 mole r a t i o was applied i n these sit u a t i o n s . 1. In these predictive situations, the subject was asked to use the volumes obtained i n the t i t r a t i o n and the known concentration of the base. After each prediction, the subject was asked to explain his prediction i r r e s p e c t i v e of whether he showed his ca l c u l a t i o n s . The above procedure, that i s , asking for predictions 83 instead of asking subjects to perform more t i t r a t i o n s was employed because of the length of the interview. The entire interview was audio-taped for subsequent t r a n s c r i p t i o n . Each interview took from 45 minutes to one hour for completion. Scoring In scoring the subtests of the CPT, 3 points were given for those subjects who selected the correct answer and reasoned i n terms of r a t i o or proportion; 2 points were awarded to those who selected the wrong answer but used r a t i o or porpor-t i o n i n t h e i r explanation. Subjects who selected the correct answer but employed additive or any other explanations or no explanation were given a score of one. A zero score was awarded to those who selected the wrong alternative and gave the wrong reasons. The maximum score obtainable on each subtest of the CPT was 21. The scoring procedure used for the Equilibrium i n the Balance Task was similar to that for the CPT. The maximum score for the Equilibrium i n the Balance Task was 9. The scoring procedure used for the VAT and the SCT was to give one point for each correct answer. The maximum score obtainable on the VAT was 15 while the maximum score for the SCT was 28. Data Analysis Analysis was done separately for the test data and the interview data. In preparation of the test data for analysis by computer, the investigator coded the scores on foreran 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 i d e n t i f i e d h i s g rade , c l a s s and s c h o o l . The s core on the i n d i v i d u a l i tems o f the t e s t s n t a k e n by each s tudent was a l s o coded . The coded da ta were key-punched and v e r i f i e d by the Key-punching S e r v i c e s o f the Computer C e n t r e a t 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 . The da ta were hand-checked f o r k e y - p u n c h i n g e r r o r s by the i n v e s t i g a t o r . Item a n a l y s i s was per formed u s i n g the LERTAP (Nel son , 1974) computer program i n o r d e r t o determine the r e l i a b i l i t i e s o f the VAT, SCT,and CPT and t o examine the performance o f the i tems i n each t e s t . The two s u b t e s t s o f the CPT were i t e m - a n a l y z e d s e p a r a t e l y s i n c e they were t r e a t e d as d i f f e r e n t v a r i a b l e s i n the s t u d y . The S t a t i s t i c a l Package f o r the S o c i a l S c i e n c e s program (Nie , e t . a l . 1975) was used t o o b t a i n the c o r r e l a t i o n between the s core s o f the s u b j e c t s -on the CPT and the E q u i l i b r i u m i n the Ba lance Task . The i n t e r v i e w da ta were a n a l y z e d by f i r s t o b t a i n i n g a complete t r a n s c r i p t o f the aud io t a p e s . The responses o f the s u b j e c t s t o each main q u e s t i o n asked i n the s tudy were grouped t o g e t h e r . F o r each q u e s t i o n , responses which were s i m i l a r were c a t e g o r i z e d t o g e t h e r . These c a t e g o r i e s were c a l l e d Response P a t t e r n s . The i n t e r v i e w p r o t o c o l s and the w r i t t e n work (obta ined from the s u b j e c t s when they were asked t o work out s o l u t i o n s t o s e v e r a l v o l u m e t r i c a n a l y s i s problems u s i n g the da ta o b t a i n e d from t h e i r t i t r a t i o n ) were a n a l y z e d t o i d e n t i f y the approach (or s t r a t e g y ) u s e d . As the s tudent s were asked t o t h i n k a l o u d when d o i n g these prob lems , the v e r b a l p r o t o c o l s 8 5 along with t h e i r written responses were used to categorize the strategy they used i n terms of a formula approach or a proportional reasoning approach. The responses obtained when the subjects were asked to predict the concentration of the acid i n three problem situations requiring 2:1 mole r a t i o s were analyzed to see whether correct predictions were made. The manual s k i l l s used by the subjects were examined to f i n d the frequency of occurrence of each s p e c i f i c s k i l l i n the interview sample. F i n a l l y , the written work of the subjects on the VAT was analyzed i n order to i d e n t i f y the conceptual errors made i n t h e i r solutions. For each subject, the in d i v i d u a l steps used to solve each question on the VAT was examined for any errors i n reasoning. Each conceptual error was recorded only once for any p a r t i c u l a r student regardless of whether he had made the same error on two or more other questions i n the tes t . Results and Discussion  Classroom Proportionality Test Item Analysis: The item c h a r a c t e r i s t i c s for the two subtests of the CPT are presented i n Table 3.. The item-subtest correlations revealed that a l l the item-test values for both the Direct Proportionality Subtest and the Inverse Proportion-a l i t y Subtest were po s i t i v e and high. This indicated that those who scored high on the subtest got the ind i v i d u a l items correct. 8 6 Table 3 Test S t a t i s t i c s of the Classroom Proportionality Test SUBTEST A - DIRECT PROPORTION Test mean = 1 7 . 1 0 Highest sicore = 2 1 . 0 0 Hoyt estimate of r e l i a b i l i t y = 0 . 7 9 Standard error of measurement = 2 1 1 0 Standard deviation = 5 . 0 0 Lowest eicore = 4 . 0 0 Test Item D i f f i c u l t y Index (%) Discrimination Index (Point B i s e r i a l ) 1 2 3 4 5 6 7 62, 6 0 , 8 7 . 8 2 . 8 7 . 8 7 . 8 2 . 0 . 7 4 0 . 8 0 0 . 5 3 0 . 6 9 0 . 8 0 0 . 5 9 0 . 6 1 SUBTEST B - INVERSE PROPORTION Test mean = 1 0 . 2 0 Highest score = 2 1 . 0 0 Hoyt estimate of r e l i a b i l i t y = 0 . 8 4 Standard error of measurement = 2 . 2 1 Standard deviation = 5. Lowest score = 1 . 0 0 9 5 Test Item D i f f i c u l t y Index (%) Discrimination Index (Point B i s e r i a l ) 8 4 2 . 5 0 . 4 9 9 4 7 . 5 0 . 6 2 1 0 3 2 . 5 0 . 7 8 1 1 3 2 . 5 0 . 7 8 1 2 5 3 . 0 0 . 8 0 1 3 3 0 . 0 0 . 8 2 1 4 2 5 . 0 0 . 8 3 87 The d i f f i c u l t y indices of the items i n the two sub-tests revealed that the items i n the Inverse Proportionality Subtest were more d i f f i c u l t than those i n the Direct Propor-t i o n a l i t y Subtest. The r e l i a b i l i t i e s for the Direct Proportionality and Inverse Proportionality Subtests were high (.79 and .84, r e s p e c t i v e l y ) . This implies that there i s an acceptable inte r n a l consistency among the items i n each subtest. V a l i d i t y : The Pearson product moment cor r e l a t i o n between the t o t a l scores of the subjects on the interview task - the E q u i l i -brium i n the Balance Task - and the Classroom Proportionality Test was calculated to be 0.91. The close correspondence between the two measures affirms that the Classroom Proportion-a l i t y Test has convergent v a l i d i t y . This implies that the Classroom Proportionality Test can be used i n place of the i n d i v i d u a l interview with reasonable confidence to measure proportional reasoning of subjects. In administering the test one hour was found to be s u f f i c i e n t . The Subconcepts Test The r e s u l t s of the item analysis of t h i s test are given i n Table 4. The item-subtest correlations revealed that almost a l l the items possessed acceptable psychometric properties. Only one item i n the subtest, D i l u t i o n Calculation, was found to have a c o r r e l a t i o n as low as 0.28. Examination of the calculations done by the subjects revealed that they did not read t h i s question c a r e f u l l y . 88 T a b l e 4 T e s t S t a t i s t i c s o f the Subte s t s i n the Subconcepts T e s t Item D i f f i c u l t y Index D i s c r i m i n a t i o n Index (%) ( P o i n t - B i s e r i a l ) SUBTEST A - CONCENTRATION CALCULATION T e s t iiiean = 3.20 S tandard 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 = 0.49 1 82.7 .65 2 93.8 .43 3 85.2 .64 4 58.0 .76 SUBTEST B - DILUTION CALCULATION T e s t mean = 2.02 S tandard d e v i a t i o n = 1 . 0 4 Hoyt e s t i m a t e o f r e l i a b i l i t y = 0.30 5 53.1 .67 6 71.6 .71 7 61.7 .56 8 76.0 .28 SUBTEST C - STOICHIOMETRY T e s t mean = 2.49 S tandard d e v i a t i o n = 1.10 Hoyt e s t imate o f r e l i a b i l i t y = 0.41 9 53.1 .65 10 81.5 .42 11 39.5 .68 12 75.3 .63 SUBTEST D - GRAM/MOLE CONVERSION T e s t mean = 3 . 1 0 S tandard 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 = .43 13 71.6 .68 14 87.7 .61 15 61.7 .68 16 88.9 .44 89 Table 4 continued Item D i f f i c u l t y Index Discrimination Index (%) (Point-Biserial) SUBTEST E - MOLE Test mean = 2.62 Standard deviation = 1.12 Hoyt estimate of r e l i a b i l i t y = .40 17 65.4 .52 18 58.0 .60 19 58.0 .67 20 80.2 .61 SUBTEST F - BALANCING EQUATIONS Test mean = 2.04 Standard deviation = 1.02 Hoyt estimate of r e l i a b i l i t y = .43 21 71.6 .75 22 46.9 .63 23 77.8 .64 24 7.4 .36 SUBTEST G - INTERPRETATION OF REACTIONS Test mean - 2.41 Standard deviation = 1.12 Hoyt estimate of r e l i a b i l i t y = 0.40 25 65.4 .59 26 49.4 .59 27 79.0 .54 28 46.9 .61 R e l i a b i l i t y (Cronbachs alpha) for composite = 0.80 9 0 The r e l i a b i l i t i e s f o r the s u b t e s t s were a l l below 0 . 5 . T h i s was a t t r i b u t e d t o the few items i n each s u b t e s t . A l s o , the low r e l i a b i l i t i e s may be p a r t i a l l y due to the na ture o f the s u b t e s t c o n t e n t . However, the r e l i a b i l i t y o f the t o t a l t e s t (Cronbach ' s composi te a lpha) was 0 . 8 0 which i n d i c a t e d t h a t a l l the s u b t e s t s may have been a s s e s s i n g the same c o n t e n t domain. The one hour a l l o c a t e d t o t h i s t e s t was found t o be s u f f i c i e n t . The V o l u m e t r i c A n a l y s i s T e s t In t h i s t e s t , one i t em was found t o have a c o r r e l a -t i o n o f 0 . 2 0 . F u r t h e r examina t ion o f t h i s i t em suggested t h a t the format o f the q u e s t i o n was l i k e l y c o n f u s i n g t o some o f the s t u d e n t s . One i t em i n the t e s t was answered c o r r e c t l y by a l l s u b j e c t s w i t h the r e s u l t t h a t t h e r e was no d i s c r i m i n a -t i o n on t h i s i t em between the s u b j e c t s who s c o r e d h i g h on the t e s t and those who s c o r e d low on the t e s t . (Table 5) The r e m a i n i n g i tems i n the t e s t were a c c e p t a b l e i n terms o f t h e i r i t em — t o t a l t e s t c o r r e l a t i o n s and d i f f i ^ c u l t y c o e f f i c i e n t s . The r e l i a b i l i t y o f the o v e r a l l t e s t was found t o be 0 . 8 7 . T h i s r e f l e c t s the i n t e r n a l homogeneity o f the i tems i n the t e s t . One hour was found t o be s u f f i c i e n t f o r t h i s t e s t . C o n c e p t u a l E r r o r s S i x types o f c o n c e p t u a l e r r o r s were found i n the s u b j e c t s ' s o l u t i o n s t o the i tems i n the V o l u m e t r i c A n a l y s i s T e s t . These were: ( 1 ) the assumption o f 1 : 1 r a t i o i n a l l 91 T a b l e 5 T e s t S t a t i s t i c s o f the V o l u m e t r i c A n a l y s i s T e s t T e s t mean = 10.26 S tandard d e v i a t i o n = 3.79 H i g h e s t s core = 15 Lowest s core = 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 tandard e r r o r o f measurement = 1.33 T e s t Item D i f f i c u l t y Index D i s c r i m i n a t i o n Index (%) ( P o i n t - B i s e r i a l ) 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 react with any base i n a r a t i o of 1:2; (3) the notion that the concentration of a stock solution of acid i s d i f f e r e n t from the concentration of the portion used i n t i t r a t i o n ; (4) the reversal of the stoichiometric mole r a t i o s ; (5) the writing of incorrect formulas; and (6) the f a i l u r e to write balanced chemical equations i n the problem solutions. The f i r s t conceptual error may be attributed to a number of possible factors including the use of incorrect formulas and the f a i l u r e to write balanced chemical equations for the reactants. The second conceptual error revealed that some of the subjects-failed to discriminate between reactions i n which H2SO4 reacted, i n a 1:2 and 1:1 mole r a t i o s , respectively, with other substances. They only considered the two protons i n H2SO4 without giving equal attention to the nature of the other reacting substances. Once again t h i s d i f f i c u l t y might be attributed to the i n a b i l i t y of some of the subjects to write correct formulas and correct balanced equations for the reacting substances. The t h i r d conceptual error revealed that the subjects were applying a d i r e c t proportional r e l a t i o n to obtain a d i f f e r e n t concentration for the t o t a l solution. These students did not see the invariance of the concentration when only part of i t has been removed. This implied that the sub-jects may not have had a good understanding of what a solution was. However, t h e i r d i f f i c u l t y could also have been viewed as being s t r u c t u r a l i n nature. That i s , there i s a part-whole relationship here. But t h i s i s ..notutheccommonsense one which implied that the whole i s greater than one of i t s parts:, since in t h i s case the concentration of the t o t a l solution was the same as the concentration i n a portion of i t . The fourth conceptual error — reversal of mole r a t i o s — suggested that the subjects who made t h i s error did not recognize the nature of the relationship between the stoichiometric c o e f f i c i e n t s and the moles of substances used i n the reaction. The f i f t h conceptual error — writing incorrect molecular formulas — suggested that the subjects who made th i s error probably did not understand the concept of valency. The written work of the subjects who made t h i s error suggested that some of them, even though they could write the formulas for the compounds involved i n the reaction, could not balance the equation. Other subjects f a i l e d to write equations i n the i r solutions. Laboratory S k i l l s i n T i t r a t i o n The analysis of the laboratory techniques of the f i v e university subjects revealed that three of the subjects who had done more than one t i t r a t i o n i n Grade 1 2 did not have any d i f f i c u l t i e s i n proceeding with the experiment. They displayed reasonably accurate s k i l l s i n the handling and reading of the burette and pipette, used a funnel i n transfer-ing solutions from the reagent bottles and exercised caution in adding the base. The other two students who had done only one t i t r a t i o n l a b i n Grade 1 2 had t o be g i v e n some h e l p . D e s p i t e the p r o p e r s k i l l s d i s p l a y e d i n some areas by most o f them o n l y one s u b j e c t r e a l i z e d the need f o r a whi te background to a i d i n o b s e r v i n g the c o l o u r changes and no s tudent washed down the a c i d s o l u t i o n a d h e r i n g t o the s i d e s o f the f l a s k t o ensure t h a t a l l the a c i d r e a c t e d . Other o b s e r v a t i o n s made were : t h r e e o f the s u b j e c t s asked f o r an e x t r a b u r e t t e t o be used i n measur ing the a c i d s o l u t i o n , some o f the s tudent s i n d i c a t e d t h a t they were not f a m i l i a r w i t h the bromthymol b l u e i n d i c a t o r and hence asked f o r p h e n o l p h t h a l e i n , two s tudent s wrote down the volumes o b t a i n e d i n the t i t r a t i o n o n l y when they were prompted by the i n v e s t i g a t o r . The I n t e r v i e w Task The a n a l y s i s o f the f i r s t p a r t o f the i n t e r v i e w p r o t o c o l s which e x p l o r e d the s u b j e c t s ' u n d e r s t a n d i n g o f t i t r a t i o n concept s r e v e a l e d a number o f areas where the s u b j e c t s d i d not seem t o have c l e a r u n d e r s t a n d i n g . In g e n e r a l , i t was found t h a t a l t h o u g h , the s u b j e c t s had a good u n d e r s t a n d i n g about c o n c e n t r a t i o n and mole s , they d i d not have a c l e a r u n d e r s t a n d i n g o f pH, i n d i c a t o r b e h a v i o u r and the s c i e n t i f i c terms used i n t i t r a t i o n . W i t h r e s p e c t t o mole s , o n l y one o f the f i v e s tudent s i n d i c a t e d t h a t the moles o f the a c i d i n a s o l u t i o n w i l l be lower when d i s t i l l e d water i s added. A l l the o t h e r q u e s t i o n s on moles 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. 9 5 Most of the subjects r e a l i z e d that the pH at the stoichiometric point for the reaction between HC1 and NaOH was 7. They also r e a l i z e d that the pH of the acid increased with the addition of the base. However, only two subjects could calculate the pH of .1 M NaOH or .1 M HC1. Also four of the subjects indicated that the pH for the reaction between acetic acid and NaOH was 7 while one student didn't know. Even though, they knew that the concentration of a solution w i l l change with addition of d i s t i l l e d water, none of them r e a l i z e d that the pH w i l l be higher. Although they seemed to have a clear notion of why indicators are used i n t i t r a t i o n , they did not seem to under-stand the mechanism of indicator action. They also did not seem to know why bromthymol blue i s used i n the t i t r a t i o n . Here, i t must be noted, that since some of the students indicated that they were more f a m i l i a r with phenolphthalein, the r e s u l t could have been d i f f e r e n t i f phenolphthalein had been used. The terminology questions indicated that only one subject could explain what endpoint and stoichiometric point meant. However, a l l of them indicated that they had heard the term stoichiometric point. The d i f f i c u l t i e s of the subjects could be attributed to the fact that such questions are not usually asked during experiments, as was indicated by two students. The analysis of the transcribed verbal protocols and the written work obtained from the subjects as they used 96 t h e i r own data to calculate the concentration of the HC1 indicated that two basic approaches — the Formula Approach and the Proportional Approach — were employed by the subjects in the study. The basic form of the Formula Approach involved the use of the formula: Molarity x Volume = Moles, to calculate the moles of base i n i t i a l l y and then use the value obtained and the mole r a t i o to f i n d the moles of acid. The concentra-ti o n of the acid was then found by a second application of the above formula but in t h i s case using values for the acid. In a variant form of the Formula Approach, one subject used a formula which avoids a double application of the formula used in the basic form of the Formula Approach, namely, molarity of base x volume of base = molarity of acid x volume of acid. Thus, t h i s variant form acutally combines some of the steps i n the basic form. In the basic form of the Proportional Approach, the subject d,i-.d not use an e x p l i c i t formula but instead uses the porportional r e l a t i o n between moles and volume to obtain the moles of base. Using the reaction mole r a t i o s , the moles of acid was then obtained. Thereafter, the same proportional r e l a t i o n between moles and volume was applied to the acid to obtain the concentration. The student who used a variant form of t h i s approach employed a proportional r e l a t i o n s h i p between the volumes and the concentrations of the acid and the base, namely, Volume of acid _ Molarity of base Volume of base Molarity of acid 9 7 This, i n a sense^is a combination of the two proportions used in the basic form of the Proportional Approach. The analysis showed that two students used the basic form of the Formula Approach while one student used the variant form of the Formula Approach. For the remaining two students, one used the basic form of the Proportional Approach while the other used the variant form. The analysis of the protocols on the subjects' prediction of the concentration of the acid from t h e i r data revealed that they had problems with reactions involving 2:1 mole r a t i o s . I t was found that while the i n i t i a l problem involving 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 subjects, three of the subjects f a i l e d to make correct predictions i n the three situations although they used the same approach which gave them the correct answer i n the i n i t i a l problem. I t was conjectured that t h i s d i f f i c u l t y might be due to the i n a b i l i t y to coordinate the stoichiometric mole r a t i o with other sources of information such as the actual moles of base used i n the t i t r a t i o n . However, t h i s i s not clear since the subjects were not asked to explain t h e i r predictions. Revisions of the Instruments  Volumetric Analysis Test Discussions with the teachers involved i n the p i l o t study confirmed the ambiguity of the item i n t h i s test which had an item - t o t a l test c o r r e l a t i o n of 0.20. This item was modified to remove the ambiguity. Also, the one item with-an item-subtest 98 c o r r e l a t i o n of zero was removed from the t e s t because of i t s h i g h d i f f i c u l t y index. I t was however, r e p l a c e d w i t h an item from the t e s t used by Duncan and Johnstone (1973). The f i n a l r e v i s e d V o l u m e t r i c A n a l y s i s T e s t (VAT) a l s o c o n s i s t e d of 15 items (Appendix A ) . I t appeared from the i n t e r v i e w r e s u l t s , d i s c u s s e d above, t h a t the number of l a b o r a t o r y t i t r a t i o n s done by the s u b j e c t s might i n f l u e n c e t h e i r l a b o r a t o r y techniques and p o s s i b l y t h e i r c a l c u l a t i o n s . As such, s u b j e c t s were asked to i n d i c a t e the number of t i t r a t i o n s they had p r e v i o u s l y done on the f r o n t page of the V o l u m e t r i c 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 T e s t (CPT) As a r e s u l t of comments r e c e i v e d from the p a r t i c i p a t -i n g t e a c h e r s some wording changes were made i n the i n s t r u c t i o n s . Emphases were a l s o p l a c e d on c e r t a i n aspects of the t e s t i n g procedure. For example, s u b j e c t s were asked to pay p a r t i c u l a r a t t e n t i o n t o t h e i r e x p l a n a t i o n s . They were a l s o asked to pay a t t e n t i o n w hile the i n v e s t i g a t o r was performing demonstrations. T h i s t e s t and the t e s t i n g procedures are presented i n Appendix B. The Subconcepts T e s t For 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 , the item with a c o r r e l a t i o n of 0.28 was r e t a i n e d . However, d u r i n g the a d m i n i s t r a t i o n of the t e s t , students were asked to read the q u e s t i o n s c a r e f u l l y . The Subconcepts T e s t (SCT) can be found i n Appendix C. 9 9 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 As a r e s u l t o f the o b s e r v a t i o n s made, s tudent s were s u p p l i e d w i t h t h r e e i n d i c a t o r s — bromthymol b l u e , p h e n o l p h -t h a l e i n ^ a n d methy l orange — from which they c o u l d choose one f o r the t i t r a t i o n . A l s o , s i n c e some o f them wanted t o use a b u r e t t e t o measure out the volume o f a c i d , an e x t r a b u r e t t e was i n c l u d e d i n the l i s t o f a p p a r a t u s . Another i t em was i n c l u d e d i n the c h e c k l i s t i n o r d e r t o f i n d out i f the s u b j e c t s r e c o r d e d t h e i r volumes as soon as they were o b t a i n e d . \ The f i n a l c h e c k l i s t , 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 , c o n s i s t e d o f 2 0 i tems (Appendix D ) . The I n t e r v i e w Task The r e v i s i o n o f the i n t e r v i e w p r o t o c o l s r e f l e c t e d i n p a r t , the a d d i t i o n a l i n d i c a t o r s g i v e n t o the s u b j e c t s . A q u e s t i o n was i n c l u d e d which r e q u i r e d the s u b j e c t t o e x p l a i n h i s c h o i c e o f i n d i c a t o r f o r the t i t r a t i o n . Another q u e s t i o n asked the s u b j e c t t o i n d i c a t e whether the o t h e r i n d i c a t o r s not chosen f o r the t i t r a t i o n c o u l d be u sed . Other q u e s t i o n s i n c l u d e d i n the i n t e r v i e w r e q u i r e d the s u b j e c t t o e x p l a i n i f t h e r e was any d i f f e r e n c e i n the r a t e o f change o f the pH a t the b e g i n n i n g o f the t i t r a t i o n and near the e n d p o i n t . W i t h the q u e s t i o n s on t e r m i n o l o g y , the q u e s t i o n on e q u i v a l e n c e p o i n t was dropped s i n c e no s u b j e c t seemed to have heard o f i t even though they had heard o f s t o i c h i o m e t r i c p o i n t . No r e v i s i o n s were made to the q u e s t i o n r e q u i r i n g s u b j e c t s t o use t h e i r t i t r a t i o n da t a to c a l c u l a t e the c o n c e n t r a -100 t i o n of the HC1. However, wi t h regards to the p r e d i c t i o n q u e s t i o n s , a f u r t h e r q u e s t i o n was i n c l u d e d to ask the s u b j e c t s 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 . To improve the concretness of the p r e d i c t i o n ; j q u e s t i o n s , l a b e l l e d s o l u t i o n s of s u l p h u r i c a c i d and sodium carbonate were i n c l u d e d i n the l i s t of s o l u t i o n s . S o l u t i o n s o f a c e t i c a c i d and ammonia were a l s o i n c l u d e d . In g e n e r a l the p i l o t i n t e r v i e w a f f o r d e d an opportu-n i t y t o assess the g e n e r a l format of the i n t e r v i e w and p r o v i d e d some ideas about the understanding and s k i l l s t h a t students b r i n g i n t o a t i t r a t i o n s i t u a t i o n . I t a l s o allowed the i n v e s -t i g a t o r t o i d e n t i f y the approaches used by the s u b j e c t s i n t h e i r c a l c u l a t i o n s . 1 0 1 CHAPTER IV METHODS OF THE STUDY I n t r o d u c t i o n In t h i s c h a p t e r , the in s t rument s u sed , the p o p u l a -t i o n o f i n t e r e s t , the sample, and the procedures employed i n da ta c o l l e c t i o n and a n a l y s e s i n the s tudy are d e s c r i b e d . S ince the p rocedure s f o r the s e l e c t i o n o f the sample o f s tudent s who p a r t i c i p a t e d o n l y i n the model t e s t i n g phase o f the s tudy (completed a l l w r i t t e n ins t ruments ) was c l o s e l y r e l a t e d to the sub-sample s e l e c t e d f o r i n t e r v i e w i n g on the t i t r a t i o n t a s k , the sampl ing procedures and data c o l l e c t i o n procedure s f o r bo th a spec t s o f the s tudy are d e s c r i b e d i n t h i s c h a p t e r . A l s o , the p r e l i m i n a r y r e s u l t s conducted on the t e s t da ta p r i o r t o the use o f p a t h a n a l y s i s are i n c l u d e d i n t h i s c h a p t e r . Ins truments S tudent da ta on 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 r e a s o n i n g , knowledge o f p r e r e q u i s i t e concept s 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 were c o l l e c t e d by a d m i n i s -1 0 2 t e r i n g the Classroom P r o p o r t i o n a l i t y Test, Subconcepts Test, and V o l u m e t r i c A n a l y s i s Test, r e s p e c t i v e l y . V o l u m e t r i c A n a l y s i s T e s t T h i s t e s t , p r e t e s t e d i n the p i l o t study and d e s c r i b e d i n the p r e v i o u s chapter, c o n s i s t e d of 1 5 items. The s u b j e c t s were 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 , volume o r mass of one r e a c t a n t from the known va l u e s of another r e a c t a n t . The t e s t measured the s u b j e c t s ' 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 . Classroom P r o p o r t i o n a l i t y T e s t The two s u b t e s t s of t h i s instrument were a l s o p r e -t e s t e d i n the p i l o t study. Each of the s u b t e s t s c o n t a i n e d seven items. 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 measured d i r e c t p r o p o r t i o n a l reasoning while the s u b t e s t , Inverse Propor-t i o n a l i t y , measured i n v e r s e p r o p o r t i o n a l reasoning. Subconcepts Test The subconcepts t e s t , a l s o p r e t e s t e d i n the p i l o t study, c o n t a i n e d seven s u b t e s t s , each of which was represented by f o u r items. Thus, the t e s t c o n t a i n e d a t o t a l of 2 8 items. T h i s t e s t measured the s u b j e c t s ' knowledge of the p r e r e q u i s i t e concepts 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 . P o p u l a t i o n The present study was an a n a l y t i c a l study p r i m a r i l y designed to i n v e s t i g a t e the d i f f i c u l t i e s t h a t s e n i o r secondary 103 s c h o o l s tudent s face when p e r f o r m i n g 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 . S i n c e v o l u m e t r i c a n a l y s i s was o n l y f o r m a l l y taught i n the Grade 12 c h e m i s t r y . c o u r s e o f s tudy i n B r i t i s h C o l u m b i a , i t was d e c i d e d t h a t s tudent s e n r o l l e d i n the Grade 12 c h e m i s t r y course i n B r i t i s h Columbia would c o n s t i t u t e the g e n e r a l p o p u l a t i o n . However, the o n l y a rea o f B r i t i s h . Columbia t h a t was p r a c t i c a l and a v a i l a b l e t o c a r r y out the s tudy was the Lower M a i n l a n d . Thus f o r the purpose o f t h i s s t u d y , the s tudent s e n r o l l e d i n Grade 12 c h e m i s t r y cour se s i n the Lower M a i n l a n d o f B r i t i s h Columbia c o n s t i t u t e d the t a r g e t p o p u l a t i o n . S i n c e Chemis t ry 12 i s an o p t i o n a l c o u r s e , n o r m a l l y o n l y s e l e c t e d by more a c a d e m i c a l l y - o r i e n t e d s t u d e n t s , these s u b j e c t s c o n s t i t u t e d a s e l f s e l e c t e d g roup . Sample In May 1979, l e t t e r s d e s c r i b i n g the o b j e c t i v e s o f the s tudy and r e q u e s t i n g p e r m i s s i o n t o use s e l e c t e d s c h o o l s i n t h e i r d i s t r i c t were sent to four S c h o o l Boards i n the Lower M a i n l a n d . Two S c h o o l Boards showed immediate i n t e r e s t i n the s t u d y . Schoo l s w i t h i n the j u r i s d i c t i o n o f these S c h o o l Boards were c o n t a c t e d by S c h o o l Board r e p r e s e n t a t i v e s . Of the 21 s c h o o l s c o n t a c t e d , 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 i n the s t u d y . The l o c a t i o n s o f the s c h o o l s were such t h a t a wide g e o g r a p h i c a l r e p r e s e n t a t i o n was o b t a i n e d i n the sample. F u r t h e r m o r e , the catchment areas o f these s c h o o l s r e p r e s e n t e d a wide d i v e r s i t y o f soc io -economic l e v e l s as a s se s sed by the S c h o o l Board o f f i c i a l s . The s tudent s i n the i n t a c t c l a s s e s i n 104 these s c h o o l s c o n s t i t u t e d the sample f o r the s t u d y . There were i n i t i a l l y 17 c l a s s e s w i t h a t o t a l s tudent e n r o l l m e n t o f 402 who wrote the f i r s t two in s t rument s (CPT and S C T ) . However, due t o l o g i s t i c a l reasons (a d e l a y i n t e a c h i n g the r e q u i r e d u n i t on v o l u m e t r i c a n a l y s i s ) o n l y e i g h t s c h o o l s w i t h a t o t a l o f 14 c l a s s e s p a r t i c i p a t e d f u l l y i n the s t u d y , l e a v i n g a t o t a l number o f s u b j e c t s o f 328.(203 boys and 125 g i r l s ) . Data C o l l e c t i o n Procedures The procedures used i n the s tudy r e l a t e d to the a d m i n i s t r a t i o n o f t e s t s t o the p a r t i c i p a t i n g s tudent s and the s e l e c t i o n o f s tudent s f o r i n t e r v i e w i n g . T e s t A d m i n i s t r a t i o n In October and November, 19 8 0", the C la s s room P r o p o r t i o n -a l i t y T e s t was a d m i n i s t e r e d by the i n v e s t i g a t o r to a l l s tudent s i n each s e l e c t e d c l a s s . The d e t a i l e d procedure f o r a d m i n i s t e r -i n g t h i s t e s t can be found i n Appendix B. F o l l o w i n g the a d m i n i s t r a t i o n o f the above t e s t , the Subconcepts T e s t was a d m i n i s t e r e d to the s tudent s i n t h e i r i n t a c t c l a s s e s t o a s ses s t h e i r knowledge o f p r e r e q u i s i t e -. c o n c e p t s . Chemis t ry t e a c h e r s p a r t i c i p a t i n g i n the study agreed to a d m i n i s t e r t h i s t e s t t o t h e i r s tudent s as a r ev i ew t e s t b e f o r e t e a c h i n g v o l u m e t r i c a n a l y s i s . A f t e r t e a c h i n g the u n i t on a c i d s and bases 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 T e s t was a d m i n i s t e r e d by the i n v e s t i g a t o r t o the s tudent s as an e n d - o f -1 0 5 unit t e s t . In administering t h i s test, the students were asked to show the steps they were using in t h e i r solution to the problems. Interview Sample One week afte r administering the f i n a l test — the Volumetric Analysis Test — the students i n each school were grouped into high, 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 to f i v e were c l a s s i f i e d as low achievers, six to ten were c l a s s i f i e d as medium achievers and those scoring from eleven to fourteen as high achievers. In each school, two students were randomly selected from each group f o r the interview. Thus, six students were selected from each school, giving a t o t a l subsample of 48 subjects for the interview task. This was done to ensure that the subjects from the o r i g i n a l sample were adequately represented. The students were interviewed i n d i v i d u a l l y in a separate room, usually the science prepara-t i o n room. However, one student i n the low achievement group in one school could not be interviewed because he was not available when other students i n that school were interviewed; t h i s l e f t a t o t a l interview 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 interviewing of the subsample, was scheduled about six to 106 seven days a f t e r the administration of the Volumetric Analysis Test. In consultation with the class teachers and the students, appointments were arranged for interviewing according to the convenience of the students. The interviews were conducted by the investigator. At the beginning of the interview, the interviewer i d e n t i f i e d himself, indicated the intent of the interview and his a f f i l i a t i o n to the University of B r i t i s h Columbia. The c o n f i d e n t i a l i t y of t h e i r responses was emphasized. The consent of each student was obtained before using the audio-tape recorder. The interviewer then proceeded with the interview according to the established protocols. The interviewer showed the student the required apparatus and solutions f o r the t i t r a t i o n task. The burette and the stand were l e f t unassembled. Whilst the subject was doing the t i t r a t i o n , the interviewer observed his laboratory techniques and completed the laboratory s k i l l s in t i t r a t i o n c h e c k l i s t . The procedure employed during the interview can be found i n Appendix E. The sequence of events involved i n the data c o l l e c -t i o n for the entire study has been summarized i n Figure 6 and Table 6. 107 SELECTION OF CHEMISTRY 12 STUDENTS IN INTACT CLASSES ADMINISTRATION OF THE CLASSROOM PRO-PORTIONALITY 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 : PATH ANALYSIS, INTERVIEW DATA ANALYSIS & ANALYSIS QF WRITTEN WORK ; : ; Figure 6 Overal l Design of Experimental Procedure 108 Table 6 Test Administration and Inter-test Breaks Tests administered p r i o r to the teaching of volumetric analysis unit CPT: DP Subtest IP Subtest 3 weeks Interval] SCT Interval during which volumetric Interval Test Interval analysis a f t e r admin- before (VA) teaching i s - i n t e r -i s taught VA te red view 2 months- 1 week VAT 7 days 5 months Inter-view period 3 days-1 week Data Analysis Data Preparation For each student, the following data set was coll e c t e d : number of laboratory t i t r a t i o n s completed, d i r e c t proportionality test score, inverse proportionality test score, subconcepts test score, and volumetric analysis test score. The number of laboratory t i t r a t i o n s completed was obtained by asking the subjects to indicate the number of t i t r a -tions they had performed since Grade 11-.on the front page of the Volumetric Analysis Test booklet. The values given by the stu-dents were checked with t h e i r teachers to ensure that no one overstated the t i t r a t i o n laboratories completed. Other background data c o l l e c t e d included school, class, and gender. After marking the Volumetric Analysis Test the investigator also categorized the subjects into a group who used the 1:1 stoichiometric ratio with understanding i n solving the volumetric analysis problems 109 and a group which did not use the stoichiometric r a t i o with understanding.* This information was converted to numerical codes and recorded on fortran statement forms. Data were key-punched and v e r i f i e d by the s t a f f at the University of B r i t i s h Columbia Computer Centre. A l l the data were hand-checked for key-punching errors by the i n v e s t i -gator. Errors were double-checked and corrected. In a l l , only f i v e errors were detected. Scoring The same scoring procedure used i n the p i l o t study (see Chapter III) f o r the proportionality tests, subsumed (prerequisite) concepts test, and volumetric analysis c a l c u l a -t i o n test were employed i n the main study. The scores f o r the two subtests of the Classroom Proportionality Test, namely, di r e c t proportionality and inverse proportionality, were kept separate since each was considered as a d i f f e r e n t variable i n the study. Preliminary Analyses of Data f o r Model Testing  R e l i a b i l i t y of Instruments Item analyses and r e l i a b i l i t y estimation were performed on the Inverse Proportionality and Direct Propor-t i o n a l i t y subtests using the computer program LERTAP * Those subjects who wrongly applied a 1:1 mole ratio to more than two problems i n the Volumetric Analysis Test were assumed to have used algorithms without understanding. 110 (Nelson, 1974). Subjects who were absent during the adminis-t r a t i o n of any of the remaining tests but who took the CPT were eliminated before the item analysis was done. A similar procedure was carried out on the Subconcept and Volumetric Analysis Test scores. Hoyt r e l i a b i l i t y index was calculated for the d i r e c t proportionality, inverse proportionality and Volumetric Analysis instruments. This r e l i a b i l i t y c o e f f i c i e n t r e f l e c t s on the int e r n a l consistency or homogeneity of the groups of items i n each test or subtest. The scores for the entire sample on the Direct Proportionality Subtest of the Classroom Proportionality Test ranged from 0 to 21. The Hoyt estimate of r e l i a b i l i t y for t h i s subtest was 0.80. The d i f f i c u l t y index and discrimination index for each of the items i n t h i s subtest i s reported i n Appendix F. Table 7 gives the mean and standard deviation. The scores for the sample on the Inverse Proportion-a l i t y subtest also ranged from 0 to 21. Hoyt 1s estimate of r e l i a b i l i t y for t h i s test was 0.93. The item s t a t i s t i c s for t h i s t e s t are given i n Appendix G. The scores for the Subconcepts Test ranged from 0 to 28. Cronbach's composite alpha was calculated for t h i s test i n order to determine how the seven subtests hung together. The value for Cronbach's alpha was 0.78. This value r e f l e c t s on the homogeneity of the items i n the t e s t . The item d i f f i c u l t y and the item-subtest correlations for the items i n t h i s test are reported i n Appendix H. I l l The s core s f o r the V o l u m e t r i c A n a l y s i s T e s t ranged from 0 t o 15. The H o y t ' s r e l i a b i l i t y e s t imate was 0.88 (see Appendix I ) . The r e l i a b i l i t i e s r e p o r t e d above f o r the i n s t r u m e n t s are comparable t o those o b t a i n e d i n the p i l o t s tudy (see Chapter I I I ) . T a b l e 7 summarizes the s t a t i s t i c s f o r the i n s t r u m e n t s . T a b l e 7 Summary T e s t S t a t i s t i c s F o r A l l 4 T e s t s DP IPI/. SCT VAT Number o f i tems 7 7 28 15 Mean 18.75 13.92 19.04 7.08 S tandard d e v i a t i o n 4.07 7.29 4.91 4.23 H o y t ' s r e l i a b i l i t y 0.80 0.93 0 .78 * 0.88 S tandard e r r o r 1.68 1.84 2.04 1.39 * C r o n b a c h ' s composi te a l p h a (Cronbach, 1951) D i f f e r e n c e s o f C l a s s e s The data f o r a l l 328 s u b j e c t s who p a r t i c i p a t e d ' i n the s tudy were sc reened t o e l i m i n a t e those who d i d not w r i t e a l l four t e s t s . T h i s l e f t a f i n a l sample o f 265 i n 14 d i f f e r e n t c l a s s e s , upon which the pa th a n a l y t i c a l p rocedures were employed. 112 The SPSS subprogram CONDESCRIPTIVE (Nie, e t . a l . 1975) was used to o b t a i n d e s c r i p t i v e s t a t i s t i c s (means and standard d e v i a t i o n s ) 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 , i n v e r s e p r o p o r t i o n a l i t y , subsumed concepts, and v o l u m e t r i c a n a l y s i s t e s t scores f o r each of the 14 c l a s s e s (see Appendix J ) . Since the s u b j e c t s belonged to c e r t a i n c l a s s e s i t was necessary to t e s t f o r p o s s i b l e d i f f e r e n c e s among the c l a s s e s f o r the d i r e c t p r o p o r t i o n , i n v e r s e p r o p o r t i o n , subsumed concepts and v o l u m e t r i c a n a l y s i s scores before c o n t i n u i n g with the a n a l y s i s . Using the computer program OWMAR maintained by the Department of Psychology (U.B.C), a m u l t i v a r i a t e t e s t was performed on the d i f f e r e n c e s i n means among the c l a s s e s and to t e s t the t e n a b i l i t y o f the assumption of homogeneity of v a r i a n c e -c o v a r i a n c e . Table 8 shows the r e s u l t s of t h i s a n a l y s i s . Table 8 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 , Sub-concepts and Volumetric A n a l y s i s Test Scores of C l a s s e s Test DF1 DF2 F-Ratio PROB 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 52 963 3.8690 0.001* 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 130 17348.1 1.4749 0.000* P < -05 113 These res u l t s indicate that the differences among the class means on a l l four variables i s probably r e a l . I t also indicates that the n u l l hypothesis that the variance-covariance matrix i s the same for a l l classes i s untenable. However, since t h i s heterogeneity may be due to differences in variances, a l l the scores were standardized within classes to a mean of zero and standard deviation of one. The standardized scores were then analyzed using the OWMAR PROGRAM. The r e s u l t s as shown i n Table 9 indicate that the transforma-t i o n was successful i n removing class differences. Consequently, a l l subsequent analyses were conducted using students' standard-ized scores and disregarding class as a factor, and the scores f o r a l l classes pooled together. Table 9 Test of Differences i n Means and Homogeneity of Dispersion among Direct Proportionality, Inverse Proportionality, sub-concepts and Volumetric Analysis Test Scores of Cl a s s e s 3 . Test DF1 DF2 F-Ratio PROB MANOVA Likelihood Ratio Test for Differences among Means 52 963 .0.0019 1.000* Bartlett-Box Homogeneity of Dispersion Test 130 17348.1 0.5969 0.999* p > .05 The scores for a l l the variables were transformed within classes to a mean of zero and a variance of one. 1 1 4 Before te s t i n g the model for the two groups of subjects, that i s those who use algorithms with and without understanding, the scores were standardized separately. Path Analysis of Data Path models and other st r u c t u r a l equation, models have been evaluated i n the past by using d i f f e r e n t procedures. The most popular of these procedures involves estimating and testing f o r the significance of the in d i v i d u a l s t r u c t u r a l parameters. In t h i s procedure, the f i t of the model as a whole to the sample data i s not evaluated - only the in d i v i d u a l s t r u c t u r a l parameter estimates are evaluated (e.g. Gimmel, 1 9 7 4 ) . The non-significant paths are usually eliminated and the r e s u l t i n g modified (or trimmed) model i s re-evaluated. In t h i s procedure, the f i t of the model as a whole i s not evaluated. As Land ( 1 9 7 3 ) has noted, the evaluation of struc-t u r a l models ( i . e . models i n which the causal directions are specified) based so l e l y on the evaluation of the in d i v i d u a l s t r u c t u r a l (path) c o e f f i c i e n t s may not always be applicable to the model as a whole es p e c i a l l y when there are more than one over-identifying r e s t r i c t i o n s ( i . e . more than one path has been eliminated). This i s because i t may be possible f o r some of the in d i v i d u a l c o e f f i c i e n t s to be s t a t i s t i c a l l y s i g n i f i c a n t while the test of the model as a whole may e i t h e r 115 produce a s i g n i f i c a n t o r i n s i g n i f i c a n t r e s u l t . A t c o t h e r t i m e s , an i n v e s t i g a t o r may e l i m i n a t e a p a r t i c u l a r pa th which i s s t a t i s t i c a l l y n o n - s i g n i f i c a n t even though t h i s pa th may be e s s e n t i a l to the model as a whole . Land (1973), t h e r e f o r e , recommends the use o f b o t h the l i k e l i h o o d r a t i o c h i - s q u a r e s t a t i s t i c which t e s t s the g o o d n e s s - o f - f i t o f the model as a whole and the i n d i v i d u a l Z (or t - ) t e s t s t o e v a l u a t e the f i t o f s t r u c t u r a l pa ths . Other r e s e a r c h e r s ( B e n t l e r , 1980; B e n t l e r and B o n e t t , 1980) have made s i m i l a r s u g g e s t i o n s . In a d d i t i o n , the examina t ion o f the r e s i d u a l c o v a r i a n c e m a t r i c e s h a s " been suggested ( e . g . B e n t l e r , 1980) . The c h i - s q u a r e s t a t i s t i c p r o v i d e s a t e s t o f the proposed model a g a i n s t the s a t u r a t e d mode l , t h a t i s , the g e n e r a l a l t e r n a t i v e t h a t the v a r i a b l e s are s i m p l y c o r r e l a t e d to an a r b i t r a r y e x t e n t . T h i s , i n a sense , i s a c h i - s q u a r e d i f f e r e n c e t e s t i n t h a t f o r the s a t u r a t e d mode l , s i n c e t h e r e i s 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 , the degrees o f freedom w i l l be ze ro w h i l e the c h i - s q u a r e v a r i a t e w i l l a l s o be z e r o . In t h i s model a p p r a i s a l , i f the c h i - s q u a r e s t a t i s t i c i s l a r g e compared w i t h the degrees o f freedom, i t i s conc luded t h a t the h y p o t h e s i z e d model does not p r o v i d e a p l a u s i b l e r e p r e s e n t a t i o n o f the system o f i n f l u e n c e s among the v a r i a b l e s i n the p o p u l a t i o n . The converse w i l l suggest t h a t the model m i r r o r s the c a u s a l p roce s se s t h a t genera ted the d a t a . In the p r e s e n t a n a l y s i s a c o m b i n a t i o n o f the above procedure s was used t o e v a l u a t e the proposed i n t e g r a t e d model f o r the 265 s u b j e c t s . The SPSS subprogram, REGRESSION (Nie , et. a l . 1 9 7 5 ) was used for t h i s analysis. This computer pro-gram estimates the str u c t u r a l parameters through equation-by-equation l e a s t squares procedure. However, programs l i k e LISREL (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 to estimate the parameters simultaneously. However, for recursive models with indepen-dent disturbances and no a p r i o r i cross-equation constraints, as i s the case i n the proposed model t h i s method reduces to the equation-by-equation least squares solution (Land, 1 9 7 3 ) . The chi-square t e s t s t a t i s t i c used to evaluate the theory i s given by Land ( 1 9 7 3 ) for recursive models with independent errors and no cross-equation constraints as G — A . K N log 0" /0"° ^ > y gg gg g ='i th where £ r 0 i s the estimate of the g disturbance variance gg ( i . e . , the mean squared residual) obtained when y^ i s regressed th on a l l antecedent variables and i s the estimate of the g gg disturbance variance obtained when y^ i s regressed on a l l antecedent variables except those which are excluded a p r i o r i from the g^1 equation. N i s , the sample size. In evaluating the ^"model, the proposed integrated model was f i r s t tested against the saturated model for 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 the-strength of the influence of one variable on another, were also obtained and tested for sig n i f i c a n c e . This proce-117 dure was used 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 w i t h under-s t a n d i n g (N = 105) and those u s i n g a l g o r i t h m s w i t h o u t under-s t a n d i n g (N = 160) . D e t a i l s o f t h i s a n a l y s i s a re p r e s e n t e d i n the next c h a p t e r . A n a l y s i s o f Data f o r the S p e c i f i c S tudent D i f f i c u l t i e s  A n a l y s i s o f C o n c e p t u a l E r r o r s I n i t i a l l y each i t em i n the V o l u m e t r i c A n a l y s i s T e s t was s c o r e d 1 f o r c o r r e c t answer and 0 f o r an i n c o r r e c t answer. The s teps used by the s u b j e c t i n h i s c a l c u l a t i o n s were then examined f o r c o n c e p t u a l e r r o r s . The procedure used t o i d e n t i f y the c o n c e p t u a l e r r o r s i n v o l v e d a c r i t i c a l examinat ion o f a l l the i tems i n the t e s t f o r each s t u d e n t . S i n c e s tudent s were asked t o show the s teps used i n each c a l c u l a t i o n , the a n a l y s i s o f the c o n c e p t u a l e r r o r s i n v o l v e d examining each s tep used by the s u b j e c t and i d e n t i f y i n g the s tep( s ) which produced the i n c o r r e c t response. ; Items f o r which no e x p l i c i t s teps were shown but o n l y the f i n a l answers were shown were e l i m i n a t e d from the a n a l y s i s . Some s u b j e c t s made e x p l i c i t the s teps used i n t h e i r computat ions i n o n l y a few i t e m s . In a l l , every s u b j e c t showed the s teps f o r a t l e a s t two i tems on the t e s t . The f o l l o w i n g problem s o l u t i o n o f f e r e d by Ben* . s e rves t o i l l u s t r a t e t h i s type o f a n a l y s i s o f the c o n c e p t u a l e r r o r s . * A l l names used i n the t h e s i s are f i c t i t i o u s . 118 Q u e s t i o n : A 500 mL s o l u t i o n o f H C l i s p repared f o r use i n a c l a s s exper iment . I f 40.0 mL of t h i s n e u t r a l i z e s 25.0 mL o f 0.10 M N a 2 C 0 3 s o l u t i o n , what i s the c o n c e n t r a t i o n o f the o r i g i n a l s o l u t i o n o f HCl? B e n ' s r e s p o n s e : Step 1: N a 2 C 0 3 + HCl Step 2: 25.0 mL 40.0 mL 0.10 m o l / L x m o l / L Step 3: 25.0 mL x 0.1 m o l / L x L 1000 mL = 2.5 x 10~ 3 mol N a 2 C 0 3 Step 4: 2.5 x 10~ 3 mol N a 2 C 0 3 x 1 mol HCl 1 mol N a 2 C 0 3 = 2.5 x 10~ 3 mol HCl Step 5: Cone. = 2.5 x 10 3 m o l . HCl x 1000 mL 40 mL L = 6.25 x 10=?mol/L Step 6: 6.25 x 10~ 2 mol x 1 x 500 mL L 40 mL = 7 .8 x 1 0 _ 1 m o l / L E x a m i n a t i o n o f B e n ' s c a l c u l a t i o n shows t h a t no b a l a n c e d e q u a t i o n i s w r i t t e n , o n l y the r e a c t a n t s are i n d i c a t e d (Step 1 ) . However, i n Step 4, Ben assumes t h a t the s t o i c h i o -m e t r i c mole r e l a t i o n between the two r e a c t a n t s i s 1 :1 . In s tep 6, Ben assumes t h a t the c a l c u l a t e d c o n c e n t r a t i o n was f o r the 40 mL o f the a c i d used i n the r e a c t i o n and hence c a l c u l a t e s 119 the molarity for the 500 ml stock solution. Three conceptual errors can therefore be i d e n t i f i e d i n Ben's solution. They are: 1) assumption of 1:1 mole r a t i o , 2) that the concentration of stock solution i s d i f f e r e n t from the concentration of the volume used i n the t i t r a t i o n , 3) use of balanced chemical equations not considered i n problem solution. The r e s u l t s of the p i l o t study indicated the presence of these and other kinds of conceptual errors which could account for the incorrect answers on the items i n the Volumetric Analysis Test. To check to see whether other people knowledge-able i n t h i s area would arr i v e at similar conceptual errors, twelve randomly selected answer booklets were given to a graduate student i n chemistry to analyze i n terms of the errors occurring i n the solutions. Before doing the analyses, four students' answer booklets were used to in s t r u c t the other analyst about the intent of the analyses. It was emphasized that only the incorrect solutions should be analysed. I t was also stressed that since for each subject a l l the items which had incorrect responses were to be analyzed, any one p a r t i c u l a r error i d e n t i f i e d for each subject should be indicated only once even though i t may occur many times. That i s , i f a subject assumed 1:1 mole r a t i o s i n 3 problems requiring 2:1 mole r a t i o s , t h i s error was indicated only once. For the analyses of the 12 booklets an agreement ranging from 75.4 to 100% was obtained with a mean percent agreement of 9 1 . 5 . 120 D i s c u s s i o n on the d i s c r e p a n c i e s i n the a n a l y s i s r e v e a l e d t h a t w h i l e f o r any one i t em on the t e s t the i n v e s t i g a t o r a n a l y z e d a l l the p o s s i b l e e r r o r s t h a t c o u l d be i d e n t i f i e d , the o t h e r a n a l y s t 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 e r r o r f o r each i t em on the t e s t . To t r y t o r e s o l v e t h i s d i s c r e p a n c y s i x more answer b o o k l e t s were a n a l y z e d . T h i s produced a p e r c e n t agreement o f 9 5 . 4 . A n a l y s e s o f I n t e r v i e w Data The i n t e r v i e w s w i t h 4 7 s t u d e n t s , u s i n g a l a b o r a t o r y -type t i t r a t i o n t a s k , genera ted data t o address s e v e r a l d i f f e r e n t a s p e c t s about s t u d e n t s ' u n d e r s t a n d i n g o f and p o t e n t i a l d i f f i c u l t i e s w i t h v o l u m e t r i c a n a l y s i s . The d i f f e r e n t types o f a n a l y s e s used on these i n t e r v i e w da ta are p r e s e n t e d i n the f o l l o w i n g s e c t i o n s w h i l e the r e s u l t s o f these a n a l y s e s are g i v e n i n Chapter V I . Manual S k i l l s i n T i t r a t i o n U s i n g the i n f o r m a t i o n from the c h e c k l i s t — 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 — the m a n i p u l a t o r y t e c h n i q u e s used by a l l the s u b j e c t s were t a b u l a t e d (Appendix K ) . The presence or absence o f each i t em on the c h e c k l i s t i n the i n t e r v i e w sample as a whole was then r e c o r d e d i n terms o f the percentage o f the sample u s i n g or not u s i n g a p a r t i c u l a r t e c h n i q u e . Ideas About the Concepts I n v o l v e d i n T i t r a t i o n The tapes c o n t a i n i n g the i n t e r v i e w data f o r the 4 7 s tudent s were t r a n s c r i b e d v e r b a t i m by the i n v e s t i g a t o r and the 121 graduate, student who assisted in the analysis of the conceptual errors i n the Volumetric Analysis Test. In tr y i n g to i d e n t i f y the ideas that subjects have about cert a i n phenomena - objects and processes of nature - from protocols obtained i n an i n t e r -view setting, d i f f e r e n t analytic techniques have been applied by d i f f e r e n t researchers (Erickson, 1975). These analytic techniques range from the selection and c l a s s i f i c a t i o n of s p e c i f i c pertinent sections of the t r a n s c r i p t (Piaget, 1929; Inhelder and Piaget, 1958) to more elaborate procedures which employ the entire t r a n s c r i p t of the interview (e.g. Witz, 1970). The analyses adopted i n the present study i s c l o s e l y related to that of Piaget and others (e.g. Anderson, 1979) . I t involved i d e n t i f y i n g the pattern of responses expressed by a group of subjects to account for each major question posed by the investigator i n the course of the interview. Questions which were used as 'starters' or 'linking questions' (Archenhold, 1979) were not analyzed. This analyses,was applied to the f i r s t part of the interview. Before i s o l a t i n g the response patterns, each sub-ject's response for each question posed i n the interview (Appendix L) was examined to i d e n t i f y the idea being expressed. A subject's idea (or notion) as used here i s a statement or explanation offered by the subject to account for a problem si t u a t i o n i n i t i a t e d by the investigator. Thus they are the responses given verbally and i n writing by the subject i n answer to questions posed by the investigator. The ideas expressed by each subject for each problem 122 s i t u a t i o n (or que s t ion ) was then examined i n r e l a t i o n t o o t h e r s u b j e c t s ' i dea s about the same q u e s t i o n . Ideas which seemed t o go t o g e t h e r or expres sed s i m i l a r i n t e n t were then grouped t o g e t h e r under what i s here r e f e r r e d t o as a Response P a t t e r n . In most c a s e s , as c o u l d be seen by compari son o f Appendix L and Tables 25, t o 2 8' (presented i n Chapter V I ) , these response p a t t e r n s are d i r e c t r e f l e c t i o n s o f the language used by the s u b j e c t s . Problem S o l v i n g Approach and P r e d i c t i v e Behav iour In the second p a r t o f the i n t e r v i e w , d u r i n g which the s u b j e c t was asked t o v e r b a l i z e h i s s o l u t i o n s , each sub j ec tS s p r o t o c o l was a n a l y z e d w i t h the o b j e c t i v e o f i d e n t i f y -i n g the approach used i n s o l v i n g the problem posed i n t h i s p a r t o f the i n t e r v i e w . The problem r e q u i r e d the s u b j e c t 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 from the data o b t a i n e d from the t i t r a t i o n . Thus , i n the a n a l y s e s o f t h i s s e c t i o n o f the i n t e r v i e w , the t r a n s c r i b e d v e r b a l p r o t o c o l s suppor ted by the s u b j e c t ' s w r i t t e n work p r o v i d e d the pr imary d a t a . The a n a l y s e s i n the main s tudy was s i m i l a r t o the one used i n the p i l o t s t u d y . E s s e n t i a l l y , the p r o t o c o l s and the w r i t t e n c a l c u l a t i o n s were examined to i s o l a t e the s teps used by the s u b j e c t t o s o l v e the p r o b l e m . To i l l u s t r a t e t h i s : : . a n a l y s i s , H a r r y ' s v e r b a l r e p o r t and h i s w r i t t e n work w i l l be u s e d . V e r b a l i z a t i o n : "We have .1M o f NaOH, we added 25.4 m i l l i l i t e r s o f i t t o the a c i d ; and moles i s c o n c e n t r a t i o n t imes volume. T h e r e f o r e 25.4 t imes .1 i s 2.54 m i l l i m o l e s o f NaOH 123 which i s the 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 tudent p o i n t s t o the NaOH and HCl 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 the s u p p l i e d paper) 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 or HCl 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 the moles i n t o m i l l i l i t e r s , no , by the m i l l i -l i t e r s ; the m i l l i c a n c e l out and you have moles per l i t e r which i s c o n c e n t r a t i o n and you have about 2.54 d i v i d e d by 25, t h a t g i v e s you c o n c e n t r a t i o n o f .101 . C o n c e n t r a t i o n e q u a l s to 0.101 m o l a r . " W r i t t e n work: NaOH + H C l = N a + + C I " + H 2 0 0.1 m o l / L NaOH x 25.4 mL = 2.54 mmol NaOH 2'li rcf31 = 0.101 M 25 mL = 0.10 M HCl From the above d a t a , the r e l e v a n t s teps f o l l o w e d by Harry a r e : ( i ) w r i t e b a l a n c e d c h e m i c a l e q u a t i o n , ( i i ) s u b s t i t u t e the volume and c o n c e n t r a t i o n o f the base i n t o a formula ( M o l a r i t y x volume = moles) t o o b t a i n the moles o f base used i n t i t r a t i o n , ( i i i ) use the r e l a t i o n between the s t o i c h i o m e t r i c r a t i o o f the r e a c t a n t s i n the.^equation and the moles o f base used i n the r e a c t i o n t o o b t a i n moles o f a c i d , ( iv) r e a r r a n g e the formula a p p l i e d p r e v i o u s l y and s u b s t i t u t e v a l u e s i n t o i t t o o b t a i n the m o l a r i t y o f the a c i d . From t h i s , Harry was d e s c r i b e d as employ ing the b a s i c form o f the Formula A p p r o a c h . 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 the da ta f o r a l l the s u b j e c t s i n v o l v e d i n the i n t e r v i e w . In a d d i t i o n to a s k i n g the s u b j e c t s t o v e r b a l i z e t h e i r s o l u t i o n s i n t h i s p a r t o f the i n t e r v i e w , the s u b j e c t s were a l s o asked t o make p r e d i c t i o n s and o f f e r reasons f o r t h e i r p r e d i c t i o n s f o r t h r e e problems i n v o l v i n g a 2:1 mole r a t i o between reactants. The protocols obtained from asking the subjects to predict the concentration of the acid i n these three related problem situations (using the data they had obtained i n the t i t r a t i o n ) were analyzed i n terms of the type of prediction and the explanation offered by the subjects. Analysis showed that the same approach that was used i n obtaining the molarity of the^hydrochloric acid i n the i n i t i a l s i t u a t i o n involving 1:1 reacting r a t i o s were used i n respond-ing to the prediction questions. I t was also found that those who i n i t i a l l y had d i f f i c u l t y i n c a l c u l a t i n g the molarity of HC1 i n the 1:1 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 as using the NC (not comprehensible) Approach made the same errors i n t h e i r calculations with the prediction questions even though they occasionally made the correct predictions and offered appropriate explanations i n some cases. A further analysis was done to show the changes i n predictions i n the three situations with respect to the „/ approach employed. This analysis attempted to provide a clear diagramatic representation of the possible conceptual paths followed by the subjects i n making t h e i r predictions. Performance on VAT and Number of Titration- Experiments Correlation analysis using the S t a t i s t i c a l Package for the Social Sciences (Nie, et. a l . , 1975) was performed on the number of t i t r a t i o n experiments done by the students p r i o r to the study and t h e i r scores on the Volumetric Analysis Test (VAT). This analysis was performed for the 265 subjects who completed a l l the t e s t s a d m i n i s t e r e d i n the s t u d y . T h i s was done to f i n d out the na ture o f the r e l a t i o n s h i p between these two v a r i a b l e s . 126 CHAPTER V RESULTS AND DISCUSSION OF MODEL TESTING Introduction In t h i s chapter, the res u l t s and interpretation of the analyses conducted to answer the research questions . related to the v a l i d a t i o n of the proposed integrated model are presented. The s t a t i s t i c a l hypotheses corresponding to these research questions are presented f i r s t . Since the preliminary analysis of the data showed that there might be a difference among the students i n terms of t h e i r use of stoichiometric r a t i o s i n calculations, the te s t of the pro-posed integrated model was performed on the two groups of subjects i d e n t i f i e d i n the study. As w i l l be shown, differences were observed among these two groups. Therefore the test of the model for the t o t a l sample was deleted. For reasons of brevity, the variables, d i r e c t proportional reasoning, inverse proportional reasoning, know-ledge of subsumed or prerequisite concepts,and performance on volumetric analysis calculations w i l l be designated by DP, IP, SC,and VAC respectively, i n t h i s chapter. 127 As used i n t h i s chapter, the term s a t u r a t e d model w i l l r e f e r to a model s i m i l a r to the proposed i n t e g r a t e d model but w i t h an assumed c a u s a l connection between d i r e c t propor-t i o n a l r e a s o n i n g 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 . Trimmed (or modified) i n t e g r a t e d model w i l l r e f e r to the f i n a l model r e s u l t i n g from the d e l e t i o n of some of the c a u s a l connections i n the proposed i n t e g r a t e d model. E v a l u a t i o n of the I n t e g r a t e d Model  S t a t i s t i c a l Hypotheses 1. Model E v a l u a t i o n f o r T o t a l Sample The hypothesized i n t e g r a t e d model accounts f o r the t o t a l sample data. However, as w i l l be shown i n t e s t i n g the second h y p o t h e s i s , d i f f e r e n c e s i n some of the path c o e f f i c i e n t s were found among the two groups. Because of these d i f f e r e n c e s the p o o l i n g of the scores from the two groups may l e a d to confounded r e s u l t s . Hence no a n a l y s i s was performed on the t o t a l sample. 2. Model E v a l u a t i o n f o r the Two Groups The two groups of s u b j e c t s c o n s i d e r e d i n t h i s study were those s u b j e c t s who used algorithms with understanding and those who used al g o r i t h m s without understanding. (a) The proposed i n t e g r a t e d model e x p l a i n s the observa-t i o n data f o r the s u b j e c t s who use algorit h m s with understanding. (b) The proposed i n t e g r a t e d model e x p l a i n s the observa-t i o n data f o r the s u b j e c t s who use algorit h m s without understanding. 128 These hypotheses f o r the two groups were t e s t e d a t the .05 l e v e l o f s i g n i f i c a n c e and c o r r e s p o n d t o r e s e a r c h q u e s t i o n s i n Chapter I . Model E v a l u a t i o n 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 Under s t and ing The a n a l y s i s o f the sample da ta (Table 10) f o r the s u b j e c t s who a p p l i e d the 1:1 mole; r a t i o i n d i s c r i m i n a t e l y y i e l d e d the parameter e s t i m a t e s r e p o r t e d i n Tab le 11. T a b l e 10 C o v a r i a n c e M a t r i x 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 Under-s t a n d i n g (N = 160) VAC SC IP DP VAC (1 .000) * 0.994 (0.405) (0.122) (-0.053) SC 0.403 (1.000) 0.994 (0.378) (0.196) IP 0.122 0.376 (1.000) 0.994 (0.386) DP -0 .053 0.195 0.383 (1.000) 0.994 C o r r e l a t i o n s i n parenthese s 129 T a b l e 11 Parameter E s t i m a t e s , S tandard 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 Without Under s t and ing (N = 160) Parameter E s t i m a t e SE CR P 4 3 0.419 0.078 5 .372* P 42 -0 .036 0.078 -0 .462 P 32 0.355 0.079 4 .494* P 3 1 0.059 0.079 0.747 P 2 1 0.386 0.073 5 .288* P < .05 2 N o t e : Model = 2.3668 (df = 1) , P > .05 The e s t i m a t e s p re sen ted i n T a b l e 11, are symbol i zed by P i j which r e p r e s e n t s the e f f e c t o f v a r i a b l e j , the i n d e p e n -dent v a r i a b l e , on v a r i a b l e i , the dependent v a r i a b l e i n the model . T a b l e 11 a l s o g i v e s the s t andard e r r o r and c r i t i c a l r a t i o s f o r each parameter e s t i m a t e . Each c r i t i c a l r a t i o i s a t - r a t i o r e p r e s e n t i n g the r a t i o o f the parameter e s t i m a t e minus the n u l l v a l u e o f zero t o i t s s t andard e r r o r . The c r i t i c a l r a t i o s p r o v i d e an i n d i c a t i o n o f the importance o f each p a r a -meter e s t i m a t e i n the proposed i n t e g r a t e d model . The v a l u e s f o r the c r i t i c a l r a t i o s suggest t h a t two p a t h s , namely, the i n f l u e n c e o f i n d i r e c t p r o p o r t i o n on performance i n v o l u m e t r i c 130 a n a l y s i s c a l c u l a t i o n s and the i n f l u e n c e o f d i r e c t p r o p o r t i o n on knowledge o f subsumed p r e r e q u i s i t e concept s are s t a t i s t i -c a l l y i n s i g n i f i c a n t and hence may not be impor tant t o the model . A l l r e m a i n i n g e s t i m a t e s are e s s e n t i a l t o the mode l . In F i g u r e 7, which i s a d iagrammatic r e p r e s e n t a t i o n o f the i n t e g r a t e d model , the n u m e r i c a l v a l u e s o f the pa th c o -e f f i c i e n t s are i n d i c a t e d on the i n d i v i d u a l p a t h s . A l s o i n d i c a t e d i n parentheses i n the f i g u r e are the s t andard e r r o r s c o r r e s p o n d i n g t o each pa th c o e f f i c i e n t . The c h i - s q u a r e t e s t o f the f i t o f the proposed 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 v a l u e o f 2.3668 w i t h 1 degree o f freedom (see Tab le 1 1 ) . T h i s v a l u e i s not s i g n i -* S tandard e r r o r s i n parenthese s F i g u r e 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 A l g o r i t h m s Without U n d e r s t a n d i n g . 131 f i c a n t at the 5% l e v e l of sig n i f i c a n c e . This indicates that the model cannot be distinguished s t a t i s t i c a l l y from the saturated model i n which there i s no over-identifying r e s t r i c -t i o n ( i . e . a l l the causal connections are assumed to be pre-sent) . This suggests that the model offers a plausible explanation of the variance-covariance matrix i n t h i s subsample. The residual matrix r e s u l t i n g from the deviation of the reproduced covariance matrix from the sample covariance matrix (for the proposed integrated model) i s given in Table 12. The table also gives the reproduced covariance matrix f o r the integrated model. Examination of t h i s matrix reveals that almost a l l the sample covariances have been explained by the proposed integrated model. The result of the chi-square test suggests that a l l the causal paths, taken together, may be important to the model as a whole. However, as the test 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 revealed, the paths between DP and SC and between IP and VAC may not be very c r u c i a l in explaining the test scores for the subjects who use algorithms without understand-ing even though when considered together with the other paths, they appear to be important. As other researchers (e.g. Kerlinger and Pedhazur, 1973, pp. 317 - 318; Heise, 1969, pp. 59 - 60) have suggested, the goals of s t r u c t u r a l modelling may include model trimming -that i s , the removal of paths which are s t a t i s t i c a l l y i n s i g n i f i c a n t from the model. This i s done in order to assess whether a more parsimonious model could 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 from the D i f f e r e n c e between the Sample C o v a r i a n c e M a t r i x and the Reproduced C o v a r i a n c e M a t r i x 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 Under s t and ing VAC SC IP DP VAC (0 .994) * 0.000 [ 0 . 0 0 0 ] + (0.403) (0.122) (0.068) SC 0.000 [0.000] (0.994) 0.000 [0.000] (0.376) (0.195) IP - 0.000 [-0.031] 0.000 "[0.000] (0.994) 0.000 [0.000] (0.383) DP -0 .121 [-0.112] 0.000 [0.050] 0.000 [0.000] (0.994) 0.000 [0.000] Reproduced c o v a r i a n c e s i n parentheses + R e s i d u a l m a t r i x f o r trimmed model i n square b r a c k e t s w e l l f o r the d a t a . F o l l o w i n g t h i s p r o c e d u r e , the pa th between i n d i r e c t p r o p o r t i o n 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 and the one between d i r e c t p r o p o r t i o n and know-ledge o f subsumed p r e r e q u i s i t e concept s were d e l e t e d from the proposed i n t e g r a t e d mode l . The r e s u l t i n g model , which i s r e f e r r e d t o as the trimmed i n t e g r a t e d model (or the Z-mode l ) , i s r e p r e s e n t e d i n F i g u r e 8. A n a l y s i s o f the da ta i n terms o f the trimmed i n t e g r a t e d model y i e l d e d the parameter e s t i m a t e s g i v e n i n F i g u r e 8 and T a b l e 13. The c r i t i c a l r a t i o s show t h a t a l l the 133 T a b l e 13 Parameter E s t i m a t e s , S tandard 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 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 o u t Under s t and ing Parameter E s t i m a t e 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 < .05 N o t e : Model yC2 = 2.0215 (df = 3 ) , P > .05 * "Standard e r r o r s i n parenthese s F i g u r e -8V Trimmed 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 o u t U n d e r s t a n d i n g 134 paths are es s e n t i a l to the model.., The chi-square goodness-of-f i t t est for t h i s modified model gave a value of 2.0215 (df = 3) which was not s t a t i s t i c a l l y s i g n i f i c a n t at the 5% l e v e l . This indicates that the trimmed integrated model also provides a plausible representation of the observed data. Further examination of i t s residuals (Table 12) revealed that the trimmed integrated model accounts for a substantial proportion of the covariances i n the matrix. Residuals close to zero are not unexpected since the trimmed integrated model assumes that these covariances are zero. Thus, i t would appear from the two sets of res u l t s that both models are plausible representa-t i o n of the data. To.further c l a r i f y t h i s s i t u a t i o n , a detailed analysis was performed to examine the t o t a l associations among the variables i n the proposed integrated model. The re s u l t s of th i s analysis are presented i n Table 14. In t h i s table, the t o t a l association between any two variables has been decomposed into 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 e f f e c t s . Spurious e f f e c t s between two variables r e f e r to the compound paths that are mathematically part of the decomposi-tio n but may be due to the presence of a common cause or the presence of causes which are correlated (Alwin and Hauser, 1975) . The t o t a l e f f e c t i s that part of the t o t a l association between two variables which i s not due to spurious e f f e c t s . The d i r e c t e f f e c t i s that part of the t o t a l e f f e c t which i s not transmitted v i a intervening variables, while the i n d i r e c t e f f e c t i s that part of a variable's t o t a l e f f e c t which i s mediated by intervening variables. 135 T a b l e 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 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 P r e - Dependent T o t a l D i r e c t I n d i r e c t E f f e c t Spu-de termined V a r i a b l e E f f e c t E f f e c t V i a r i o u s V a r i a b l e s IP SC SC/IP E f f e c t SC VAC 0.419 0.419 - - -0 .014 IP 0.113 -0 .036 - 0.149 0.009 DP 0.068 - -0 .014 0.025 0.057 IP SC 0.355 0.355 - - 0.023 DP 0.196 0.059 0.137 - _ DP IP 0.386 0.386 — — — — The e f f e c t s d e c o m p o s i t i o n i n Tab le 14 was o b t a i n e d u s i n g W r i g h t ' s (1934) procedures and the fundamental theorem o f p a t h a n a l y s i s (Duncan, 1966) . The v a l u e s o b t a i n e d from these procedures were c o n f i r m e d by u s i n g the s i m p l e r procedure due to A l w i n and Hauser (1975). The r e s u l t s shown i n T a b l e 14 show t h a t the i n d i r e c t e f f e c t o f DP on SC (0.137) i s f a r g r e a t e r than i t s d i r e c t e f f e c t (0 .059 ) . T h i s i n d i r e c t e f f e c t accounts f o r about 70% o f the t o t a l e f f e c t . T h i s i s even g r e a t e r f o r the trimmed i n t e g r a t e d model (see F i g u r e 8 ) . The e f f e c t s a n a l y s i s a l s o r e -v e a l e d t h a t f o r the s u b j e c t s who use 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 , 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 may be suppres-s i n g the e f f e c t o f 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 on v o l u m e t r i c 136 a n a l y s i s c a l c u l a t i o n s . T h i s i s shown by the v a l u e o f -0 .014 f o r the m e d i a t i n g e f f e c t o f 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 . However, i n an i n t e r a c t i v e system, when one v a r i a b l e i s found t o suppress the e f f e c t o f a n o t h e r , the conver se s i t u a t i o n c o u l d a l s o be s a i d t o e x i s t s i n c e s u p p r e s s i o n i s " f u l l y symmetric or m u t u a l " (Cohen and Cohen, 1975, p p . 89-90) . The c h o i c e o f the d i r e c t i o n o f the s u p p r e s s i o n becomes then one o f meaningfu lnes s (Cohen and Cohen, 1975, pp . 89-90. A p p l i e d t o the p r e s e n t p rob lem, i t c o u l d be argued t h a t 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 the suppres sor v a r i a b l e . T h i s i s suggested by the n e g a t i v e v a l u e (-0.143) o b t a i n e d f o r the d i r e c t e f f e c t o f 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 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 when t h i s pa th i s i n c l u d e d i n the proposed i n t e g r a t e d mode l . Because o f t h i s s u p p r e s s i n g e f f e c t o f 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 on the i n f l u e n c e o f 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 (IP) 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 (VAC), the d i r e c t e f f e c t o f IP on VAC i s n e g a t i v e . A l s o the i n d i r e c t e f f e c t o f 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 on VAC v i a IP i s n e g a t i v e because o f t h i s s u p p r e s s i o n . T h i s may be impor tan t s i n c e some o f these s u b j e c t s were found i n the i n t e r v i e w t o i n c o r r e c t l y app ly 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 between m o l a r i t y and volume o f s o l u t i o n when 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 p rob lems , even though they c o u l d rea son i n terms o f i n v e r s e p r o p o r t i o n . A l s o , those s u b j e c t s i n t h i s group who o b t a i n e d the c o r r e c t r e a c t i n g r a t i o s had d i f f i c u l t y i n s o l v i n g the problems because they ' r e v e r s e d 1 the mole r a t i o s i n t h e i r c a l c u l a t i o n s . That i s , these s u b j e c t s 137 i n c o r r e c t l y assumed an inverse r e l a t i o n between stoichiometric c o e f f i c i e n t and moles of the reacting substances. The substantive meaning of the deleted paths i n the trimmed integrated model i s that d i r e c t proportional reasoning does not influence knowledge of prerequisite concepts d i r e c t l y and that inverse proportional reasoning does not have a d i r e c t influence on performance i n volumetric analysis c a l c u l a t i o n s . This seems to be contrary, i n some sense, to some of the the o r e t i c a l analyses performed on some chemical concepts. For example Herron (1975) analyzed some chemical concepts includ-ing volumetric analysis calculations and concluded that students w i l l have d i f f i c u l t i e s i n doing volumetric analysis-type calculations because of the inverse proportionality involved. He implied by his analysis that successful perfor-mance on such a problem w i l l require the a b i l i t y to reason using i n d i r e c t proportion. The empirical r e s u l t obtained for the trimmed integrated model suggests that i n d i r e c t proportional reasoning does not have any d i r e c t e f f e c t on performance i n volumetric analysis c a l c u l a t i o n s . Its influence i s mainly through the knowledge of the subsumed concepts. Even the re s u l t for the proposed integrated model reveals, as already noted above, that the mediating e f f e c t of subsumed concepts on volumetric analysis calculations i s very substantial. This r e s u l t i s understandable i n view of the suggestion i n the l i t e r a t u r e that the content of a task may intera c t with reasoning a b i l i t y of the subjects on the task (e.g. Lunzer, 1975) . 138 Ingle and Shayer (19 71) also analyzed the mole concept and i t s associated calculations and concluded that since the mole concept involves proportional ( r e a l l y , d i r e c t proportional) reasoning, high school students who can't reason in t h i s way w i l l f i n d i t d i f f i c u l t to comprehend these concepts. This analysis implies a d i r e c t e f f e c t of d i r e c t proportional reasoning. The above empirical r e s u l t f o r the trimmed int e -grated model indicates, however, that d i r e c t proportional reasoning does not have any d i r e c t e f f e c t on the prerequisite concepts (which includes the mole concept). I t treats i t s ef f e c t on subsumed concepts so l e l y as being transmitted v i a inverse proportional reasoning. I t could be that, since d i r e c t proportional reasoning precedes and i s subsumed by inverse proportional reasoning, most of the e f f e c t of d i r e c t proportional reasoning or subsumed concepts i s transmitted through inverse propor-t i o n a l reasoning as soon as t h i s l a t t e r reasoning a b i l i t y i s developed. Also, i t seems reasonable to hypothesize that with tasks involving solely d i r e c t proportions or simple r a t i o s , the unmediated influence of d i r e c t proportional reasoning may be considerable ( i . e . d i r e c t proportional reasoning w i l l be displayed i n the solution to the task). But where formulas or algorithms could be applied t h i s d i r e c t e f f e c t may become small. S i m i l a r l y , the d i r e c t influence of inverse proportional reasoning may become important i n tasks requiring i t s use. However, where there i s a formula which 139 could be applied as i n the case of volumetric analysis c a l c u l a -tions, t h i s d i r e c t e f f e c t may diminish while i t s e f f e c t through the application of the formula may become substantial. Also the above re s u l t s indicate that i n both the o r i g i n a l model and the trimmed model, d i r e c t proportional reasoning has a d i r e c t influence on inverse proportional reasoning. This is.consistent with the r e s u l t s obtained by others (Rogers, 1977). It also suggests that d i r e c t propor-t i o n a l reasoning precedes inverse proportional reasoning i n a c q u i s i t i o n . I t suggests further that inverse proportional reasoning influences knowledge of subsumed prerequisite concepts which i n turn affects performance on volumetric analysis c a l c u l a t i o n s . The s i g n i f i c a n t e f f e c t of subsumed prerequisite concepts compared with inverse proportional reasoning on performance i n volumetric analysis calculations, i s not unexpected since i n the studies by Wheeler and Kass (1977) and G r i f f i t h s (1979) subsumed concepts (or content) were found to be more important than the reasoning a b i l i t y of the subjects. I t has been suggested that i n the evaluation of s t r u c t u r a l models, apart from the computation of s t a t i s t i c s and the examination of residuals, the theory and meaningful interpretation should dictate the choice of model. From the above discussion of the r e s u l t s , i t seems the trimmed i n t e -grated model would be preferable to the proposed integrated model on the grounds of meaningfulness and parsimony. I t has been argued that, 1 4 0 " the p o t e n t i a l f o r r e f i n i n g o r t r imming a t h e o r y , and thus making the 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 the 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 from the 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 . " (He i se , 1 9 6 9 , pp . 5 9 - 6 0 ) . I t i s seen a l s o t h a t the trimmed i n t e g r a t e d model i s c o n s i s t e n t w i t h the k i n d s o f f o r m u l a t i o n s u s u a l l y made i n the s c i e n c e e d u c a t i o n l i t e r a t u r e i n terms o f the e x p l a n a t o r y v a r i a b l e s ( s t r u c t u r e and content ) used to r a t i o n a l i z e s t u d e n t s ' u n d e r s t a n d i n g o f s c h o o l concept s (Wheeler and Kass , 1 9 7 7 ; H e r r o n , 1 9 7 5 ; J ohns tone , MacDonald and Webb, 1 9 7 7 ) . However, cogn i sance i s t aken o f the f a c t t h a t f o r these s u b j e c t s 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 s u p p r e s s i n g the e f f e c t o f 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 r e s u l t s f o r the " sub jec t s who tended to use i n c o r r e c t s t o i c h i o m e t r i c r a t i o s i n t h e i r c a l c u l a t i o n s seem t o suggest (as e x p l a i n e d above) t h a t , w i t h these s u b j e c t s the a b i l i t y t o rea son 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 the t r u e e f f e c t o f t h e i r 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 a b i l i t y 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 t c o u l d be argued t h a t s i n c e some o f these s u b j e c t s (as borne out by the i n t e r v i e w da ta d i s c u s s e d l a t e r ) . a s s u m e 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 i n s t e a d o f an i n v e r s e one between m o l a r i t y and volume even though they can rea son u s i n g i n v e r s e p r o p o r t i o n , t h i s a b i l i t y t o rea son u s i n g i n v e r s e p r o p o r t i o n tends t o have a s l i g h t n e g a t i v e i n f l u e n c e on t h e i r 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 . I t i s suggested t h a t t h i s b e h a v i o u r o f the s u b j e c t s may r e f l e c t on the s t r u c t u r a l c o m p l e x i t y o f v o l u m e t r i c a n a l y s i s problems — t h a t i s , the f a c t t h a t i t i n v o l v e s both 141 d i r e c t and inverse re 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 applicable between which two variables. Again, i t i s suggested here that the strong influence of prerequisite concepts on volumetric analysis calculations for these subjects may mean that the mediating influence of prerequisite concepts may be esse n t i a l to t h e i r performance on the calculations since they tend to get confused over which st r u c t u r a l r e l a t i o n they have to use. I t could be that some of these subjects employ formulas i n t h e i r solutions with an understanding of the basic concepts involved thereby increasing the influence of the subconcepts on performance on volumetric analysis c a l c u l a t i o n s . Model Evaluation for Subjects Using Algorithms With Understanding The v a l i d a t i o n of the proposed integrated model for subjects using algorithms with understanding gave the parameter estimates, standard errors, and c r i t i c a l r a t i o s presented i n Table 15. These values were obtained from the analyses of the sample variance-covariance matrix for t h i s subsample (Table 16). As the values indicate, with the exception of the path co-e f f i c i e n t between d i r e c t proportional reasoning and knowledge of subsumed prerequisite concepts, a l l other parameters are important to the proposed integrated model. That i s , only d i r e c t proportional reasoning has s t a t i s t i c a l l y i n s i g n i f i c a n t d i r e c t influence on knowledge of subsumed prerequisite concepts. 142 T a b l e 15 Parameter E s t i m a t e s , S tandard 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 U n d e r s t a n d i n g . (N' —105) Parameter ' E s t i m a t e SE CR P 4 3 0.214 0.104 2 .058* P 42 0.208 0.104 2 .000* P 32 0.429 0.095 4 .515* P 3 1 0.123 0.095 1.295 P 2 1 0.439 0.088 4 .989* P < .05 Note : Model % = 0.9183 (df = 1 ) , P > 0.05 F i g u r e 9 g i v e s the i n t e g r a t e d model and the parameter e s t i m a t e s . . A t e s t o f the model as a whole f o r these s u b j e c t s u s i n g the c h i - s q u a r e g o o d n e s s - o f - f i t t e s t y i e l d e d a c h i - s q u a r e v a l u e o f 0.9183 w i t h 1 degree o f freedom (Table 15 ) . T h i s v a l u e i s not s i g n i f i c a n t a t the ^05 l e v e l o f s i g n i f i c a n c e , s u g g e s t i n g t h a t the model i s r e p r e s e n t a t i v e o f the p roce s se s u n d e r l y i n g the sample d a t a . The 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 e v i a t i o n o f the reproduced c o v a r i a n c e m a t r i x from the sample c o v a r i a n c e m a t r i x i s g i v e n i n T a b l e 17. The e x a m i n a t i o n o f t h i s m a t r i x shows t h a t v i r t u a l l y no r e s i d u a l s are l e f t b e h i n d . 143 T a b l e 16 C o v a r i a n c e M a t r i x 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 Under-s t a n d i n g (N = 105) VAC SC IP DP VAC SC IP DP (1.000) 0.990 0.311 0.308 0.182 (0.315) (1.000) 0.990 0.478 0.308 (0.312) (0.483) (1.000) 0.990 0.435 (0.184) (0.311) (0.439) (1.000) 0.990 * C o r r e l a t i o n s are i n parenthese s F i g u r e 9: Proposed 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 U n d e r s t a n d i n g . 144 T a b l e 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 C o v a r i a n c e M a t r i x and the Reproduced C o v a r i a n c e M a t r i x 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 VAC SC IP DP VAC SC IP DP (0 .990) * 0.000 [0.000]+ 0.000 [0.000 [ 0.000 [0.000] 0.026 [0.047] (0.311) (0.990) 0.000 [0.000] 0.000 [0.000] 0.000 [0.098] (0.308) (0.478) (0.990) 0.000 [0.000] 0.000 [0.000] (0.156) (0.308) (0.435) (0.990) 0.000 [0.000] Reproduced c o v a r i a n c e s i n parentheses + R e s i d u a l m a t r i x f o r the trimmed model i n square b r a c k e t s A l t h o u g h the r e s i d u a l s and the c h i - s q u a r e t e s t i n d i c a t e t h a t a l l the paths among the v a r i a b l e s may h e l p i n e x p l a i n i n g the c o v a r i a n c e s i n the sample d a t a , the t e s t o f the i n d i v i d u a l paths shows t h a t the p a t h between DP and SC may not count v e r y much i n t h i s e x p l a n a t i o n . The next s tage i n the a n a l y s i s t h e r e f o r e , was t o a p p l y model t r imming procedures t o f i n d i f a trimmed model c o u l d be o b t a i n e d which i s more r e p r e s e n t a t i v e o f the sample data than the proposed i n t e g r a t e d mode l . S i n c e the pa th • 145 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 o f sub-sumed concepts ;was not s i g n i f i c a n t , i t was d e l e t e d and the model r e - a n a l y z e d . The e s t i m a t e s o b t a i n e d are p r e s e n t e d i n T a b l e 18. T a b l e 18 Parameter E s t i m a t e s , S tandard 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 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 U n d e r s t a n d i n g Parameter E s t i m a t e SE CR P 4 3 0.214 0.104 2 .058* P 42 0.207 0.104 2 .000* P 32 0.483 0.086 5 .616* P 2 1 0.439 0.088 4 .989* N o t e : Model %  2 = 1.5859 (df = 2 ) , P > .05 P < .05 A diagrammatic r e p r e s e n t a t i o n o f the trimmed model i s g i v e n i n F i g u r e 10. As i n d i c a t e d i n T a b l e 18, a l l the parameter e s t i m a t e s a re s i g n i f i c a n t a t the f i v e p e r c e n t l e v e l , sugge s t ing t h a t a l l the paths a re p r o b a b l y r e a l . The c h i - s q u a r e t e s t f o r t h i s trimmed model suggests t h a t the. trimmed model a l s o accounts r e a s o n a b l y w e l l f o r the sample d a t a . The examina t ion o f the r e s i d u a l s f o r the trimmed i n t e g r a t e d model (g iven i n T a b l e 17) 146 * Standard errors i n parentheses Figure 10: Trimmed Model for Subjects Using Algorithms with Understanding shows that part of the covariance between SC and DP i s l e f t unexplained. However, t h i s remaining covariance constitutes only about one t h i r d of the t o t a l association between these two variables. Thus, i t could be argued that the trimmed integrated model also provides a reasonable explanation of the covariances i n the sample data. A detailed analysis of the eff e c t s of each variable through the decomposition of the t o t a l associations among the variables yielded the values i n Table 19. This table shows that both the d i r e c t and i n d i r e c t e f f e c t of d i r e c t proportional reasoning on subsumed concepts are substantial (0.123 and 0.188 147 Table 19 Effe c t s Analysis i n the Integrated Model for Subjects Using Algorithms With Understanding Pre- Dependent Total Direct Indirect E f f e c t Spu-determined Variable E f f e c t E f f e c t Via rious Variables IP SC SC/IP "Effect SC VAC 0.214 0.214 - 0.101 IP 0.300 0.208 - .092 - 0.012 DP 0.112 - .046 0.026 0.040 -IP SC 0.429 0.429 - 0.054 DP 0.311 0.123 0.188 -DP IP 0.439 0.439 - - - -resp e c t i v e l y ) . However, the i n d i r e c t e f f e c t through inverse proportional reasoning accounts for about 60% of the t o t a l e f f e c t (0.311). Using the c r i t e r i a of parsimony and the fact that the i n d i r e c t e f f e c t accounts for a greater proportion of the covariance, the trimmed integrated model was retained. However, because the path between IP and VAC (Table 18) i s r e l a t i v e l y small, i t was decided to delete t h i s path i n addition to the path between DP and SC i n order to determine whether a much more ov e r - i d e n t i f i e d model (similar to the one for the subjects using algorithms without understanding i . e . a Z-model) could be obtained which could also provide a reasonable explanation of the data for t h i s subsample. 148 Analysis of the data for these subjects i n terms of 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 that a l l the three paths are esse n t i a l to the Z-model. The chi-square test of the goodness-o f - f i t gave a value of 2.6655 (df = 3 ) . This value was i n s i g n i f i c a n t at the 5% l e v e l , suggesting that the Z-model may also provide a reasonable representation of the model. However, examination of the residual variance-covariance matrix (Table 21) revealed that quite a substantial proportion of the covariance between VAC and IP was unexplained. Further-more, from Table 19, i t could be seen that the d i r e c t influence of inverse proportional reasoning on volumetric analysis calculations i s greater than i t s influence through prerequisite concepts. As such,'the Z-model was rejected in.favour of 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 Errors (SE) and C r i t i c a l Ratios (CR) for Subjects using Algorithms With Understanding (N = 105) Parameter Estimate SE CR P 43 0.314 0.093 3 .376* P *32 0.483 0.086 5 .616* P 2 1 0.439 0.088 4 .989* * P < .05 Note: Model X 2 = 2.6655 (df = 3 ) , P > .05 149 * Standard errors i n parentheses Figure 11: The Further Trimmed Model for Subjects Using Algorithms With Understanding Table 21 . . .Residual Matrix for the Next Trimmed Model VAC SC IP DP VAC SC IP DP 0.000 0.000 0.158 0.116 0.000 0.000 0.098 0.000 0.000 0.000 150 S u b s t a n t i v e l y , the r e s u l t s 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 w i t h u n d e r s t a n d i n g 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 r e a s o n i n g has a s t r o n g i n f l u e n c e 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 which i n t u r n has a s t r o n g i n f l u e n c e on p r e r e q u i s i t e c o n c e p t s . These r e s u l t s a re s i m i l a r t o those r e p o r t e d 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 w i t h o u t u n d e r s t a n d i n g . The r e s u l t s a l s o i n d i c a t e t h a t knowledge o f p r e r e q u i s i t e concept s i n f l u e n c e s 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 l t h o u g h t h i s i n f l u e n c e i s moderate . F u r t h e r m o r e , f o r these s u b j e c t s , the a b i l i t y to rea son u s i n g i n v e r s e p r o p o r t i o n may have a moderate d i r e c t i n f l u e n c e on t h e i r 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 . T h i s r e s u l t i s d i f f e r e n t f o r the one r e p o r t e d 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 w i t h o u t u n d e r s t a n d i n g (see p r e v i o u s s e c t i o n ) . The i m p l i c a t i o n i s t h a t f o r these s u b j e c t s who use a p p r o p r i a t e s t o i c h i o m e t r i c r a t i o s i n t h e i r c a l c u l a -t i o n s , t h e i r 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 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 to rea son u s i n g i n v e r s e p r o p o r t i o n . I t c o u l d be argued t h a t the a b i l i t y t o i d e n t i f y the c o r r e c t s t o i c h i o m e t r i c r a t i o i n a r e a c t i o n may i n v o l v e the a b i l i t y t o ba l ance and i n t e r p r e t e c h e m i c a l e q u a t i o n s . F u r t h e r -more, the a b i l i t y t o i d e n t i f y how t h i s r a t i o i s c o r r e c t l y a p p l i e d i n c a l c u l a t i o n s may i n v o l v e some i n s i g h t i n t o the p r o p o r t i o n a l r e l a t i o n s h i p s i n v o l v e d . As s u c h , i t seems r e a s o n a b l e t h a t f o r s u b j e c t s u s i n g the a p p r o p r i a t e s t o i c h i o -m e t r i c r a t i o , t h e i r 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 depends 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 and subsumed c o n c e p t s . 151 The r e s u l t of the model va l i d a t i o n for these subjects also indicate that d i r e c t proportional reasoning influences knowledge of subsumed concepts mainly through i t s e f f e c t on inverse proportional reasoning. This i s similar to the r e s u l t obtained for the subjects who use algorithms without understand-ing (see the previous section for the discussion 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 res u l t s and discussions to the research hypotheses stated at the beginning of the chapter, the following conclusions can be drawn. 1., For the subjects using algorithms with understanding, even though both the proposed integrated model and the trimmed integrated model offered reasonable explanation of t h e i r sample data, the trimmed integrated model was retained as tenable on the grounds of parsimony and meaningfulness and the fact that the path between d i r e c t proportional reasoning and subsumed concepts contributed l i t t l e to the explanation of t h e i r performance. 2. For the subjects using algorithms without understanding, the trimmed integrated model (or Z-model) was accepted in favour of the proposed integrated model even though both appeared to account for the observation data for these subjects. The acceptance of the trimmed integrated model was made on the grounds of parsimony, meaningfulness and the fact that the d i r e c t e f f e c t of d i r e c t proportional reasoning on subsumed concepts and that of inverse proportional reasoning on performance on volumetric analy-ses calculations were found to contribute l i t t l e to the explanation of the sample data for these subjects. 152 CHAPTER VI RESULTS AND DISCUSSION OF DATA ON SPECIFIC STUDENT DIFFICULTIES I n t r o d u c t i o n In t h i s c h a p t e r , the r e s u l t s f o r the f o u r q u e s t i o n s on s p e c i f i c s tudent d i f f i c u l t i e s posed i n the second p a r t o f the s tudy are p r e s e n t e d and d i s c u s s e d . In a d d i t i o n , the c o r r e l a t i o n between 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 and the p r i o r number o f t i t r a t i o n s performed by the s u b j e c t s i s r e p o r t e d and d i s c u s s e d . The c h a p t e r c o n c l u d e s w i t h a d i s c u s s i o n o f how the c o n c e p t u a l e r r o r s and some o f the i n t e r v i e w r e s u l t s bear on the v a l i d a t e d pa th models f o r the two groups o f s u b j e c t s . In r e p o r t i n g these r e s u l t s , the r e s e a r c h q u e s t i o n , and where a p p r o p r i a t e the s t a t i s t i c a l h y p o t h e s i s , precede the r e s u l t s and i n t e r p r e t a t i o n . The numbering o f the r e s e a r c h q u e s t i o n s cor re sponds t o the numbering used i n Chapter I . 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 understand some of the d i f f i c u l t i e s encountered by the students while doing 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 i n the VAT, the w r i t t e n work d i s p l a y e d by the students was examined. The s p e c i f i c r e s e a r c h q u e s t i o n addressed was: 3. What conc e p t u a l e r r o r s are. made by students on the V o l u m e t r i c A n a l y s i s Test? A n a l y s i s of the steps used i n s o l v i n g the problems i n 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 fundamental misconceptions h e l d by the students. The a n a l y s i s i n d i c a t e d t h a t 7 students d i d not make any e r r o r on the V o l u m e t r i c A n a l y s i s T e s t while the remaining 280 s u b j e c t s made at l e a s t one e r r o r . Table -22 l i s t s the conceptual d i f f i c u l t i e s t h a t students had w i t h 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 . Only those d i f f i c u l t i e s or e r r o r s which were encountered by a t l e a s t t h r e e percent of the s u b j e c t s who took t h i s t e s t are r e p o r t e d i n the t a b l e . The t a b l e a l s o shows the number and percentage of s u b j e c t s who made each e r r o r . The major conceptual e r r o r made by the s u b j e c t s i s i n assuming t h a t i n every r e a c t i o n one mole of one substance w i l l be needed to r e a c t completely with one mole of the o t h e r . I t c o u l d be i n f e r r e d t h a t these s u b j e c t s performed t h e i r c a l c u l a t i o n s without t a k i n g i n t o account the s t o i c h i o m e t r i c r e l a t i o n s between the r e a c t i n g substances. Thus, they used the 1:1 r e a c t i n g r a t i o i n d i s c r i m i n a t e l y i n t h e i r c a l c u l a t i o n s . Some of these students used t h i s r a t i o c o n s i s t e n t l y i n a l l 154 T a b l e 22 C o n c e p t u a l E r r o r s on the V o l u m e t r i c A n a l y s i s T e s t * E r r o r , . „_ - , „ Number o f No, C o n c e p t u a l E r r o r S u b j e c t s 1. Assumption o f 1:1 mole r a t i o 201 70.0 2. C o n c e n t r a t i o n o f s tock 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 used i n t i t r a t i o n 99 34.5 3. I n c o r r e c t formula o r formula weight 85 2 9.6 4. Assumption o f 2:1 mole r a t i o s i n problems 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^ always r e a c t s i n a 1:2 r a t i o ) 12 4.2 5. R e v e r s a l o f s t o i c h i o m e t r i c mole r a t i o s 45 15.7 6. Assume d i r e c t p r o p o r t i o n between c o n c e n t r a t i o n and volume of s o l u t i o n s 37 12.9 7. E q u a t i n g c o n c e n t r a t i o n w i t h amount (moles) o f substance 29 10.1 8. C a l c u l a t i o n o f 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 55 19.2 9. A d d i n g volumes o f a c i d and base 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 v e r s a 17 5.9 10. D i f f i c u l t i e s i n u s i n g mass t o c a l c u l a t e c o n c e n t r a t i o n o r v i c e v e r s a 45 15.7 11. Use o f b a l a n c e d c h e m i c a l equa-t i o n s not c o n s i d e r e d i n problem s o l u t i o n 173 60.3 T o t a l number o f s u b j e c t s who wrote t h i s t e s t = 2 8 7 155 problems while others used i t i n a few problems. The predominance of t h i s conceptual error could be attributed to the f a i l u r e on the part of the students to write balanced chemical equations or correct chemical formulas. The recog-n i t i o n and the use of the appropriate stoichiometric r a t i o i n volumetric analysis calculations requires knowledge of the cor r e c t l y balanced chemical equation between the reactants; t h i s i n turn requires knowledge of the formulas for the reactants. However, since Table 22 shows that quite a large number of students did not write chemical equations (60.3%) or correct formulas (29.6%), i t i s not surprising that the indiscriminate use of a l : l : ; r a t i o was prevalent. In the second conceptual error, the subjects viewed the molarity calculated from the volumes used i n the reaction to be a f r a c t i o n of the concentration of the o r i g i n a l solution. This i s exemplified by Pat's solution to the following problem: If 25.0 mL of 0.5 M HCl neutralized 30.0 mL of a 250.0 mL stock solution of NaOH, what i s the molarity of the 250.0 mL stock solution? Pat's solution: 0.5 mol L 25 mL m->r i T I m —=— X TrTrTn 7" X = .0125 mol HCl L 1000 mL .0125 1 mol NaOH L 1000 mL = 0.42 mol NaOH X 1 mol HCl X 30 X L L 0.42 mol 1 250 mL _ 3.5 mol L X 30 X L From t h i s solution one can see that Pat and other students who made similar errors did not see the concentration of the solu-1 5 6 t i o n as remaining invariant i n t h i s problem. I t seems that these students employed a d i r e c t proportional r e l a t i o n s h i p between the concentration of the volume used i n t i t r a t i o n and the concentration of the stock solution. This error implied that thesecstudents may have had some misunderstanding of what a solution was since they applied a proportional r e l a t i o n where i t was not needed. The application of t h i s r e l a t i o n may be attributed to the fact that a part-whole relationship i s involved here. However, t h i s r e l a t i o n i s not the common-sense one which suggests that the whole i s greater than one of i t s parts and which allows the use of mathematical opera-tions. However, the invariance of the concentration seemed ;to be more problematic when the subjects were;.asked to perform calculations than when asked to give q u a l i t a t i v e predictions. During the interview, almost a l l the interview subjects indicated that the concentration of the stock solution would remain invariant when you remove a portion of i t (see Table 25, Question 3 ) . Error number 3 , incorrect formula and or formula mass, could be attributed to a number of basic misunderstand-ings. F i r s t l y , a b i l i t y to calculate the correct molar or formula mass of a chemical substance seemed to require know-ledge of t h e i r formulas. Secondly, knowledge of the symbols for the atoms and the formulas for ionic species such as carbonate ion i s needed together with a knowledge of valencies before one can write an accurate formula. I t .is possible that the students who made t h i s error lacked some of t h i s basic knowledge. 157 E r r o r number 4, u s i n g a 2:1 mole r a t i o where a 1:1 r a t i o was r e q u i r e d , was d e t e c t e d i n about 4% o f the s tudent s and was p a r t i c u l a r l y e v i d e n t i n the problem i n v o l v i n g the r e a c t i o n o f H 2 S 0 4 w i t h N a 2 C 0 3 . I t seems t h a t s i n c e , the ... s tudent s were a l r e a d y f a m i l i a r w i t h the r e a c t i o n between H 2 S 0 4 and NaOH i n which the mole r a t i o i s 1 :2, they assumed t h a t H 2 S 0 4 would r e a c t w i t h N a 2 C 0 3 i n a 1:2 mole r a t i o . Once a g a i n , t h i s 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 o f the s tudent s t o w r i t e 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 and c o r r e c t f o r m u l a s . F o r example, some o f the s tudent s were noted.".to assume the carbonate i o n t o be u n i v a l e n t . Those who r e v e r s e d the s t o i c h i o m e t r i c mole r a t i o s i n t h e i r c a l c u l a t i o n s ( e r r o r number 5 ) , d i d not seem t o have any problems i n c a l c u l a t i n g the moles o f the s t andard sub-s tance ( i . e . the substance w i t h known m o l a r i t y and volume) used i n the r e a c t i o n . However, i t appears t h a t i n c a l c u l a t i n g the moles o f the o t h e r substance i n the r e a c t i o n whose concen-t r a t i o n or volume was t o be f o u n d , they became confused over whether they had to d i v i d e o r m u l t i p l y the moles o f the s t a n d a r d substance by the s t o i c h i o m e t r i c r a t i o . That i s , these s u b j e c t s d i d nbt..seem t o r e c o g n i z e the na ture o f the r e l a t i o n s h i p between the c o e f f i c i e n t s i n the b a l a n c e d e q u a t i o n and the a c t u a l moles o f subs tances used i n the r e a c t i o n . They t h u s , p r o b a b l y a p p l y an i n v e r s e r e l a t i o n w i t h o u t knowing the r e a l r e l a t i o n even though they seem t o know how t o o b t a i n the c o e f f i c i e n t s . F o r those who made e r r o r number 6, t h a t i s assumed 158 d i r e c t proportion between concentration and volume of reacting solutions, i t could be argued that they attempted to es t a b l i s h an equality between the r a t i o s of the volumes and molarities of the acid and base involved i n the reaction. However, i n so doing, they assumed that the concentration varied d i r e c t l y with the .volume of the solution. Having to decide whether there i s a d i r e c t or i n d i r e c t proportional relationship before doing the calculations rather than using the formula, can be looked upon as a more conceptually demanding approach and, therefore, l i k e l y to lead to more errors. In the next conceptual error which was detected i n the solutions of about 10% of the subjects (error number 7), the subjects seemed to confuse molarity with moles of substance. Part of Sid's solution i l l u s t r a t e s t h i s error. 25 mL of acid x .1 M = 2.5 M NaOH It's not clear whether t h i s d i f f i c u l t y was due to improper assignments of units to the variables involved as was found by Rowell and Dawson (1980) on stoichiometric c a l c u l a t i o n s . If i t was, then probably the use of dimensional analysis (factor l a b e l l i n g method) could help i n removing t h i s d i f f i c u l t y . However, i t could be that the students have learnt c e r t a i n formulas which have no proper meanings for them and hence makes i t d i f f i c u l t for them to co r r e c t l y r e c a l l and apply these algorithms. Error number 8 made by about 19% of the subjects who calculated the moles of a substance ir r e s p e c t i v e of i t s purity, was detected only i n solutions to the t h i r d problem 159 in the Volumetric Analysis Test (see Appendix A). The subjects who made t h i s error attempted i n the i n i t i a l stages of t h e i r solution to calculate the number of moles of BafOH^. Without taking into account that i t was impure, the subjects divided the given weight of the impure substance by i t s molar mass. These subjects did not, therefore, seem to r e a l i z e that the moles of a substance i n a given weight of an impure substance can be calculated only when the substance can be i s o l a t e d into i t s pure state. Thus, they did not appear to show much under-standing of when a p a r t i c u l a r memorized algorithm was applicable. The next conceptual error (number 9), adding volumes of acid and base together i n c a l c u l a t i n g the unknown concentra-tion or vice versa, was detected i n the solutions of about 6% of the subjects. I t could be that these subjects did not d i f f e r e n t i a t e the problems included i n the Volumetric Analysis Test from problems they've done in class which involved c a l c u l a t i n g the concentration of an acid solution from a t i t r a t i o n graph. In t h i s l a t t e r problem, students may be asked to calculate the concentration of the remaining acid af t e r a c e r t a i n amount of base has been added. To do t h i s , the student has to calculate the moles of acid remaining aft e r the addition of a s p e c i f i e d volume of base and then divide t h i s by the t o t a l volume of solution ( i . e . volume of acid + volume of added base) i n order to f i n d the concentration of the remaining acid. I t seemed the students who made t h i s error employed t h i s method in c a l c u l a t i n g the concentration 160 a t the s t o i c h i o m e t r i c p o i n t (which i s what the problems i n the V o l u m e t r i c A n a l y s i s T e s t are a b o u t ) . F o r example, George , i n s o l v i n g q u e s t i o n 2 on the t e s t (see Appendix A) w r o t e : M o l a r i t y o f a c i d = 0.50 M x 36 = 0.32 M 56 A l s o , J o h n i n c a l c u l a t i n g the volume of a c i d i n q u e s t i o n 1 (see Appendix A) wro te : 20 x .02 = 6.7 mL H „ S O . .06 ^ John seemed t o extend t h i s t o s i t u a t i o n s i n v o l v i n g the c a l c u -l a t i o n o f the volume. Thus , here he combined the m o l a r i t i e s t o o b t a i n a composi te m o l a r i t y f o r the t o t a l s o l u t i o n . T h i s e r r o r , t h e r e f o r e , i l l u s t r a t e s an i n d i s c r i m i n a t e a p p l i c a t i o n o f l e a r n e d a l g o r i t h m s . A s u b s t a n t i a l number o f the s tudent s a l s o encountered d i f f i c u l t i e s w i t h problems r e q u i r i n g the c a l c u l a t i o n o f the mass o f a substance or the c a l c u l a t i o n o f c o n c e n t r a t i o n from a g i v e n mass o f the substance ( e r r o r number 10 ) . T h i s e r r o r c o u l d be a t t r i b u t e d t o four b a s i c m i s u n d e r s t a n d i n g s d e t e c t e d i n the s u b j e c t s ' s o l u t i o n s . a) d e f i n i n g m o l a r i t y as grams per m i l l i l i t r e , b) m u l t i p l y i n g grams by molar mass to o b t a i n moles o f the subs t ance , c) d i v i d i n g molar mass by grams o f substance to o b t a i n mole s , and, d) u s i n g e q u i l i b r i u m c o n s t a n t e q u a t i o n s . I t i s e v i d e n t from the above t h a t (a ) , (b) and (c) a g a i n , i l l u s t r a t e i n c o r r e c t r e c a l l and a p p l i c a t i o n o f memorized formulas w h i l e (d) i l l u s t r a t e s the i n d i s c r i m i n a t e a p p l i c a t i o n 161 of l e a r n e d f o r m u l a s . I t should',.be noted t h a t g e n e r a l l y these s tudent s were a b l e t o s o l v e problems which d i d not r e q u i r e the use o f , o r the c a l c u l a t i o n o f , the mass o f the subs t ance . However, as soon as they c o n f r o n t e d problems i n v o l v i n g masses, they s t a r t e d a p p l y i n g i n c o r r e c t a l g o r i t h m s . As e x p l a i n e d under the f i r s t c o n c e p t u a l e r r o r , c o n c e p t u a l e r r o r number 11 c o u l d be a t t r i b u t e d to the i n a b i l i t y to w r i t e c o r r e c t formulas f o r the c h e m i c a l substances i n v o l v e d i n the r e a c t i o n l e a d i n g t o an i n a b i l i t y to w r i t e c o r r e c t b a l a n c e d e q u a t i o n s . Some o f the s tudent s were a b l e to w r i t e the e q u a t i o n s but c o u l d not ba l ance them c o r r e c t l y . Other e r r o r s which were made by l e s s than 3% o f the s u b j e c t s and w h i c h , t h e r e f o r e , were not i n c l u d e d i n T a b l e 25 i n c l u d e d the f o l l o w i n g : d i v i d i n g c o n c e n t r a t i o n by moles t o o b t a i n volume, and m u l t i p l y i n g moles by volume t o o b t a i n c o n c e n t r a t i o n . A g a i n , these e r r o r s i l l u s t r a t e d - t h a t the s u b j e c t s who made them p r o b a b l y c o u l d not r e c a l l the c o r r e c t formulas r e q u i r e d f o r the c a l c u l a t i o n s . I t seems from the above d i s c u s s i o n s t h a t a p a r t from the two c o n c e p t u a l d i f f i c u l t i e s ( e r r o r numbers 5 and 6, T a b l e 22) which r e l a t e t o the l o g i c a l s t r u c t u r e o f the prob lems , a l l the o t h e r d i f f i c u l t i e s r e l a t e t o a m i s u n d e r s t a n d i n g o f the c o n t e n t a r e a . A l t h o u g h , the above a n a l y s i s seems t o s u b s t a n -t i a t e H e r r o n ' s (1975) c l a i m t h a t v o l u m e t r i c a n a l y s i s problems may be v e r y demanding, i t does not i n d i c a t e t h a t t h i s d i f f i -c u l t y i s due to the i n a b i l i t y o f the s u b j e c t s t o handle i n v e r s e p r o p o r t i o n s However, i t c o u l d be argued t h a t p a r t o f the 162 d i f f i c u l t y r e l a t e s to the p o t e n t i a l presence o f b o t h d i r e c t and i n v e r s e p r o p o r t i o n i n the same prob lem. I t i s suggested t h a t s i n c e s tudent s r a r e l y concep-t u a l i z e the w o r l d i n c h e m i c a l terms on t h e i r own (Kass, 1981) , the d i f f i c u l t i e s encountered by some of the s tudent s c o u l d be a t t r i b u t e d t o the l a c k o f a p p l i c a t i o n of. the concept s t o an every day c o n t e x t i n which these i d e a s m a n i f e s t t h e m s e l v e s , d u r i n g the c l a s s r o o m d i s c u s s i o n s o f these c o n c e p t s . F i n a l l y , r e l a t i n g the above d i s c u s s i o n s t o the r e s e a r c h q u e s t i o n s t a t e d i n t h i s s e c t i o n , i t c o u l d be c o n c l u d e d t h a t the c o n c e p t u a l e r r o r s made by the s u b j e c t s i n c l u d e e r r o r s r e l a t e d t o b o t h the l o g i c a l s t r u c t u r e o f the problems ( i . e . the requ i rement o f 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 ) and the b a s i c concept s such as b a l a n c i n g o f e q u a t i o n s and w r i t i n g f o r m u l a s , r e q u i r e d by the p rob lems . Manual S k i l l s i n T i t r a t i o n To i d e n t i f y the s p e c i f i c d i f f i c u l t i e s t h a t s tudent s have when p e r f o r m i n g l a b o r a t o r y t i t r a t i o n s , an a n a l y s i s was done on the da ta from the c h e c k l i s t used to i d e n t i f y these d i f f i c u l t i e s d u r i n g the t i t r a t i o n . The s p e c i f i c r e s e a r c h q u e s t i o n addres sed was: Research Quest ion 4: What p r a c t i c a l l a b o r a t o r y s k i l l s do s u b j e c t s d i s p l a y i n a t i t r a t i o n exper iment? The r e s u l t s o f the a n a l y s i s o f the l a b o r a t o r y s k i l l s used by the s u b j e c t s are p r e s e n t e d i n T a b l e 23 . The t a b l e a l s o 163 Table 2 3 Manual Skills i n Titration Total no. Type of Skills of students Percent A. Skills in Using Lab Equipment 1. Skills displayed in the handling and reading of burette a) Inadequate 17 36.2 b) Adequate 12 25.5 c) Superior 18 38.3 2. Skills displayed in the handling and reading of pipette a) Inadequate 3 17.6 b) Adequate 6 35.3 c) Superior 8 47.1 B. Skills i n Performing Lab Techniques 3. Does the student need help in setting up the experiment? a) Yes 9 19.1 b) No 38 80.9 4. Is an indicator added to the solution before titration? a) Yes 46 97.9 b) No 1 2.1 5. Is caution exercised near the endpoint? a) Yes 27 57.4 b) No 20 42.6 6. Does the student overshoot the endpoint? a) Yes 23 48.9 b) No 24 51.1 7. Is the flask constantly shaken during the titration? a) Yes 39 83.0 b) No 8 17.0 8. Is a white background used to detect colour changes? a) Yes 18 38.3 b) No 29 61.7 164 Table23 Continued ... Type of S k i l l s Total no. of students Percent 9. Is the insi d e of the f l a s k or beaker rinsed with d i s t i l l e d water during the t i t r a t i o n ? a) Yes b) No 9 38 19.1 80.9 10. Is the f i r s t t i t r a t i o n regarded as f i n a l ? a) Yes b) No 32 15 68.1 31.9 11. Are data recorded immediately a f t e r readings are made? a) Yes b) No 26 21 55.3 44.7 12. To how many decimal places i s the burette read? a) 2 b) 1 c) 0 13. To obtain volume of acid student uses: a) Pipette b) Burette c) Measuring cy l i n d e r 14. The ind i c a t o r chosen f o r t i t r a t i o n i s : a) Phenolphthalein b) Bromthymol blue c) Methyl orange d) A l l 3 indi c a t o r s 15. Container used f o r t i t r a t i o n : a) Conical f l a s k b) Beaker 1 34 12 17 6 24 31 13 2 1 28 19 2.1 72.3 25.5 36.2 12.8 51.0 66.0 27.7 4.3 2.1 59.6 40.4 16. Is funnel used f o r t r a n s f e r r i n g solutions into burette a) Yes b) No 25 22 53.2 46.8 165 gives the t o t a l number and the percentage of subjects demon-str a t i n g each s p e c i f i c s k i l l . For example, about 38% of the subjects demonstrated accurate s k i l l s i n the handling of a burette. Table 24 gives a similar information for the i n t e r -view subjects when they were grouped according to the p r i o r number of laboratory t i t r a t i o n s performed. The three groups are those who have completed 0 to 2,3 to 5 and 6 to 12 laboratory t i t r a t i o n s . The large percentage of subjects showing at least an adequate s k i l l i n the use of the burette may be ascribed to the fact that invariably in any t i t r a t i o n conducted i n the schools, a burette i s used. As such, even.those who have done less than f i v e t i t r a t i o n s seemed-to know to-some extent that they have to (i) rinse the burette with d i s t i l l e d water and the base to 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 burette. However, i t seems from Table 24, that doing more t i t r a t i o n s may allow these ideas to become more entrenched as part of the students' laboratory techniques. The few students selecting the pipette for use i n measuring the volume of the base, imply that, i n general, students may not be conversant with the operation of the pipette. I t could be that they do not use i t i n t h e i r labora-tory work. Thus they tend to use the measuring cylinder with which they are familar even though they were t o l d to be as accurate i n t h e i r work as possible. It seems (Table 24) that those who have done less than 5 t i t r a t i o n s tend to avoid the use of the pipette i n t h e i r work. 166 Table 24 -Manual S k i l l s Displayed by Subjects C l a s s i f i e d According to the number of Lab T i t r a t i o n s performed since Grade 1 1 * " SKILLS Number of Lab T i t r a t i o n s 0-2 (N=17) 3-5 (N=13) 6-12 (N=17) 1. S k i l l s displayed i n . the handling and reading of burette a) Inadequate 8(47.1) 7(53.8) 3(17.6) b) Adequate 3(17.6) 5(38.5) 3(17.6) c) Superior 6(35.3) 1(7 .7) 11(64.7) 2 . . . S k i l l s displayed i n handling and read-ing of pipette a) Inadequate 1 (5 .9) 1(7 .7) 1(5 .9) b) Adequate 1 (5 .9) 2(15.4) 3(17.6) c) Superior 1 (5 .9) 0 7(41.2) 3. Does the student need help i n setting up the experiment? a) Yes 5(29.4) 3(23.1) 1(5 .9) b) NO 12 (70.6) : 10 (76.9) 16 (94 .1) 4. Is an indicator added to the solution before t i t r a t i o n ? a) Yes 16 (94.1) 13 (100.0). 17 (100. b) No 1.(5.9) 0 0 5. Is caution exercised % „ . near the endpoint? *" a) Yes 10(59.9) 6(46.2) 11(64.7) b) No 7(41.2) 7(53.8) 6(35.3) 6. Does the student overshoot the end-point? a) Yes 8 (47.1) 9 (69.2) 6 (35 b) No 9(52.9) 4(30.8) 11(64, 7. Is the fla s k constant-l y shaken during the t i t r a t i o n ? . ' . a) Yes 12(70.6) 12(92.3) 15(88.2) b) NO 5(29.4) 1(7 .7) 2(11.8) 0) 3) 7) 167 SKILLS Number of Lab T i t r a t i o n s . 0-2 (N=17) 3-5.. (N=13) 6-12 (N=17) 8. Is a white background used to detect colour changes? a) Yes 6(35.3) 3(23.1) 9(52.9) b) No 11(64.7) 10(76.9) 8(47.1) 9. Is the inside of the flask or beaker rinsed with d i s t i l l e d water during the t i t r a t i o n ? a) Yes 6 (35.3) 1 (7.7) 4 (23.5) b) No 11(64.7) 12(92.3) 13(17.6) 10. Is the f i r s t t i t r a -t i o n regarded as f i n a l ? a) Yes 13(76.5) 10(76.9) 9(52.9) b) No 4(23.5) 3(23.1) 8(47.1) 11. Are data recorded immediately aft e r readings are made? a) Yes 11(64.7) 9(69.2) 6(35.3) b) No 6(35.3) 4(30.8) 11(64.7) 12. To how many decimal places i s the burette read? a) 2 0 0 1 (5.9) b) 1 13(76.5) 9(69.2) 12(70.6) c) 0 4 (23.5) 4 (30.8) 4 (23.5) 13. To obtain volume of acid, student uses a) Pipette 3(17.6) 3(23.1) 11(64.7) b) Burette 4(23.5) 0 2(11.8) c) Measuring cylinder 10(58.8); 10 (76.9) 1 4 (23.5) 14. The indicator chosen for t i t r a t i o n i s a) Phenolphthalein 9(52.9) 10(76.9) 12(70.6) b) Bromothymol blue 7(41.2) 3(23.1) 3(17.6) c) Methyl orange 1(5.9) 0 1(5.9) d) A l l 3 indicators 0 0 1(5.9) 168 Number o f Lab T i t r a t i o n s 0-2 3-5 6-12 SKILLS (N=17) (N=13) (N=17) 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 6(35.3) 8(61.5) 14(82.4) b) Beaker 11(64.7) 5(38.5) 3(17.6) 16. F u n n e l used f o r t r a n s f e r r i n g s o l u -t i o n i n t o b u r e t a) Yes 6(35.3) 4(30.8) 15(88.2) b) No 11(64.7) 9(69.2) 2(11.8) * Percentage o f t o t a l s u b j e c t s i n p a r e n t h e s i s 169 T h i s seems to be a l s o t r u e f o r the subjects.when they were gi v e n the o p t i o n to use c e r t a i n equipment (e.g. f u n n e l and c o n i c a l f l a s k ) i n t h e i r experiment. For example, the many students who chose the beaker f o r the t i t r a t i o n had d i f f i c u l t i e s i n s w i r l i n g or s t i r r i n g the s o l u t i o n i n the beaker. Those who t r i e d -to ;swirl i t s p i l l e d some of the s o l u -t i o n w h ile some of those who s t i r r e d w i t h a g l a s s rod c o u l d not mix the s o l u t i o n w e l l . Those who s p i l l e d p a r t of the s o l u t i o n w h ile s w i r l i n g the beaker d i d not attempt to repeat the t i t r a t i o n run i n order to o b t a i n a more ac c u r a t e endpoint. I t seems as i f the c h o i c e to use the beaker was d i c t a t e d by the f a c t t h a t t h i s was what they had been u s i n g i n t h e i r l a b o r a t o r y work. Here too, i t seems t h a t experience w i t h t i t r a t i o n ( i . e . the-vnumber of p r i o r t i t r a t i o n s ) i n f l u e n c e s the c h o i c e of the beaker or c o n i c a l f l a s k . S i m i l a r l y , the c h o i c e of f u n n e l f o r t r a n s f e r i n g s o l u t i o n s seemed to be i n f l u e n c e d by the experience of the student with l a b o r a t o r y t i t r a t i o n s . The f a i l u r e on the p a r t of most of the s u b j e c t s (68.1%) to repeat t h e i r t i t r a t i o n s may suggest t h a t these students may have the n o t i o n t h a t one o b s e r v a t i o n of a phenomenon may adequately r e p r e s e n t the phenomenon. Tables 2 3 and 2 4 a l s o show t h a t i n performing the t i t r a t i o n , almost a l l students were ab l e to s e t up the experiment, added drops of the i n d i c a t o r t o the a c i d i n the flask;.before commencing the a d d i t i o n of base and c o n s t a n t l y shook the t i t r a t i n g f l a s k d u r i n g the t i t r a t i o n . However, almost h a l f of the s u b j e c t s overan the endpoint because they 1 7 0 d i d not e x e r c i s e c a u t i o n i n the a d d i t i o n o f base as they approached the e n d p o i n t and d i d not a l s o use a whi te background to d e t e c t the c o l o u r changes i n the s o l u t i o n . A l s o , about h a l f o f the s u b j e c t s d i d not r e c o r d t h e i r r e a d i n g s u n t i l they were asked to do so . In r e c o r d i n g t h e i r r e a d i n g s , i t i s noteworthy t h a t about a q u a r t e r o f the s u b j e c t s d i d not read the b u r e t t e t o even one d e c i m a l p l a c e . T h i s i s d e s p i t e the f a c t t h a t the b u r e t t e i s g raduated i n such a way t h a t the f i r s t d e c i m a l p l a c e c o u l d be o b t a i n e d a c c u r a t e l y w h i l e the second d e c i m a l p l a c e c o u l d be e s t i m a t e d and the u n c e r t a i n t i e s c a l c u l a t e d . I t seemed most o f the s tudent s (72.3%) a v o i d e d e s t i m a t i n g the u n c e r t a i n t y i n v o l v e d by r e c o r d i n g t h e i r r e s u l t s to o n l y one d e c i m a l p l a c e . I t c o u l d a l s o be t h a t t h i s i s the normal p r a c t i c e i n the s c h o o l s . A l a r g e number o f the s u b j e c t s (80%) d i d not see the need to use the d i s t i l l e d water s u p p l i e d i n the washbot t l e t o wash down the a c i d d r i p p i n g on the s i d e s o f the f l a s k o r beaker i n t o the main body o f the s o l u t i o n . T h i s suggests t h a t they have p r o b a b l y never employed i t i n t h e i r t i t r a t i o n s . Some o f the t e a c h e r s i n v o l v e d i n the s tudy i n d i c a t e d t h a t the s tudent s are no t a l l o w e d t o use the wash b o t t l e s because they use i t t o s q u i r t water on each o t h e r i n the l a b o r a t o r y . The c h o i c e of- i n d i c a t o r f o r the t i t r a t i o n i s d i s c u s s e d i n the next s e c t i o n . Thus , i t seems t h a t above a c e r t a i n degree o f „" ~ .' invo lvement i n l a b o r a t o r y t i t r a t i o n s , the s u b j e c t s psychomotor s k i l l s improve w h i l e below t h i s number, these s k i l l s are not w e l l d e v e l o p e d . In g e n e r a l i t c o u l d be c o n c l u d e d from the above r e s u l t s t h a t the adequacy o f the m a n i p u l a t o r y s k i l l s o f the s u b j e c t s i n the s tudy v a r i e d . That i s , w h i l e a p p r o p r i a t e m a n i p u l a t i v e s k i l l s were shown i n some a r e a s , e . g . s e t t i n g up the exper iment , add ing i n d i c a t o r t o the a c i d b e f o r e t i t r a t i o n and shak ing the f l a s k c o n s t a n t l y , i n a p p r o p r i a t e s k i l l s were d i s p l a y e d i n o t h e r a r e a s , e . g . o v e r s h o o t i n g the e n d p o i n t and not f l u s h i n g the a c i d s o l u t i o n on the i n s i d e w a l l s o f the f l a s k . I t c o u l d be t h a t t h i s v a r i a t i o n r e f l e c t s the r e l a t i v e emphases p l a c e d by t e a c h e r s on these t e c h n i q u e s . I t ' s been s t a t e d e l sewhere (Doran, 1978) t h a t ' the p r e c i s e r e l a t i o n s h i p o f s tudent l a b o r a t o r y - a c t i v i t i e s t o the g o a l s o f s c h o o l s c i e n c e cour se s i s not c l e a r l y d e f i n e d . . . Each s c i e n c e t e a c h e r . . . d i f f e r s i n the emphasis he g i v e s t o the s t u d e n t ' s equipment m a n i p u l a t i o n and l a b o r a t o r y t e c h n i q u e s . As s u c h , t h e d i f f e r i n g s k i l l s d i s p l a y e d by 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 m a n i p u l a t i v e t e c h n i q u e s deemed by the t e a c h e r to be i m p o r t a n t . G i v e n t h a t one o f the impor tant o b j e c t i v e s s t a t e d : f o r l a b o r a t o r y work by c h e m i s t r y t e a c h e r s (Ker r , 1963; Thompson, 1975; Gunning and Johns tone , 1976) i s t o h e l p s tudent s to deve lop a p p r o p r i a t e l a b o r a t o r y s k i l l s , i t i s a l i t t l e s u r p r i s i n g t h a t s tudent s i n t h i s s tudy l a c k a number o f impor tan t s k i l l s . However, g i v e n t h a t these s t u d i e s were conducted e l sewhere and not i n Canada, i t may be i n t e r e s t -i n g t o f i n d out the k i n d s o f aims t h a t c h e m i s t r y t e a c h e r s i n B r i t i s h Columbia may have f o r p r a c t i c a l work i n the s c h o o l s . I t may be t h a t one o f the main aims f o r l a b o r a t o r y work i s to keep the s tudent s o c c u p i e d . I f t h a t ' s the ca se , then i t i s not s u p p r i s i n g i f the s tudent s d i s p l a y inadequate s k i l l s . 172 Concepts i n T i t r a t i o n To o b t a i n a much broader v iew of the d i f f i c u l t i e s s tudent s have w i t h v o l u m e t r i c a n a l y s i s b e s i d e s those i d e n t i f i e d i n t h e i r c a l c u l a t i o n s on VAT, s t u d e n t s ' u n d e r s t a n d i n g o f the concept s i n v o l v e d i n an a c t u a l l a b t i t r a t i o n s i t u a t i o n was e x p l o r e d . T h i s p r o v i d e d a much wider p e r s p e c t i v e on the d i f f i c u l t i e s encountered by the s t u d e n t s . The s p e c i f i c r e s e a r c h q u e s t i o n addres sed i n t h i s s e c t i o n i s : Research Quest ion."5: What d i f f i c u l t i e s do s u b j e c t s have w i t h concept s i n v o l v e d i n an a c t u a l t i t r a t i o n exper iment? The r e s u l t s o f the a n a l y s i s o f s t u d e n t s ' d i f f i c u l t i e s i n u n d e r s t a n d i n g t i t r a t i o n concept s are grouped a c c o r d i n g t o whether the d i f f i c u l t i e s are m a i n l y i n the areas o f pH, c o n c e n -t r a t i o n , i n d i c a t o r b e h a v i o u r , o r s c i e n t i f i c t e r m i n o l o g y . F o r each a rea the r e s u l t s from the i n t e r v i e w q u e s t i o n s are p r e s e n t e d i n t a b u l a r form f o l l o w e d by d i s c u s s i o n . S tudent D i f f i c u l t i e s w i t h pH The r e s u l t s o f the a n a l y s i s o f the i n t e r v i e w q u e s t i o n s r e l a t i n g to pH are g i v e n i n T a b l e 2 5. The response p a t t e r n s f o r q u e s t i o n s 1 and 2 i n d i c a t e t h a t a l a r g e p r o p o r t i o n o f the s u b j e c t s succeeded i n c a l c u l a t e i n g the pH o f the a c i d i c and b a s i c s o l u t i o n s used i n the t i t r a t i o n by a p p l y i n g the mathemat ica l d e f i n i t i o n o f pH. Those 1 7 3 T a b l e 2 5' Quest ions and Response P a t t e r n s D e r i v e d From I n t e r v i e w Data i n the A r e a o f pH — ._ Number o f Response P a t t e r n „ . • , . ^ S tudents Quest ion 1: The sodium h y d r o x i d e s o l u t i o n you are u s i n g i n the t i t r a t i o n i s O . l M , what w i l l be i t s pH? 1. The pH w i l l be 13 s i n c e pH = - l o g [H + ] and pH = 14-POH 32 68.1 2. I d o n ' t know .- 5 10.6 3. The pH w i l l be 1 3 6.4 4. The pH w i l l be 13 s i n c e a n t i l o g o f d i s s o c i a t i o n c o n s t a n t g i v e s POH 1 2.1 5. pH w i l l be 0.1 because c o n c e n t r a t i o n i s 0.1 1 2.1 6. O t h e r s a 5 10.6 Quest ion 2: I f the h y d r o c h l o r i c a c i d you are u s i n g i s O . l M , what w i l l be i t s pH? 1. pH i s n e g a t i v e l o g a r i t h m o f the hydrogen i o n c o n c e n t r a t i o n so i t w i l l be 1 35 74.5 2. I d o n ' t know 6 12.8 3. pH of 1 i s the h i g h e s t f o r an a c i d so i t w i l l be 1 1 2.1 4. pH w i l l be a n t i l o g o f d i s s o c i a t i o n c o n s t a n t which i s 1 1 2.1 5. pH w i l l be 0.1 2 4.3 6. O t h e r a 2 4.3 T h i s response p a t t e r n r e p r e s e n t s s tudent s answers which were e i t h e r n o n - i n t e r p r e t a b l e or v e r y u n u s u a l . 174 Table 2 5 continued _ „ .. Number of „ Response Pattern „. , . % ^ Students Question 5: As you're adding sodium hydroxide solution to the acid i n the flas k , what do you think happens to the pH of the solution i n the flask? 1. The pH w i l l increase toward pH of 7 40 85.1 2. The pH w i l l decrease 5 10.6 3. I t w i l l stay the same 2 4.3 Question 6: How does the pH change at the i n i t i a l stages of the t i t r a t i o n compare to the change near the stoichiometric point for a given volume of added base? 1. The t i t r a t i o n curve shows that i t changes slowly at the beginning but shoots up near the stoichiometric point 7 14.9 2. It w i l l be greater near the s t o i c h i o -metric point because there w i l l be smaller amount of H + ions remaining near the stoichiometric point. 16 34.0 3. I t w i l l be slower near the s t o i c h i o -metric point because the moles of hydrogen ions w i l l be less compared to the i n i t i a l stages. 6 12.8 4. The pH change w i l l be the same because a drop i s a drop, i t changes the same way/ d 14 29.8 5. The pH w i l l l e v e l o ff near the s t o i c h i o -metric point 1 2.1 6. I don't know 3 6.4 Question 7: What w i l l be the pH at the point when there are equal amounts of acid and base? 1. The pH w i l l be 7 42 89.4 2. The pH w i l l be less than 7 5 10.6 One person gave t h i s response but did not know why. Of the 5 students who indicated that the pH w i l l decrease with the addition of base, 2 gave response (2), 1 gave response (3), 1 gave response (5) and 1 gave response (6). 175 Table 2 5 continued T, —» . i Numoer of Response Pattern „. , . . % ^ Students Question 8: When you overrun the endpoint of the t i t r a t i o n , what w i l l ,the pH.be? 1. The pH w i l l be more than 7 37 78.7 2. The pH w i l l be lower than 7 4 8.5 3. The pH remains at 7 3 6.4 4. Others e 3 6.4 Question 12: If you use acetic acid instead of HCl i n the t i t r a t i o n against NaOH, what w i l l be the pH at the s t o i c h i o -metric point? 1. I t w i l l be greater than pH 7 because acetic acid i s a weak acid 14 29.8 2. The pH w i l l be 7 since 7 i s the neutral point when they have reacted completely 23 48.9 3. The pH w i l l be 7 since NaOH i s strong 1 2.1 4. The pH w i l l be less than 7 since acetic acid i s a weak acid 2 4.3 5. The pH w i l l be below 7 because of the r a t i o of concentration of base to acid : and the value 1 2.1 6. pH w i l l be zero because water has pH of zero 1 2.1 7. The pH w i l l be 4 since 4 i s the stoichiometric point for NaOH + HCl 1 2.1 8. The pH w i l l be below 7 because acetic i s a strong acid 1 2.1 9. pH w i l l be 1 because pH does not depend on the nature of the acid 1 2.1 10. I don't know 2 2.1 e For example a) pH w i l l remain at 1 b) pH w i l l be 13 since i t ' s just addition of base to H2O c) pH w i l l remain at 0 1 7 6 Table 2 5 continued Number .of „ Response Pattern Students Question 1 3 : If ammonia i s reacted with HCl i n the t i t r a t i o n , what w i l l be the pH at the stoichiometric point? 1. The pH w i l l be 7 2 1 4 4 . 7 2. More than 7 3 6.4 3. Less than 7 16 3 4 . 0 4. I don't know 7 1 4 . 9 who indicated that they did not know what the pH was attempted i n i t i a l l y to r e c a l l the d e f i n i t i o n but were not successful. Others simply were confused about the d e f i n i t i o n of pH. When asked about the e f f e c t of the added base on the pH of the acid during the t i t r a t i o n (Question 5 ) , almost a l l the students ( 8 5 . 1 % ) knew that the pH of the acid w i l l increase, However, there were a few students who did not seem to have a clear understanding. Those who said that the pH w i l l decrease, appeared to be confused over whether acids have pH ranges above or below the neutral pH of 7. The two students who gave the l a s t response pattern seemed to think that the pH i s a constant of a solution and'hence undisturbed by the addition of other substances. These students therefore, seem to have a confused notion of pH. A follow-up to the above was to ask the students to indicate whether there were any differences in the rate of change of pH of the acid with each drop of base at the i n i t i a l 177 s tages o f the t i t r a t i o n and any p o i n t t h e r e a f t e r (Question 6 ) . A s u b s t a n t i a l number o f the s u b j e c t s responded t h a t the r a t e o f change i n pH w i l l e i t h e r be the same throughout the t i t r a -t i o n o r i t w i l l be s lower near the endpo in t o f the t i t r a t i o n . As s tudent V e x p l a i n s , " I t ( i . e . the r a t e o f change i n pH) w i l l be the same; i t must be because they r e a c t the same way so i t d o e s n ' t mat ter when you added the d r o p , so the pH s h o u l d go up s t e a d i l y " . S i m i l a r l y , s tudent I e x p l a i n s t h a t " S i n c e t h e r e are more H + i o n s a t the b e g i n n i n g , the r e a c t i o n w i l l be e a s i e r a t the b e g i n n i n g than a t the end so t h e r e w i l l be a r a p i d change i n pH a t the b e g i n n i n g but i t w i l l s low down near the e n d . " Thus , i n response p a t t e r n 4 which i s e x e m p l i f i e d by V ' s e x p l a n a t i o n , the s tudent s p r o b a b l y reason t h a t s i n c e each drop o f base has the same c o n c e n t r a t i o n and moles o f NaOH, the change i n pH s h o u l d be the same throughout the t i t r a t i o n . In response p a t t e r n 3, the s t u d e n t s , response i s d i a m e t r i c a l l y opposed t o response p a t t e r n 2, a l t h o u g h the s t u d e n t s ' e x p l a n a -t i o n s f o r b o t h p a t t e r n s are the same. I t may be noted however, t h a t the s t u d e n t s who gave response p a t t e r n s .4 and 3 p r o b a b l y do not r e a l i z e t h a t a r a p i d change i n pH o f the s o l u t i o n a t the s t o i c h i o m e t r i c p o i n t i s nece s s a ry f o r the e f f i c i e n t o p e r a -t i o n o f i n d i c a t o r s i n t i t r a t i o n . Those who gave the f i r s t response p a t t e r n responded by drawing the t i t r a t i o n curve but d i d not e x p l a i n why the curve shoots up near the s t o i c h i o m e t r i c p o i n t . - S i n c e chemistry i s f u r t h e r removed from everyday l i f e than-the o t h e r s c i e n c e s (Kass, 1981) , t h i s may suggest t h a t these 1 7 8 students encounter d i f f i c u l t i e s i n tr a n s l a t i n g t h e i r observa-tions into a mental picture of the molecules and the ions i n solution, subsequent to any verbal explanations. The students 1 responses to Question 7 show that a l -most a l l of them r e a l i z e d that when -a strong acid reacts with a strong base i n equal amounts the pH of the r e s u l t i n g solution w i l l be 7 . It i s int e r e s t i n g that even though some students indicated that the pH w i l l be lower than 7, no one said that i t w i l l be higher than 7 . Those who indicated that the pH w i l l be lower than 7 seem to be confused about how the pH operates - whether pH 7 represent a c i d i c or basic solutions. S i m i l a r l y , most of the students r e a l i z e d that when they over-shoot the endpoint of the t i t r a t i o n , t h e pH of the solution w i l l be higher than 7 (Question 8 ) . These students seem to r e a l i z e that the addition of excess base to the f i n a l solu-t i o n w i l l increase the concentration of the OH ions..and hence the pH. However, a few students gave a variety of responses. Here too, those who indicated that the pH w i l l be below 7 seem to confuse the a c i d i c and basic ranges of pH. Those who indicated that the pH w i l l remain at 7 seem to reason that as soon as the neutral pH i s reached, addition of excess base w i l l not a l t e r the pH. The response patterns given by the students for the t i t r a t i o n of acetic acid against NaOH (Question 1 2 ) can-.be:.further categorised into four groups: those who indicated that the pH w i l l be (a) above pH 7 ( 2 9 . 8 % ) , (b) at pH 7 ( 5 1 % ) , (c) below pH 7 ( 1 4 . 8 % ) , and those (d) who didn't know what the pH w i l l be ( 4 . 3 % ) . The large f r a c t i o n of subjects who indicated that ~a neutral ,pH w i l l be achieved appeared to equate the acetic acid-sodium hydroxide reaction with the: hydrochloric acid-sodium hydroxide reaction. Some of the students who gave t h i s response pattern indicated also that acetic acid i s a weak acid but t h i s did not deter them from saying that the pH w i l l be 7. They did not seem to invoke the notion of hydrolyses of the acetate ion formed during the reaction. This i s shown by the following response from student DD. "The r e s u l t w i l l be the same. The pH w i l l be 7 because i t i s the same as when we used NaOH and HCl; the only difference i s HCl i s strong acid and acetic acid i s weak acid and so the difference i s we only have to use less NaOH to t i t r a t e the acetic acid". Such responses show that the students have the notion that every acid-base reaction r e s u l t s i n a neutral solution regardless of the nature of the acid. The subjects who indicated that the pH w i l l be below 7 gave a wide variety of reasons none of which showed a high frequency of occurrence i n the interview sample. This probably indicates the d i f f e r e n t ideas that these students have about pH. I t also shows t h e i r misunderstanding of the underlying chemical concepts. Similar response patterns were given by the students when questioned on the stoichiometric pH for the reaction between aqueous ammonia and hydrochloric acid (Question 13). This supports the notion that students assume every acid-base t i t r a t i o n to r e s u l t - i n a neutral solution i r r e s p e c t i v e of the r e l a t i v e strength's of the acid and base involved. 180 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 response p a t t e r n s f o r the q u e s t i o n s p e r t a i n -i n g t o the concept o f c o n c e n t r a t i o n are p r e s e n t e d i n T a b l e 26. In the t h i r d q u e s t i o n , v i r t u a l l y a l l the s tudent s seem to r e a l i z e i n one way o r another t h a t the c o n c e n t r a t i o n w i l l remain i n v a r i a n t . However, the na ture o f t h e i r responses i s no tewor thy . W h i l e the f i r s t response p a t t e r n which was g i v e n by 66% of the s tudent s i s p r o b a b l y based on- the knowledge o f the p r e p a r a t i o n o f s o l u t i o n s , the second response p a t t e r n seems to be based on the a c t i o n s performed by the s u b j e c t . The t h i r d response p a t t e r n which showed up i n the responses o f 25.5% o f the s tudent s d e a l s w i t h the r e l a t i o n s h i p among the amount o f s o l u t e , the volume and the c o n c e n t r a t i o n o f the s o l u t i o n . T h i s l a t t e r response p a t t e r n may suggest t h a t these s u b j e c t s may have a b e t t e r u n d e r s t a n d i n g o f the c h e m i s t r y i n v o l v e d than those who gave the f i r s t two response p a t t e r n s . S i m i l a r response p a t t e r n s were g i v e n f o r the e f f e c t o f d e c r e a s i n g volume of the s o l u t i o n on the moles o f a c i d i n the r e m a i n i n g s tock s o l u t i o n (Question 4 ) . H e r e , 55.3% a t t r i b u t e d the decrease i n the amount o f a c i d t o t h e i r a c t i o n s w h i l e o n l y a few (12.8%) r e l a t e d the decrease t o the re la t ion : - : s h i p among volume, amount o f s o l u t e and c o n c e n t r a t i o n o f s o l u t i o n . I t i s suggested t h a t these l a t t e r s tudent s may be b e t t e r a b l e t o r e l a t e t h e i r o b s e r v a t i o n s ( i . e . decrease i n volume o f s tock) to the r e l e v a n t v a r i a b l e s i n v o l v e d — 181 Table 2 6 Questions and Response Patterns Derived from Interview Data in the Area of Concentration r ^ ^ r , ^ -o^i-A-^m Number of Response Pattern Students Question 3: You took 25 mL of hydrochloric acid from the stock solution i n the b o t t l e . Do you think the 25 mL taken out from the stock solution w i l l have any e f f e c t on the concentration of the remaining stock solution? Why? 1. The concentration w i l l be the same because concentration i s the same throughout the entire solution 31 66.0 2. I t w i l l stay the same because you're not adding or taking anything 3 6.4 3. I don't think i t w i l l change because I'm taking equal proportion of acid and water 12 25.5 4. I don' t know .- 1 2.1 Question 4: W i l l the volume taken from the stock solution a f f e c t the number of moles of acid i n the remaining stock solution? Why? 1. I t w i l l be less because you're taking something (moles or volume) out of the stock solution 26 55.3 2. The concentration i s the same, the volume changes so the number of moles w i l l decrease 6 12.8 3. The number of moles stays the same 15 31.9 Question 9: During the t i t r a t i o n you used d i s t i l l e d water to flus h down the acid on the sides of the flas k , what e f f e c t w i l l t h i s water have on the concentration of the solution i n the flask? 1. I t w i l l decrease the concentration since the volume w i l l be greater 26 55.3 Reasons given are explained i n the text 182 Table 26 continued ... T, - r , j - j . Number of „ Response Pattern „, , , % c Students Question 9 ,continued ... 2. It w i l l decrease because the moles w i l l stay the same while volume w i l l increase 11 23.4 3. Concentration w i l l stay the same because water i s neutral 7 14.9 4. Concentration w i l l be the same but I don't know why 1 2.1 5. It w i l l increase due to the e q u i l i -brium conditions 1 2.1 6. It w i l l decrease because water w i l l add more 0H~ ions 1 2.1 Question 10: How w i l l the added water a f f e c t the number of moles of acid i n the flask? 1. I t w i l l stay the same because water i s neutral 18 38.3 2. The concentration w i l l decrease but the volume w i l l increase to make the number of moles the same 6 12.8 3. I t w i l l stay the same because you're not changing the i n i t i a l number of moles 15 31.9 4. I t w i l l stay the same but I don't know why 3 6.4 5. I t w i l l decrease because the concen-t r a t i o n w i l l decrease 4 8.5 6. It w i l l decrease but I'm not sure why 1 2.1 Question 11 : How w i l l the pH be affected by the added water? 1. The pH w i l l increase because the concentration w i l l decrease 7 14.9 2. The pH w i l l increase because water i s basic 1 2.1 3. I t w i l l increase thelpH since pH of water i s higher 1 2.1 183 Table 2 6 continued ^ Number of 0 Response Pattern Students Question 11 continued ... 4. The pH w i l l stay the same because d i s t i l l e d water i s neutral 19 40.4 5. The pH w i l l stay the same because concentration of water i s constant 1 2.1 6. The pH w i l l stay the same because i t i s constant 1 2.1 7. The pH never changes because i f i t does, i t w i l l d i s t o r t the c a l c u l a t i o n of the concentration 1 2.1 8. The pH w i l l stay the same but I don't know why 2 4.3 9. The pH w i l l decrease because the concentration w i l l decrease 10 21.3 10..': The pH w i l l decrease because water i s neutral 1 2.1 11. I'm not sure how the pH w i l l be affected 3 6.4 s p e c i f i c a l l y , the a b i l i t y to rel a t e t h e i r observations to and manipulate the formula: molarity x volume = moles of substance. However, quite a substantial proportion of the subjects (31.9%) maintained i n c o r r e c t l y that the moles w i l l remain unchanged. The reasons given by the students for t h i s response pattern included the following: because i t ' s the same solution or the same concentration, because moles i s the same and because you're not adding any substance. These responses may be due to a lack of adequate comprehension about the d i s t i n c t i o n between the concentration and molescof substances in solution. 184 The response p a t t e r n s under Quest ions 9, 10 and 11, r e f l e c t the s tudent s u n d e r s t a n d i n g o f the e f f e c t o f add ing d i s t i l l e d water on r e s p e c t i v e l y , the c o n c e n t r a t i o n , moles and a c i d i t y (pH) o f the a c i d s o l u t i o n i n the t i t r a t i o n . In the f i r s t response p a t t e r n i n Quest ion 9, the s u b j e c t s a t t r i b u t e the decrease r e s u l t i n g from the a d d i t i o n o f water t o o n l y .ithe i n c r e a s e d volume - no r e f e r e n c e i s made to the amount o f a c i d even though they may be w e l l aware o f i t s i m p o r t a n c e . In the second response p a t t e r n however, the s u b j e c t s e x p l i c i t l y e x p l a i n the decrease i n terms o f the r e l a t i o n s h i p s among the volume, c o n c e n t r a t i o n and moles o f a c i d . Those who 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 w i l l s t ay the same p r o b a b l y v iew i:-... i n c r e a s e i n c o n c e n t r a t i o n as r e s u l t i n g from the a d d i t i o n o f s o l u t e s or s o l u t i o n s h a v i n g excess H + o r OH i o n s . As s u c h , s i n c e water has an e q u a l p r o p o r t i o n o f H + and OH i o n s , they i n d i c a t e t h a t l i t ' , .would no t have any e f f e c t on the c o n c e n t r a -t i o n . W h i l e these s tudent s v iewed water as a n e u t r a l and a n o n - r e a c t i n g subs tance , a t l e a s t two s tudent s thought the H + and O H - i o n s i n water r e a c t w i t h the H + i o n s o f the a c i d t o change the c o n c e n t r a t i o n . On the q u e s t i o n o f the e f f e c t o f d i s t i l l e d water on the moles o f a c i d used i n the t i t r a t i o n (Question 10 ) , the response p a t t e r n s o f the s tudent s demonstrate t h a t about 89% under s tood the r e l a t i v e i n v a r i a n c e o f the moles o f a c i d i n a s o l u t i o n (as opposed to i t s volume o r c o n c e n t r a t i o n ) . The few s tudent s who i n d i c a t e d t h a t c t h e amount o f a c i d w i l l decrease seemed unable t o d i s t i n g u i s h moles from 185 concentration. However, a closer look at the response patterns of the subjects who c o r r e c t l y answered the question shows the variety of reasons used to j u s t i f y t h e i r answers. It ranges from a simple "I don't know" (Response pattern 4) to a more complex explanation embodying the relationship among the concentration, moles and volume of the solution (Response pattern 2). It would seem that those who gave response pattern 3 could be assumed to understand the i n t e r - r e l a t i o n s h i p s among the concentration, volume and moles, however further probing did not seem to e l i c i t any new information. The following protocol exemplifies t h i s : Student: No, i t won't because you are not taking away any moles of HCl and you are not adding any. Investigator: But you are adding water, and you mentioned that the concentration w i l l decrease, so how w i l l the water a f f e c t the moles of acid? Student: The water doesn't have any e f f e c t on the moles, lit.:only increases the volume. The question on the e f f e c t of water on the a c i d i t y of the solution (Question 11) seemed to pose some d i f f i c u l t i e s for the subjects. With the exception of 2 students who indicated that the pH w i l l increase because water i s basic or that water has a higher pH than the acid, a l l the others who said that the pH w i l l increase reasoned from the formal d e f i n i t i o n of pH as the negative logarithm of the hydrogen ion concentration. I t could be argued that for these students, the formal d e f i n i t i o n of pH seems to be meaningful to them i n the sense that they saw i t s a p p l i c a b i l i t y to the present problem. 186 Those who indicated that the pH w i l l decrease reasoned mainly that t h i s was the r e s u l t of the decrease i n concentration. Thus, they could not transfer t h e i r knowledge2 of the formal d e f i n i t i o n of pH to bear on t h i s problem even though they were able to calculate the pH of the acid and the base when supplied with the molarities of the solutions. They just assumed a d i r e c t r e l a t i o n between concentration and pH. Thus, they seem to have d i f f i c u l t i e s i n demonstrating a q u a l i t a t i v e understanding of the mathematical rel a t i o n s h i p between concentration and pH. The subjects who indicated that the pH w i l l remain constant gave a variety of reasons to j u s t i f y t h e i r responses (Response Patterns 3 through 7). The most common explanation was that since water contains equal amounts of OH and H + ions i t wouldn't a f f e c t the hydrogen ions already i n the acid and so the pH w i l l remain the same. I t seems that, just l i k e those students who said that the pH w i l l decrease, these stue.. dents have a rote d e f i n i t i o n of pH which becomes meaningless when i t becomes necessary to apply i t i n a novel s i t u a t i o n . Student D i f f i c u l t i e s with Indicator Behaviour Student response patterns to questions related to indicator behaviour are given i n Table 27. In Question 14, with the exception of one student, a l l the students, i n one way or the other, seem to know the function of the indicator i n the t i t r a t i o n of HCl against NaOH. Those who gave response pattern 4 indicated that i f i n the 187 T a b l e 27 Quest ions and Response P a t t e r n s D e r i v e d from I n t e r v i e w Data i n the A r e a o f I n d i c a t o r Behav iour Response P a t t e r n S t S d e n t ^ Quest ion 14: Why i s an i n d i c a t o r needed i n t h i s t i t r a t i o n ? 1. To i n d i c a t e when the m o l e s o f base equa l s moles o f a c i d 24 51.1 2. To i n d i c a t e when the s o l u t i o n i s n e u t r a l 12 25.5 3. To i n d i c a t e when the s o l u t i o n has reached a c e r t a i n pH range 7 14.9 4. To show when the s o l u t i o n i s b a s i c 3 6.4 5. I d o n ' t know 1 2.1 Quest ion 15: You noted t h a t the i n d i c a t o r had d i f f e r e n t c o l o u r s i n d i f f e r e n t mediums, how does the i n d i c a t o r change c o l o u r ? 1. The i n d i c a t o r i s a weak a c i d o r base w i t h two d i f f e r e n t l y c o l o u r e d forms i n e q u i l i b r i u m . The p r o p o r -t i o n o f these forms i n s o l u t i o n determines the c o l o u r o f the s o l u t i o n 15 31.9 2. The i n d i c a t o r r e a c t s w i t h the H + o r 0H~ i o n s t o form c o l o u r e d compounds 7 14.9 3. I t ' s something to do w i t h e l e c t r o n s i n s o l u t i o n 2 4.3 4. I c a n ' t e x p l a i n 21 44.7 5. Other 2 4.3 Quest ion 16: Do you t h i n k the amount o f i n d i c a t o r added w i l l a f f e c t the f i n a l volume o f base used i n the t i t r a t i o n ? Why? 1. Y e s , the i n d i c a t o r w i l l r e a c t w i t h the base t o a f f e c t the volume of the base 10 21.3 2. I t w i l l a f f e c t the volume o f base because the i n d i c a t o r w i l l i n c r e a s e the pH o f the a c i d 2 4.3 3. I t w i l l a f f e c t i t but I d o n ' t know why 1 2.1 188 Table ,2 7 continued . . Response Pattern Number of Students Question 16 continued ... 4. I don't think so because i t only-makes the colour more perceptible 5. No because an indicator i s only there to show the endpoint 6. I don't think so because the indicator i s a weak acid 7. I don't know 11 17 1 3 3 23.4 36.1 Z6.4 6.4 Question 17: Why did you choose t h i s (particular) indicator for the t i t r a t i o n ? Indicator Chosen Response Pattern Number of Students Bromthymol blue Phenolph-thal e i n 1. Because i t changes between pH 6 to pH 7.6 which covers the neutral pH range 8 2. Because i t changes between pH 6-7.6 which covers pH 7 of the base 1 3. Because i t gives a d i s t i n c t colour 2 4. Because IJm f a m i l i a r with i t 2 1. Because pH range f a l l s on v e r t i c a l section of the graph 2 2. Because i t changes i n the basic range 3 3. Because i t gives a d i s t i n c t colour 4 4. Because the acid i s i n the t i t r a t i n g f l a sk 2 5 . ,Because I'm f a m i l i a r with i t 16 6. I don't know 2: 17.0 2.1 4.3 4.3 4.3 6.4 8.5 4.3 34.1 4.3 189 T a b l e 2.7 c o n t i n u e d I n d i c a t o r Chosen Response P a t t e r n Number o f S tudents Quest ion 17 c o n t i n u e d . Pheno lph-t h a l e i n M e t h y l orange A l l t h r e e i n d i c a t o r s 7. Because the r e a c t i o n i s com-p l e t e d between pH 8 and 10 8. Because p h e n o l p h y t h a l e i n i s i n the n e u t r a l pH range 1. Because the a c i d i s i n the f l a s k 2. Because i t g i v e s a d i s t i n c t c o l o u r 1. To make sure t h a t a t l e a s t one o f them i n d i c a t e s the e n d p o i n t 1 1 1 1 2.1 2.1 2.1 2.1 2.1 Quest ion 18: I f a c e t i c a c i d i s used i n the t i t r a t i o n i n s t e a d o f H C l which o f the 3 i n d i c a t o r s (bromothymolLblue, p h e n o l p h -t h a l e i n and methy l orange) w i l l you use i n the t i t r a t i o n ? Why? Bromthymol" b l u e P h e n o l p h -t h a l e i n 3, 4, Because pH o f f i n a l s o l u -t i o n w i l l be around 7 Because the s t o i c h i o m e t r i c p o i n t w i l l be around pH 6 Because I 'm f a m i l i a r w i t h i t I d o n ' t know 1. Because i t changes i n the b a s i c range 2. Because i t ' s pH range c o i n -c i d e s w i t h the s t o i c h i o m e t r i c p o i n t 3. Because pH a t e q u i v a l e n c e p o i n t i s 7 4. Because i t i s a weaker a c i d than a c e t i c a c i d 5. Because i t i s used when a c i d i s i n the f l a s k 6. Because I 'm f a m i l i a r w i t h i t 7. I don '.t know 1 1 1 14 3 1 2 7 1 14.9 2.1 2.1 2.1 2.1 29.8 6.4 2.1 4.3 14.9 2.1 190 Tab le 2 7 c o n t i n u e d I n d i c a t o r Chosen Response P a t t e r n Number o f S tudents Quest ion 18 c o n t i n u e d 1 M e t h y l orange Because a c e t i c a c i d i s a weak a c i d 2. Because i t i s used when the a c i d i s i n the f l a s k 3. I d o n ' t know 1 1 4.3 2.1 2.1 None 1. I d o n ' t know 6.4 Other s I ' l l use bo th methy l orange and bromthymol b l u e (no reason) 2.1 Quest ion 19: I f ammonia i s used i n the t i t r a t i o n which i n d i c a t o r out o f the t h r e e w i l l you choose? 1. Bromothymol b l u e 2. P h e n o l p h t h a l e i n 3. M e t h y l orange 4. I d o n ' t know 9 21 13' 4 19.1 44.7 27.7 8.5 Quest ion 20: Do you t h i n k you c o u l d have used any o f the o t h e r i n d i c a t o r s i n t h i s t i t r a t i o n ? Why? I n d i c a t o r not used i n T i t r a t i o n Response P a t t e r n Number o f S tudents M e t h y l orange C o u l d be used because pH range w i l l f a l l on v e r t i c a l s e c t i o n o f graph C o u l d be used because HCl i s s t r o n g e r a c i d than methyl orange C o u l d be used because i t ' s an i n d i c a t o r 1 4 4.4 2.2 8.9 191 T a b l e 2 7 c o n t i n u e d . . . I n d i c a t o r not used i n Response P a t t e r n Number,of % T i t r a t i o n S tudents Quest ion 20 c o n t i n u e d . . . M e t h y l 4. C o u l d be used because c h o i c e orange o f i n d i c a t o r depends on whether y o u ' r e f i n d i n g the c o n c e n t r a t i o n o f the a c i d o r base i n the f l a s k 3 6.7 5. No because pH of methyl orange i s f a r too low 27 60.0 6. No because pH o f methyl orange d o e s n ' t span the a c i d and b a s i c range 1 2.2 7. No, I 'm not f a m i l i a r w i t h methy l orange 5 11.1 8. No, but I d o n ' t know why 1 2.2 9. I d o n ' t know 1 2.2 Pheno lph- 1. I t h a l e i n pH does not span the a c i d i c and b a s i c range 1 6.7 2. I c a n ' t because the c o l o u r i s not d i s t i n c t 1 6.7 3. C a n ' t because pH i s h i g h 7 46.7 4. C o u l d be used because i t s range w i l l f a l l on v e r t i c a l s e c t i o n o f graph 1 6.7 5. C o u l d be used because c h o i c e o f i n d i c a t o r depends on whether a c i d o r base i s i n the f l a s k 1 6.7 6. C o u l d use i t because pH i s c l o s e enough t o 7 2 13.3 7. I d o n ' t know 2 13.3 Bromthymol 1. C o u l d be used because i t s pH b l u e range w i l l f a l l on v e r t i c a l s e c t i o n o f graph 3 9.1 192 T a b l e 2 7 c o n t i n u e d I n d i c a t o r not used i n Response P a t t e r n Number o f % T i t r a t i o n S tudents Quest ion 20 c o n t i n u e d . . . Bromothymol 2. C o u l d have used i t because b l u e pH goes over 7 4 12.1 3. C o u l d have used i t because i t ' s an i n d i c a t o r 4 12.1 4. C o u l d have used i t s i n c e c h o i c e o f i n d i c a t o r depends on whether a c i d o r base i n the f l a s k 2 6.1 5. C o u l d be used because H C l i s s t r o n g e r than bromothymol b l u e 1 3.0 6. C o u l d have used i t because pH range i s around s t o i c h i o m e t r i c p o i n t 5 15.2 7. No, because I 'm not f a m i l i a r w i t h i t 3 9.1 8. C a n ' t because pH range i s low 9 27.3 9. C a n ' t because pH i s near n e u t r a l range 1 3.0 10. I d o n ' t know 1 3.0 t i t r a t i o n a base i s b e i n g added from t h e b u r e t t e t o the a c i d , then an i n d i c a t o r s h o u l d be chosen which changes i n the b a s i c pH range . These s t u d e n t s , t h e r e f o r e , do not seem t o have a c o r r e c t c o n c e p t i o n o f the f u n c t i o n o f an i n d i c a t o r . With r e g a r d t o those who i n d i c a t e d t h a t the f i n a l s o l u t i o n w i l l be n e u t r a l , examina t ion o f t h e i r responses t o f o l l o w - u p q u e s t i o n s r e l a t i n g t o 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 s between H C l and NaOH, NH-, and H C l , and a c e t i c a c i d 193 and NaOH seem to i n d i c a t e g e n e r a l l y t h a t they r e g a r d every a c i d - b a s e r e a c t i o n t o r e s u l t i n a n e u t r a l s o l u t i o n i r r e s p e c t i v e of the . .nature o f the r e a c t a n t s . D e s p i t e t h i s c o n f u s i o n i t c o u l d be i n f e r r e d t h a t the s u b j e c t s g e n e r a l l y under s tand why i n d i c a t o r s are used i n a c i d - b a s e r e a c t i o n s i n v o l v i n g s t r o n g a c i d s and s t r o n g ba se s . However, as the responses on Quest ion 15 shows, q u i t e a l a r g e percentage o f the s u b j e c t s e i t h e r d i d not know how the i n d i c a t o r used i n the t i t r a t i o n changed c o l o u r o r they c o n j u r e d up some o t h e r e x p l a n a t i o n s . In the f i r s t response p a t t e r n , t h e s tudent s r e a l i z e t h a t the c o l o u r o f the i n d i c a t o r i s de termined by the p r o p o r t i o n o f the u n d i s s o c i a t e d form and the d i s s o c i a t e d form (or con juga te a c i d or b a s e ) . In the second response p a t t e r n , s tudent s seem t o have the n o t i o n t h a t the hydrogen i o n s i n the s o l u t i o n complexes w i t h the i n d i c a t o r t o form c o l o u r e d compounds. In the t h i r d response p a t t e r n r e f e r e n c e i s made t o the t r a n s f e r o f e l e c t r o n s or the e f f e c t o f l i g h t on e l e c t r o n s i n s o l u t i o n as the cause o f the c o l o u r changes . When asked about the e f f e c t o f u s i n g a l a r g e volume o f the i n d i c a t o r (Question 16 ) , o n l y about a q u a r t e r o f the s u b j e c t s i n d i c a t e d t h a t the i n d i c a t o r w i l l a f f e c t the t i t r a -t i o n r e s u l t s . These s tudent s seem t o r e a l i z e t h a t s i n c e the i n d i c a t o r i s a weak a c i d o r base i t w i l l r e a c t t o i n c r e a s e ( i f i t ' s a weak ac id ) o r decrease ( i f i t ' s a weak base) the volume of the base . The l a r g e number o f s tudent s who i n d i c a t e d t h a t the 194 increased volume of the indicator wouldn't a f f e c t the t i t r a -t i on r e s u l t s , reasoned i n two main ways. One group (23.4%) reasoned that the indicator only makes the colour deeper and hence more perceptible. Thus, these students appealed only to the sense data (their observations) without try i n g to under-stand the possible underlying chemical processes. A second group (36.1%) reasoned that the indicator i s only used to indicate when the reaction i s complete and as such shouldn't a f f e c t the volume of added base. These students also seem to address themselves b a s i c a l l y to the function of the indicator. Some of these students indicated by t h e i r responses that the indicator i s a neutral substance or, even i f i t i s an acid, i t ' s so weak that i t doesn't have s u f f i c i e n t hydrogen ions to make a difference. I t could be stated i n general that although quite a number of students r e a l i z e d that an indicator i s a weak acid, they did not r e a l i z e that i f used i n large amounts i t would af f e c t the volume of base needed i n the t i t r a t i o n . This i s despite the fact that most know...that, weak acids l i k e acetic acid react with sodium hydroxide. The inadequate conceptions that the students have of the behaviour of acid-base i n d i c a -tors i n t i t r a t i o n can also be noticed i n t h e i r selection of indicators for the t i t r a t i o n of HCl against NaOH and the reasoning behind t h e i r choice (Question 17). It's notable that 66.1% of the interview subjects selected phenolphthalein while 27.7% used bromthymol blue. Only 4.2% chose methyl orange for the t i t r a t i o n . Looking back at the response patterns for Question 7, one would have expected that most of the students would select bromthymol blue for the task. However, t h i s wasn't the case. I t i s intere s t i n g also, that most of the students explained t h e i r choices i n terms of t h e i r experiences ( f a m i l i a r i t y ) with the indicators. Of the subjects who chose phenolphthalein 71% indicated i n some way that t h e i r selection was based on t h e i r laboratory experiences. That i s , either they are fa m i l i a r with i t , they know i t ' s used when acid i s put i n the t i t r a t i n g f l a sk or because i t gives a d i s t i n c t colouration. These students did not relate t h e i r choice to any chemical c r i t e r i a . Only a few (4.3%) indicated that even though i t s pH range was beyond pH7, they could s t i l l use i t because of the wide range of pH change near the st o i c h i o -metric point of the reaction. Of those who selected bromthymol blue as an in d i c a -tor, 31% seemed to indicate that t h e i r selection was based on th e i r laboratory experiences. The rest explained t h e i r choices by r e l a t i n g the pH range of the indicator to the pH at the stoichiometric point for the reaction between HCl and NaOH. The two subjects who chose methyl orange also reasoned i n terms of t h e i r laboratory experiences. Thus, i n a l l about 64% of the t o t a l interview sample used p a r t i c u l a r indicators for the t i t r a t i o n of HCl with NaOH because of t h e i r experiences with them. These students, used the indicators without showing an understanding of why they are used ( i . e . without appealing to the chemical processes involved). The students did not r e a l i z e that the sharp increase i n pH near the stoichiometric point makes i t possible for any of the three indicators to be used in the t i t r a t i o n of strong acids with strong bases (Toon and E l l i s , 1973) . This may be r e f l e c t e d i n t h e i r lack of understanding of how the pH of the acid changes during t i t r a t i o n . The above trend i n the choice of indicators noted for the t i t r a t i o n of HCl with NaOH can also be found to some extent i n the choices made by the students i n the t i t r a t i o n of acetic acid against sodium hydroxide (Question 18) and the t i t r a t i o n of ammonia against sodium hydroxide (Question 19 ) . With the NaOH - acetic acid reaction, there seems to be a s l i g h t s h i f t both i n the proportion selecting a p a r t i c u l a r indicator and the reasons offered by the students to explain t h e i r choice. Instead of r e l a t i n g i t to t h e i r laboratory experiences, half of the students who made the correct choice ( i . e . chose phenolphthalein) explained t h e i r selection by r e l a t i n g the pH range of the indicator to the pH at the stoichiometric point for the reaction between acetic acid and sodium hydroxide. These students alluded to the strength of the acid and base and the ensuing hydrolysis to explain the pH of the f i n a l solution. However, s t i l l an appreciable propor-tion of those who chose phenolphthalein (33%) indicated that t h e i r choice was based on t h e i r laboratory experiences. In fact, those who said they were f a m i l i a r with phenolphthalein indicated that they had not used the other indicators to any appreciable extent so they just stuck to what they've been using i n the lab. As one student explained: "I w i l l probably 197 use the same indicator, phenolphthalein. This i s what I used in c l a s s " . Also those who indicated that phenolphthalein i s used when the acid used i n the t i t r a t i o n i s placed i n the flask instead of the burette, seem to think that i f the acid i s i n the f l a s k , then i t suggests that one wants a basic solution whereas the opposite w i l l mean one wants an a c i d i c solution as the f i n a l solution. That i s , the purpose of t i t r a t i o n i s to obtain an a c i d i c or basic solution. These students therefore applied i r r e l e v a n t variables l i k e the position of the acid i n the t i t r a t i o n to explain t h e i r choice instead of looking at variables l i k e the strength of the reactants, the hydrolysis r e s u l t i n g from t h e i r reaction and the probable a c i d i t y of the product of hydrolysis. Those who made an incorrect choice by selecting bromthymol blue mostly reasoned that the pH at the s t o i c h i o -metric point w i l l be 7 even though some of them knew that acetic acid was a weak acid. The same trend was noted i n the t i t r a t i o n of ammonia against hydrochloric acid (Question 19). Those who chose methyl orange reasoned that since acetic acid i s a weak acid and sodium hydroxide i s a strong base, the equilibrium of the hydrolysis w i l l s h i f t toward the acid side of the pH scale. This shows an inadequate grasp of the concept of hydrolysis. The lack of adequate knowledge about the selection of indicators for t i t r a t i o n was also noted when the students were asked to indicate i f the other two indicators not used i n the t i t r a t i o n of HCl against NaOH could have been used i n 198 the t i t r a t i o n (Question 20). About 75,6%.of those who did hot select methyl orange indicated that they couldn't have used i t while only 22.; 2% stated that they could have used i t . Those who said that i t can't be used, explained mainly by saying that the pH of the methyl orange was far lower than the pH at the stoichiometric point or they had not used i t i n t h e i r lab work. Although some of these students seemed to r e a l i z e the importance of the stoichiometric point, they f a i l e d to recog-nize the wide range of pH change near the stoichiometric point. The main reasons offered by those who indicated that i t could be used were that methyl orange was an indicator and the choice of indicator depends on whether acid or base i s in the the burette. That i s , these subjects appealed to t h e i r lab experiences for an explanation. About 60% of those who did not select phenolphthalein stated that they couldn't have used i t i n the t i t r a t i o n and t h e i r major reason was that the pH range of phenolphthalein was high. These students, also did not recognize the wide range of pH change at the stoichiometric point for strong acid-strong base reactions. A similar arguement can be used for the 26.7% who indicated that they could have used phenolph-th a l e i n by reasoning that i t s pH range was close to pH 7 or that the acid was i n the f l a s k . With regard to those who did not select bromthymol blue, 39.4% indicated that i t can't be used i n the t i t r a t i o n because i t s pH range i s low or because they haven't used i t often i n the lab. However, 57.6% stated that they could have used i t in the the t i t r a t i o n . These students reasoned by saying that i t s pH range was around that of the stoichiometric point, that i t ' s an indicator, or that the pH range i s higher than 7. Only a few students (9.1%) indicated that i t could be used because of the wide range of pH change near the ; . : i :. stoichiometric point - from about pH 3.0 to pH 10, even though many subjects drew t i t r a t i o n curves to explain the changes i n pH during t i t r a t i o n (see Question 6 ) . The picture portrayed by the response patterns on Question 20 i s that most students b l i n d l y perform t i t r a t i o n s without showing adequate understanding of why they use certain indicators. There also seems to be an inadequate application or transfer of knowledge learnt i n other situations (e.g. t i t r a t i o n curves) to the task given to them. In .general, i t seems that an adequate understanding of indicator behaviour, as i t i s r e f l e c t e d i n Questions 14, 15, 16, 17, 18, 19 and 20, w i l l require an understanding of: a) the nature of indicators ( i . e . what kind of substances they are), b) the relationship between the pH at stoichiometric point for the reaction and the pH range of the indicator, c) .:.and the changes i n pH during t i t r a t i o n ( i . e . a clear understanding of t i t r a t i o n curves). The knowledge of the pH at..the;:stoichiometric point of the reaction w i l l i n turn require knowledge of: i) the nature of the acid and base involved in the t i t r a t i o n - i . e . whether they are strong or weak, and 200 i i ) whether t h e i r s a l t undergoes hydrolysis. Even though a l l these concepts have been taught i n cl a s s , the students often did not seem to apply them i n t h e i r explan