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Identification of students' errors made in the solution of linear equations Koe, Carryl Diane 1989

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IDENTIFICATION OF IN THE SOLUTION  STUDENTS' ERRORS MADE OF LINEAR EQUATIONS by  CARRYL DIANE KOE B.Sc, M.A.,  U n i v e r s i t y of C a l i f o r n i a ( D a v i s ) , 1967 U n i v e r s i t y of B r i t i s h Columbia, 1979  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF EDUCATION  in  THE FACULTY OF GRADUATE STUDIES Mathematics and S c i e n c e E d u c a t i o n  We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d  THE UNIVERSITY OF BRITISH COLUMBIA May, ©  1989  C a r r y l Diane Koe,  1989  In  presenting  degree freely  at  this  the  thesis  in  University of  partial  fulfilment  of  of  department  this or  publication of  thesis for by  his  or  requirements  British Columbia, I agree that the  available for reference and study. I further  copying  the  representatives.  Library shall make  It  this thesis for financial gain shall not  is  granted  by the  understood  of  AUr/t£/y#r/(LS #rtl> S?/&X>CJS- £l> V<?/hT/OnJ  The University of British Columbia Vancouver, Canada  /  DE-6 (2/88)  it  that  head  of  copying  my or  be allowed without my written  permission.  Department  an advanced  agree that permission for extensive  scholarly purposes may be her  for  ABSTRACT  T h i s study d e s c r i b e s t h e development and v a l i d a t i o n o f a d i a g n o s t i c checklist  i n t e n d e d t o a s s e s s s t u d e n t s ' understandings  of beginning  a l g e b r a i c concepts and i d e n t i f y t h e e r r o r s made on one-, two-, and m u l t i s t e p l i n e a r e q u a t i o n s i n one v a r i a b l e with i n t e g r a l c o e f f i c i e n t s and solutions. scope  ( 1 ) textbook  The study o c c u r r e d i n f i v e phases:  o f concepts and sequence o f e q u a t i o n s ;  (2)  analysis for  error-categorization  scheme development based on p r e v i o u s r e s e a r c h ; ( 3 ) c o n s t r u c t i o n , t e s t i n g , and s e l e c t i o n o f e q u a t i o n s based on s y s t e m a t i c v a r i a t i o n o f complexity, s t r u c t u r a l format the instrument  and n u m e r i c a l s i g n and magnitude;  f o r format and wording  changes;  (5)  (4) i n i t i a l  t e s t i n g of  t e s t i n g of t h e f i n a l  i n s t r u m e n t and i n v e s t i g a t i o n o f t h e nature, frequency,  and i n t e r -  r e l a t i o n s h i p s among t h e concepts h e l d and the e r r o r s which o c c u r r e d .  The  c h e c k l i s t used a s e m i - s t r u c t u r e d i n t e r v i e w t e c h n i q u e i n c o n j u n c t i o n with error-analysis.  The r e s u l t s i n d i c a t e d t h a t t h e c h e c k l i s t  was a u s e f u l  d i a g n o s t i c t o o l and p r o v i d e d many i n s i g h t s i n t o the i n t e r r e l a t i o n s h i p s among s t u d e n t s ' concepts, e r r o r s , and achievement l e v e l s .  Implications f o r  i n s t r u c t i o n i n a l g e b r a were e x p l o r e d , and s u g g e s t i o n s f o r f u t u r e instrument development were made.  iii TABLE  OF  CONTENTS  Page ABSTRACT  i i  LIST OF TABLES  vi  ACKNOWLEDGEMENT  v i i  CHAPTER I  II.  STATEMENT OF THE PROBLEM  1  Background o f t h e Problem  1  Purpose and Nature o f t h e Study J u s t i f i c a t i o n o f t h e Study Research Q u e s t i o n s  3 4 5  REVIEW OF RELATED  7  LITERATURE  D i a g n o s t i c Models Algebraic Processing Processing of Linear Equations Summary o f D i a g n o s t i c Models D i a g n o s t i c Methods Paper-and-Pencil Tests Error-Analysis Semi-Structured Interviews Summary o f D i a g n o s t i c Methods Types o f E r r o r s i n A l g e b r a i c D i a g n o s i s Conceptual E r r o r s Variable Expressions Equality Meaning o f E q u a t i o n s I d e n t i t y Elements Numerical Magnitude E q u a t i o n S t r u c t u r e and Complexity Summary o f C o n c e p t u a l E r r o r s Procedural Errors Order o f O p e r a t i o n s Algebraic Properties Summary o f P r o c e d u r a l E r r o r s  7 7 9 11 11 11 12 18 20 21 22 23 26 28 30 32 33 34 35 36 36 37 38  iv Resultant Errors Computational E r r o r s . Sign Errors Mechanical/Perceptual Errors Random E r r o r s Incomplete Work Summary and Recommendations III  IV  V  38 38 40 41 42 42 43  DESIGN OF THE STUDY  47  Phase 1: Content A n a l y s i s of F i r s t - Y e a r A l g e b r a Textbooks . Phase 2: Development of the E r r o r - C a t e g o r i z a t i o n Scheme . Phase 3: C o n s t r u c t i o n and S e l e c t i o n of E q u a t i o n s . . . . . C o n s t r u c t i o n of E q u a t i o n s S e l e c t i o n of E q u a t i o n s Subjects Procedures . . . . . Analyses Phase 4: Refinement of the Instrument Subjects Procedures Analyses . Phase 5: T e s t i n g of the F i n a l Instrument Subjects Procedures Analyses : Summary of Design  47 49 50 52 55 55 55 57 59 59 59 60 60 61 61 62 63  RESULTS  64  Summary of Phases 3 and 4 Phase 5: Instrument T e s t i n g Subjects Procedures P r e s e n t a t i o n of R e s u l t s Achievement L e v e l s and E r r o r s I d e n t i f i c a t i o n of Systematic and Common E r r o r s . . . Instrument V a l i d i t y C o n c e p t s , V o c a b u l a r y , and E r r o r s C o m p l e x i t y / S t r u c t u r e / M a g n i t u d e E f f e c t s on E r r o r s . . Summary of R e s u l t s  64 68 68 72 73 73 81 83 85 91 94  DISCUSSION  96  Review of P r o c e d u r a l Design R a t i o n a l e f o r the Diagnostic Assessment of the Diagnostic  Checklist Checklist  for Algebra for Algebra  . . . . . . . .  96 100 103  V  S t a t i s t i c a l Properties Validity • Reliability Diagnostic Considerations Diagnostic Categories S y s t e m a t i c and Common E r r o r s E r r o r s R e l a t i n g t o P r e v i o u s Research Summary o f Assessment o f t h e C h e c k l i s t S i g n i f i c a n c e o f the Study L i m i t a t i o n s o f t h e Study S u g g e s t i o n s f o r F u t u r e Research REFERENCES Appendix A: Phase 2: Appendix B: Phase 3: Appendix C: Phase 4: Appendix D: Phase 5: Appendix E: Diagnostic  103 103 104 105 105 10 6 107 110 I l l 113 115 119  E r r o r - C a t e g o r i z a t i o n Scheme Development E q u a t i o n C o n s t r u c t i o n , T e s t i n g , and Selection Wording and Format Changes S t a t i s t i c a l A n a l y s e s of E r r o r Correlations Checklist for Algebra  126 132 166 172 198  vi LIST  OF  TABLES  Table  Page  1  Number o f R e s e a r c h e r s F i n d i n g E r r o r s  46  2 3  Types o f L i n e a r E q u a t i o n s P r e s e n t e d i n 15 Textbooks ...... Phase 4: Comparison of High and Low A c h i e v e r s ' E r r o r F r e q u e n c i e s on t h e Instrument I n d i v i d u a l S u b j e c t s ' Values on Demographic, IQ, and M a t h e m a t i c a l Achievement V a r i a b l e s Phase 5: Comparison o f High, Average, and Low A c h i e v e r s ' C o n c e p t u a l Understandings on t h e Instrument Phase 5: Comparison o f High, Average, and Low Achievers' E r r o r F r e q u e n c i e s on t h e Instrument R e l a t i o n s h i p Between Text P o s i t i o n and Mean P-Values of E q u a t i o n Types The R e l a t i o n s h i p o f Numerical Magnitude w i t h F i n a l Answers and Rule-Based Sign E r r o r s . . . . . .  51  4 5 6 7 8  66 70 74 75 86 92  Vll  ACKNOWLEDGEMENTS I would l i k e t o extend my a p p r e c i a t i o n t o t h e many people who a s s i s t e d me i n t h e c o m p l e t i o n  of t h i s  study.  Dr. D a v i d R o b i t a i l l e s e r v e d as my a d v i s o r and sounding p a t i e n t guidance  board.  and u n t i r i n g d e v o t i o n t o e x c e l l e n c e i n prose  His  and r e s e a r c h  has made t h e success o f t h i s endeavour p o s s i b l e . Dr. James S h e r r i l l s e r v e d as my r e s e a r c h c o n s u l t a n t , q u e r y i n g t h e developmental  a s p e c t s ; e n s u r i n g t e r s e n e s s and  c o n s i s t e n c y o f the r e s e a r c h  paradigm and t h e data p r e s e n t a t i o n . Dr. Robert through  Conry a c t e d as s t a t i s t i c a l c o n s u l t a n t , p r o v i d i n g  guidance  t h e maze o f a n a l y s e s and t h e demands o f a q u a l i t a t i v e study based  on q u a n t i t a t i v e f o r m u l a t i o n s . S p e c i a l thanks go t o Dr. Douglas Edge and Dr. Gaalen c o n s u l t e d on d i a g n o s t i c and c o n c e p t u a l a p s e c t s ,  E r i c k s o n who  respectively.  I am i n d e b t e d t o t h e s t a f f and students who p a r t i c i p a t e d i n t h e study, and made p o s s i b l e t h i s Lastly, difficult present  investigation.  I wish t o thank my f a m i l y f o r t h e i r p a t i e n c e throughout  days and weeks.  re-writes.  T h e i r f a i t h gave me t h e courage f o r t h e ever  S p e c i a l thanks t o my husband, Gerry,  encouragement and s u p p o r t .  many  for h i s  1  CHAPTER STATEMENT D i a g n o s t i c procedures i n beginning algebra  OF  I  THE PROBLEM  need t o be developed  (Davis & Cooney, 1978).  out, a s t u d e n t ' s b e g i n n i n g ,  t o h e l p i d e n t i f y e r r o r s made As K r u t e t s k i i  (1976) p o i n t e d  " e x t e r n a l " e r r o r s must be i d e n t i f i e d and  e r a d i c a t e d b e f o r e they become " i n t e r n a l , mental o p e r a t i o n s . " Students operations.  o f t e n c r e a t e t h e i r own r u l e s f o r p e r f o r m i n g a l g e b r a i c S i m i l a r l y , they may a l t e r t h e new p r o c e d u r e s  t o accommodate p r e v i o u s erroneous erroneous  understandings  These  of e a r l y a l g e b r a i c notions are very r e s i s t a n t t o Furthermore,  formed i n b e g i n n i n g a l g e b r a may account  i n l a t e r mathematical  work (Matz,  these e r r o r s at t h e i r  onset.  Background The  ( K i e r a n , 1983).  (Rosnick & Clement, 1 9 8 0 ) .  change a t a l a t e r date understandings  conceptions  they a r e l e a r n i n g  1980).  erroneous  f o r many e r r o r s  Hence, i t i s important  found  to identify  of the Problem.  s o l u t i o n o f l i n e a r e q u a t i o n s i n one v a r i a b l e has a c e n t r a l r o l e i n  f i r s t - y e a r algebra courses.  W i t h i n the a l g e b r a c u r r i c u l u m , t h i s t o p i c  the b a s i s f o r t h e s o l u t i o n o f e q u a t i o n s with two o r more v a r i a b l e s . s t u d e n t s have a thorough  foundation i n the s o l u t i o n of l i n e a r  l a t e r p r o g r e s s i n a l g e b r a may be impeded.  forms  Unless  equations,  T h e r e f o r e , i n v e s t i g a t i o n o f the  e q u a t i o n - s o l v i n g p r o c e s s , i n terms o f a r i t h m e t i c and a l g e b r a i c e x p r e s s i o n s , would seem t o be an important  s t e p i n d i a g n o s i n g e r r o r s which a r e made i n  beginning algebra. In t h e 1920s, s t u d i e s o f f i r s t - y e a r a l g e b r a p l a c e d emphasis on product, or r e s u l t a n t , e r r o r s .  A wide range o f e r r o r s was i d e n t i f i e d ;  including  e r r o r s w i t h a r i t h m e t i c , s i g n s , l i k e terms, and exponents, as w e l l as incomplete  solutions,  c a r e l e s s e r r o r s , and u n c l a s s i f i e d e r r o r s  (Pease, 1929;  2  Wattawa, 1927) . product,  U n f o r t u n a t e l y , f o r purposes  o r r e s u l t a n t , e r r o r s were i d e n t i f i e d without  underlying  two-step,  ( K i e r a n , 1983;  s t u d e n t s who  were adept  Wagner, R a c h l i n , &  Jensen,  at s o l v i n g one-step,  t o s o l v e e q u a t i o n s of a more complex form.  example, s t u d e n t s c o u l d s o l v e 7 x + l = 5 0 ,  but not 3 x - 5 + 4 x + 6 = 5 0 .  and  operations  (e.g. "Add  In s o l v i n g  transition  t o o b t a i n the  (Booth & Hart,  algebraic  solution.").  1982).  academically able  S t u d i e s are needed which f o c u s on  t r a n s i t i o n between mental a r i t h m e t i c and a l g e b r a i c p r o c e d u r e s . bring to l i g h t  e r r o r s made by s t u d e n t s i n s o l v i n g l i n e a r e q u a t i o n s  i n one  s t u d i e s a c h i e v e d the g o a l s f o r which they were d e s i g n e d .  Such focus  conceptual  variable.  These  However, f o r  o f the p r e s e n t study, most of them f a i l e d t o c o n t r o l one  or more of  1) f a c t o r s which might i n f l u e n c e e i t h e r the n a t u r e or  f r e q u e n c y o f the e r r o r s ; 2) e q u a t i o n c o m p l e x i t y : e q u a t i o n s were i n t e r s p e r s e d ; i n t e r s p e r s e d with i n t e g e r s ; fractions;  the  the reasons u n d e r l y i n g p a r t i c u l a r p r o c e d u r a l e r r o r s .  D u r i n g the p a s t twelve y e a r s many s t u d i e s have f o c u s s e d on  the f o l l o w i n g :  (e.g.  from mental a r i t h m e t i c t o a l g e b r a i c o p e r a t i o n s a p p a r e n t l y  poses a major s t u m b l i n g b l o c k f o r many o t h e r w i s e adolescents  r a t h e r than  the  -1 t o both s i d e s t o o b t a i n the e q u i v a l e n t e q u a t i o n  7x=49; then m u l t i p l y both s i d e s by one-seventh  purposes  even  For  e q u a t i o n , s t u d e n t s seemed t o be u s i n g mental a r i t h m e t i c s k i l l s  "What p l u s 1 makes 5 0 ? " "What times 7 makes 49?")  may  their  e q u a t i o n s of the form ax+b=c, where a, b, and c are whole numbers  and a ^ O , were unable  The  i n v e s t i g a t i o n of  have f o c u s s e d on the p r o c e s s e s used by s t u d e n t s i n s o l v i n g e q u a t i o n s .  They have n o t e d t h a t  first  these  causes.  More r e c e n t i n v e s t i g a t o r s 1984)  of the p r e s e n t study,  3) types of numbers:  confounding  one-,  two-,  and m u l t i - s t e p  r a t i o n a l numbers were  e r r o r s i n algebra with e r r o r s i n  4) the magnitude or s i g n s of the numbers, and the placement of  3  the v a r i a b l e i n r e l a t i o n t o e i t h e r the e q u a l i t y symbol o r numeric c o n s t a n t s . T h i s f a i l u r e t o c o n t r o l f a c t o r s which may have i n f l u e n c e d t h e type o f c o n c e p t u a l e r r o r s made by s t u d e n t s makes i t d i f f i c u l t the r e s u l t s o f these r e c e n t s t u d i e s t o t h e p r e s e n t Purpose  The purpose  and  Nature  of  the  to interpret  and apply  study. Study  o f t h e p r e s e n t study was t o develop and v a l i d a t e a  diagnostic checklist  (Diagnostic  Checklist  for Algebra)  t h a t would i d e n t i f y  e r r o r s made by b e g i n n i n g a l g e b r a s t u d e n t s when s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e .  I t would d i s t i n g u i s h among t h e c o n c e p t u a l , p r o c e d u r a l , and  r e s u l t a n t e r r o r s made by students when s o l v i n g such e q u a t i o n s ; thus, a l l o w i n g t h e i n v e s t i g a t i o n o f t h e nature and frequency o f t h e e r r o r s . A s e m i - s t a n d a r d i z e d i n t e r v i e w t e c h n i q u e was u t i l i z e d t o f a c i l i t a t e investigation.  T h i s t e c h n i q u e was used w i t h i n t h e c o n f i n e s o f a d i a g n o s t i c  i n s t r u m e n t which was i n p a r t a c h e c k l i s t o f mastery in part a checklist  l e v e l s o f concepts and  o f e r r o r s which o c c u r r e d when s o l v i n g l i n e a r  of a s p e c i f i e d d i f f i c u l t y The  such  equations  level.  f i r s t p a r t o f t h e instrument was used t o i n v e s t i g a t e t h e c o n c e p t u a l  framework o f a f i r s t - y e a r a l g e b r a student and was i n t h e form o f a checklist.  I t was used t o i d e n t i f y the s t a t u s o f a p a r t i c u l a r  concept  w i t h i n t h e s t u d e n t ' s c o g n i t i v e framework.  concept  was determined  (mastery,  partial,  A mastery  remedial).  i n s t r u m e n t was i n t h e form of a s e t o f e q u a t i o n s . t o i n v e s t i g a t e p r o c e d u r a l and r e s u l t a n t e r r o r s .  algebraic  l e v e l f o r each  The second p a r t o f the  These e q u a t i o n s were used The s e t of e q u a t i o n s was  f o r m u l a t e d t o c o n t r o l c o m p l e x i t y and s t r u c t u r e , as w e l l as n u m e r i c a l and magnitude.  sign  An e r r o r - c a t e g o r i z a t i o n scheme was developed t o f a c i l i t a t e  i d e n t i f i c a t i o n o f e r r o r s found by p r e v i o u s r e s e a r c h e r s i n a l g e b r a .  The  e r r o r - c a t e g o r i z a t i o n scheme was used i n c o n j u c t i o n w i t h t h e f o r m u l a t e d s e t  •  4  of e q u a t i o n s . The  a n a l y s i s o f t h e d i a g n o s t i c data from Grade 8 , 9 , and 1 0 s t u d e n t s was  d e s c r i p t i v e i n nature.  The reason f o r t h i s was t w o - f o l d :  t e c h n i q u e s employed and sampling s t r u c t u r e d i n t e r v i e w techniques  techniques u t i l i z e d .  diagnostic  Use o f semi-  i n c r e a s e d t h e time spent  diagnosing  i n d i v i d u a l s t u d e n t s , which i n t u r n l i m i t e d t h e number o f s t u d e n t s whose e r r o r s were i n v e s t i g a t e d .  To i n c o r p o r a t e a s t r a t i f i e d  the l i m i t e d number o f s t u d e n t s used, school.  t h e sample was drawn from a s i n g l e  The use o f s t a t i s t i c a l t e s t s r e q u i r i n g randomly s e l e c t e d data  the b e g i n n i n g a l g e b r a p o p u l a t i o n was not f e a s i b l e . used  random sample w i t h i n  Hence, t h e data  from  analysis  i n t h e study was d e s c r i p t i v e i n n a t u r e . Validity  i s s u e s were addressed throughout  f i n a l t e s t i n g of the instrument. textbook  t h e f o r m a t i o n , refinement, and  The sequencing  o f e q u a t i o n s was based on  a n a l y s i s and s t a t i s t i c a l t e c h n i q u e s were used t o p r o v i d e  validity.  structural  I n t e r a c t i o n e f f e c t s were a n a l y z e d t o p r o v i d e c o n s t r u c t v a l i d i t y .  Comparison o f r e s u l t s with measures o f a l g e b r a i c achievement, IQ, and l a t e r grades p r o v i d e d measures o f c o n c u r r e n t and p r e d i c t i v e Justification  of  the  validity.  Study  S e v e r a l r e s e a r c h e r s have noted t h e importance  of e a r l y diagnosis of  e r r o r s i n v o l v i n g t h e e q u a t i o n - s o l v i n g p r o c e s s e s used by s t u d e n t s i n beginning algebra Rosnick  (Davis & Cooney, 1 9 7 8 ; Hart,  1981, 1983; Krutetskii, 1 9 7 6 ;  & Clement, 1 9 8 0 ; Wagner e t a l . , 1 9 8 4 ) . An instrument  h e l p e x p l o r e s t u d e n t s ' understandings  o f a l g e b r a i c concepts  t o i d e n t i f y s t u d e n t s ' e r r o r s i n a l g e b r a i c procedures, e r r o r s a s s o c i a t e d w i t h computation, While many p a p e r - a n d - p e n c i l  was needed t o and vocabulary,  and t o determine  other  s i g n s , and t r a n s c r i b i n g .  a l g e b r a t e s t s e x i s t , they do not meet t h e  d i a g n o s t i c needs s t a t e d above. ' U s i n g p a p e r - a n d - p e n c i l t e s t s ,  correct  5  s o l u t i o n s may be reached through  erroneous  methods.  S i m i l a r l y , no paper-and-  p e n c i l t e s t s d i s t i n g u i s h among e r r o r t y p e s , n o r i d e n t i f y e r r o r s which occur d u r i n g t h e i n t e r m e d i a t e steps when s o l v i n g l i n e a r e q u a t i o n s .  However, by  u t i l i z i n g semi-structured interview techniques within a d i a g n o s t i c instrument,  such d i a g n o s t i c needs, c r e a t e d by t h e complex nature o f a l g e b r a ,  can be met. . L i s t e n i n g t o a student s o l v e an e q u a t i o n aloud, and i n q u i r i n g about t h e r e a s o n i n g p r o c e s s e s used, may permit e a r l y i d e n t i f i c a t i o n o f erroneous  conceptions  s t u d e n t ' s reasons  (Kilpatrick,  1 9 6 7 ) and f a c i l i t a t e  f o r h o l d i n g such c o n c e p t i o n s  understanding  (Kieran, 1 9 8 3 ; Hart,  of the  1983).  Thus, s e m i - s t r u c t u r e d i n t e r v i e w s seem t o meet t h e d i a g n o s t i c needs t o a g r e a t e r degree than do p a p e r - a n d - p e n c i l Hart  (1983)  expressed  tests.  concern about t h e l a c k o f a p p l i c a b i l i t y o f  q u a l i t a t i v e r e s e a r c h t o o t h e r s t u d e n t s and t e a c h i n g s i t u a t i o n s . when q u a l i t a t i v e r e s e a r c h u t i l i z e s equation complexity,  instruments which c o n t r o l t h e f a c t o r s of  s t r u c t u r a l format,  and n u m e r i c a l magnitude,  i d e n t i f i c a t i o n o f t h e s y s t e m a t i c e r r o r s made by a p a r t i c u l a r  student and t h e  common e r r o r s made among groups o f s t u d e n t s s h o u l d be p o s s i b l e 1982;  Hart,  1981),  r e g a r d l e s s o f changes i n s e t t i n g .  Such  (Anderson,  identification  should allow a p p l i c a b i l i t y of f i n d i n g s t o other i n s t r u c t i o n a l Furthermore,  However,  settings.  t h e i d e n t i f i c a t i o n o f s y s t e m a t i c and common e r r o r s i s n e c e s s a r y  t o p r o v i d e a f i r m f o u n d a t i o n f o r both t h e t e a c h i n g and t h e r e m e d i a t i o n of the e q u a t i o n - s o l v i n g p r o c e s s .  I n v e s t i g a t i o n of t h e concepts,  p r o c e s s e s , and  c o n t e x t i n which e r r o r s a r e made s h o u l d not be o v e r l o o k e d i n a l g e b r a i c diagnosis. Research S e v e r a l c r i t e r i a were important Diagnostic  Checklist  for Algebra.  Questions i n t h e development and v a l i d a t i o n o f t h e  First,  the d i a g n o s t i c u t i l i t y  of t h e  i n s t r u m e n t was c o n s i d e r e d t o be o f paramount importance.  Second, t h e  i d e n t i f i c a t i o n o f t h e n a t u r e and frequency o f e r r o r s o c c u r r i n g p a r t i c u l a r student was c r i t i c a l .  Third,  for a  the i d e n t i f i c a t i o n of systematic  and common e r r o r s made by s t u d e n t s i n s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e was i m p o r t a n t .  F o u r t h , the e s t a b l i s h m e n t o f t h e v a l i d i t y o f the  i n s t r u m e n t i n terms o f o t h e r measures o f achievement was n e c e s s a r y . the i n t e r a c t i o n e f f e c t s among e r r o r types and c o n t r o l f a c t o r s investigation. for  Hence, t h e f o l l o w i n g  instrument v a l i d i t y ,  required  f i v e c r i t e r i a were c o n s i d e r e d n e c e s s a r y  t h e development o f a sound d i a g n o s t i c  utility,  instrument  i n algebra:  criteria  following  diagnostic  error i d e n t i f i c a t i o n , interaction effects  among e r r o r s , and i n t e r a c t i o n e f f e c t s among e r r o r s and c o n t r o l The  Fifth,  factors.  r e s e a r c h q u e s t i o n s were f o r m u l a t e d t o determine i f the  f o r instrument development had been met:  1.  Does t h e Diagnostic Checklist for Algebra a s s i s t i n the i d e n t i f i c a t i o n o f t h e nature and frequency o f t h e e r r o r s made by a p a r t i c u l a r student?  2.  Does t h e Diagnostic Checklist for Algebra a s s i s t i n the i d e n t i f i c a t i o n of systematic e r r o r s f o r p a r t i c u l a r students and common e r r o r s among groups o f s t u d e n t s ?  3.  I s t h e Diagnostic  4.  Are s t u d e n t s ' understandings o f a l g e b r a i c concepts and v o c a b u l a r y r e l a t e d t o t h e k i n d s o f e r r o r s they make i n t h e s o l u t i o n o f l i n e a r e q u a t i o n s i n one v a r i a b l e ?  5.  Do t h e f a c t o r s o f e q u a t i o n complexity, s t r u c t u r a l format, and n u m e r i c a l magnitude i n t e r a c t i n terms o f t h e n a t u r e and f r e q u e n c y o f t h e e r r o r s produced?  Checklist  for Algebra  valid?  CHAPTER REVIEW  The  OF  II  RELATED  review o f t h e l i t e r a t u r e focuses  LITERATURE  on c r i t i c a l  development o f a d i a g n o s t i c instrument i n a l g e b r a .  f a c t o r s i n the It i s divided into  three  s e c t i o n s : d i a g n o s t i c models, d i a g n o s t i c methods, and t y p e s o f e r r o r s i n algebraic diagnosis. conceptualizing 1980;  The f i r s t  section presents  algebraic processing  Bernard & B r i g h t ,  1982) .  two models f o r  and d i a g n o s i n g  algebraic errors  The second s e c t i o n reviews p a p e r - a n d - p e n c i l  t e s t s , e r r o r - a n a l y s i s , and s e m i - s t r u c t u r e d  interviews.  examines t h e t y p e s of e r r o r s which have been r e p o r t e d researchers. research  (Matz,  The t h i r d by t h r e e  section  or more  The purposes o f t h e review o f the l i t e r a t u r e a r e t o p r o v i d e a  foundation  f o r the formulation  of a diagnostic  l i n e a r e q u a t i o n s i n one v a r i a b l e and t o put the c u r r e n t  instrument f o r study i n  perspective. Diagnostic  Models  S e v e r a l models have been proposed t o e x p l a i n e r r o r s made i n a l g e b r a (Bernard & B r i g h t , Elrick,  1979;  1982;  Davis,  1979,  Booth, 1 9 8 1 ; Matz, 1 9 8 0 ) .  just c i t e d e x p l i c i t l y conceptualize process:  algebraic processing  equations  (Bernard & B r i g h t ,  1980;  Matz  McKnight, Parker &  However, o n l y two o f t h e models  errors involved i n the equation-solving  (Matz, 1 9 8 0 ) and p r o c e s s i n g  1982).  These models p r o v i d e d  c a t e g o r i z i n g e r r o r s made i n s o l v i n g a l g e b r a i c Algebraic  Davis,  of l i n e a r a framework f o r  equations.  Processing  (1980)  i n d i c a t e d that  students f a l l v i c t i m t o one or more of t h r e e  g e n e r a l t y p e s of e r r o r s : t o new  1) unwarranted e x t r a p o l a t i o n  problem s i t u a t i o n s ; 2)  generalization  of known, v a l i d  unjustified extrapolation  t o another; and  3)  of  one  i n a b i l i t y t o change from one  u n d e r s t a n d i n g of a concept t o the next h i g h e r l e v e l .  rules  level  of  These t h r e e g e n e r a l  e r r o r t y p e s form a t h e o r e t i c a l f o u n d a t i o n f o r i n v e s t i g a t i n g e r r o r s made i n s o l v i n g one  variable  Extrapolation  equations.  e r r o r s can  occur i n one  of two  d i s t r i b u t i v e or r e p e a t e d a p p l i c a t i o n . The over a d d i t i o n Treating  the  ways:  d i s t r i b u t i v e law  i s an example of a v a l i d e x t r a p o l a t i o n square root  generalized  [e.g.  of m u l t i p l i c a t i o n a(b+c)=ab+ac].  symbol as i f i t were something which c o u l d  be  d i s t r i b u t e d over a d d i t i o n p r o v i d e s an example of a common erroneous (e.g. Va  extrapolation law  + b = Va+Vb).  The  i n c o r r e c t use  p r o v i d e s an example of r e p e a t e d a p p l i c a t i o n s  (ax+by)/(a+b)=x+y], where the terms of the obtain  the  numerator and  answer.  denominator and  Matz noted t h a t  s o l u t i o n s t o problems u s i n g (e.g.  student a p p l i e s  16/64=1/4 done by  then adds the p a r t i a l  generalization  Students know A*0  has  sometimes confuse m u l t i p l i c a t i o n and  addition.  between the  "x"  operations  ( i . e . the  r e s u l t s to correct  generalizable  6's). t o another o f t e n  However, s t u d e n t s  They f a i l  symbol vs the  S t u d e n t s ' i n a b i l i t y t o change from one  occurs  the v a l u e of 0 or A depending  on whether * r e p r e s e n t s m u l t i p l i c a t i o n or a d d i t i o n .  two  ax/a=x t o  s t u d e n t s sometimes a r r i v e at  c a n c e l l i n g the of one  cancellation  c a n c e l l a t i o n to i n d i v i d u a l  t e c h n i q u e s which are not  Unjustified extrapolation i n i d e n t i t y confusions..  [e.g.  of the  "+"  to d i s t i n g u i s h symbol).  l e v e l of u n d e r s t a n d i n g of a  concept  t o t h e next h i g h e r l e v e l a l s o seems t o r e s u l t  i n errors.  i n s t a n c e i n a r i t h m e t i c , when 7 and 8 a r e c o n c a t e n a t e d concept  o f p l a c e v a l u e and i m p l i c i t  denotes i m p l i c i t  multiplication.  both t h e o l d and new concepts  addition.  The student  For  t o form 7 8 t h e r e i s a  However, i n a l g e b r a , Ix i s r e q u i r e d t o keep and use  i n o r d e r t o understand  e x p r e s s i o n s such as  78x.  E r r o r s sometimes o c c u r "as the r e s u l t (s) o f r e a s o n a b l e , unsuccessful, situation"  attempts  t o adapt  of knowledge, i t i s important conceptions Kieran,  p r e v i o u s l y a c q u i r e d knowledge t o a new  1 9 8 0 , p. 1 5 4 ) .  (Matz,  although  In o r d e r t o diagnose  these m i s a p p l i c a t i o n s  t o a c q u i r e an u n d e r s t a n d i n g  involved i n students' errors  o f t h e erroneous  (Herscovics, 1 9 7 9 ; Herscovics &  1 9 8 0 ; Skemp, 1 9 8 0 ) .  Processing  Bernard performance  of  Linear  & Bright  Equations  (1982)  suggested  t h a t any model o f e q u a t i o n - s o l v i n g  should:  ... be a m u l t i - l e v e l account r a n g i n g a c r o s s (a) p e r c e p t i o n and i n t e r p r e t a t i o n o f a l g e b r a i c symbolism, (b) c o n c e p t u a l u n d e r s t a n d i n g o f t h e p r o b l e m - s o l v i n g t a s k , (c) a p p l i c a t i o n o f i n t e l l e c t u a l o p e r a t i o n s and p r o c e s s e s , and (d) the development of s t r a t e g i e s and g e n e r a l methods f o r s o l v i n g any e q u a t i o n of a p a r t i c u l a r type (p. 4 ) . They then proposed  a h e i r a r c h i c a l schema f o r modeling  contains s i x l e v e l s : tactic.  e q u a t i o n - s o l v i n g which  t a s k , method, s t r a t e g y , p r o c e s s , o p e r a t i o n , and  Each o f t h e s e l e v e l s may be a source o f a l g e b r a i c e r r o r s .  example, when p r e s e n t e d w i t h t h e t a s k o f s o l v i n g an e q u a t i o n 3(x+5)=27,  For  such as  a student may choose a s t e p - b y - s t e p method, a p p l y a s t r a t e g y of  removing b r a c k e t s through  the process of d i s t r i b u t i n g ,  and then use t h e  o p e r a t i o n o f m u l t i p l i c a t i o n with t h e r e s u l t o f w r i t i n g "3x" (Bernard & B r i g h t , adapted from F i g u r e 3, p. 37). At each l e v e l o f performance method, s t r a t e g y , p r o c e s s , o p e r a t i o n , t a c t i c )  (task,  s t u d e n t s have an o p p o r t u n i t y  t o make e r r o r s . When a student  i s p r e s e n t e d with a t a s k o f s o l v i n g an e q u a t i o n ,  knowledge o f t h e concepts  prior  o f v a r i a b l e and e q u a l i t y ( K i e r a n , 1981a, 1981b,  1983;  Wagner, 1981a) as w e l l as a r i t h m e t i c o p e r a t i o n s  1981)  a c t as i n p u t f a c t o r s f o r the t a s k .  (Englehardt & Wiebe,  Many p r o c e s s i n g f a c t o r s  when c h o o s i n g t h e methods, s t r a t e g i e s ,  and p r o c e s s e s  the t a s k .  from t h e i n c o r r e c t  Sometimes c o n f u s i o n s r e s u l t  interact  used i n a c c o m p l i s h i n g ordering of  a l g e b r a i c o p e r a t i o n s i n v o l v i n g a d d i t i o n and m u l t i p l i c a t i o n p r o p e r t i e s of equality  (Davis & Cooney, 1978), d i s t r i b u t i v e e r r o r s  of o p e r a t i o n s  (Rosnick & Clement, 1980) .  t a c t i c s necessary  f o r completion  When p e r f o r m i n g  o f the t a s k , output  as v i s u a l and p e r c e p t u a l c o n f u s i o n s  (Matz, 1980) and order t h e o p e r a t i o n s and  e r r o r s may occur,  such  (Matz, 1980) and c a r e l e s s mistakes  (Englehardt & Wiebe, 1981) . Bernard  & Bright  a l g e b r a i c concepts,  (1982) i n d i c a t e d t h a t s t u d e n t s ' understandings o f symbolism, o p e r a t i o n s and s t r a t e g i e s must be e x p l o r e d t o  p r o v i d e comprehensive d i a g n o s i s .  The i n t e r a c t i o n among these  understandings  may a f f e c t a s t u d e n t ' s p e r c e p t i o n o f the t a s k and t h e methods a v a i l a b l e t o accomplish  that task.  Too o f t e n s t u d i e s on e r r o r s i n e q u a t i o n - s o l v i n g  emphasize t h e r e s u l t a n t e r r o r s with some focus on p r o c e d u r a l e r r o r s and little,  i f any, concern  f o r conceptual e r r o r s .  instrumental understanding  Such emphasis l e a v e s o n l y an  o f s t u d e n t s ' e r r o r s and "may focus a t t e n t i o n on  aspects of equation  s o l v i n g t h a t do not suggest  (Bernard & B r i g h t , 1 9 8 2 , p. 2 1 ) .  remediation  Hence, i t i s important  techniques"  t o explore the  i n t e r r e l a t i o n s h i p s among c o n c e p t u a l , p r o c e d u r a l , and r e s u l t a n t e r r o r s t o p r o v i d e a thorough u n d e r s t a n d i n g  o f t h e nature  o f t h e e r r o r s which  students  make and t o p r o v i d e a b a s i s f o r r e m e d i a t i o n . Summary  Matz  of  Diagnostic  (1980)  Models  and Bernard  & Bright  (1982)  d e s c r i b e d models o f a l g e b r a i c  l e a r n i n g t h a t p r o v i d e a framework w i t h i n which a l g e b r a i c e r r o r s might be explored.  They emphasized t h e importance o f p r i o r knowledge, o r d e r i n g of  concepts,  processing a b i l i t y ,  p a r t i a l and f i n a l  results.  and accuracy  of r e t r i e v a l  T h e i r work u n d e r l i n e s t h e importance o f  a s s e s s i n g i n p u t and p r o c e s s , as w e l l as output, Diagnostic  Paper-and-pencil have been developed  i n o b t a i n i n g both  i n diagnosis.  Methods  tests, error-analysis,  and s e m i - s t r u c t u r e d i n t e r v i e w s  t o i n v e s t i g a t e students' understanding  of algebra.  However, each o f t h e s e methods has weaknesses, as w e l l as s t r e n g t h s . Paper-and-Pencil  Petrosko  (1978)  Tests  reviewed  the s t r e n g t h s and weaknesses o f 1 2 2  s t a n d a r d i z e d a l g e b r a t e s t s a v a i l a b l e a t t h e secondary  level.  These t e s t s  i n c l u d e d achievement and d i a g n o s t i c t e s t s , but not a p t i t u d e t e s t s , e v a l u a t e d on 3 9 c r i t e r i a by a t l e a s t two r a t e r s . i n c l u d e d a s h o r t statement procedures  r e g a r d i n g sources  Almost h a l f o f t h e t e s t s  f o r items. • However, a c t u a l  used f o r s e l e c t i o n of items were p r e s e n t e d  t e s t manuals.  and were  i n o n l y t e n of the  C o r r e l a t i o n s with o t h e r t e s t s , e x p e r i m e n t a l  uses o f the t e s t ,  and t h e o r e t i c a l support were not p r o v i d e d i n 110 o f t h e 122 t e s t s . Concurrent  and p r e d i c t i v e v a l i d i t y s t u d i e s were not mentioned i n any o f the  t e s t manuals.  While  20 of t h e 122 t e s t s p r o v i d e d a c c e p t a b l e  consistency r e l i a b i l i t y c o e f f i c i e n t s was  p r o v i d e d f o r any o f the t e s t s .  were " d e f i c i e n t  (r>0.70), Petrosko  no o t h e r form o f r e l i a b i l i t y (1978) noted t h a t many t e s t s  i n b a s i c aspects of t e s t q u a l i t y "  Most s t a n d a r d i z e d mathematics t e s t s do l i t t l e student w i l l  "perform p o o r l y , " o r has performed  (p.  146).  more than p r e d i c t t h a t a  poorly, i n the p a r t i c u l a r  areas t e s t e d (Swanson, Schwartz, G i n s b e r g & Kossan, 1981) . t e s t s p r o v i d e an a c c u r a t e r e c o r d of s t u d e n t s ' work, but f a i l thought  processes.  "Paper-and-pencil  tests  internal  Paper-and-pencil to reveal their  ... a r e so o f t e n m i s i n t e r p r e t e d  t h a t t h e y need t o be supplemented with o t h e r means o f e v a l u a t i o n " (Peck and Jencks,  1974, p. 5 4 ) . P a p e r - a n d - p e n c i l  regarding input e r r o r s ,  and almost  t e s t s p r o v i d e minimal  no i n f o r m a t i o n r e g a r d i n g p r o c e s s  and e r r o r s made when working through t h e i n t e r m e d i a r y s t e p s results)  l e a d i n g t o the f i n a l  information  solution.  errors  (partial  These d e f i c i e n c i e s l i m i t t h e  d i a g n o s t i c e f f e c t i v e n e s s of paper-and-pencil  tests.  Error-Analysis The purpose o f e r r o r - a n a l y s i s i s t h r e e f o l d : t o determine frequency,  the type,  and source o f e r r o r s made i n p e r f o r m i n g a l g e b r a i c t a s k s .  The  e a r l y s t u d i e s i n a l g e b r a f o c u s e d on t h e nature and frequency o f e r r o r s . More r e c e n t work has f o c u s e d on t h e kinds o f r u l e s t h a t s t u d e n t s generate i n c r e a t i n g these Wattawa  errors.  (1927) r e c o r d e d the e r r o r s made on t e s t s ,  w r i t t e n work, and o r a l  responses i n a n i n t h - g r a d e noted  a l g e b r a c l a s s f o r a p e r i o d of t h r e e months.  s e v e r a l mistakes r e l a t e d to s o l v i n g l i n e a r equations  coefficients 13.8%,  and  (arithmetic:  4 1 . 4 % , a l g e b r a i c laws: 6.9%).  incorrect operations: (17.2%)  fractions  and q u a d r a t i c s  (3.5%).  with  He  integral  1 7 . 2 % , sign errors:  Other e r r o r s o c c u r r e d r e l a t i n g t o Many of the a r i t h m e t i c and  sign  (10.8%).  e r r o r s were a r e s u l t of f a u l t y r e a d i n g or c o p y i n g (1929)  i n v e s t i g a t e d the e r r o r s of 3 5 0 f i r s t - y e a r a l g e b r a  students  in four d i f f e r e n t  s c h o o l s u s i n g a t e s t of the f o u r b a s i c o p e r a t i o n s  with  Pease  numbers, monomials, and p o l y n o m i a l s . s u b j e c t s had d i f f i c u l t y monomials. difficulties numeric  He  He  found t h a t almost 25% of  with m u l t i p l y i n g by zero, e i t h e r with numbers or  a l s o found t h a t about 20% of the s u b j e c t s  i n adding  coefficient  and  subtracting  (e.g. -aH—a) .  monomials  with  experienced no  apparent  S u b t r a c t i n g a p o s i t i v e monomial from a  n e g a t i v e monomial a l s o produced about a 20% e r r o r r a t e .  Removing b r a c k e t s  p r e c e d e d by a minus s i g n produced about a 25% e r r o r r a t e f o r both and  numbers  polynomials. Pease p r o v i d e d a summary of e r r o r types a c r o s s o p e r a t i o n s  monomials and p o l y n o m i a l s . (31%),  l i t e r a l number (5%),  arithmetic exponent  zero  These e r r o r types were s i g n  (e.g. a+a=a, 6 % ) , (4%),  combination  careless errors  (8%),  (23%),  of terms  unclassified  f o r numbers, process (4%), (11%),  and  (8%).  Randall 56  the  (1955)  high schools.  r e p o r t e d on the common e r r o r s made by Two  or t h r e e students  t h e i r mathematics t e a c h e r s  1 3 1 students  were chosen from each s c h o o l  from by  as b e i n g the b e s t g e n e r a l mathematics students  in  that p a r t i c u l a r school.  The t e s t a d m i n i s t e r e d  the f o u r b a s i c o p e r a t i o n s , Students  l i n e a r equations,  t o these  factoring,  involved  and word problems.  had common e r r o r s o f s u b t r a c t i n g t h e l a r g e r number from t h e s m a l l e r  number, combining u n l i k e terms, and knowing procedures unknown.  students  A l l o t h e r common e r r o r s found  were not a p p l i c a b l e t o t h e present  r e l a t e d t o f r a c t i o n s o r decimals and  study.  Suggestions  the r e s u l t s by s t r e s s i n g a r i t h m e t i c accuracy, g e n e r a l mathematics s t u d e n t s ,  f o r f i n d i n g the  were made t o improve  by t e a c h i n g some a l g e b r a t o  and by emphasizing mathematical t h i n k i n g  through t h e use o f word problems. Hilling-Smith the a b i l i t y  (1979) t e s t e d 105 t h i r t e e n y e a r - o l d s from t h e t o p 90% o f  range u s i n g t e n q u e s t i o n s d e v i s e d t o i n v e s t i g a t e t h e i r  understanding  of beginning  algebra.  He found t h e f o l l o w i n g types o f e r r o r s :  o r d e r o f o p e r a t i o n s done l e f t t o r i g h t exponents  5  (5x with x , 24%; z  ( i e . 3+Q =3Q,  17%);  3  (60%); c o n f u s i n g c o e f f i c i e n t s  with 3z, 22%);  adding u n l i k e terms  c o n f u s i o n o f i n e q u a l i t y symbols  a b s t r a c t and g e n e r a l i z e  (64%).  Suggestions  with  (20%); and i n a b i l i t y t o  were made f o r t e a c h i n g  algebra  u s i n g apparatus and demonstrations so t h a t a b s t r a c t i o n s would be more c o n c r e t e l y c o n c e p t u a l i z e d , and f o r emphasizing t h e c o r r e c t r e a d i n g o f mathematics f o r s t u d e n t s . Tatsuoka, Bierenbaum, Ginsburg, students  & Kossan  (1980) t e s t e d 127 Grade 8  on a d d i t i o n o f i n t e g e r s a f t e r t h r e e weeks o f i n s t r u c t i o n .  Two  e r r o r s r e l a t i n g t o t h e a b s o l u t e v a l u e o f t h e numbers were i d e n t i f i e d . Students  e i t h e r added t h e a b s o l u t e v a l u e s q f t h e numbers o r s u b t r a c t e d t h e  number with s m a l l e r a b s o l u t e v a l u e from t h e number w i t h l a r g e r a b s o l u t e  value.  S i x e r r o r s were found which r e l a t e d t o t h e s i g n o f t h e answer.  Students e i t h e r used t h e s i g n o f t h e l a r g e r o r s m a l l e r  number, always used a  p o s i t i v e o r n e g a t i v e s i g n , o r always used t h e s i g n o f t h e f i r s t number.  A "sign X absolute value e r r o r vector"  Four items were r e q u i r e d  o r second  was used t o c r e a t e  t o determine a s t u d e n t ' s r u l e u n i q u e l y .  r e s u l t s o f t h e study i n d i c a t e d t h a t  s t u d e n t s most f r e q u e n t l y  df t h e number w i t h t h e l a r g e r a b s o l u t e v a l u e and t h a t  of v a l i d rules using  the rules  other o p e r a t i o n s ,  n e g a t i v e s make a p o s i t i v e , " were not i n c l u d e d .  The  used the s i g n students  g e n e r a t e d were based on t h e type o f i n s t r u c t i o n they r e c e i v e d . involving extrapolation  items.  Rules such as "two  The l a c k o f i n c l u s i o n of  such r u l e s may have l i m i t e d t h e a p p l i c a b i l i t y o f t h e r e s u l t s t o c l a s s r o o m situations. Cote  (1981) s t u d i e d t h e o p e r a t i o n s o f a d d i t i o n  integers.  The f o c u s was on t h e c o n f u s i o n s t h a t  symbols and t h e s i g n o f t h e numbers. a d d i t i o n and s u b t r a c t i o n direct add  subtraction  of integers.  F i f t y - f i v e s t u d e n t s were taught Three major t y p e s o f e r r o r s  errors,  S u b j e c t s were c a t e g o r i z e d  Results  i n d i c a t e d that  direct,  17 from 45 t o g i v e  (e.g. -17 an answer  by the number and types o f e r r o r s  o f the 30 s u b j e c t s  18 were c l a s s i f i e d as d i r e c t s u b t r a c t i o n ,  were u n c l a s s i f i e d .  occurred:  (e.g. -17+49 meant  and problems o f i n t e g r a t i o n  - 45 meant change 45 t o -45 but then s u b t r a c t  made.  a r i s e between t h e o p e r a t i o n  (e.g. 17-38 meant 38-17), s u b s t i t u t i o n  17 and 49 and make i t n e g a t i v e ) ,  of 2 8 ) .  and s u b t r a c t i o n o f  Of t h e 15 s u b j e c t s  who made l e s s than 10 9 as s u b s t i t u t i o n and 3  who made more than t e n e r r o r s ,  3 s u b s t i t u t i o n , and 6 i n t e g r a t i o n .  they  The h i g h e r a c h i e v e r s  6 were  d i d not  make i n t e g r a t i o n e r r o r s .  Ten s u b j e c t s  r e s u l t s suggested t h a t t h e d i f f e r e n c e s achievers  made no e r r o r s .  i n the e r r o r patterns  may p r o v i d e i n s i g h t i n t o t h e d i f f i c u l t i e s  with i n t e g e r s .  Cote's  (1981)  o f h i g h and low  which low a c h i e v e r s  have  The r e s u l t s r e i n f o r c e d t h e importance o f c o n s i d e r i n g t h e  s i g n , magnitude and o r d e r o f p r e s e n t a t i o n  when u s i n g  integers  i n algebraic  diagnosis. (1982)  Anderson  number, i n t e g e r ,  used e r r o r - a n a l y s i s t o compare t h e e r r o r s made on whole  and p o l y n o m i a l e x p r e s s i o n s .  incorrect-operation  She found s i g n  errors,  e r r o r s , d i s t r i b u t i o n e r r o r s , and exponent e r r o r s t o be  common among many s t u d e n t s .  However, students made t h e s e e r r o r s o n l y  context of e i t h e r arithmetic  or algebra,  In f o u r  reports  but not c o n c o m i t a n t l y .  o f a study on a l g e b r a i c  1984b, 1984c, 1986) d e s c r i b e d  i n the  diagnosis,  Sleeman  (1984a,  t h e use o f a computer-based Leeds M o d e l l i n g  System t o d i a g n o s e e r r o r s o f 14- and 1 5 - y e a r - o l d a l g e b r a  students.  A high  p e r c e n t a g e o f t h e 1 4 - y e a r - o l d s ' e r r o r s were undiagnosed by t h i s method. I n t e r v i e w s were s u b s e q u e n t l y used t o determine t h e a c t u a l errors.  There were f o u r d i s t i n c t  i n which a c o r r e c t inappropriate  classes  of errors  operation;  2) parsing, 2 + 3x = 5x);  misreads o r m i s c o p i e s ; and 4) random.  l e a r n i n g algebra  1) m a n i p u l a t i v e , by an  i n which t h e student m i s g e n e r a l i z e s the 3) c l e r i c a l , The f i r s t  support t h e t h e o r e t i c a l framework o f a l g e b r a i c Sleeman  found:  r u l e e i t h e r has one stage o m i t t e d o r r e p l a c e d  problem s i t u a t i o n (e.g.  (1980).  reasons f o r the  (1984b) f u r t h e r suggested t h a t  i n which t h e student  two o f these e r r o r types  e r r o r s proposed by Matz "the d i f f i c u l t i e s i n  have been g r e a t l y u n d e r - e s t i m a t e d "  (p.  35).  (1987)  Movshovitz-Hadar, Z a s l a v s k y , & Inbar  i n v e s t i g a t e d the e r r o r s made  by h i g h s c h o o l s t u d e n t s i n the Mathematics M a t r i c u l a t i o n Examination in  I s r a e l t o a l l s t u d e n t s i n the n o n - s c i e n t i f i c stream at the end of the  Grade 1 1 y e a r .  c o n s e c u t i v e years the r e s e a r c h e r s a n a l y z e d 18 open-  For two  items c o n s i s t i n g of 1 3 g e n e r a l i t e m types among which  ended t e s t  e q u a t i o n s was  o n l y one  item type.  The  based  six  e r r o r c a t e g o r i e s and t h e i r percentages  data  (2%,  1%),  f o r both y e a r s were:  m i s i n t e r p r e t e d language 4)  (17%,  18%),  3)  (0%, 2 % ) ,  purpose of the model was  and  6)  1)  invalid  32%),  (25%, 2 7 % ) .  technical error  The  misused  logically  (34%,  d i s t o r t e d theorem or d e f i n i t i o n  unverified solution  determine  developed  on the e m p i r i c a l data g a t h e r e d and not on p r e v i o u s r e s e a r c h .  ( 2 2 % , 2 0 % ) , 2)  inference  linear  c a t e g o r y system which they  was  5)  given  The  to provide i n s i g h t i n t o students' e r r o r s to  i f they occur a c r o s s areas of mathematical  t o p i c s or are  a s s o c i a t e d with p a r t i c u l a r l e a r n i n g or t e a c h i n g s t y l e s . The e r r o r s a s s o c i a t e d with l i n e a r e q u a t i o n s i n one v a r i a b l e f e l l the c a t e g o r i e s of misused data, m i s i n t e r p r e t e d language, definition, The  and t e c h n i c a l e r r o r .  Four coders were used  distorted f o r each s u b j e c t .  coders c a t e g o r i z e d e r r o r s w i t h an i n t e r - r a t e r r e l i a b i l i t y  d e t e r m i n i n g the c a t e g o r y i n t o which an e r r o r was  classified,  w r i t t e n work of the student, i n c l u d i n g " c r o s s e d - o u t , (Movshovitz-Hadar  e t a l . , 1 9 8 7 , p. 5)  was  analyzed.  of 0 . 9 1 .  The  untidy parts" T h i s meant t h a t o n l y  authors a l s o found " c o r r e c t  from e r r o r s t h a t c a n c e l l e d one  another"  (p. 7 ) .  In  o n l y the  the n a t u r e and f r e q u e n c y of the e r r o r s were c a t e g o r i z e d , without the u n d e r l y i n g cause.  into  determining  s o l u t i o n s t h a t arose  There was  no i n f o r m a t i o n  18  p r o v i d e d r e g a r d i n g the frequency cause of the double reasons  e r r o r s was  of these types of c o r r e c t s o l u t i o n s .  not i n v e s t i g a t e d .  f o r s t u d e n t s making these e r r o r s was  The  Hence, a d i a g n o s i s of the  not p r o v i d e d through  the use  of  p o s t - f a c t o a n a l y s i s of s t u d e n t s ' w r i t t e n work. E r r o r - a n a l y s i s p r o v i d e s a wealth  of v e r y p r e c i s e i n f o r m a t i o n .  the i n f o r m a t i o n i s o f t e n overwhelming. frequency  of i n d i v i d u a l e r r o r s ,  The  focus tends  foundations  (Anderson,  1982) .  Semi-Structured  the  r a t h e r than on the i n t e r r e l a t i o n s h i p s of the  e r r o r s and t h e i r u n d e r l y i n g c o n c e p t u a l f o u n d a t i o n . conceptual  t o be on  However,  These u n d e r l y i n g  cannot be i d e n t i f i e d by e r r o r - a n a l y s i s alone  Interviews  D i a g n o s t i c i a n s have noted t h a t i n t e r v i e w s p r o v i d e i n s i g h t  into  the  c o n c e p t u a l f o u n d a t i o n s of e r r o r s which i s l a c k i n g i n o t h e r d i a g n o s t i c approaches  (Hart, 1981;  K i e r a n , 1983;  Wagner et a l . , 1984;  However, i n t e r v i e w s are time-consuming and (Carpenter, Blume, Herbert, Anick, s u b j e c t s ' e x p l a n a t i o n s of how r e f l e c t i o n of thought  and  speech  slows the response i n t e r v i e w e r may lower  processes  (Davis, 1980) .  rate  (Kilpatrick,  Opper, 1977) .  In a d d i t i o n ,  not be a v a l i d  Procedures  Davis e t a l . , 1979).  1967).  also influence results  used by  ( H e r s c o v i c s , 1979).  between  the  Wording changes  (Opper, 1977) .  s u b j e c t s ' e x p l a n a t i o n s of t h e i r  students  Verbalization  I n t e r a c t i o n with  of " t h i n k - a l o u d " t e c h n i q u e s  limitations,  i s questionable  be i n f l u e n c e d by the m e d i a t i o n  (Behr e t a l . , 1976;  the r e l i a b i l i t y  d e s p i t e these  & Pimm, 1982;  they s o l v e d problems may  when s o l v i n g problems o r a l l y may thought  reliability  Sleeman, 1986).  own  However,  thought  19  processes  are b e l i e v e d t o p r o v i d e more data and more r e l i a b l e data  paper-and-pencil  tests  (Anderson,  1982;  Hart,  r e l i a b i l i t y and v a l i d i t y need t o be addressed  1983).  than  Hence, the i s s u e s of  i n the development of a  d i a g n o s t i c interview technique. S t r u c t u r e d i n t e r v i e w s which p r e s e n t the same q u e s t i o n s seem t o s t a b i l i z e s t u d e n t s ' responses, 1983; words.  Wagner e t a l . , 1984).  thereby  reliability  (Hart,  S t r u c t u r e d i n t e r v i e w s reduce v a r i a b i l i t y  However, the r i g i d i t y of c o m p l e t e l y  their usefulness.  improving  i n the same order  s t r u c t u r e d i n t e r v i e w s may  i n cue limit  E f f e c t i v e d i a g n o s i s cannot i g n o r e i n d i v i d u a l d i f f e r e n c e s .  There needs t o be a compromise between c o m p l e t e l y free-associational questioning.  structured interviews  and  S e m i - s t r u c t u r e d i n t e r v i e w s seem t o p r o v i d e  such a compromise. S e m i - s t r u c t u r e d i n t e r v i e w s p r e s e n t the same t a s k t o each student, f o l l o w - u p q u e s t i o n s can be more f l e x i b l e  (Wagner e t a l . , 1984).  but  Semi-  s t r u c t u r e d i n t e r v i e w s a l l o w f o r m u l a t i o n of hypotheses about the nature of a student's d i f f i c u l t i e s ,  t e s t i n g of hypotheses,  s y s t e m a t i c n a t u r e of e r r o r s  and v e r i f i c a t i o n of the  (Swanson e t a l . , 1981;  Wagner, 1981b).  S e m i - s t r u c t u r e d i n t e r v i e w s seem t o i n c r e a s e the r e l i a b i l i t y  from f r e e -  a s s o c i a t i o n a l q u e s t i o n i n g and yet s t i l l p r o v i d e u s e f u l d i a g n o s t i c i n f o r m a t i o n about i n d i v i d u a l  students.  I t s h o u l d be noted t h a t s e m i - s t r u c t u r e d i n t e r v i e w s a l s o have limitations.  When u s i n g s e m i - s t r u c t u r e d i n t e r v i e w s , Wagner e t a l . (1984)  reported s e v e r a l occurrences " e x p r e s s i o n , " and  "equation."  of c o n f u s i o n among the terms The  "variable,"  " i n v e s t i g a t o r s were u n s u c c e s s f u l i n  20  f o r m u l a t i n g q u e s t i o n s t h a t would demonstrate a l l of these incisively"  (p. 39).  There i s a need f o r t e c h n i q u e s  within semi-structured interviews that w i l l and p r o v i d e i n s i g h t One  study,  i d e n t i f y conceptual  which can be u t i l i z e d with  interviews i s detailed error-analysis,  1980;  t o be i n c o r p o r a t e d  i n t o the i n t e r r e l a t i o n s h i p s among d i f f e r e n t  such t e c h n i q u e  several researchers  confusions  (Anderson,  Wagner e t a l . , 1984).  1982;  confusions errors.  semi-structured  and t h i s approach has been used by  Sleeman, 1984a, 1986;  However, except  Tatsuoka  et a l . ,  i n the case of the Wagner  the use of s e m i - s t r u c t u r e d i n t e r v i e w s o c c u r r e d a f t e r the r e s u l t s of  the e r r o r - a n a l y s i s . p r o c e s s e s may an e r r o r .  Students' p o s t - f a c t o v e r b a l i z a t i o n s of t h e i r  be i n f l u e n c e d by memory-decay and the knowledge t h a t they made  Hence, such v e r b a l i z a t i o n s may  s t u d e n t s ' thought  not be v a l i d r e p r e s e n t a t i o n s of  p r o c e s s e s when i n i t i a l l y  solving equations.  i m p e r a t i v e t h a t s e m i - s t r u c t u r e d i n t e r v i e w s be conducted  w i t h the c o n t i n u i t y of s t u d e n t s ' thought of e r r o r - a n a l y s i s ,  processes.  It i s  d u r i n g the a c t u a l  e q u a t i o n - s o l v i n g p r o c e s s ; w h i l e , e n s u r i n g t h a t p r o b i n g does not  techniques  thought  interfere  By combining the  s e m i - s t r u c t u r e d i n t e r v i e w s , and j u d i c i o u s  p r o b i n g w i t h i n a d i a g n o s t i c instrument,  complementary and  confirming  i n f o r m a t i o n s h o u l d be p r o v i d e d r e g a r d i n g the nature of the e r r o r s students make i n s o l v i n g l i n e a r e q u a t i o n s Summary  of  Diagnostic  (Anderson,  Semi-structured  variable.  Methods  Used alone, p a p e r - a n d - p e n c i l instruments  i n one  1982;  t e s t s are not e f f e c t i v e as d i a g n o s t i c  Peck & Jencks,  1974;  Swanson e t a l . , 1981).  i n t e r v i e w s seem t o p r o v i d e a procedure  which i s e f f e c t i v e i n  21  diagnosing again,  e r r o r s made i n s o l v i n g e q u a t i o n s  interviews  used alone  mathematical c o n t e n t . which i n c o r p o r a t e s interviews.  (Wagner e t a l . , 1984).  do not p r o v i d e  sufficient  e r r o r - a n a l y s i s techniques  with  semi-structured  Such a d i a g n o s t i c procedure may be seen as u t i l i z i n g  performance on a l g e b r a i c t a s k s .  "Used s e p a r a t e l y , kinds  s t r u c t u r i n g of the  There i s a need t o develop a d i a g n o s t i c procedure  q u a n t i t a t i v e and q u a l i t a t i v e methods t o a n a l y z e and  students'  When f o c u s e d  both  u n d e r s t a n d i n g s of  As p o i n t e d out by F i r e s t o n e  q u a l i t a t i v e and q u a n t i t a t i v e s t u d i e s p r o v i d e  of information.  But,  (1987),  different  on the same i s s u e , q u a l i t a t i v e and  q u a n t i t a t i v e s t u d i e s can t r i a n g u l a t e - t h a t i s , use d i f f e r e n t methods t o assess  the robustness or s t a b i l i t y of f i n d i n g s . "  q u a n t i t a t i v e and q u a l i t a t i v e methods o f d i a g n o s i s would improve i t s d i a g n o s t i c u t i l i t y .  The i n c l u s i o n o f w i t h i n t h e same instrument  However, no d i a g n o s t i c  instruments  were found which i n c o r p o r a t e d both q u a n t i t a t i v e and q u a l i t a t i v e a n a l y s e s . Hence, t h e r e  i s a need f o r t h e development o f such an Types  The  of  Errors  in  Algebraic  types of e r r o r s i n a l g e b r a i c diagnosis  instrument.  Diagnosis  are presented  the c a t e g o r i z a t i o n scheme developed f o r use i n t h e Diagnostic Algebra. and  This presentation provides  an o r d e r e d  reference  according to Checklist  a r e s e a r c h b a s i s f o r t h a t development  f o r each c a t e g o r y  and s u b - c a t e g o r y .  The  d e f i n i t i o n s used i n t h e d e v e l o p e d c a t e g o r i z a t i o n scheme f o r e r r o r s a r e presented. pertinent  for  Each e r r o r - t y p e i s then p r e s e n t e d  with  i t s review o f t h e  literature.  Three main c a t e g o r i e s f o r e r r o r s i n a l g e b r a  were f o r m u l a t e d  i n t o an  22  e r r o r - c a t e g o r i z a t i o n scheme based on the work, of C a r r y e t a l . (1980),  Booth  (1981),  errors  and B e r n a r d  & Bright  (1982).  As these r e s e a r c h e r s suggested,  s h o u l d be viewed i n terms of p r e v i o u s knowledge, p r o c e s s e s  used, and  obtained.  (knowledge of  concepts  The  t h r e e main c a t e g o r i e s are c a l l e d C o n c e p t u a l  and v o c a b u l a r y a s s o c i a t e d with a l g e b r a i c e q u a t i o n s  v a r i a b l e , e q u a l i t y , expressions, equations, identities,  and i n v e r s e s ) , P r o c e d u r a l  l i n e a r equations  coefficients,  including  like  terms,  ( e r r o r s made i n i n the s o l u t i o n of  with i n t e g r a l c o e f f i c i e n t s i n v o l v i n g a d d i t i o n and  m u l t i p l i c a t i o n p r o p e r t i e s of e q u a l i t y , o r d e r of o p e r a t i o n s , and d i s t r i b u t i v e p r o p e r t y ) , and R e s u l t a n t  (computational  and  from the f o u r b a s i c o p e r a t i o n s , m e c h a n i c a l / p e r c e p t u a l and  incomplete  results  work).  the  sign errors arising  errors,  These e r r o r s c o u l d occur d u r i n g the  random e r r o r s ,  intermediate  s t e p s , as w e l l as i n the f i n a l s o l u t i o n s of l i n e a r e q u a t i o n s  i n one  variable. A d e t a i l e d d i s c u s s i o n of the e r r o r - c a t e g o r i z a t i o n scheme and a t a b u l a r p r e s e n t a t i o n of the r e s e a r c h e r s ' f i n d i n g s of p a r t i c u l a r e r r o r s i s p r o v i d e d i n Appendix A. presented  A summary of the r e s e a r c h f i n d i n g s on e r r o r s i n a l g e b r a i s  i n T a b l e 1 at the c o n c l u s i o n of t h i s  Conceptual Englehardt  Errors & Wiebe  from incomplete,  s o l v i n g equations  (1981) d e f i n e d c o n c e p t u a l e r r o r s as " e r r o r s a r i s i n g  absent,  m a t h e m a t i c a l concepts  concepts  chapter.  or i n c o r r e c t u n d e r s t a n d i n g  and p r i n c i p l e s "  (p. 14).  The  of u n d e r l y i n g c h o i c e of steps used i n  can be i n f l u e n c e d by the l e v e l of u n d e r s t a n d i n g  of v a r i a b l e (Wagner, 1981a), l i k e terms  (Anderson,  1982),  of the and  23  e q u a l i t y symbol  ( K i e r a n , 1981a).  The meaning of e q u a t i o n s  and the use o f i d e n t i t i e s and i n v e r s e s (Matz,  1980)  (Kieran, 1 9 8 3 )  a l s o i n f l u e n c e d some of  the p r o c e s s e s used by s t u d e n t s i n s o l v i n g e q u a t i o n s . V a r i a b l e  Students concept  seem t o have an extremely i n c o n s i s t e n t u n d e r s t a n d i n g of the  of v a r i a b l e .  Some view a v a r i a b l e t o be a l e t t e r of the alphabet,  w h i l e o t h e r s view i t as a replacement  f o r a hidden number.  argued t h a t u n d e r s t a n d i n g o f the concept o f v a r i a b l e may age.  Kieran  Kiichemann  be a  f u n c t i o n of  (1983) and Wagner (1981a) r e p o r t e d u n d e r s t a n d i n g of the  of v a r i a b l e t o be r e l a t e d t o the n o t a t i o n used.  Siford  (1978)  (1981) found  concept that  Grade 10 s t u d e n t s were unable t o accept a v a r i a b l e as a g e n e r a l unknown. They thought  of v a r i a b l e s o n l y as r e p r e s e n t i n g one  I t i s important  i n d i a g n o s i s not o n l y t o determine  s p e c i f i c unknown number. a s t u d e n t ' s view of  v a r i a b l e , but a l s o t o understand the s t a b i l i t y of t h i s view. view o f v a r i a b l e may  A student's  i n f l u e n c e the types of e r r o r s made when s o l v i n g  equations. Kiichemann  (1978) and Booth  (1981) r e p o r t e d t h a t the degree  of  u n d e r s t a n d i n g o f the concept of v a r i a b l e seemed t o i n t e r a c t with both  the  type and number o f o p e r a t i o n s i n v o l v e d i n s i m p l i f y i n g a p a r t i c u l a r expression or s o l v i n g a p a r t i c u l a r equation.  Kiichemann  (1978) found tha't  the o p e r a t i o n s of s u b t r a c t i o n and d i v i s i o n p r o v i d e d more i n s i g h t  into  the  t e n a c i t y w i t h which s t u d e n t s would c l i n g t o t h e i r p a r t i c u l a r view of variable.  The number of o p e r a t i o n s and the o c c u r r e n c e s of the  w i t h i n the e q u a t i o n s caused s h i f t i n g views of the concept  variable  of v a r i a b l e  (Booth,  1981).  D a v i s e t a l . (1979) a l s o noted these phenomena and suggested  that mathematically without  s t r o n g e r s t u d e n t s may be a b l e t o d i s c u s s e q u a t i o n s  a c t u a l l y s o l v i n g them.  Such a b i l i t y seemed t o enable t h e h i g h e r  a c h i e v i n g s t u d e n t s t o make t h e t r a n s i t i o n from a r i t h m e t i c t o a l g e b r a more readily  (Davis & McKnight,  Rosnick  & Clement  1979).  (1980) and Rosnick  (1981) found t h a t even c o l l e g e  s t u d e n t s d i d not have a good u n d e r s t a n d i n g  o f t h e concept  of variable.  Both  s t u d i e s r e p o r t e d t h a t over o n e - t h i r d o f t h e c o l l e g e s t u d e n t s were unable t o s o l v e t h e " P r o f e s s o r and Student" q u e s t i o n c o r r e c t l y f o r t h e statement:  ( i . e . Write an e q u a t i o n  There a r e s i x times as many s t u d e n t s as p r o f e s s o r s ) .  These s t u d e n t s seemed t o view v a r i a b l e as t h e o b j e c t i t s e l f S wherever you see t h e word students i n t h e s e n t e n c e ) . Kuchemann's objects.  (1978) concerns  This  Wagner  supports  r e g a r d i n g students v i e w i n g v a r i a b l e s as merely  However, i t c a l l s i n t o q u e s t i o n h i s c o n t e n t i o n t h a t  understanding  and  (e.g. s u b s t i t u t e  students'  o f v a r i a b l e i s a f u n c t i o n o f age.  (1981a) p r e s e n t e d 15 middle  s c h o o l s t u d e n t s , median age 13 y e a r s ,  15 h i g h s c h o o l s t u d e n t s , median age 16.5 y e a r s , w i t h a c o n s e r v a t i o n task  i n v o l v i n g a b i l i t y t o d i s t i n g u i s h whether t h e a l p h a b e t i c a l r e p r e s e n t a t i o n of a v a r i a b l e was a c r i t i c a l  f a c t o r i n the s o l u t i o n o f simple l i n e a r  7xW+22=109 7xN+22=109 Students  were asked t h i s q u e s t i o n :  I f you were t o f i g u r e out what W s h o u l d be t o make t h i s statement t r u e ( p o i n t i n g t o the f i r s t equation) and what N s h o u l d be t o make t h i s statement t r u e ( p o i n t i n g t o t h e second e q u a t i o n ) , which would be l a r g e r , W o r N? How can you t e l l ? (p. 109).  equations:  25  Most s t u d e n t s seemed t o r e a c t t o t h i s q u e s t i o n change of l e t t e r was an e n t i r e l y new  considered  problem.  i n one  of two  ways: the  i r r e l e v a n t or the change of l e t t e r  implied  Wagner found t h a t the a b i l i t y t o r e c o g n i z e  and N were d i f f e r e n t names f o r the  same number was  exposure t o a l g e b r a i c concepts, but  that  W  s i g n i f i c a n t l y r e l a t e d to  not t o the age  of t h e  student.  This  suggests t h a t u n d e r s t a n d i n g of v a r i a b l e i s r e l a t e d t o a l g e b r a i c e x p e r i e n c e , r a t h e r than  age.  Wagner n o t e d t h a t the wording of her q u e s t i o n Some s t u d e n t s may order  have f e l t t h a t one  l e t t e r was  r a t h e r than mathematical p r o p e r t i e s .  1981a) e q u a t i o n s , t h e s e two  but  changed the q u e s t i o n  e q u a t i o n s the  value  through a 1-1  of the  the  year o l d s who  l e t t e r s was  the  same.  (1978) f i n d i n g s . Clement's  seemed c o n t r a r y Yet,  had  no  misleading.  to alphabetical  used Wagner's  format: "Are  v e r t i c a l matching of symbols  same v a l u e  l a r g e r due  Kieran  f a c t t h a t young, a l g e b r a i c a l l y i n e x p e r i e n c e d had  deliberately  same or d i f f e r e n t ? " (p. 161).  w i t h t h i s wording change, 12-13 t h a t the  was  the  s o l u t i o n s to  Kieran formal  (1977,  found t h a t  algebra  knew  Their usual explanation  was  (See e q u a t i o n s on p. 24).  The  students knew t h a t the  letters  t o both Wagner's  (1981a) and  Kuchemann's  t h e i r method of a n a l y s i s s u p p o r t e d Rosnick &  (1980) c o n t e n t i o n  t h a t students view v a r i a b l e s as o b j e c t s .  view of t h e s e c o n t r a d i c t o r y f i n d i n g s , f u r t h e r e x p l o r a t i o n of  In  students'  u n d e r s t a n d i n g of the concept of v a r i a b l e seems i n d i c a t e d . To  summarize, s t u d i e s  confusion  seemed t o c e n t r e  actual object  i n v o l v i n g u n d e r s t a n d i n g of v a r i a b l e i n d i c a t e d t h a t around v i e w i n g the v a r i a b l e as r e p r e s e n t i n g  or l e t t e r of the alphabet, r a t h e r than a magnitude.  an  These  26  erroneous  c o n c e p t i o n s were not l i m i t e d t o b e g i n n i n g a l g e b r a s t u d e n t s , but  p e r s i s t e d even a t c o l l e g e l e v e l s , with age and a l g e b r a i c background a c t i n g as p o s s i b l e m o d i f y i n g  factors.  Expressions  The  s i m p l i f i c a t i o n o f e x p r e s s i o n s forms a major component o f t h e  equation-solving process.  S e v e r a l r e s e a r c h e r s have i d e n t i f i e d t h e  a p p l i c a t i o n o f t h e d i s t r i b u t i v e p r o p e r t y and t h e combination t o be t h e two most prominent s i m p l i f i c a t i o n procedures  a s s o c i a t e d with  e r r o r s found i n t h e s o l u t i o n o f a l g e b r a i c e q u a t i o n s . of t h e concepts  o f u n l i k e terms  As such,  understanding  i n v o l v e d i n s i m p l i f i c a t i o n o f e x p r e s s i o n s may i n f l u e n c e t h e  t y p e s o f e r r o r s made i n l i n e a r e q u a t i o n s i n one v a r i a b l e . Anderson  (1982) s t u d i e d t h e e r r o r p a t t e r n s o f 200 Grade 9 and 10  s t u d e n t s on t h r e e t e s t s d e a l i n g with a l g e b r a i c e x p r e s s i o n s , a r i t h m e t i c e x p r e s s i o n s , and s i n g l e o p e r a t i o n s with i n t e g e r s .  She found t h a t v e r y few  s t u d e n t s made both a r i t h m e t i c and r e l a t e d a l g e b r a i c e r r o r s .  F o r example,  s t u d e n t s c o r r e c t l y s o l v e d 697-(494+387), but i n c o r r e c t l y wrote a-(b+c) as a -b+c.  T h i s type o f e r r o r seemed t o suggest  a l g e b r a i c manipulations  without  u n d e r l y i n g mathematical  properties.  s t u d e n t s used saw  r e l a t i n g these m a n i p u l a t i o n s  i n h e r study, Anderson  " l e t t e r s " as p r e c i s e l y t h a t .  t h a t s t u d e n t s may perform t o the  A f t e r i n t e r v i e w i n g 16 o f t h e 200 (1982) found t h a t most s t u d e n t s  They r e f e r r e d t o " x  2  " as "2x ' s . "  just This  l e d them t o make mistakes  i n m a n i p u l a t i n g v a r i a b l e s which they d i d not make  i n m a n i p u l a t i n g numbers.  Kuchemann (1978) and K i e r a n  t h a t t h e degree o f u n d e r s t a n d i n g  o f these' concepts  (1983) a l s o  suggested  i n f l u e n c e d the processes  27  used by s t u d e n t s i n e q u a t i o n - s o l v i n g , Another e x p l a n a t i o n  f o r the  and hence i n f l u e n c e d  inconsistency  of Anderson's  between a l g e b r a i c and a r i t h m e t i c e r r o r s may be t h a t items were not at the  same l e v e l  results.  of d i f f i c u l t y .  (1982)  results  a r i t h m e t i c and a l g e b r a i c  A r i t h m e t i c items  usually  i n v o l v e d l a r g e r numbers than d i d the c o r r e s p o n d i n g a l g e b r a i c i t e m s . use  This  o f l a r g e r numbers may have caused s t u d e n t s t o w r i t e out t h e i r work  explicitly,  r a t h e r than j u s t  arithmetic errors  (e.g.  answers,  x +x =x  thus  r e d u c i n g the  frequency of  was a common a l g e b r a i c e r r o r ,  but  231+231=231 was not a common a r i t h m e t i c e r r o r ) . • Another p o s s i b i l i t y the  l a r g e r numbers h e l p e d s t u d e n t s t o focus  simply t r y i n g to give Anderson systematic  (1982)  an answer  (Booth,  claimed that  on the p r o c e s s  1981).  systematic  for that p a r t i c u l a r e r r o r .  errors exist  A total  e r r o r s were made by 7 9 of the 200 s t u d e n t s i n the were a p p l i c a b l e t o the p r e s e n t multiplication  [e.g.  -5(2p  study.  in algebra.  [e.g.  of  30 d i f f e r e n t  study.  and a d d i t i o n  (e.g.  [e.g.  (17.x +2)-(12x +9)=29x  Surprisingly,  [e.g.  27b -10=17h  basic  fact  errors,  t o be s y s t e m a t i c  systematic ten  for  5r H—21r =16r  and 5 r +-3(7r -2)=12r  -2], found  errors.  also investigated  ),  -11].  and i n c o r r e c t w r i t i n g of o p e r a t i o n symbols were not  In a d d i t i o n t o s y s t e m a t i c (1982)  available  -6(13a +8)=-78a +8 and -6(13a +8)=13a-48],  and i n c o r r e c t o p e r a t i o n e r r o r s l i k e term e r r o r s  A  Of these o n l y  These i n v o l v e d s i g n e r r o r s  -7)=-10p -35]  p a r t i a l - d i s t r i b u t i v e errors  that  r a t h e r than  e r r o r was d e f i n e d as an e r r o r made i n more than 50% of  opportunities  is  e r r o r s made by i n d i v i d u a l s t u d e n t s ,  common e r r o r s made among groups of  Anderson  students.  A  28  common e r r o r was  1 5 such common e r r o r s , t h r e e of which o c c u r r e d i n 20% or more  Anderson found of the sample. 2  ax ;  These t h r e e e r r o r s were ax +-bx  and ax +bx  =abx.  Of the twelve  remaining  i n v o l v e d p o l y n o m i a l e x p r e s s i o n s of degree one present  10 o r more s t u d e n t s .  d e f i n e d as a s y s t e m a t i c e r r o r made by  investigation.  They were  ax +bx  =(b-a)x,  (|b|>|a|); ax  common e r r o r s , o n l y  (1982)  two  and were thus p e r t i n e n t t o the  =(a+b)x  2  and a(bx +c)=abx  D e s p i t e the n u m e r i c a l magnitude i n c o n s i s t e n c i e s among her Anderson's  +x=  +c.  tests,  study r e p r e s e n t e d a unique attempt t o e x p l o r e  the  r e l a t i o n s h i p between s i m p l i f y i n g e x p r e s s i o n s artd s o l v i n g e q u a t i o n s .  Many  common e r r o r s o c c u r r e d i n the s i m p l i f i c a t i o n of e x p r e s s i o n s w i t h i n the equations.  This occurrence  r e i n f o r c e s the need f o r i n v e s t i g a t i n g  conceptions  of the e x p r e s s i o n - s i m p l i f i c a t i o n  students',  process.  Equality Some s t u d e n t s p e r c e i v e the e q u a l i t y symbol t o be somewhat l i k e operation. as 5 = 2 + 3 ,  an  F o r example, when d i r e c t e d t o read and e x p l a i n a sentence s t u d e n t s would respond  t h a t i t was  backwards and  r e - w r i t e the  sentence  as 2+3=5 and then s t a t e t h a t 2 p l u s 3 makes 5 .  been due  t o f a m i l i a r i t y with the second a r i t h m e t i c sentence,  word "makes" and the i n a b i l i t y t o read the f i r s t s t u d e n t s p e r c e i v e d the e q u a l i t y symbol t o be (Behr, Erlwanger, o p e r a t i o n was  & N i c h o l s , 1 9 7 6 , p. 1 7 ) .  sentence  While t h i s may  suggest  T h i s n o t i o n of the e q u a l i t y symbol as an o p e r a t o r may " e r r o r , " but may  t h a t the something"  n o t i o n o f e q u a l i t y as  a l s o supported by the work of Denmark, Barco, not  have  the use of the  "an i n d i c a t i o n t o do The  such  & Voran  represent  an  (1976) . an  a f f e c t the nature of the e r r o r s produced when e q u a t i o n s  are  29  p r e s e n t e d with v a r i a b l e s on both s i d e s .  Thus, t h e c o n c e p t i o n o f t h e  e q u a l i t y symbol as an o p e r a t i o n r e q u i r e s i n v e s t i g a t i o n i n d i a g n o s i s o f e r r o r s i n l i n e a r equations Behr e t a l . ( 1 9 7 6 )  i n one v a r i a b l e .  noted some d i s c r e p a n c y between w r i t t e n and o r a l  p r e s e n t a t i o n s o f q u e s t i o n s i n v o l v i n g t h e e q u a l i t y symbol 3+2=2+3).  Some s t u d e n t s  i n Grades 1 -  presented  i n o r a l form.  questions  i n w r i t t e n form.  (e.g. A = 2 + 3 , 3 = 3 ,  6 were a b l e t o make sense of q u e s t i o n s  However, they were unable  t o understand  The d i s c r e p a n c y between s t u d e n t s '  t h e same  understanding  of t h e e q u a l i t y symbol i n w r i t t e n as opposed t o o r a l p r e s e n t a t i o n may bear f u r t h e r i n v e s t i g a t i o n , p a r t i c u l a r l y a t t h e secondary If beginning algebra students  still  level.  view t h e e q u a l i t y symbol as an  o p e r a t i o n , they may be c o n f u s e d when working with e q u a t i o n s  i n algebra.  T h i s o p e r a t i o n a l view o f e q u a l i t y may cause c o n f u s i o n when they a r e r e q u i r e d t o use c o n s e c u t i v e e q u a l e x p r e s s i o n s i n s o l v i n g w r i t t e n e q u a t i o n s Bright,  1982) .  In a d d i t i o n , Matz  (1980)  c l a i m s t h a t t h e r e a r e two d i s t i n c t  e q u a l i t y symbol i n a l g e b r a which may s u b s t a n t i a l l y extend notion of e q u a l i t y . and  (Bernard &  uses of the  the a r i t h m e t i c  He r e f e r r e d t o the two uses o f e q u a l i t y as " t a u t o l o g y "  "constraint equation."  In a " t a u t o l o g y , " the student performs a c h a i n  of t r a n s f o r m a t i o n s on an e x p r e s s i o n t o o b t a i n i t s s i m p l e s t form.  Each  subsequent e x p r e s s i o n i s the same as the p r e v i o u s one. In " c o n s t r a i n t e q u a t i o n s , " t h e student performs t r a n s f o r m a t i o n s on both  s i d e s o f the  e q u a l i t y symbol, m a i n t a i n i n g e q u i v a l e n c e between the two s i d e s o f the equation,  but not between each s t e p o f t h e t r a n s f o r m a t i o n .  The problem f o r  30  s t u d e n t s i n b e g i n n i n g a l g e b r a i s t h a t t h e y must s i m p l i f y each s i d e of the e q u a t i o n , keeping the l e f t itself,  thus u t i l i z i n g  s i d e equal to i t s e l f  and the r i g h t  s i d e equal t o  the t a u t o l o g i c a l sense of e q u a l i t y and then s w i t c h t o  the c o n s t r a i n t use i n o r d e r t o s o l v e the e q u a t i o n , p e r f o r m i n g operations- so t h a t the l e f t  s i d e i s e q u a l t o the r i g h t s i d e at a l l times, but  not e q u a l t o i t s e l f .  While  some may  i n n a t u r e , the d i f f i c u l t i e s  definitely  t h i n k t h e s e d i f f e r e n c e s t o be  semantic  e x p e r i e n c e d by s t u d e n t s i n s w i t c h i n g from  s i m p l i f y i n g e x p r e s s i o n s t o s o l v i n g e q u a t i o n s can be seen on a d a i l y b a s i s i n beginning algebra classrooms.  T h i s d i s t i n c t i o n between the two p o s s i b l e  views of the e q u a l i t y symbol may  appear u s e f u l i n the classroom,  does r e q u i r e e m p i r i c a l v a l i d a t i o n . e r r o r s produced  In  As such the r e l a t i o n s h i p between the  when u s i n g e q u a l i t y i n a " t a u t o l o g i c a l " sense and a  " c o n s t r a i n t e q u a t i o n " sense need f o r t h e s e  requires i n v e s t i g a t i o n to c l a r i f y  the  alleged  distinctions.  summary, r e s e a r c h has shown t h a t the o p e r a t i o n a l use of e q u a l i t y i n  a r i t h m e t i c forms the c e n t r a l reason f o r many s t u d e n t s ' i n a b i l i t y equations  ( K i e r a n , 1983) .  r a t h e r than a r e l a t i o n , level  however, i t  Misunderstandings  Understanding  confounded by e q u a t i o n c o m p l e x i t y . s t u d e n t s may expressions Meaning  of e q u a l i t y as an o p e r a t i o n ,  seem t o remain with s t u d e n t s , even at the  (Rosnick & Clement, 1980).  to solve  college  of e q u a l i t y i s f u r t h e r  As q u e s t i o n s become more complex  change t h e i r minds r e a d i l y r e g a r d i n g the e q u a l i t y of  two  (Denmark e t a l . , 197 6). of  Equations  L i n e a r e q u a t i o n s a r i s e as mathematical  models of problems i n the  real  31  world  (Krutetskii,  1976).  Students  t e n d t o t r e a t e q u a t i o n s as a b s t r a c t ,  u n i n t e l l i g i b l e e n t i t i e s t o be d e a l t with u s i n g a s e t of memorized without  conceptual underpinnings  (Kieran, 1 9 8 0 ) .  of what an e q u a t i o n r e p r e s e n t s may process  understanding  s i g n i f i c a n t l y a f f e c t the e q u a t i o n - s o l v i n g  (Kieran, 1 9 8 3 ) .  Kieran reported that, p r i o r to i n s t r u c t i o n , one  T h i s l a c k of  procedures  of t h r e e procedures  number f a c t s ,  seventh grade n o v i c e s  t o s o l v e l i n e a r e q u a t i o n s i n one  substitution,  variable:  used  known  or " i n v e r t " - l e a v e the v a r i a b l e on the l e f t  and  t r a n s p o s e a l l numbers t o the r i g h t s i d e of the e q u a t i o n u s i n g i n v e r s e operations.  A f t e r i n s t r u c t i o n which was  symmetric procedure 1983,  p. 1 6 8 ) ,  "very s t r o n g l y o r i e n t e d toward the  of p e r f o r m i n g the same o p e r a t i o n on both s i d e s "  the r e s u l t s f o r n o v i c e s appeared  those of the Grade 8 - 1 1 e x p e r t s . the same f o r both  While  (known number f a c t s ,  t o be  (Kieran,  i n sharp c o n t r a s t t o  the f o u r types of procedures substitution,  inversing,  o p e r a t i o n on both s i d e s ) , the frequency of use of each type of i n d i c a t e d that experts s t r o n g l y p r e f e r r e d i n v e r s i n g  were  and same procedure  ( 1 5 2 i n s t a n c e s compared  t o 1 9 i n s t a n c e s of the o t h e r 3 p r o c e d u r e s ) , while n o v i c e s o n l y m a r g i n a l l y p r e f e r r e d the same i n v e r s i n g procedure i n s t a n c e s of the o t h e r p r o c e d u r e s ) .  ( 7 7 i n s t a n c e s compared t o 63  Of g r e a t e r s i g n i f i c a n c e was  the  t h a t t e a c h i n g of a p a r t i c u l a r method o n l y i n f l u e n c e d those s t u d e n t s a l r e a d y had a view of v a r i a b l e compatible whose view of v a r i a b l e was procedures,  While  who students  incompatible with t h e i r e q u a t i o n - s o l v i n g  changed t h e i r procedures  when such a change was  with t h a t method.  fact  t o match t h e i r view of v a r i a b l e ,  i n d i r e c t c o n t r a d i c t i o n t o the method b e i n g  even  taught.  32  Kieran's  (1983) f i n d i n g s i l l u s t r a t e  the importance  of examining  the  i n t e r r e l a t i o n s h i p s which e x i s t between c o n c e p t i o n s and procedures,  as w e l l  as the n e c e s s i t y of d e t e r m i n i n g , not o n l y s t u d e n t s ' views of v a r i a b l e ,  but  a l s o t h e i r c o n c e p t i o n s of the meaning of e q u a t i o n s . Identity  Matz  Elements  (1980) suggested  t h a t s t u d e n t s were l i k e l y t o make e r r o r s when a  g e n e r a l i z a t i o n i n v o l v e d s p e c i f i c numerical values.  He  further  contended  t h a t the i d e n t i t y elements f o r a d d i t i o n and m u l t i p l i c a t i o n were examples of such c r i t i c a l numbers.  To i l l u s t r a t e ,  the student l e a r n e d t h a t n *  number) = n and t h a t n * ( s p e c i a l number) = 0 .  I t was  (special  the c h o i c e of the  o p e r a t o r * t o g e t h e r with the s p e c i a l number, 0 or 1 which determined answer.  the  Some s t u d e n t s never seemed t o l e a r n t h a t both the o p e r a t i o n and  s p e c i a l number were c r i t i c a l .  These s t u d e n t s responded  n t o n * 1 and 0 t o  n * 0, unheeding of the f a c t t h a t * might be + i n both i n s t a n c e s . learned these i d e n t i t i e s together randomly produced  ( i e . n x 1 = n,  the  n x 0 = 0 ,  Others  n + 0 = n)  and  e i t h e r 0 or n f o r n * 0.  Other e r r o r s p o s s i b l y r e l a t e d t o the i d e n t i t y elements o c c u r r e d i n examples such as 2n  -  2n  = n, where the student e x p l a i n e d t h a t t h e r e  "zero n, but then wrote n, because 0 * n - n. s t u d e n t s may zero."  e x p l a i n : "There's  These examples suggest  elements may  In w r i t i n g 3x + x = 3x,  was some  n o t h i n g i n f r o n t of the o t h e r x, so i t ' s j u s t t h a t g e n e r a l i z a t i o n s i n v o l v i n g the  be a source of e r r o r i n a l g e b r a , but no  identity  studies reported  a n a l y s i s of the i d e n t i t y elements i n a l g e b r a i c e r r o r s .  33  Numerical  The  Magnitude  magnitude of the numbers used i n an e q u a t i o n may  students'  equation-solving  ability.  Herscovics  also  influence  (1979) s t r e s s e d :  For s m a l l v a l u e s of N, s t u d e n t s have no d i f f i c u l t y i n s o l v i n g simple e q u a t i o n s such as x+a=b, x-a=b, ax=b, x/a=b a l t h o u g h a-x=b and a/x=b seem somewhat more d i f f i c u l t . I t i s when the numbers used are l a r g e enough or when m u l t i p l e o p e r a t i o n s appear i n the e q u a t i o n or when terms i n the unknown appear on both s i d e s (ax+b=cx+d) t h a t mental a r i t h m e t i c u s u a l l y ceases and a l g e b r a i c methods come i n t o t h e i r own (p. 110) .. The  p o i n t where a r i t h m e t i c a l g o r i t h m s  be,  i n p a r t , a f u n c t i o n of n u m e r i c a l magnitude. Booth  focus  (1981) a l s o found t h a t the use  students'  and  to increase errors.  and  a l g e b r a i c procedures b e g i n  of l a r g e numbers  across  She  computational e r r o r s  Anderson  (Herscovics,  (1982).  Such c o n t r o l may  1979), y e t  encourage the use  opposed t o a r i t h m e t i c , methods f o r s o l v i n g e q u a t i o n s  reduced p r o c e s s a  significant  of a l g e b r a i c ,  (Herscovics,  The  c o n t r o l of n u m e r i c a l magnitude i s f u r t h e r s u p p o r t e d by the  the  S t r a t e g i e s and  Booth & Hart  E r r o r s i n Secondary Mathematics  (p. 4 ) .  or q u e s t i o n s  do not  r e s u l t s of  (SESM) p r o j e c t i n which  r e a d i l y extend to more complex  i n which the numbers are l a r g e or n o n - i n t e g e r "  These f i n d i n g s f u r t h e r emphasize the  magnitude i n d i a g n o s i s .  as  1979).  (1982) i n d i c a t e d t h a t s t u d e n t s employ "child-methods" which  "are of l i m i t e d a p p l i c a b i l i t y and questions  across  Large numbers seemed  C o n t r o l of n u m e r i c a l magnitude would appear t o be  concern i n d i a g n o s i s .  s o l u t i o n of  documented r e s u l t s which were c o n s i s t e n t  time, s u p p o r t i n g  may  (n>100) seemed to  a t t e n t i o n on the p r o c e s s e s i n v o l v e d i n the  a l g e b r a i c equations. subjects  end  importance of n u m e r i c a l  34  The numbers used as c o e f f i c i e n t s may a b i l i t y to solve equations.  Kieran  d i f f e r e n t i a l l y influence  (1981b) r e p o r t e d t h a t a l t h o u g h  l a r g e c o e f f i c i e n t s proved t o be no hindrance  the t w o - o p e r a t i o n  equations"  e q u a t i o n i s 12x + 216  = 468.  (p. 164).  "...  An example of a  complexity  two-operation  E r r o r frequency seems t o v a r y d i r e c t l y  the i n t e r a c t i o n between n u m e r i c a l magnitude and e q u a t i o n c o m p l e x i t y . i n t e r a c t i o n between n u m e r i c a l magnitude and e q u a t i o n c o m p l e x i t y further  the  at a l l f o r e q u a t i o n s of the  form ax = b and x + a = b , they tended t o i n c r e a s e s l i g h t l y the of  students'  with This  bears  investigation.  I t would seem t h a t the magnitude of the numbers used i n e q u a t i o n s i n f l u e n c e s the p r o c e s s e s chosen by s t u d e n t s i n s o l v i n g e q u a t i o n s . magnitude may  Numerical  a l s o be i n t e r a c t i n g with the c o m p l e x i t y of e q u a t i o n s .  such, c o n t r o l of n u m e r i c a l magnitude may  be a c r i t i c a l  As  factor i n algebraic  diagnosis. Equation  Structure  and  Complexity  E q u a t i o n s t r u c t u r e and c o m p l e x i t y may solve equations.  Pettito  ( s t r u c t u r a l format) format.  The  also influence students' a b i l i t y  (1979) d e f i n e d s t r u c t u r a l d i f f i c u l t y  (easiest s t r u c t u r a l d i f f i c u l t y  the v a r i a b l e a p p e a r i n g i n the r i g h t numerator  (a/b=x/c),  u n f a m i l i a r forms had the v a r i a b l e a p p e a r i n g i n the r i g h t [a/b=c/(x+d)] and  level  i n terms of student f a m i l i a r i t y with e q u a t i o n s  f a m i l i a r equations  i n both denominators  [a/x=b/(x+c)].  while  in ratio  level)  had  the  denominator  Nine  ninth-grade  s t u d e n t s were a b l e t o s o l v e the f a m i l i a r e q u a t i o n s over t h r e e l e v e l s of numerical d i f f i c u l t y .  Only h a l f of the students who  to  c o u l d s o l v e the  first  35  (n<5)  and  second  (n<ll) l e v e l s of u n f a m i l i a r n u m e r i c a l l y  c o u l d a l s o s o l v e the t h i r d l e v e l i n a b i l i t y t o s o l v e the n u m e r i c a l l y a r i t h m e t i c e r r o r s , but  t o an  (p. 74).  In some cases the  question.  In o t h e r  Pettito (1979)  (n<57).  difficult  d i f f i c u l t y f o r the Despite  l a c k of the use  of the  l i m i t e d by  complexity, in analyzing  s t r u c t u r a l format and e r r o r s i n the of  students' w i t h age students'  factors.  numerical  of s t u d i e s  reviewed  Unfortunately,  Pettito's  a possible  f i n d i n g s supported  c o m p l e x i t y may  interact.  Kieran's  Equation  n u m e r i c a l magnitude need t o be c o n t r o l l e d  s o l u t i o n of a l g e b r a i c  Conceptual  limitation  level.  the m a j o r i t y  However, her  t h a t magnitude and  equations.  Errors  i n f l u e n c e of c o n c e p t u a l  requires  only  second l e v e l of n u m e r i c a l  l a c k of adequate c o n t r o l and  structure interaction.  (1981b) c o n t e n t i o n  The  The  the  informal  the apparent impact of both s t r u c t u r a l format and  (1979) f i n d i n g s are  Summary  a s u c c e s s f u l approach"  fractions.  second s t r u c t u r a l d i f f i c u l t y  f a i l e d t o c o n t r o l or manipulate these two  further investigation.  e r r o r s on the e q u a t i o n - s o l v i n g The  process  c o n t r a d i c t o r y f i n d i n g s surrounding  u n d e r s t a n d i n g of v a r i a b l e and  i n t e r a c t i o n of t h a t  and mathematical background r e q u i r e c l a r i f i c a t i o n .  understanding The  e f f e c t of  views of the e q u a l i t y symbol on the procedures used i n s o l v i n g  e q u a t i o n s needs t o be  "the  to  s t u d e n t s were unable even t o attempt  magnitude on the s o l u t i o n of equations,  numeric by  not  cases students were unable t o modify the  the  equations  concluded that  problem i s due,  i n a b i l i t y to organize  procedures i n v o l v e d i n obtaining equivalent t o t h i s study was  difficult  considered  (Kieran,  1983)  along  w i t h the  r o l e of  i d e n t i t i e s and i n v e r s e s  (Matz, 1980).  Complexity,  s t r u c t u r a l format, and  n u m e r i c a l magnitude need t o be c o n t r o l l e d w i t h i n t h e e q u a t i o n s diagnosis.  S t u d i e s need t o be developed  t h e s e f a c t o r s comparing c o n c e p t i o n s obtained.  It i s c r i t i c a l  conceptions,  equations  which s y s t e m a t i c a l l y v a r y each of  h e l d , procedures  used, and r e s u l t s  f o r d i a g n o s t i c i a n s t o understand  u n i n f l u e n c e d by i n v e s t i g a t i o n o f t h e i r reasons  p a r t i c u l a r procedures  used i n  students' for selecting  o r o b t a i n i n g p a r t i c u l a r r e s u l t s when s o l v i n g l i n e a r '  i n one v a r i a b l e .  Procedural  Errors  P r o c e d u r a l e r r o r s i n a r i t h m e t i c a r e " e r r o r s d e r i v e d from s y s t e m a t i c and i n a p p r o p r i a t e a l g o r i t h m i c procedures"  (Engelhardt & Wiebe, 1981, p. 14). I  a l g e b r a , p r o c e d u r a l e r r o r s encompass t h e e r r o r s o c c u r r i n g i n s i m p l i f y i n g a r i t h m e t i c e x p r e s s i o n s and m i s u s i n g t h e r u l e s f o r o r d e r o f o p e r a t i o n s , and extend t o e r r o r s , i n s i m p l i f y i n g a l g e b r a i c e x p r e s s i o n s , distributive errors,  including  partial  s i g n e r r o r s with the d i s t r i b u t i v e p r o p e r t y and  combining u n l i k e terms. Order The left  of  Operations  tendency o f s t u d e n t s t o perform  a r i t h m e t i c operations i n order  from  t o r i g h t may i n f l u e n c e the types o f e r r o r s they make i n s o l v i n g  equations.  Many s t u d e n t s view a r i t h m e t i c o p e r a t i o n s as a s t r i n g o f  o p e r a t i o n s t o be performed i n a l e f t Kieran,  1979).  Kieran  to right  sequence  (Hilling-Smith,  197 9  (1979) found t h a t i n c r e a t i n g a r i t h m e t i c i d e n t i t i e s ,  s u b j e c t s would w r i t e " o p e r a t i o n by o p e r a t i o n , as they were t h i n k i n g them, • and were keeping  a r u n n i n g t o t a l . a s they went a l o n g "  (p. 2 ) .  F o r example,  37  s t u d e n t s would w r i t e 4x2-3=5+10+3 as an i d e n t i t y . t h a t b r a c k e t s be i n s e r t e d around 5+10, e n t i r e l y unnecessary. first,  They thought  many s t u d e n t s thought  understanding  suggestion  this  was  t h a t the b r a c k e t e d numbers had t o appear  i n o r d e r to' c o r r e s p o n d t o t h e i r l e f t  v a r i a t i o n of l e f t  When g i v e n the  t o r i g h t sequencing may  to r i g h t  sequencing.  be r e q u i r e d t o  Systematic  determine  of a l g e b r a i c o p e r a t i o n s i n e r r o r - a n a l y s i s .  Algebraic  Properties  M i s a p p l i c a t i o n of the a d d i t i v e and m u l t i p l i c a t i v e p r o p e r t i e s of e q u a l i t y may  result  i n errors involving coefficients,  terms. Two  i n v e r s e s , or combining  of the common e r r o r s found among both f i r s t -  and  second-year  a l g e b r a s t u d e n t s were adding o p p o s i t e s i n a p p r o p r i a t e l y (-a+ax =x combining is  -1/a)  multiplicative  i n v e r s e and a d d i t i v e i n v e r s e  (Davis & Cooney, 1978) .  when s o l v i n g a l g e b r a i c e q u a t i o n s The  tendency  (Carry e t a l . , 1980;  Booth & Hart  Lewis,  (1982) i n t e r v i e w e d a p p r o x i m a t e l y Science  When p r e s e n t e d w i t h an i t e m  70  (CSMS)  c o n t i n u e d t h e i r answer t o x+y=z, s t a t i n g : " I f you add two number so I suppose adding two Students  involving  team made x g o a l s  the o t h e r y g o a l s , some s t u d e n t s were a b l e t o w r i t e down x+y,  9).  1980).  be an outcome of o v e r -  the t o t a l number of s o c c e r g o a l s and the f a c t t h a t one  (p.  errors  had made e r r o r s which were a l s o made by a l a r g e number of  s t u d e n t s on t h e CSMS A l g e b r a t e s t .  another  and  (the r e c i p r o c a l of a  s t u d e n t s from the Concepts i n Secondary Mathematics and p r o j e c t who  )  Even c o l l e g e s t u d e n t s made s i m i l i a r  t o make a l g e b r a i c e r r o r s may  g e n e r a l i z a t i o n of r u l e s .  unlike  but  then  numbers you  l e t t e r s g i v e s you another  and  get  letter"  not o n l y seek c l o s u r e when adding l e t t e r s , but a l s o have a  38  tendency t o add 4 and 3n and t o o b t a i n e i t h e r 7 o r In.  Matz  that  t h e r u l e t o f i t the  s t u d e n t s who make mistakes o f t e n t r y t o g e n e r a l i z e  problem  (e.g.  s i n c e 3 + 4 = 7 then 3 + 4r> = 7n) .  Summary  of  Procedural generalize  Procedural  Errors  e r r o r s seem t o a r i s e from t h e tendency o f s t u d e n t s t o  arithmetic  situations-.  (1980) noted  r u l e s and i n a p p r o p r i a t e l y a p p l y them t o a l g e b r a i c  I t would appear t h a t t h e r u l e s most o f t e n g e n e r a l i z e d  to the d i s t r i b u t i v e property.  I t i s c r i t i c a l t o understand  students'  p r o c e d u r e s i n s o l v i n g a l g e b r a i c e q u a t i o n s so t h a t m i s a p p l i c a t i o n s can be  pertain  of rules  detected.  Resultant  Errors  Resultant  e r r o r s a r e those e r r o r s which o c c u r i n w r i t i n g o r s t a t i n g a  r e s u l t e i t h e r i n t h e p a r t i a l s o l u t i o n o f a problem o r i n i t s f i n a l They a r i s e from c o m p u t a t i o n a l , s i g n , mechanical, p e r c e p t u a l , errors,  as w e l l as from an i n a b i l i t y  Resultant  e r r o r s s h o u l d be e x p l o r e d  answer.  o r random  t o complete t h e s o l u t i o n o f e q u a t i o n s . a t each stage o f t h e e q u a t i o n - s o l v i n g  p r o c e s s , as s t u d e n t s may make e r r o r s and s t i l l  obtain  a correct f i n a l  answer  (Anderson, 1982). Computational  Errors  Computational e r r o r s i n v o l v e any o f t h e f o u r b a s i c o p e r a t i o n s subtraction,  multiplication, division).  made i n b a s i c operation. due  f a c t s , f a u l t y algorithms,  Considerable information  F o r each o p e r a t i o n , or i n c o r r e c t choice  e x i s t s regarding  (addition,  e r r o r s may be of that  computational e r r o r s ,  t o t h e f r e q u e n c y o f t h e i r i n v e s t i g a t i o n (Brown & Burton, 1978; CSMS,  39  described i n Hart,  1981;  limited discussion  of the more s a l i e n t  Roberts  (1968),  SESM, d e s c r i b e d i n Booth,  u s i n g 148 Grade 3 s t u d e n t s ,  c a t e g o r i e s : wrong o p e r a t i o n , algorithm,  obvious  and random response.  starting point,  Engelhardt  basic  fact  (7%),  identity  established  (1968)  f o u r major e r r o r  defective  framework as a  He a n a l y z e d the e r r o r s and i d e n t i f i e d  (38%), g r o u p i n g  incorrect operation  (22%), i n a p p r o p r i a t e i n v e r s i o n  (4%) , d e f e c t i v e a l g o r i t h m (18%), incomplete  (1%),  and zero  (6%).  most common e r r o r s made by s t u d e n t s .  Basic fact  These d a t a e v i d e n c e d t h a t  (1981) and E n g l e h a r d t  eight  (21%),  algorithm  e r r o r s appeared to be  the e r r o r s made i n a r i t h m e t i c were c o m p u t a t i o n a l i n n a t u r e . by E n g l e h a r d t & Wiebe  only a  provided.  computational e r r o r ,  Using Robert's  Therefore,  (1977) t e s t e d 198 Grade 3 and Grade 6 students on  an 8 4 - i t e m c o m p u t a t i o n a l t e s t . types:  results is  1984) .  the  over h a l f  Later  studies  (1982) c o n f i r m e d these e r r o r  classifications. Graeber & W a l l a c e e r r o r s at the Grade 3, Instruction  (IPI)  (1977) 4,  (p.  8).  s y s t e m a t i c a d d i t i o n and s u b t r a c t i o n  and 7 l e v e l s u s i n g I n d i v i d u a l l y P r e s c r i b e d  pretests.  i n c r e a s e d w i t h grade l e v e l . or more of the  identified  They r e p o r t e d t h a t both s y s t e m a t i c e r r o r s An e r r o r was c l a s s i f i e d  i n d i v i d u a l ' s t e s t items e v i d e n c e d the  S e v e r a l s y s t e m a t i c e r r o r s were i d e n t i f i e d  as s y s t e m a t i c i f  same e r r o r p a t t e r n "  across  grade  These i n c l u d e d o m i t t i n g c a r r y i n g i n a d d i t i o n or b o r r o w i n g i n adding a l l d i g i t s  irrespective  of p l a c e v a l u e ,  In a d d i t i o n t o s y s t e m a t i c e r r o r s , also  reported.  "three  levels.  subtraction,  and i g n o r i n g the tens column.  random e r r o r s and incomplete  work were  40  Computational e r r o r s have been r e p o r t e d i n s o l v i n g l i n e a r equations i n algebra Wattawa, 1927). i n algebra  Their  as a major source o f d i f f i c u l t y  (Davis & Cooney, 1978; Pease, 1929;  i d e n t i f i c a t i o n forms a c r i t i c a l  t o p r o v i d e d i s t i n c t i o n s between a l g e b r a i c  However, no s y s t e m a t i c e x p l o r a t i o n  aspect o f d i a g n o s i s  and a r i t h m e t i c  of computational e r r o r s  errors.  forlinear •  e q u a t i o n s i n one v a r i a b l e was found i n the l i t e r a t u r e . Sign  Errors  Sign e r r o r s may i n v o l v e  frequently  occur i n t h e s o l u t i o n o f l i n e a r e q u a t i o n s .  any o f t h e f o u r b a s i c o p e r a t i o n s ,  misapplication  of sign  They rules  (e.g. two n e g a t i v e s make a p o s i t i v e when adding) o r misuse o f t h e d i s t r i b u t i v e property  [e.g.  - 3 ( x -5)= -3x -15].  Tatsuoka e t a l . (1980) r e p o r t e d  that  i n s t r u c t i o n a l approaches  correlated  w i t h r u l e - b a s e d s i g n e r r o r s . That i s t o say, i f a t e a c h e r s t a t e d a r u l e f o r one  operation,  operation.  then s t u d e n t s would use i t i n a p p r o p r i a t e l y  D a v i s e t a l . (1979), u s i n g  14-item v e r s i o n  o f Tatsuoka's  approximately one-third  an 8-item m o d i f i c a t i o n  (1977) i n t e g e r  o f t h e answers c o u l d be e x p l a i n e d  error.  e q u a l s two.  The o r d e r does not matter. I f you see a  smaller  you j u s t  subtract  number from t h e l a r g e r number."  of i n c o r r e c t a p p l i c a t i o n o f p r e v i o u s l y  al.  by t h e "symmetric  F o r example a student might s t a t e : Three minus f i v e  D a v i s e t a l . (1979) suggested t h a t  empirical  o f an e a r l i e r  a d d i t i o n t e s t , found t h a t  subtraction"  the  w i t h another  s t u d e n t s may make s i g n e r r o r s because learned  rules.  e v i d e n c e o f t h i s have been p r o v i d e d by Matz  (1980), r e s p e c t i v e l y .  Cote  (1981) r e p o r t e d  T h e o r e t i c a l and (1980) and Tatsuoka e t  s i m i l a r f i n d i n g s and f u r t h e r  noted, t h a t some s t u d e n t s viewed i n t e g e r a r i t h m e t i c as merely unrelated rules.  a c o l l e c t i o n of  When d i a g n o s i n g a s t u d e n t ' s e r r o r s i n s o l v i n g  the p a r t i c u l a r combination  equations,  o f s i g n s and magnitudes o f t h e numbers i n v o l v e d  i n t h e s e l e c t e d e q u a t i o n s may not a l l o w t h e student t o make a m i s t a k e . i s p a r t i c u l a r l y t r u e f o r some o f t h e more complex s t u d e n t - g e n e r a t e d (Tatsuoka e t a l . , 1980).  This  rules  Hence, t h e s y s t e m a t i c v a r i a t i o n o f t h e s i g n s and  magnitude o f numbers i s n e c e s s a r y t o f a c i l i t a t e m i s a p p l i e d or student-generated  rules.  the i d e n t i f i c a t i o n of the  Without such f o r m u l a t i o n o f  d i a g n o s t i c e q u a t i o n s , many r u l e s , which randomly produce c o r r e c t and incorrect  answers, depending on t h e n u m e r i c a l s i g n s and magnitudes i n v o l v e d ,  may go u n n o t i c e d . Mechanical/Perceptual  Students  Errors  c o n s i s t e n t l y seem t o make a s m a l l percentage o f  m e c h a n i c a l / p e r c e p t u a l e r r o r s when s o l v i n g e q u a t i o n s . e r r o r s a r e " s y s t e m a t i c and non-systematic v i s u a l or motor-related d i f f i c u l t i e s misalignment  Mechanical/perceptual  e r r o r s appearing t o r e s u l t  such as i l l e g i b l e  o f numbers, o r m i s c o p y i n g / m i s r e a d i n g  (Engelhardt & Wiebe, 1981, p. 14). Wattawa  from  number f o r m a t i o n ,  digits  o r symbols"  (1927) found t h a t 26% o f t h e  e r r o r s made by a n i n t h - y e a r a l g e b r a c l a s s were due t o c o p y i n g o r r e a d i n g mistakes.  More r e c e n t l y , Davis &• Cooney  (1978) r e p o r t e d a 9% e r r o r r a t e  among f i r s t - y e a r and a 13% e r r o r r a t e among second-year i n v o l v i n g miscopying  o r m i s r e a d i n g items.  algebra  students  Such e r r o r s need t o be c o n s i d e r e d  when d i a g n o s i n g e r r o r s o c c u r r i n g i n a l g e b r a .  Random  Errors  W i t h i n t h e l i t e r a t u r e , many r e s e a r c h e r s r e f e r t o " c a r e l e s s " e r r o r s . is  It  an i l l - d e f i n e d c a t e g o r y and, hence, t h e more w e l l - d e f i n e d c a t e g o r y o f  "random" e r r o r s has been u s e d . i n t h e p r e s e n t study. d e s c r i b e d as "non-systematic  Random e r r o r s may be  e r r o r s appearing t o r e s u l t  inattention to relevant stimuli"  from momentary  (Engelhardt & Wiebe, 1981, p. 15). They  can be d i s t i n g u i s h e d from o t h e r e r r o r s by t h e i r n o n - s y s t e m a t i c  n a t u r e as  w e l l as t h e responses  students g i v e when q u e s t i o n e d as t o how a p a r t i c u l a r  answer was o b t a i n e d .  U s u a l l y when t h i s type o f e r r o r i s p o i n t e d out t o the  s t u d e n t s , t h e y respond: the e r r o r  "How s i l l y !  I know b e t t e r , " and p r o c e e d t o c o r r e c t  immediately.  Wagner e t a l . (1984) e x p l a i n e d t h a t . I t i s not c l e a r what c a u s e s - s t u d e n t s t o make l o w - l e v e l e r r o r s on problems t h a t p r e s e n t h i g h - l e v e l d i f f i c u l t y . Perhaps i t ' i s s t r i c t l y a c c i d e n t a l , perhaps i t r e s u l t s from a temporary o v e r l o a d i n s h o r t - t e r m memory, o r perhaps i t r e f l e c t s a r e v e r s i o n t o an e a r l i e r " n a i v e " r u l e i n t h e f a c e o f a h i g h e r l e v e l problem t h a t , f o r t h e moment, e x a c t s f u l l c o n c e n t r a t i o n (p. 31) . While to  random e r r o r s seem t o r e p r e s e n t a low l e v e l type' o f e r r o r , they need  be c o n s i d e r e d when d i a g n o s i n g e r r o r s , i n a l g e b r a . Incomplete  Incomplete algebra.'  Work  work appears  Wattawa  as an e r r o r c a t e g o r y i n many s t u d i e s i n v o l v i n g  (1927) found t h a t 5% of. s t u d e n t s f a i l e d t o complete the  s o l u t i o n s of equations at the beginning of the year. i n c r e a s e d t o 9% by t h e end o f t h e y e a r . 12%  r a t e o f incomplete  solutions.  The analogous  Davis & Cooney  (197 8)  However, s l i g h t l y over  figure  reported a  90% o f t h i s  43  i n c o m p l e t e work o c c u r r e d i n e q u a t i o n s c o n t a i n i n g f r a c t i o n s . if  such a h i g h percentage  I t i s unclear  o f e r r o r s would o c c u r i n t h e s o l u t i o n o f l i n e a r  e q u a t i o n s i n one v a r i a b l e c o n t a i n i n g o n l y i n t e g e r s .  I n v e s t i g a t i o n of  s t u d e n t s ' incomplete work may be c r i t i c a l t o u n d e r s t a n d i n g e r r o r s made i n s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e . Summary  a n d Recommendations  S e v e r a l f a c t o r s seemed t o i n f l u e n c e t h e n a t u r e and frequency o f e r r o r s r e p o r t e d by r e s e a r c h e r s i n b e g i n n i n g a l g e b r a .  Students' views o f a v a r i a b l e  as an o b j e c t and o f e q u a l i t y as an o p e r a t i o n seem c e n t r a l t o s t u d e n t s ' i n a b i l i t y t o solve l i n e a r equations.  However, t h e r e a r e c o n t r a d i c t o r y  f i n d i n g s r e g a r d i n g t h e r e l a t i o n s h i p o f t h e s e two concepts t o age and mathematical  background.  f a c t t h a t both concepts  These c o n t r a d i c t i o n s a r e f u r t h e r compounded by the i n t e r a c t with e q u a t i o n c o m p l e x i t y .  findings necessitate further investigation. researchers,  s i m i l a r erroneous  understandings  Such r e s e a r c h  As i n d i c a t e d by s e v e r a l o f t h e meaning o f e x p r e s s i o n s  and e q u a t i o n s have l e d s t u d e n t s t o make p r o c e d u r a l e r r o r s i n v o l v i n g o r d e r o f o p e r a t i o n s , t h e d i s t r i b u t i v e p r o p e r t y , combining  o f u n l i k e terms,  and the  a d d i t i o n and m u l t i p l i c a t i o n p r o p e r t i e s of e q u a l i t y . Magnitude o f numbers used i n e q u a t i o n s , placement  o f s i g n e d numbers  w i t h i n e q u a t i o n s , and t h e c o m p l e x i t y o f t h e e q u a t i o n i n terms o f t h e number of o c c u r r e n c e s o f t h e v a r i a b l e and the number o f o p e r a t i o n s have a l l been r e p o r t e d t o be s i g n i f i c a n t equations.  While  f a c t o r s a f f e c t i n g student performance  i t was apparent  i n solving  from most s t u d i e s t h a t n u m e r i c a l magnitude  i n f l u e n c e d r e s u l t s , t h e r e seemed t o be two v e r y d i f f e r e n t views o f why t h i s  44  occurred.  Some s t u d i e s  suggested t h a t  l a r g e r numbers a f f e c t e d s t u d e n t s '  a b i l i t y t o r e t r i e v e c o r r e c t processes f o r s o l v i n g equations. i n d i c a t e d that  Other  l a r g e r numbers f o r c e d s t u d e n t s t o focus on t h e p r o c e s s ,  r a t h e r than g u e s s i n g and s u b s t i t u t i n g v a l u e s i n t o t h e e q u a t i o n . issues  studies  a r e f u r t h e r confounded by t h e tendency  operations i n a l e f t  to r i g h t order.  magnitude and t h e e q u a t i o n s ' s t r u c t u r e  These  of students t o perform  The i n t e r r e l a t i o n s h i p s among n u m e r i c a l and c o m p l e x i t y have been l a r g e l y  unexplored. Computational  e r r o r s seem t o be a s i g n i f i c a n t source o f d i f f i c u l t y f o r  some s t u d e n t s when s o l v i n g l i n e a r e q u a t i o n s . The r e l a t i o n s h i p between c o m p u t a t i o n a l e r r o r s and a l g e b r a i c Furthermore,  errors requires  investigation.  t h e r e seems t o be a p o s i t i v e c o r r e l a t i o n  (r=0.52, E n g l e h a r d t &  Wiebe, 1981) between s t u d e n t s ' u n d e r s t a n d i n g o f a r i t h m e t i c c o m p u t a t i o n a l e r r o r s they make. determine  concepts and the  There have been no i n v e s t i g a t i o n s t o  i f such a c o r r e l a t i o n e x i s t s between u n d e r s t a n d i n g o f a l g e b r a i c  concepts and t h e p r o c e d u r a l o r r e s u l t a n t e r r o r s made i n s o l v i n g e q u a t i o n s . Mechanical, perceptual,  and random e r r o r s may be a f u n c t i o n of  i n a t t e n t i o n t o t h e t a s k a t hand.  In t h e case of t h e f i r s t  two e r r o r s ,  s t u d e n t s u s u a l l y miscopy o r m i s r e a d t h e q u e s t i o n o r some aspect o f t h e i r work. In t h e case o f random e r r o r s , t h e r e i s some e v i d e n c e t o suggest s t u d e n t s may be a t t e n d i n g t o h i g h e r l e v e l concepts and procedures i n s o l v i n g e q u a t i o n s and s u b s e q u e n t l y make a mistake. of i n a t t e n t i o n r e q u i r e  careful consideration  s u g g e s t e d t h a t mechanical  that  involved  The d i f f e r e n t causes  i n diagnosis.  Radatz  (1979)  e r r o r s may be due t o " l a c k o f i n t e r e s t o r t o  45  diversion"  (p. 1 8 ) .  While o v e r l a p  between m e c h a n i c a l / p e r c e p t u a l and random  e r r o r s may occur, f o r t h e purposes o f the p r e s e n t study t h e d i s t i n c t i o n between t h e s e e r r o r t y p e s was m a i n t a i n e d . A summary within  of t h e number o f i n v e s t i g a t o r s who found a p a r t i c u l a r e r r o r -  the e r r o r - c a t e g o r i z a t i o n  study i s p r e s e n t e d i n T a b l e 1 . g r o u p i n g s , from 1 9 2 2 - 1 9 7 4  scheme developed f o r use i n t h e p r e s e n t The f i n d i n g s were s e p a r a t e d i n t o two  and from 1 9 7 5 - 1 9 8 8 .  As can be seen, the focus i n  f  the  first  group was on r e s u l t a n t e r r o r s  compared t o c o n c e p t u a l e r r o r s (8 i n s t a n c e s  errors that  the highest  (5 instances  over 8 c a t e g o r i e s ) .  over 12  over 8 c a t e g o r i e s )  categories) and p r o c e d u r a l  While the second group s t i l l  frequency o f e r r o r s were r e s u l t a n t e r r o r s  over 12 c a t e g o r i e s ) , reported  (33 instances  (48  found  instances  e r r o r s were more e v e n l y d i s t r i b u t e d among t h e types  i n Table 1 ( i e . conceptual -  procedural - 39 instances  48 i n s t a n c e s  over 8 c a t e g o r i e s ) .  over 8  The  categories;  error-categorization  scheme d e v e l o p e d f o r use i n t h i s study p r o v i d e d an o p p o r t u n i t y t o view the changes i n r e s e a r c h errors  Detailed  f i n d i n g s of  f o r i n d i v i d u a l i n v e s t i g a t o r s may be found i n Appendix A.  Bernard & Bright categories occurring The  f o c u s over the p a s t seven decades.  (1982)  i n d i c a t e d that  across a v a r i e t y of studies regarding  an a p p r o p r i a t e  error-categorization  "the s i m i l a r i t y o f e r r o r  suggests t h a t  categorization  convergence may be  o f these e r r o r s "  scheme used i n t h e p r e s e n t study attempts  convergence and p r o v i d e s a framework i n which t o view p r e v i o u s regarding  the conceptual, procedural,  and r e s u l t a n t e r r o r s t h a t  make when s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e .  (p. 1 4 ) . this  research students  46  Table 1 Number o f Researchers F i n d i n g E r r o r - C a t e g o r i z a t i o n Scheme E r r o r s  Type o f E r r o r :  CONCEPTUAL Variable Expression Equality Equation Coefficient L i k e Terms Inverses I d e n t i t y Elements PROCEDURAL Zero Annexation I d e n t i t y Confusion L i k e Terms ( c o n j o i n i n g ) Partial Distributive Order o f O p e r a t i o n s + Property of = x Property of = Coefficient Errors RESULTANT Computational add/subtract multiply/divide basic facts faulty algorithm wrong o p e r a t i o n Sign add/subtract multiply/divide rule-based distributive Other mechanical/perceptual random incomplete work  Number o f Researchers 1925-1974 1975-1988 0 3 0 1 0 0 0  9 9 5 6 5 8 3 3  1 0 1 3 0 2 1 0  3 3 7 9 4 4 4 5  3 3 2 3 3  6 5 5 4 9  2 3 0 2  3 2 3 3  5 4 3  4 1 3  1  CHAPTER DESIGN  OF  III  THE  STUDY  The purpose o f the study was t o develop a d i a g n o s t i c c h e c k l i s t i d e n t i f y the e r r o r s made by f i r s t - y e a r a l g e b r a s t u d e n t s equations  i n one v a r i a b l e .  occurred i n five  in solving linear  Development and v a l i d a t i o n of the  checklist  phases:  1.  analysis  of f i r s t - y e a r a l g e b r a textbooks  2. 3. 4.  development of an e r r o r - c a t e g o r i z a t i o n scheme based on p r e v i o u s research, c o n s t r u c t i o n and s e l e c t i o n of e q u a t i o n s , refinement of the i n s t r u m e n t ,  5.  testing  o f the f i n a l  f o r content  r e s e a r c h e r s as i n f l u e n c i n g s t u d e n t s '  i d e n t i f i e d by p r e v i o u s  errors in solving l i n e a r equations.  Because the d e s i g n of the c h e c k l i s t was c o n t i n g e n t Phase 1 and Phase 2, phases  and sequence,  instrument.  Each phase was d e s i g n e d t o address f a c t o r s  the  to  upon the r e s u l t s  of  b o t h the d e s i g n a s p e c t s and the f i n d i n g s f o r these two  are r e p o r t e d i n t h i s  c h a p t e r p r i o r t o p r e s e n t a t i o n o f the r e s t  of  design. Phase  Content  1:  Analysis  of  First-Year  An a n a l y s i s o f 15 f i r s t - y e a r a l g e b r a textbooks content  f o r the d i a g n o s t i c  instrument.  Algebra  Textbooks  was undertaken t o  The textbooks  selected  were  p u b l i s h e d between  1978 and 1984 and r e p r e s e n t e d the major f i r s t - y e a r  algebra textbooks  used i n North A m e r i c a .  concepts, results  v o c a b u l a r y , types  of t h i s  analysis,  of e q u a t i o n s ,  select  Four a s p e c t s were a n a l y z e d : and i n s t r u c t i o n a l sequence.  The  t o g e t h e r w i t h f i n d i n g s from p r e v i o u s r e s e a r c h ,  formed the b a s i s f o r the content and sequence o f the d i a g n o s t i c i n s t r u m e n t . The textbooks d i s c u s s e d the concepts of v a r i a b l e s , e x p r e s s i o n s ,  equality,  and meaning of e q u a t i o n s  Diagnostic  Checklist  concepts).  The  concepts  p r e s e n t e d through The  concept  for Algebra  (See Appendix E, S e c t i o n A of the  f o r the items a s s o c i a t e d with  of v a r i a b l e s and e x p r e s s i o n s were u s u a l l y  d e f i n i t i o n s , examples, and sometimes  of e q u a l i t y was  counter-examples.  most o f t e n p r e s e n t e d i m p l i c i t l y  as t h a t of a  b a l a n c e between the l e f t - s i d e and r i g h t - s i d e of an e q u a t i o n . the textbooks  explicitly  u s u a l l y done i m p l i c t l y through  e q u a t i o n s and asked  and sentences.  A few textbooks  The  word problems, cue words and  gave  students  f o r an e x p l a n a t i o n of t h e i r meaning.  The v o c a b u l a r y a s s o c i a t e d with c o e f f i c i e n t s and l i k e terms was o f t e n i n t r o d u c e d at the same time as e x p r e s s i o n s . through  of  relation  and t r a n s i t i v i t y .  a l t h o u g h many textbooks d i d i n c l u d e i n t r o d u c t i o n s through some work with phrases  Only two  i n t r o d u c e d e q u a l i t y as an e q u i v a l e n c e  w i t h the p r o p e r t i e s of symmetry, r e f l e x i v i t y , meaning of e q u a t i o n s was  these  Again,  t h i s was  the use of b o l d type, examples, and counter-examples.  v o c a b u l a r y a s s o c i a t e d with a d d i t i v e and m u l t i p l i c a t i v e  most done  The  i n v e r s e s was  most  o f t e n i n t r o d u c e d w i t h the a d d i t i o n and m u l t i p l i c a t i o n p r o p e r t i e s of equality.  F o r i d e n t i t y elements most of the textbooks  p r e s e n t a t i o n s through the same two  textbooks  gave  examples of the p r o p e r t i e s of one  and  non-rigorous zero.  which i n t r o d u c e d e q u a l i t y as an e q u i v a l e n c e  However, relation  a l s o i n t r o d u c e d i n v e r s e s and i d e n t i t y elements as p a r t of the p r o p e r t i e s of the r e a l number system Checklist  for Algebra  t h i s vocabulary).  (See Appendix E, S e c t i o n B of the  Diagnostic  f o r the items used t o i n v e s t i g a t e u n d e r s t a n d i n g  of  In g e n e r a l , t h e textbooks i n t r o d u c e d 12 types o f e q u a t i o n s t o students p r i o r t o t h e i n t r o d u c t i o n o f r a t i o n a l numbers as e i t h e r c o e f f i c i e n t s o r solutions. Checklist  A l l twelve t y p e s o f e q u a t i o n s were i n c l u d e d i n t h e for Algebra  Diagnostic  because a l g e b r a i c e r r o r s seem t o be s p e c i f i c t o the  p a r t i c u l a r e q u a t i o n s used  (Davis & Cooney, 197 8 ) .  As suggested by C o l l i s  (1975), t h e e q u a t i o n s occur i n t h e d i a g n o s t i c instrument i n t h e mean rank p o s i t i o n as determined  from t h e t e x t b o o k s .  These twelve e q u a t i o n types  t o g e t h e r w i t h t h e i r o r d i n a l p o s i t i o n a r e p r e s e n t e d i n T a b l e 2. Phase  2:  Development  of  the  Error-Categorization  Scheme  The e r r o r - c a t e g o r i z a t i o n scheme was developed t o f a c i l i t a t e t h e i d e n t i f i c a t i o n o f t h e s p e c i f i c e r r o r s which a student might make when s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e , reviewed  i n Chapter  2.  and was based on t h e l i t e r a t u r e  To reduce t h e p r o b a b i l i t y o f i n c l u s i o n o f e r r o r s  which were s p e c i f i c t o a p a r t i c u l a r study, o n l y those e r r o r s found by t h r e e o r more r e s e a r c h e r s were i n c l u d e d . E r r o r s were grouped  i n t o t h r e e major c a t e g o r i e s :  p r o c e d u r a l , and r e s u l t a n t .  conceptual,  The c o n c e p t u a l c a t e g o r y i n c l u d e d the f o u r  concepts and f o u r v o c a b u l a r y items mentioned e a r l i e r i n t h e textbook analysis.  E r r o r s i d e n t i f i e d within the procedural category included  a n n e x a t i o n o f zero, i d e n t i t y c o n f u s i o n , i n c o r r e c t combining  of l i k e  terms,  p a r t i a l d i s t r i b u t i v e e r r o r s , e r r o r s made i n a p p l y i n g t h e r u l e s f o r o r d e r of o p e r a t i o n s , t h e a d d i t i o n and m u l t i p l i c a t i o n p r o p e r t i e s o f e q u a l i t y , and e r r o r s i n o b t a i n i n g the c o e f f i c i e n t of the v a r i a b l e . c a t e g o r y was p a r t i t i o n e d i n t o t h r e e s u b - c a t e g o r i e s :  The r e s u l t a n t computational,  sign,  and o t h e r e r r o r s .  Computational  e r r o r s i n c l u d e d e r r o r s i n any of t h e f o u r  b a s i c o p e r a t i o n s , b a s i c f a c t s , d e f e c t i v e a l g o r i t h m s , o r t h e use o f wrong operations.  S i g n e r r o r s were e r r o r s i n p o s i t i v e o r n e g a t i v e s i g n s with any  of t h e f o u r b a s i c o p e r a t i o n s , r u l e - b a s e d e r r o r s i n d e t e r m i n i n g t h e s i g n , or sign errors using the d i s t r i b u t i v e property. mechanical  Other  errors included  and p e r c e p t u a l problems, random e r r o r s , o r incomplete  work.  O v e r a l l , 2 0 p r o c e d u r a l and r e s u l t a n t e r r o r s were i n c l u d e d i n t h e e r r o r categorizationscheme.  These 2 0 e r r o r s r e p r e s e n t e d a c o n s o l i d a t i o n o f the  type o f e r r o r s found by p r e v i o u s r e s e a r c h e r s .  The name chosen f o r each of  t h e s e 2 0 e r r o r s r e p r e s e n t e d the name most o f t e n used  i n previous research.  A d e t a i l e d d i s c u s s i o n o f t h e e r r o r - c a t e g o r i z a t i o n scheme i s p r e s e n t e d i n Appendix A,  and i n c l u d e s a t a b u l a r p r e s e n t a t i o n of t h e e r r o r s found by  each r e s e a r c h e r which o c c u r r e d i n t h e Phase  3:  Construction  e r r o r - c a t e g o r i z a t i o n scheme.  and  Selection  of  Equations  The purpose o f t h e t h i r d phase o f t h e study was t o c o n s t r u c t a s e t o f e q u a t i o n s and s u b s e q u e n t l y d i a g n o s t i c instrument.  s e l e c t a p o r t i o n o f these f o r i n c l u s i o n i n the  P r e v i o u s r e s e a r c h e r s had u t i l i z e d e q u a t i o n s by  e i t h e r s e l e c t i n g from p a r t i c u l a r c h a p t e r s o f a c u r r e n t textbook Cooney, 1 9 7 8 ) o r by "maximizing the e q u a t i o n s "  (Anderson,  (Davis &  t h e number o f m a n i p u l a t i o n s made i n s o l v i n g  1982).  Such s e l e c t i o n p r o c e d u r e s  q u e s t i o n - s p e c i f i c e r r o r s , as many a s p e c t s o f the c o m p o s i t i o n  may produce o f an e q u a t i o n  have been r e p o r t e d t o i n f l u e n c e t h e nature and frequency o f t h e e r r o r s made by s t u d e n t s i n s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e . include:  equation complexity  (Booth & Hart,  These a s p e c t s  1 9 8 2 ) , n u m e r i c a l magnitude  Table 2 Types o f L i n e a r Equations P r e s e n t e d i n 15  Equation Types I  .  General Format  Textbooks  Ordinal Position i n Textbooks  One-Step  Addition Subtraction Multiplication Division  x+a=b* x-a=b ax=b x/a=b  1 2 3 4  ax+b=c  5  ax+bx=c ax+b=cx a(x+b)=c**  6 7 8  Two-Step  II.  L i k e Terms Numerical Variable One S i d e o f Both S i d e s of = Parentheses I l l .  Multi-Step  Combined N u m e r i c / V a r i a b l e One S i d e o f = Both S i d e s o f = Parentheses One S i d e o f = Both S i d e s o f =  ax+b+cx+d=e ax+b=cx+d  9 10  a(x+b)=cx+d a(x+b)=c(x+d)  11 12  a,b,c,d,e,x a r e a l l i n t e g e r s , a*0. Based on t h e c o n s t r a i n t s o f the study t h e s e e q u a t i o n s were l i m i t e d t o those c o n t a i n i n g i n t e g r a l r o o t s , as w e l l as c o e f f i c i e n t s . ** where c i s d i v i s i b l e by a.  (Herscovics,  1979), placement of the v a r i a b l e  the e q u a t i o n  (Pettito,  al.,  To c o n t r o l f o r such i n f l u e n c e s , the s y s t e m a t i c v a r i a t i o n of  1980).  ( K i e r a n , 1983), s t r u c t u r e of  1979), and o r d e r of s i g n p r e s e n t a t i o n (Tatsuoka e t  t h e s e a s p e c t s of the e q u a t i o n s c o n s t r u c t e d f o r use i n the d i a g n o s t i c instrument  ensured  understandings,  t h a t e r r o r s were r e p r e s e n t a t i v e of s t u d e n t s '  erroneous  and not simply a r e f l e c t i o n of the type of e q u a t i o n  presented. Construction  The Algebra  of  Equations  c o n s t r u c t i o n of e q u a t i o n s  f o r use  i n the Diagnostic  Checklist  for  employed s y s t e m a t i c v a r i a t i o n of the s i g n s of the numbers, the  a b s o l u t e v a l u e of the numbers i n r e l a t i o n t o each o t h e r , and  the  r e l a t i o n s h i p of the v a r i a b l e t o both the numbers and the e q u a l i t y Such v a r i a t i o n was  a c h i e v e d through the c r e a t i o n of an  numerical-magnitude-by-structural-format e q u a t i o n types i d e n t i f i e d i n the textbook e q u a t i o n s were developed  equation-type-by-  g r i d of e q u a t i o n s .  The  a n a l y s i s were u t i l i z e d .  Sixteen  Within  o r d e r of n u m e r i c a l s i g n and r e l a t i v e . m a g n i t u d e of  numbers were s y s t e m a t i c a l l y v a r i e d as suggested by Tatsuoka T h i s r e s u l t e d i n the c o n s t r u c t i o n of 192 s t r u c t u r a l format  12  f o r each e q u a t i o n type c o n s i s t i n g of f o u r l e v e l s  of n u m e r i c a l magnitude and f o u r c a t e g o r i e s of s t r u c t u r a l format. t h e s e 4-by-4 g r i d s ,  symbol.  equations:  e t a l . (1980).  12 e q u a t i o n types x 4  c a t e g o r i e s x 4 n u m e r i c a l magnitude l e v e l s .  The  4-by-4  g r i d s f o r each of the twelve types of e q u a t i o n s and the accompanying magnitude and s i g n r u l e s used w i t h i n each g r i d are p r e s e n t e d i n Appendix B, Part  1.  53  Four l e v e l s of n u m e r i c a l magnitude were d e f i n e d , u s i n g a b s o l u t e v a l u e . Numerical Magnitude (Nl) (N2) (N3) (N4)  Integer In| 20< |n| 50< |n| 100< |n|  C u t - o f f v a l u e s were based on the work of P e t t i t o  Values < 20 < ' 50 < 100 < 1000  (1979) and Anderson  (1982) . E q u a t i o n s which s a t i s f i e d the n u m e r i c a l magnitude c o n s t r a i n t s and s i g n and n u m e r i c a l magnitude r e l a t i o n s h i p s as o u t l i n e d by the d e s i g n s p e c i f i e d i n Appendix A were not always p o s s i b l e t o c o n s t r u c t . example, c o n s i d e r an e q u a t i o n of the form ax +b=cx  i n which  the rules  For  |c|>|a| and  the s i g n s of a and c were o p p o s i t e , and yet each of a, b, and c were of second was  n u m e r i c a l magnitude  (20<|n|< 50).  p o s s i b l e ( i e . |"26|>|24| and  l e v e l would r e q u i r e s o l u t i o n s  |~26  For example, w h i l e 24x +50=~26x  + "24|<50), a l l e q u a t i o n s  at t h i s  of -1 or 1 i n o r d e r t o s i m u l t a n e o u s l y meet  the magnitude, as w e l l as s i g n and i n t e g r a l c o n s t r a i n t s .  Hence, f o r a l l  e q u a t i o n s where t h i s type of problem o c c u r r e d , as many n u m e r i c a l v a l u e s as p o s s i b l e were kept w i t h i n the d e s i g n a t e d l i m i t s , w i t h i n the c o r r e c t was  or the c o e f f i c i e n t s were  l i m i t s once b r a c k e t s were removed  [e.g. ~ 9 (x +8)=8(x +~9)  an N_3_ e q u a t i o n when the b r a c k e t s were removed] . S t r u c t u r a l format  solving  (Pettito,  v a r i a b l e appears  of e q u a t i o n s a l s o i n f l u e n c e s performance i n e q u a t i o n -  1979).  Some students e x p e r i e n c e d i f f i c u l t y when the  e i t h e r a f t e r the number  the r i g h t of the e q u a l i t y symbol  ( H e r s c o v i c s & K i e r a n , 1980)' or t o  (Behr e t a l . , 1976;  Wagner, 1981a) .  54  Therefore,  systematic  v a r i a t i o n o f t h e placement o f v a r i a b l e s w i t h  t o b o t h t h e e q u a l i t y symbol and t h e c o n s t a n t s  within  respect  an e q u a t i o n i s  n e c e s s a r y . These v a r i a t i o n s r e q u i r e d t h e c r e a t i o n o f f o u r c a t e g o r i e s o f structural  format:  S t r u c t u r a l Format 1 ( S I ) : E q u a t i o n s i n which t h e v a r i a b l e f i r s t o c c u r r e d on t h e l e f t s i d e o f t h e e q u a l i t y symbol, f o l l o w e d by numbers (eg. x +15=8). S t r u c t u r a l Format 2 (S2): E q u a t i o n s i n which t h e v a r i a b l e f i r s t o c c u r r e d on t h e l e f t s i d e . o f t h e e q u a l i t y symbol, but o c c u r r e d on t h e r i g h t s i d e of the numbers (eg. 15+x = 8 ) . S t r u c t u r a l Format 3 (S3>: E q u a t i o n s i n which t h e v a r i a b l e f i r s t o c c u r r e d on t h e r i g h t s i d e o f t h e e q u a l i t y symbol, f o l l o w e d by numbers (eg. 8=x +15) . S t r u c t u r a l Format 4 (S4): E q u a t i o n s i n which t h e v a r i a b l e f i r s t o c c u r r e d on t h e r i g h t s i d e o f t h e e q u a l i t y symbol, and o c c u r r e d on t h e r i g h t s i d e of the numbers (eg. 8=15+x ) . In t h o s e e q u a t i o n s where t h e r e (Equation step,  were two o c c u r r e n c e s o f t h e v a r i a b l e  Types 6, 7, and 9 - 12, See Table 2, page 51), t h e i n t e r m e d i a t e  where t h e v a r i a b l e s were combined i n t o one v a r i a b l e term, was used t o  determine t h e s t r u c t u r e o f t h e e q u a t i o n .  An example  o f such a L e v e l 1  s t r u c t u r e would be: 13 (x - 3) 13x - 39 15x In t h e f i r s t  line,  = = =  -2x - 9 -2x - 9 30  t h e v a r i a b l e s on both s i d e s o f t h e e q u a t i o n were  f o l l o w e d by numbers, and i t was assumed coefficient  that  s t u d e n t s would t r y t o make t h e  o f t h e combined v a r i a b l e terms p o s i t i v e .  a f t e r removing p a r e n t h e s e s i n t h e above example,  That i s t o say,  s t u d e n t s would add 2x t o  b o t h s i d e s o f t h e e q u a t i o n t o make 15x and then add 39 t o b o t h s i d e s of t h e e q u a t i o n t o make 30.  55  Selection  of  Equations  There were f o u r f a c t o r s i n v o l v e d i n the s e l e c t i o n of the e q u a t i o n s f o r use  i n the Diagnostic  Checklist  for Algebra.  The  e f f e c t of s y s t e m a t i c v a r i a t i o n of the c o m p l e x i t y  first  was  t o determine the  of an e q u a t i o n , i t s  s t r u c t u r e , and the magnitude of the numbers used i n the e q u a t i o n on e r r o r s produced by s t u d e n t s . v a l i d a t i o n of the developed  The  t o ensure t h a t the e q u a t i o n s  necessary  The  t h i r d was  from 1 9 2 to a more p r a c t i c a l l e v e l .  l e v e l s of achievement i n a l g e b r a . was  t o p r o v i d e a measure of  e r r o r - c a t e g o r i z a t i o n scheme.  reduce the number of e q u a t i o n s f o u r t h was  second was  to  The  s e l e c t e d would d i s t i n g u i s h among  In o r d e r t o c o n t r o l f o r these f a c t o r s , i t  to a s c e r t a i n s u b j e c t s ' achievement i n a l g e b r a , t o  the c o n s t r u c t e d e q u a t i o n s ,  the  and t o determine the nature  administer  of the e r r o r s  produced i n s o l v i n g them. Subjects  All  f o u r academic Grade 9 mathematics c l a s s e s i n one  were used i n t h i s phase.  The  f i r s t - y e a r algebra students  junior high school  s u b j e c t s r e p r e s e n t e d the e n t i r e p o p u l a t i o n of  i n the s c h o o l .  Absenteeism r e s u l t e d i n a l o s s  of s i x of t h e 8 6 s u b j e c t s . Procedures  In the S p r i n g of 1 9 8 3 , a l l s u b j e c t s were a d m i n i s t e r e d the Lankton Year Algebra  Test  (Lankton,  levels i n algebra.  1 9 6 5 , Form E ) t o determine t h e i r achievement  High a c h i e v e r s were d e f i n e d as s u b j e c t s s c o r i n g one  more s t a n d a r d d e v i a t i o n s above the mean on the Lankton. d e f i n e d as those  First-  s c o r i n g one  Low  or  a c h i e v e r s were  or more s t a n d a r d d e v i a t i o n s below the mean.  The  Lankton  administration, tool.  c o n s i s t s o f 5 0 items, r e q u i r e s 40 minutes f o r and was  A l g e b r a One  assessment  The c o n t e n t o f the t e s t i n c l u d e s d e f i n i t i o n o f terms and meaning of  s i g n s and symbols  (8 i t e m s ) , fundamental  and e x t r a c t i n g r o o t s (13  designed* as a mid-year,  algebraic operations, factoring  ( 1 2 i t e m s ) , e q u a t i o n s and  inequalities  i t e m s ) , a l g e b r a i c e x p r e s s i o n s and formulas, f u n c t i o n s , v a r i a t i o n  problem  solving  half r e l i a b i l i t y  (8 i t e m s ) , and g r a p h i c r e p r e s e n t a t i o n (9 i t e m s ) . c o e f f i c i e n t s f o r Form E ranged from 0 . 8 2 t o 0 . 8 6  and  Split(Lankton,  1965).  The  1 9 2 e q u a t i o n s needed t o be a d m i n i s t e r e d t o s u b j e c t s t o  their difficulty  A b a l a n c e d L a t i n Square d e s i g n was  o f the 1 9 2 e q u a t i o n s t o one o f f o u r t e s t - f o r m s  assignment Chapter  level.  9).  determine  used f o r  (See Winer, 1 9 7 1 ,  Each t e s t - f o r m c o n s i s t e d of 48 items, f o u r from each of the  twelve e q u a t i o n types  (See Appendix B, P a r t 2 ) .  The b a l a n c e d assignment  of  e q u a t i o n s t o t e s t - f o r m s a l s o ensured the b a l a n c e d d i s t r i b u t i o n of n u m e r i c a l magnitude and s t r u c t u r a l format throughout  the 48 e q u a t i o n s .  the n u m e r i c a l - m a g n i t u d e - b y - s t r u c t u r a l - f o r m a t g r i d was times throughout balanced  Each c e l l of  represented three  each t e s t - f o r m , although f o r d i f f e r e n t e q u a t i o n t y p e s .  assignment  The  of e q u a t i o n s t o t e s t - f o r m s a l s o e n a b l e d a n a l y s i s of  the e f f e c t o f i t e m p r e s e n t a t i o n and c o n t r o l l e d f o r i n d i v i d u a l  differences  among the f o u r a l g e b r a c l a s s e s . All  s u b j e c t s were g i v e n one  of the f o u r t e s t - f o r m s d u r i n g the  mathematics c l a s s f o l l o w i n g a d m i n i s t r a t i o n o f the Lankton.  T h i s immediate  a d m i n i s t r a t i o n of the e q u a t i o n s h e l p e d t o e l i m i n a t e any e f f e c t  that  i n s t r u c t i o n might have had on t h e s o l v i n g o f the e q u a t i o n s . regarding  r e s u l t s on t h e Lankton was g i v e n t o s u b j e c t s  administration equally  of the equation test-forms.  d i s t r i b u t e d within  the four classes  No feedback  prior to  The f o u r t e s t - f o r m s were involved  i n t h e study.  Analyses An 1972).  i t e m a n a l y s i s was made of each t e s t - f o r m As p o s t - f a c t o  categories  using  LERTAP  (Nelson,  a n a l y s i s o f t h e e r r o r s was made, t h e twenty e r r o r  used i n t h e d i a g n o s t i c  instrument were reduced t o t e n t o  f a c i l i t a t e t h e i n v e s t i g a t i o n o f t h e g e n e r a l nature o f t h e e r r o r s  occurring.  The  errors  o f zero annexation and i d e n t i t y c o n f u s i o n were not i n c l u d e d i n  the  a n a l y s i s due t o t h e d i f f i c u l t y o f t h e i r i d e n t i f i c a t i o n without  interviewing  the student.  One o f t h e t e n c a t e g o r i e s  was r e s e r v e d f o r  c o r r e c t answers, as t h e o c c u r r e n c e o f c o r r e c t answers was t o be checked i n the  f i n a l instrument.  T o t a l c o r r e c t answers f o r each s u b j e c t  could  c o r r e l a t e d w i t h t h e i n d i v i d u a l r e s u l t s on the Lankton t o ensure of achievement r e s u l t s . as  similarity  Each e r r o r made on a p a r t i c u l a r e q u a t i o n was coded  follows: a b c d e f g h i j  The  a l s o be  = = = = = = = = = =  R i g h t Answer Other ( m e c h a n i c a l / p e r c e p t u a l , random) Incomplete s o l u t i o n s Computational e r r o r s ( a l l f i v e e r r o r s combined) S i g n e r r o r s ( a l l except d i s t r i b u t i v e ) S i g n e r r o r on d i s t r i b u t i v e p r o p e r t y Misuse o f + o r x p r o p e r t y o f e q u a l i t y C o e f f i c i e n t e r r o r s ( f o r +, -, x, o r + and o r d e r o f o p e r a t i o n s ) Combining u n l i k e terms Partial distributive  comments i n parentheses a r e t h e names g i v e n t o t h e e r r o r s  i n the e r r o r -  c a t e g o r i z a t i o n scheme p r e s e n t e d i n Appendix A which were i n c l u d e d i n each c a t e g o r y o f t h e LERTAP a n a l y s i s .  The LERTAP a n a l y s i s t r e a t e d each  e q u a t i o n as i f i t were p a r t o f a m u l t i p l e - c h o i c e t e s t ,  with  error  c a t e g o r i e s s e r v i n g as d i s t r a c t o r s . Other  than t h e two e r r o r s mentioned above, a l l e r r o r c a t e g o r i e s  i d e n t i f i e d f o r use i n t h e d i a g n o s t i c instrument or  o c c u r r e d as e r r o r s f o r one  more o f t h e s u b j e c t s t e s t e d on the e q u a t i o n s .  T h i s l e n t some measure of  f a c e v a l i d i t y t o t h e e r r o r - c a t e g o r i z a t i o n scheme. Items used  i n the f i n a l  d i a g n o s t i c instrument  were chosen from t h e 192  e q u a t i o n s u s i n g t h e r e s u l t s from t h e LERTAP a n a l y s i s t o g e t h e r with of  the i n t e r a c t i o n e f f e c t s of complexity of the equations,  magnitude, and s t r u c t u r a l format. of  numerical  A d e t a i l e d p r e s e n t a t i o n of the r e s u l t s  t h e s e i n t e r a c t i o n e f f e c t s may be found i n Appendix B, P a r t 3.  the l a t t e r a n a l y s i s , each c e l l format  analysis  Based on  o f the n u m e r i c a l - m a g n i t u d e - b y - s t r u c t u r a l -  g r i d was i n c l u d e d for. each c o m p l e x i t y l e v e l .  Such i n c l u s i o n meant  t h e r e would be 16 items i n c l u d e d f o r each l e v e l o f complexity,  making a  t o t a l o f 48 e q u a t i o n s t o be .included i n t h e d i a g n o s t i c i n s t r u m e n t . Within these c o n s t r a i n t s ,  items were s e l e c t e d whose c o r r e c t answer had  a p o s i t i v e c o r r e l a t i o n w i t h t h e t o t a l s c o r e on the t e s t - f o r m , and whose incorrect  answers had a n e g a t i v e c o r r e l a t i o n with t h e t o t a l s c o r e on the  t e s t - f o r m . To ensure difficulty  level  t h a t t h e e r r o r s i d e n t i f i e d were not i n f l u e n c e d by item  ( i e . easy items would o n l y r e s u l t i n random e r r o r s and few  p r o c e d u r a l e r r o r s would occur; d i f f i c u l t incomplete  items would m a i n l y  result i n  work, and, a g a i n , v e r y few p r o c e d u r a l e r r o r s would occur) ,  items  were s e l e c t e d from t h e above group of e q u a t i o n s whose p - v a l u e s were c l o s e s t to  0.5.  From t h e s e s e l e c t e d e q u a t i o n s , o n l y t h o s e were used whose  c o r r e l a t i o n s w i t h t h e Lankton were a l s o p o s i t i v e . together with t h e i r s t a t i s t i c a l  The f i n a l e q u a t i o n s ,  c o r r e l a t o n s , a r e p r e s e n t e d i n Appendix B,  Part 4 . When comparisons were made between t h e e q u a t i o n s e c t i o n s o f t h e Lankton and t h e f i n a l Algebra  4 8 e q u a t i o n s s e l e c t e d f o r use i n t h e Diagnostic  t h e c o r r e l a t i o n s ranged from 0 . 6 1 t o 0 . 8 6 .  suggested acceptable concurrent v a l i d i t y  These  Checklist  for  correlations  w i t h t h e e q u a t i o n p o r t i o n s of t h e  Lankton. Phase  4:  Refinement  of  the  Instrument  The f o u r t h phase o f the study was used t o r e f i n e t h e wording and format of  the d i a g n o s t i c instrument.  to  clarify  i n wording  items, i n c l u s i o n o f h i n t s f o r d i a g n o s t i c i a n s o f p o s s i b l e  which might s t i l l the  T h i s refinement i n c l u d e d changes  need t o be made f o r f u r t h e r c l a r i f i c a t i o n ,  cues  and changes t o  format which would a l l o w f o r d i a g n o s t i c comments and t a b u l a t i o n s .  a l s o s e r v e d t o i n v e s t i g a t e t h e time needed  It  f o r completion of the diagnostic  instrument. Subjects  Four s u b j e c t s , one male and one female from each o f t h e h i g h and low achievement groups, were randomly s e l e c t e d from t h e e i g h t y s u b j e c t s used i n the  Phase  3 o f t h e study.  Procedures  The f o u r s u b j e c t s were a d m i n i s t e r e d the Diagnostic  Checklist  for  Algebra  i n a clinical-interview setting.  determine  i f t h e wording  p a r t i c u l a r l y important  They were a l s o q u e s t i o n e d t o  o f t h e instrument was c l e a r .  T h i s was  i n the Conceptual-expressions p o r t i o n of the  d i a g n o s t i c i n s t r u m e n t , where s t u d e n t s had t o show t h e i r u n d e r s t a n d i n g o f a r i t h m e t i c and a l g e b r a i c e x p r e s s i o n s through t h e m a n i p u l a t i o n o f c o n c r e t e materials.  A f t e r each response,  s u b j e c t s were asked t o e x p l a i n t h e thought  p r o c e s s e s t h e y used i n d i s p l a y i n g u n d e r s t a n d i n g o f concepts and v o c a b u l a r y in algebra  (Kilpatrick,  1967).  Analyses  Video-tapes of  o f t h e d i a g n o s t i c i n t e r v i e w s were viewed  t o determine  common c o n f u s i o n s u r r o u n d i n g any aspect o f t h e d i a g n o s t i c  Each s u b j e c t viewed  instrument.  h i s o r h e r own v i d e o - t a p e with t h e i n t e r v i e w e r .  was q u e s t i o n e d r e g a r d i n g responses, e r r o r s , and apparent  areas  Each  nervousness.  On  the b a s i s o f o b s e r v a t i o n s o f these f o u r s t u d e n t s , changes were made t o t h e wording  o f t h e Diagnostic  Checklist  for Algebra  t o c l a r i f y meaning and  reduce t e s t i n g time, as w e l l as changes t o t h e placement to  minimize  d i s t r a c t i o n s f o r the subjects.  of  F o r a complete  video-equipment  d e s c r i p t i o n of  the changes which o c c u r r e d i n t h e c h e c k l i s t d u r i n g Phase 4, r e f e r t o Appendix C. Phase  The purpose q u e s t i o n s posed  5:  Testing  of  the  Final  Instrument  o f t h e f i n a l phase o f t h e study was t o address t h e r e s e a r c h i n Chapter  1.  The f i f t h phase o f t h e study i n v o l v e d  a d m i n i s t r a t i o n o f t h e r e f i n e d d i a g n o s t i c instrument t o a l a r g e r group o f students.  Each was g i v e n t h e Diagnostic  Checklist  for Algebra  i n an  individual,  clinical-interview.  The n a t u r e and f r e q u e n c y o f e r r o r s made on  the d i a g n o s t i c i n s t r u m e n t were i n v e s t i g a t e d . h i g h , average, and low a c h i e v e r s .  Comparisons were made between  The r e l a t i o n s h i p s among c o n c e p t s ,  v o c a b u l a r y , e r r o r p a t t e r n s , and e q u a t i o n - s o l v i n g p r o c e d u r e s were e x p l o r e d .  Subjects A stratified  random sample o f 3 6 s u b j e c t s was s e l e c t e d from the  academic mathematics c l a s s e s i n the same j u n i o r h i g h s c h o o l used i n Phases 3 and 4 .  S u b j e c t s who had been r e t a i n e d and thus exposed t o e i t h e r Phase 3  o r Phase 4 were o m i t t e d from the sample. 9,  1 0 ) , a l g e b r a i c achievement l e v e l  The s t r a t a were:  grade l e v e l  (high, average, low) and gender  (8,  (male,  female). Procedures  A l l academic mathematics s t u d e n t s were g i v e n the Lankton Algebra  Test  First-Year  (Lankton, 1 9 6 5 , Form E ) i n a group s e s s i o n t o determine t h e i r  a l g e b r a i c achievement l e v e l .  Students were then a s s i g n e d t o one of  e i g h t e e n groups a c c o r d i n g t o grade l e v e l (high/average/low), and gender  (8/9/10),  (male/female).  achievement  level  Two s u b j e c t s were randomly  s e l e c t e d from each o f 18 groups. P a r e n t a l consent forms were g i v e n t o each of the 3 6 s u b j e c t s . Appendix D, P a r t 1 ) .  (See  I f p a r e n t a l consent t o p a r t i c i p a t e i n the d i a g n o s t i c  i n t e r v i e w s was not o b t a i n e d , another s u b j e c t was randomly s e l e c t e d from the a p p r o p r i a t e group. diagnostic  A l l p a r e n t a l consent forms were r e t u r n e d p r i o r t o the  interviews.  The  Concepts  in Secondary  Mathematics  and Science  (CSMS) t e s t  (Booth &  Hart, 1982) was a d m i n i s t e r e d t o a l l s u b j e c t s i n a s e p a r a t e group s e s s i o n p r i o r t o any i n t e r v i e w s . study was t o determine v a r i a b l e used  The purpose o f t h e i n c l u s i o n o f t h i s t e s t i n t h e  whether o r not t h e two q u e s t i o n s on u n d e r s t a n d i n g of  i n t h e Diagnostic  Checklist  for Algebra  1981a) a d e q u a t e l y d e s c r i b e d s t u d e n t s ' u n d e r s t a n d i n g t e s t c o n s i s t s o f 51 items and was based Kuchemann  (-1978).  understanding  (based on Wagner, of variable.  on t h e work o f C o l l i s  The CSMS  (1975) and  R e s u l t s on t h e CSMS would be compared t o s t u d e n t ' s  o f t h e concept  o f v a r i a b l e and t h e e r r o r s made r e l a t i n g t o  v a r i a b l e s i n s o l v i n g equations  (e.g. combining  u n l i k e terms).  A L a t i n - S q u a r e d e s i g n was used t o s e l e c t t h e date and time o f i n t e r v i e w s f o r each s u b j e c t t o c o n t r o l f o r t h e e f f e c t s o f grade, achievement, and gender.  The i n t e r v i e w s extended  over a six-week p e r i o d  because o f i n s t r u c t i o n a l and e x t r a - c u r r i c u l a r demands, as w e l l as absenteeism.  A l l s u b j e c t s were i n d i v i d u a l l y i n t e r v i e w e d , and i n t e r v i e w s  were v i d e o - t a p e d .  P r e l i m i n a r y d i a g n o s t i c r e s u l t s were d i s c u s s e d with each  student a t t h e c o n c l u s i o n o f t h e i n t e r v i e w . Analyses Comparisons were made with a p p r o p r i a t e items from t h e Lankton s t u d e n t s ' mathematics marks t o e s t a b l i s h c o n c u r r e n t v a l i d i t y . d i s t r i b u t i o n s were used t o examine t h e e f f e c t s o f e q u a t i o n s t r u c t u r a l format,  and n u m e r i c a l magnitude.  also calculated for a l l error categories. age,  mathematical  achievement - Lankton,  and with  Frequency  complexity,  Frequency d i s t r i b u t i o n s were E r r o r s were c o r r e l a t e d with IQ,  Marks, CSMS - and o t h e r e r r o r s .  63  The  two t e a c h e r s o f t h e 36 s u b j e c t s were i n t e r v i e w e d r e g a r d i n g t h e  e r r o r s made by t h e s t u d e n t s . determine  The purpose o f t h e t e a c h e r i n t e r v i e w s was t o  i f e r r o r s made on t h e d i a g n o s t i c instrument  e r r o r s which s t u d e n t s made i n t h e c l a s s r o o m  Summary  of  Study  were s i m i l i a r t o  setting.  Design  P r e v i o u s r e s e a r c h has i n d i c a t e d a need f o r d i a g n o s t i c t o o l s a t the secondary  l e v e l t o e x p l o r e e r r o r s i n a l g e b r a ; t h e p r e s e n t study was  d e s i g n e d t o meet t h i s need u s i n g the framework o f t h e " i n p u t - p r o c e s s o u t p u t " model o f l e a r n i n g textbooks  (Booth,  1981).  i d e n t i f i e d twelve types o f e q u a t i o n s commonly taught  y e a r a l g e b r a c o u r s e s , as w e l l as concepts i n f l u e n c e procedures one v a r i a b l e . for  algebra  in first-  and v o c a b u l a r y which might  used and e r r o r s made i n s o l v i n g l i n e a r e q u a t i o n s i n  E q u a t i o n s were c o n s t r u c t e d which r e f l e c t e d r e s e a r c h  e q u a t i o n complexity,  format.  Scrutiny of f i r s t - y e a r  n u m e r i c a l magnitude and s i g n ,  An e r r o r - c a t e g o r i z a t i o n scheme was developed  e r r o r s made i n s o l v i n g e q u a t i o n s .  and used t o i d e n t i f y combined the  of semi-structured interviews,  m i n i m i z i n g t h e n e g a t i v e a s p e c t s o f each t e c h n i q u e . d e v e l o p t h e d i a g n o s t i c instrument  and s t r u c t u r a l  The d i a g n o s t i c instrument  r i g o r o f e r r o r - a n a l y s i s with the f l e x i b i l i t y  concerns  The procedures  used t o  were s e l e c t e d t o i n c r e a s e t h e p r o b a b i l i t y  t h a t e r r o r s i d e n t i f i e d would be r e p r e s e n t a t i v e o f t h e types o f e r r o r s made by s t u d e n t s i n f i r s t - y e a r a l g e b r a c o u r s e s .  CHAPTER  IV  RESULTS The  r e s u l t s presented are the pertinent  fourth,  and f i f t h phases of t h e study.  f i n d i n g s from t h e t h i r d ,  The outcomes of t h e  item a n a l y s i s  and i n t e r a c t i o n e f f e c t s i n v o l v e d i n t h e s e l e c t i o n o f e q u a t i o n s a r e summarized from Phase 3 o f t h e study. procedures  The r e s u l t s o f t h e refinement  used i n Phase 4 o f t h e study a r e d i s c u s s e d .  The n a t u r e and  f r e q u e n c y o f t h e e r r o r s made by students u s i n g the Diagnostic Algebra  i n Phase 5 a r e p r e s e n t e d .  Checklist  for  However, t h e i r p r e s e n t a t i o n i s  o r g a n i z e d around t h e r e s e a r c h q u e s t i o n s , r a t h e r than s p e c i f i c i n t e r v i e w results.  Examples from i n d i v i d u a l i n t e r v i e w s are d e t a i l e d t o c l a r i f y  findings. Summary  of  Phases  3  and  4  Phase 3 i n v o l v e d s e l e c t i o n o f e q u a t i o n s from t h e 192 c o n s t r u c t e d equations  (12 e q u a t i o n types by 4 s t r u c t u r a l formats by 4 n u m e r i c a l  magnitudes) f o r use i n t h e d i a g n o s t i c i n s t r u m e n t .  The p o s s i b l e  e f f e c t s among c o m p l e x i t y o f t h e e q u a t i o n s , s t r u c t u r a l format,  interaction  and n u m e r i c a l  magnitude was a concern from p r e v i o u s r e s e a r c h which r e q u i r e d e m p i r i c a l clarification.  A n a l y s i s of the r e s u l t s of the equation t e s t  i n d i c a t e d t h a t t h e r e was a s t r o n g i n t e r a c t i o n e f f e c t between format  and n u m e r i c a l magnitude  (See Appendix B, P a r t 3 ) .  i n c l u d e a l l o f the' r e s u l t i n g c e l l s o f t h i s 4 x 4 e q u a t i o n c o m p l e x i t y f o r complete  diagnosis.  forms structural  I t was d e c i d e d t o  g r i d a t each  l e v e l of  This n e c e s s i t a t e d the  s e l e c t i o n o f 48 e q u a t i o n s f o r i n c l u s i o n i n t h e Diagnostic  Checklist  for  Algebra. The  f o u r t h phase o f t h e study r e f i n e d t h e wording  and format o f t h e  i n s t r u m e n t , and i n v e s t i g a t e d i t s u s e f u l n e s s i n i d e n t i f y i n g s y s t e m a t i c e r r o r s o f i n d i v i d u a l s and common e r r o r s among groups o f s t u d e n t s . r e s u l t s o f t h e wording  changes a r e p r e s e n t e d i n Appendix C.  f r e q u e n c y o f t h e e r r o r s produced  The  The nature and  by t h e f o u r s u b j e c t s i n v o l v e d i n Phase 4  are p r e s e n t e d here. The two low a c h i e v e r s seemed t o have almost l e v e l s o f s t r u c t u r e and magnitude.  equal d i f f i c u l t y across a l l  The one e x c e p t i o n was with e q u a t i o n s  which were s t r u c t u r e d w i t h t h e .variable t o t h e l e f t constants  of t h e e q u a l i t y and  ( S I ) . Twelve o f t h e 28 e q u a t i o n s on which low a c h i e v e r s made  e r r o r s were SI e q u a t i o n s .  These e q u a t i o n s were t w i c e as d i f f i c u l t  two low a c h i e v e r s as were any o t h e r l e v e l o f s t r u c t u r a l format.  for'the  F i v e of  the 11 e q u a t i o n s on which h i g h a c h i e v e r s made e r r o r s were e q u a t i o n s o f n u m e r i c a l magnitude f o u r (N4:  100<In|<1000).  Looking a t t h e s t r u c t u r a l  f o c u s o f t h e s e e l e v e n e q u a t i o n s , i t seemed t h a t e q u a t i o n s where t h e v a r i a b l e was t o t h e r i g h t o f t h e c o n s t a n t s difficulty  (7 o f t h e 11 equations)  (S2 & S4) caused t h e most  f o r high achievers.  These d i f f e r e n c e s  i n t h e p a t t e r n s o f e r r o r s seemed t o i n d i c a t e t h e need f o r i n c l u s i o n of t h e 4-by-4 m a g n i t u d e - b y - s t r u c t u r e  g r i d a t each e q u a t i o n l e v e l .  Thus, t h e f i n a l  i n s t r u m e n t c o n t a i n e d 48 e q u a t i o n s . S u b j e c t s showed d i f f e r e n c e s i n t h e n a t u r e and frequency o f t h e i r c o n c e p t u a l , p r o c e d u r a l , and r e s u l t a n t e r r o r s . u n d e r s t a n d i n g o f concepts and v o c a b u l a r y  They were r a t e d f o r t h e i r  (See Table 3, C o n c e p t u a l ) .  A  Table Phase Achievers'  Error  Types  4:  3  Comparison  Error  of  Frequencies  High on  and  the  Achievement High  Low Instrument  Levels Low  Conceptual Variable  M*  Expressions Equality Equations Coefficients Like  Terms  Inverses Identities  P  M/P  M/P  M  M  M/P  R  R  R  P/R  R  M/P  P  M  P/R  Procedural Zero  Annexation  Identity Like  Confusion  Terms  Partial Order  Distributive  of  Operations  2  0  0  1  0  6  0  0  0  0  +  Property  of  =  0  9  x  Property  of  =  4  4  0  0  Add/Subtract  1  2  Multiply/Divide  3  5  Basic  1  5  Coefficient  Resultant Computational  facts  Faulty Wrong  algorithm operation  2  2  0  2  Sign Add/Subtract  2  2  Multiply/Divide  3  7  Rule  3  3  0  1  Mechanical/percept  0  4  Random  0  0  Incomplete  0  1  F i n a l Answer  14  29  based  Distributive  Other  *  M=Mastery P=Partial R=Remedial  r a t i n g o f "mastery"  (M) meant t h a t both s u b j e c t s  c o u l d f u l l y e x p l a i n the  item, i n c l u d i n g an a b i l i t y t o demonstrate t h e i r knowledge. "mastery/partial" that the other "partial"  (M/P) meant t h a t one s u b j e c t  had o n l y  partially  demonstrated t h e g i v e n concept represented  had  (e.g. they may have known t h a t a l e t t e r but had f o r g o t t e n t h e word  A r a t i n g o f " p a r t i a l / r e m e d i a l " (P/R) meant t h a t one s u b j e c t  p a r t i a l l y mastered t h e concept and t h a t t h e o t h e r  assistance  f o r the given  both s u b j e c t s  A r a t i n g of  only p a r t l y explained or  any number as a p l a c e h o l d e r ,  "variable").  had mastered t h e concept and  mastered t h e concept.  (P) meant t h a t both s u b j e c t s  A r a t i n g of  concept.  required  A r a t i n g of "remedial"  remedial  (R) meant t h a t  d i d not have any b a s i c u n d e r s t a n d i n g o f t h e g i v e n  required remedial  concept and  assistance.  Only two concepts were below a p a r t i a l mastery l e v e l f o r t h e two high achievers, achievers  while there  were s i x concepts below p a r t i a l mastery f o r t h e low  (See T a b l e 3, C o n c e p t u a l ) .  type o f p r o c e d u r a l  error.  The boy had one i n s t a n c e  e r r o r , and t h e g i r l had f o u r i n s t a n c e s property  of equality.  achievers  Each o f the h i g h  o f a zero  d i v i d e d among t h e two s u b j e c t s ,  (See T a b l e 3, P r o c e d u r a l ) .  errors,  The h i g h a c h i e v e r s  and t h e i r e r r o r s were almost evenly  sign i n the r e s u l t a n t category.  annexation  e r r o r s made by t h e low  the boy was t h e one t o make an i d e n t i t y element c o n f u s i o n one  made one  o f e r r o r s with t h e m u l t i p l i c a t i o n  However, t h e p r o c e d u r a l  were almost e v e n l y  achievers  except  that  between zero and  made no r e s u l t a n t - o t h e r  d i v i d e d between computation and  F o r low a c h i e v e r s ,  computation and s i g n  e r r o r s seemed t o i n v o l v e m a i n l y m u l t i p l i c a t i o n and d i v i s i o n .  They a l s o  made s e v e r a l c o p y i n g mistakes i n v o l v i n g s i g n s  (See Table 3 under  mechanical/perceptual). In g e n e r a l , errors  high achievers  made almost s i x times as many r e s u l t a n t  ( 2 9 ) as p r o c e d u r a l e r r o r s  made o n l y  (5).  Low a c h i e v e r s ,  on t h e o t h e r hand,  s l i g h t l y over t h r e e times as many r e s u l t a n t e r r o r s  procedural errors  ( 2 0 ) . These d i f f e r e n c e s  ( 6 3 ) as  i n both t h e frequency, as w e l l  as t h e nature', o f t h e e r r o r s t h a t were made by s t u d e n t s o f d i f f e r e n t achievement l e v e l s may have i n s t r u c t i o n a l i m p l i c a t i o n s  and may  pinpoint  some o f t h e common e r r o r s found among s t u d e n t s . There were i n d i c a t i o n s t h a t t h e d i a g n o s t i c systematic errors within Preliminary  subjects  instrument d i d i d e n t i f y  and common e r r o r s a c r o s s  analyses i n d i c a t e d that  subjects.  the instrument had p o t e n t i a l f o r  diagnosing students' errors i n algebra. Phase  Instrument  5:  Testing  In t h e f i f t h phase o f t h e study, t h e e r r o r s made by students were a n a l y z e d and t h e r e l a t i o n s h i p s among these e r r o r s were i n v e s t i g a t e d . analyses involved algebra  comparison between a s t u d e n t ' s achievement l e v e l i n  and t h e e r r o r p a t t e r n s  results.  These  They a l s o i n v o l v e d  found among concepts, p r o c e d u r e s , and  t h e comparison of e r r o r s  l e v e l s o f e q u a t i o n complexity,  found a t d i f f e r e n t  s t r u c t u r a l format, and n u m e r i c a l magnitude.  Subjects  The (n = 1 4 3 ,  e n t i r e academic Grade 8 , Grade 9 , and Grade 1 0 p o p u l a t i o n s 1 4 1 , 8 8 , respectively)  i n the junior high school  t h i r d and f o u r t h phases were a d m i n i s t e r e d t h e Lankton  used i n the  First-Year  Algebra  69  test.  F o r each grade l e v e l ,  s u b j e c t s who  s c o r e d one  or more  standard  d e v i a t i o n s above the mean were c l a s s i f i e d as high a c h i e v e r s , those one  standard  s c o r e d one  d e v i a t i o n of the mean as average a c h i e v e r s , and those  or more s t a n d a r d  d e v i a t i o n s below the mean as low  Subjects  were a s s i g n e d t o c e l l s based on gender  (8/9/10)  and achievement l e v e l  Two  grade  Any  eighteen  i n c l u d e d i n the s e l e c t i o n process. IQ,  P u p i l Record C a r d s .  Table  4 presents  T h i s i n f o r m a t i o n was  Within  obtained  the t a b l e , the f i r s t  from  study  the  and mean mathematics marks f o r each s u b j e c t  i n the f i n a l i n t e r v i e w s .  cells  s u b j e c t s r e t a i n e d i n the  had been i n v o l v e d i n the t h i r d or f o u r t h phases of the  demographic data,  level  (high/average/low).  f o r i n c l u s i o n i n the d i a g n o s t i c i n t e r v i e w s .  were not  who  achievers.  (male/female),  s u b j e c t s were randomly s e l e c t e d from each of these  s c h o o l who  within  included  individual  f o u r s u b j e c t s at each  grade l e v e l were h i g h a c h i e v e r s , the next f o u r were average a c h i e v e r s , the  last  f o u r were low  achievers.  There were s i x students  whose IQs  seemed c o n t r a - i n d i c a t e d by  achievement l e v e l placement on the Lankton due  and  4).  (See Table  their  T h i s may  have been  i n p a r t t o the f a c t t h a t the o n l y IQ s c o r e s a v a i l a b l e f o r students  Otis-Lennon  (Form M ), which i s a group a d m i n i s t e r e d  IQ t e s t g i v e n t o  was  subjects  upon e n t r y at the Grade 8 l e v e l i n the s e l e c t e d j u n i o r h i g h s c h o o l , and such t h e i r  IQ s c o r e s may  not have been r e p r e s e n t a t i v e of t h e i r  capabilities  i n the s c h o o l s e t t i n g .  At the Grade 10 l e v e l ,  a c h i e v e r had  a comparatively  (131)  (90).  h i g h IQ  and one  However, each made 2 9 e r r o r s on the e q u a t i o n  had  one  the  as  true male average  a comparatively  low  IQ  p o r t i o n of the d i a g n o s t i c  Table  Phase  Grade  5:  4  Selected Data  for  Subjects  Gender  Acre  Mark  Lankton  10  F  15  124  B+  29  3  10  F  15  109  B  27  3  10  M  15  125  B-  26  4  10  M  15  118  B  25  3  10  CSMS  10  F  16  120  B-  21  3  -10  F  16  106  A-  19  2  10  M  15  131  A-  21  4  10  M  16  90  C+  14  2  10  • F  16  103  C-  12  3  10  F  16  106  C  11  2  10  M  16  100  B  12  3  10  M  16  102  C  10  2  9  F  15  113  B  28  3  9  F  16  117  B-  27  4  9  M  14  119  B-  28  3  9  M  15  118  B+  27  3  9  F  14  115  C+  20  3  9  F  15  105  C  20  2  •  9  M  15  105  B  20  3  9  M  14  117  C+  20  2  9  F  14  110  B  11  2  9  F  14  104  C+  11  1  9  M  15  98  C  13  2  9  M  17  97  c  11  3  8  F  14  105  B  22  1  8  F  14  108  C+  20  2  .  8  M  15  97  . C  20  2  8  M  14  98  C  19  2  8  F  13  93  C+  13  1  8  F  14  104  C  11  1  8  M  13  106  B-  17  2  8  M  14  102  C  10  2  8  F  14  105  B-  5  1  8  F  14  125  C  5  1  8  M  14  99  C-  7  2  8  M  14  96  C  4  1  At the  each next  subjects  grade four are  level,  the  subjects low  first  are  achievers  four  average as  subjects achievers,  defined  by  the  are  high  and the Lankton.  achievers, last  four  N=36  i n s t r u m e n t , which was near t h e average o f 30 e r r o r s f o r t h a t grade and achievement  level.  At t h e Grade  r e l a t i v e l y low IQ s c o r e s  8 l e v e l , t h e two male h i g h a c h i e v e r s had  (97, 98) and low average marks, y e t t h e f i r s t made  65 e r r o r s and t h e second made 57 e r r o r s on t h e d i a g n o s t i c instrument, which was  near t h e average o f 60 e r r o r s f o r t h a t grade and achievement  S i m i l a r l y , one o f t h e Grade IQ  level.  8 female average a c h i e v e r s had a r e l a t i v e l y low  (93), and a somewhat h i g h e r mark than e x p e c t e d (C+), but made 71 e r r o r s  on t h e e q u a t i o n p o r t i o n o f t h e d i a g n o s t i c instrument compared t o an average of 70 e r r o r s f o r t h a t grade and achievement low a c h i e v e r a t t h e Grade  level.  C o n v e r s e l y , one female  8 l e v e l had a r e l a t i v e l y h i g h IQ (125).  However,  t h i s s u b j e c t made 95 e r r o r s on the e q u a t i o n p o r t i o n o f t h e d i a g n o s t i c i n s t r u m e n t , which was p r e c i s e l y t h e average of 95 e r r o r s f o r h e r grade and achievement  l e v e l . '.  .  .  These d i s c r e p a n c i e s between IQ, achievement performance on d i a g n o s t i c c h e c k l i s t relationships.  F o r example,  l e v e l , marks, and  r e s u l t e d i n some o t h e r c u r i o u s  t h e average IQ f o r t h e low Grade  8s (106) was  h i g h e r than t h e average IQ f o r e i t h e r t h e h i g h o r average Grade  8 groups  (102, 101). The marks f o r a l l groups seemed low, and a t t h e Grade  8 level  were b a r e l y d i s t i n g u i s h a b l e between achievement  levels.  The CSMS r e s u l t s  d i d seem t o d i s t i n g u s h between both achievement  l e v e l and grade l e v e l with  the e x c e p t i o n o f t h e h i g h a c h i e v i n g 9s and 10s which were i n d i s t i n g u i s h able.  These d i s c r e p a n c i e s w i t h i n t h e sample may have had some i n f l u e n c e on  the r e s u l t s i n t h e f i n a l phase o f t h e study. A f t e r s e l e c t i o n , p o t e n t i a l s u b j e c t s met as a group d u r i n g t h e l u n c h  hour f o l l o w i n g a d m i n i s t r a t i o n o f t h e Lankton. the nature  o f student  p a r t i c i p a t i o n were c a r e f u l l y e x p l a i n e d .  were g i v e n a p a r e n t a l consent form  (See Appendix D, P a r t  to return t h i s p r i o r t o t h e i r interview. participate,  Subjects  1) and were asked  Three s u b j e c t s were unable t o  two because of a l a c k of p a r e n t a l consent, and one because of  long-term i l l n e s s . as  The purpose o f t h e study and  New s u b j e c t s from t h e same c e l l s were randomly s e l e c t e d  replacements.  Procedures A p p r o x i m a t e l y two weeks p r i o r t o t h e f i r s t  i n t e r v i e w i n e a r l y May,  s u b j e c t s were g i v e n a group a d m i n i s t r a t i o n of t h e Concepts Mathematics  and Science  l e v e l of understanding administered  (CSMS) t e s t .  in  T h i s t e s t was d e s i g n e d  o f t h e concept o f v a r i a b l e .  Secondary t o measure  Any absentees were  t h e CSMS i n t h e next two days, so t h a t a l l s u b j e c t s had  completed both t h e Lankton and t h e CSMS p r i o r t o t h e . d i a g n o s t i c i n t e r v i e w s . A Latin-Square  Design was used t o s e l e c t t h e date and time o f  i n t e r v i e w s t o c o n t r o l f o r gender, grade l e v e l ,  and achievement  However, some i n t e r v i e w times had t o be r e - s c h e d u l e d  level.  because o f  i n s t r u c t i o n a l demands i n t h e classroom,  e x t r a - c u r r i c u l a r sports  conflicts,  occurred  storage  and absenteeism.  room.  Interviews  A l l i n t e r v i e w s were  At t h e b e g i n n i n g  i n a small, audio-visual  video-taped.  o f each i n t e r v i e w , t h e purpose o f t h e i n t e r v i e w and  v i d e o - t a p i n g was r e - e x p l a i n e d t o each student form was reviewed.  activity  and t h e student  questionnaire  At t h e end o f t h e i n t e r v i e w s s u b j e c t s were g i v e n  feedback r e g a r d i n g t h e nature  o f t h e i r e r r o r s on t h e d i a g n o s t i c  instrument.  Questions  r e g a r d i n g t h e study were answered.  A f t e r s c h o o l hours,  r e m e d i a t i o n was p r o v i d e d upon r e q u e s t . Presentation  The  of  Results  r e s u l t s o f t h e i n t e r v i e w s a r e p r e s e n t e d i n terms o f t h e r e s e a r c h  q u e s t i o n s asked.  The e r r o r s made by s t u d e n t s o f d i f f e r e n t  achievement  l e v e l s were a n a l y z e d and s y s t e m a t i c and common e r r o r s were i d e n t i f i e d .  The  r e l a t i o n s h i p s o f e r r o r s t o demographic data, IQ, and mathematical achievement v a r i a b l e s were e x p l o r e d . made a t d i f f e r e n t  The n a t u r e and frequency o f e r r o r s  l e v e l s o f e q u a t i o n complexity,  s t r u c t u r a l format, and  n u m e r i c a l magnitude were compared. Achievement  Krutetskii  Levels  and  Errors  (1976) noted d i f f e r e n c e s among s t u d e n t s i n t h e i r a b i l i t y t o  s o l v e a l g e b r a i c problems.  "Good problem  s o l v e r s " were a b l e t o e x p l a i n what  t h e y were d o i n g i n terms o f both concepts and p r o c e d u r e s .  "Poor  problem  s o l v e r s " seemed more p r o c e d u r e - o r i e n t e d and were unable t o v e r b a l i z e reasons f o r what they were d o i n g o r t h i n k i n g . r e f e r r e d t o word problems,  While K r u t e t s k i i ' s work  s i m i l i a r f i n d i n g s c o u l d occur i n the s o l v i n g of  l i n e a r e q u a t i o n s i n one v a r i a b l e .  The f o l l o w i n g r e s e a r c h q u e s t i o n was  asked: 1.  Does t h e Diagnostic Checklist for Algebra a s s i s t i n the i d e n t i f i c a t i o n o f t h e nature and frequency o f the e r r o r s made by a p a r t i c u l a r student?  In k e e p i n g w i t h t h e i n v e s t i g a t i o n s done by K r u t e t s k i i s t u d e n t s were grouped  (1976), t h e  a c c o r d i n g t o achievement l e v e l t o f a c i l i t a t e  i n v e s t i g a t i o n o f t h e n a t u r e and frequency o f the e r r o r s made.  An average  l e v e l of u n d e r s t a n d i n g  f o r each concept  achievement l e v e l at each grade l e v e l  portion within  definitions).  E r r o r frequency r e s u l t s from the  each achievement l e v e l at each grade l e v e l .  This  level,  equation  by May  of  diagnostic  content covered p r i o r t o Christmas  but t h i s content had not been attempted  category  separation  grade l e v e l and achievement l e v e l seemed n e c e s s a r y as the investigated  each  (Refer t o page 67 f o r the r a t i n g of  of the c h e c k l i s t are p r e s e n t e d i n Table 6 f o r each e r r o r  instrument  the  of the c h e c k l i s t are p r e s e n t e d i n T a b l e 5 w i t h i n  conceptual portion  understanding  and v o c a b u l a r y i t e m from  at the Grade 9  i n the Grade 8  classes. As i n the f o u r t h phase of the study, the e r r o r s low a c h i e v e r s d i f f e r e d i n q u a n t i t a t i v e t h r e e e r r o r types Table  6)].  T a b l e 5) ,  [conceptual  r e m e d i a l was  counted  p r o c e d u r a l , and r e s u l t a n t  f o r conceptual understandings  as 2., p a r t i a l as 1 ,  a c h i e v e r s had a 28% e r r o r r a t e , average and low a c h i e v e r s had a 47% e r r o r r a t e .  and mastery as Q_.  a c h i e v e r s had a 32% e r r o r The  reasons  f o r these  i n e r r o r r a t e were i n v e s t i g a t e d and r e s u l t s i n d i c a t e d t h a t of both the concepts  and  and q u a l i t a t i v e a s p e c t s a c r o s s a l l  (See Table 5 ) ,  To compute the e r r o r r a t e s  among h i g h , average  and v o c a b u l a r y s e c t i o n s  contributed  (See (See High  rate,  differences  c e r t a i n aspects  t o the  differences. There were f o u r d i f f e r e n t a s p e c t s of the concepts •diagnostic  c h e c k l i s t which r e l a t e d t o s t u d e n t s ' achievement l e v e l s .  were the a b i l i t y of s t u d e n t s t o : 1) meaning, 2 )  s e c t i o n of the  explain  concretely  These  i d e n t i f y the v a r i a b l e and e x p l a i n i t s  the meaning of d i v i s i o n i n e x p r e s s i o n s  Table 5 Phase 5: Comparison o f High, Average, and Low A c h i e v e r s ' C o n c e p t u a l Understandings on the Instrument  Understandings*  Achievement L e v e l s * * Average  High  Conceptual Variable Expressions EqualityEquation Coefficient L i k e Terms Inverses I d e n t i t y Elements  .8  9  P P P R R R P P  M P M P P/R M/P M/P M/P  10  M M/P M P P/R R M M  Low  8  9  10  8  9  10  P/R P/R P/R R R R R P/R  M/P P M/P P/R P R P M/P  M/P P M P P P M M  R P/R P/R R R R P/R P/R  M/P P/R P R P R M/P P  P/l R P R R R P P  Grand N = 3 6 The c o n c e p t u a l p o r t i o n of the d i a g n o s t i c c h e c k l i s t i s p r e s e n t e d i n terms of u n d e r s t a n d i n g of the concepts and v o c a b u l a r y a s s o c i a t e d w i t h s o l v i n g l i n e a r e q u a t i o n s . (Refer t o page 67 f o r d e f i n i t i o n s . ) M= Mastery P= P a r t i a l R= Remedial As ranked on the Lankton  First-Year  Algebra  Test  76  Table 6 Phase 5: Comparison o f High, Average, and Low A c h i e v e r s ' E r r o r F r e q u e n c i e s on t h e Instrument  E r r o r Types 8  E r r o r F r e q u e n c i e s f o r Achievement L e v e l s * High Average Low 9 10 8 9 10 8 9 10  P rocedural  Zero A n n e x a t i o n 2 I d e n t i t y Confusion 0 L i k e Terms 9 Partial Distributive 0 Order o f O p e r a t i o n s 3 • 7 + Property, o f = x Property of = 0 Coefficient 0  3 2 9 1 1 13 2 2  2 1 15 1 1 4 4 1  4 6 13 0 0 12 0 1  4 4 8 0 1 12 6 2  4 3 0 1 0 2 5 5  2 0 4 1 8 15 1 0 1 0 5 . 14 4 0 1 6  20 13 4 2 6 0 5 9 7 ' 11  13 3 8 6 8  24 9 7 2 22  27 10 6 22 19  10 11 6 7 11  15 18 4 6 14  28 8 13 8 ' 21  31 7 8 7 23  3 4 12 0 0 16 0 1  Resultant Computational  Add/Subtract Multiply/Divide Basic Facts Faulty Algorithm Wrong O p e r a t i o n S ign  Add/Subtract Multiply/Divide R u l e Based Distributive  33 4 21 2  10 4 6 0  12 2 3 0  22 7 20 0  38 2 15 0  17 5 5 0  13 13 12 1  35 11 11 0  31 4 14 0  9 3 104 135  20 9 5 23  10 6 3 23  5 3 101 135  25 7 37 59  9 10 0 32  4 1 119 156  15 11 18 95  27 9 23 75  Other  Mechanical/Percept Random Incomplete Final  Answer  Grand N = 36 *  As r a n k e d on t h e Lankton  First-Year  Algebra  Test  c o n t a i n i n g b l a n k s and v a r i a b l e s , multiplication  i n expressions containing variables,  of e q u a t i o n s with and without an e q u a t i o n . almost  3) e x p l a i n c o n c r e t e l y t h e meaning o f and 4) c r e a t e meaning  c o n t e x t as w e l l as g i v e n t h e meaning-create  H i g h - a c h i e v i n g s t u d e n t s were a b l e t o e x p l a i n these t h i n g s  without  e x c e p t i o n and r e g a r d l e s s o f grade  level;  w h i l e t h e o p p o s i t e was t r u e f o r l o w - a c h i e v i n g s t u d e n t s .  F o r example, when  e x p l a i n i n g t h e meaning o f d i v i s i o n i n e x p r e s s i o n s c o n t a i n i n g v a r i a b l e s (x/4), one h i g h - a c h i e v i n g Grade 8 e x p l a i n e d : "Well, x c o u l d be 12, then 12 d i v i d e d by 4 means 3 groups o f 4." of 4 pennies  D u r i n g t h i s e x p l a n a t i o n she put groups  t o g e t h e r and noted 3 groups.  F o r t h e same q u e s t i o n ,  a l o w - a c h i e v i n g Grade 8 s t a t e d : " I can make up any number f o r x. 3 d i v i d e d by 4."  D u r i n g t h i s e x p l a n a t i o n she put out 3 pennies  added one more penny t o these 3. the Grade 10 l e v e l , such mastery.  Some average  I t means and then  a c h i e v e r s showed mastery at  w h i l e those o f lower grade l e v e l s d i d not t e n d t o show  When asked the " i d e n t i f y v a r i a b l e " q u e s t i o n : "What do you  c a l l t h e x. i n 5+x? What does i t s t a n d f o r ? " , an a v e r a g e - a c h i e v i n g Grade 10 correctly stated: "It's number."  a l e t t e r - umm v a r i a b l e .  I t c o u l d mean any  However, f o r t h e same q u e s t i o n , an a v e r a g e - a c h i e v i n g Grade 8  commented: "The ' p i ' ?  An answer?"  Some p o r t i o n s of the v o c a b u l a r y s e c t i o n of the d i a g n o s t i c instrument a l s o produced concepts  major d i f f e r e n c e s among l e v e l s o f achievement.  s e c t i o n , t h e r e were f o u r v o c a b u l a r y ' i t e m s  involved.  As with the These were  the v o c a b u l a r y a s s o c i a t e d w i t h : 1) a d d i t i v e i n v e r s e s , 2) m u l t i p l i c a t i v e inverses,  3) t h e a d d i t i v e i d e n t i t y element, and 4) t h e m u l t i p l i c a t i v e  78  i d e n t i t y element. One  o f t h e i n t e r e s t i n g f i n d i n g s f o r t h e Grade 10s was t h e f a c t t h a t the  f o u r average a c h i e v i n g students  had p a r t i a l mastery o f naming and  c o l l e c t i n g l i k e terms, w h i l e the h i g h - a c h i e v i n g s t u d e n t s 5).  d i d not (See Table  However, i n l a t e r work i n e q u a t i o n - s o l v i n g , t h e h i g h - a c h i e v i n g  students  were a b l e t o c a r r y out a l g e b r a i c procedures  combining l i k e terras,  and two o f them even commented t h a t they had no i d e a why they had e a r l i e r m i s s e d naming and g r o u p i n g spontaneously procedures  " l i k e terms."  The h i g h a c h i e v e r s used t h e word  i n t h e i r e q u a t i o n - s o l v i n g procedures,  and i t seemed t o be the  which t r i g g e r e d t h e meaning o f l i k e terms f o r them.  The h i g h  a c h i e v e r s s t i l l made numerous e r r o r s u s i n g l i k e terms i n s o l v i n g equations (33 compared t o average a c h i e v e r s 21), but 19 o f these were s e l f - c o r r e c t e d during solution,  while  f o r average a c h i e v e r s these mistakes with l i k e terms  went by u n n o t i c e d . P r o c e d u r a l e r r o r s i n v o l v i n g t h e a d d i t i o n p r o p e r t y o f e q u a l i t y were much h i g h e r f o r t h e Grade 9 h i g h a c h i e v e r s and average a c h i e v e r s than corresponding  Grade 10s (Refer t o Table  f o r the  6 ) . However, t h i s was not t h e case  f o r low a c h i e v e r s , who made a l a r g e number o f e r r o r s a t both t h e Grade 9 and Grade 10 l e v e l s .  An example o f a t y p i c a l e r r o r made by Grade 9 low  a c h i e v e r s was: 4x  = 7x - 36  7x - 4x = 7x - 7x - 36 3x  = -36  x  = -12  79  In the second l i n e , the r i g h t s i d e o f the e q u a t i o n was However, when he s a i d "You first  on the l e f t  about  why  s u b t r a c t Ix  done c o r r e c t l y .  from both s i d e s , " he wrote the  s i d e and then the "7x " i n f r o n t of i t .  "-"  When q u e s t i o n e d  he wrote what he d i d , he i n d i c a t e d a g a i n t h a t you have t o  s u b t r a c t the same t h i n g from both s i d e s .  He seemed o b l i v i o u s t o the  of c o m m u t a t i v i t y of the o p e r a t i o n of s u b t r a c t i o n and was  unconcerned  lack about  the c o r r e c t n e s s o f the answer. Some f i n d i n g s f o r the p r o c e d u r a l and r e s u l t a n t e r r o r s seemed o p p o s i t e to expectations.  For example, at the Grade 8 l e v e l the low group had the  fewest e r r o r s f o r zero annexation, a d d i t i o n / s u b t r a c t i o n , b a s i c f a c t s , based, m e c h a n i c a l / p e r c e p t u a l and random.  rule-  However, a n a l y s i s o f the  p r o t o c o l s o f t h e s e low a c h i e v e r s showed t h a t they tended not t o depend on memory f o r any work, but r a t h e r wrote out e x p l i c i t l y  any work d e a l i n g with  zero a n n e x a t i o n or b a s i c f a c t s , even t o the p o i n t o f making a column of five  6s t o add them r a t h e r than r e c a l l i n g the b a s i c f a c t ,  T h i s tendency  of l o w - a c h i e v i n g Grade 8s t o w r i t e out t h e i r  f i v e times s i x . computational  work, d i d not seem t o c o n t i n u e at the Grade 9 and 10 l e v e l s f o r low a c h i e v e r s , and a c t u a l l y became a way achievers.  t o d i s t i n g u i s h between low and h i g h  I t i s the h i g h a c h i e v e r s at the Grades 9 and 10 l e v e l s  who  tended t o w r i t e out t h e i r c o m p u t a t i o n a l and p r o c e d u r a l work and the a c h i e v e r s who  d i d not.  I t s h o u l d a l s o be noted t h a t some e r r o r s d i d not  o c c u r f o r the l o w - a c h i e v i n g Grade 8s because questions at a l l ,  low  they d i d not attempt  w h i l e o t h e r Grade 8s attempted  and then f a i l e d t o complete  the s o l u t i o n .  some  them, made some e r r o r s ,  S i m i l a r examples are found f o r  the  low Grade 9s f o r t h e same reason. As might be expected, f i n a l answers b e i n g e i t h e r c o r r e c t o r i n c o r r e c t  p r o v i d e d c l e a r d i s t i n c t i o n s among h i g h , average, and low a c h i e v e r s of e r r o r s made i n s o l v i n g e q u a t i o n s . 10  l e v e l s , c o r r e c t l y answered 3 0 % ,  High a c h i e v e r s  respectively. answered  Low a c h i e v e r s  a t t h e Grade 8, 9, and  88%, and 88%, r e s p e c t i v e l y .  Grade 8, 9, and 10 s t u d e n t s c o r r e c t l y answered 3 0 % ,  i n terms  Average  69%, and 83%,  a t t h e Grade 8, 9, and 10 l e v e l s c o r r e c t l y  19%, 51%, and 61%, r e s p e c t i v e l y .  Frequency o f r e s u l t a n t  s i g n e r r o r s f o r d i v i s i o n and m u l t i p l i c a t i o n  seemed t o d i s t i n g u i s h among achievement l e v e l s .  When a  high-achieving  s t u d e n t made an e r r o r i t was o f t e n a c o n f u s i o n with a c o r r e c t another o p e r a t i o n  (e.g.  rule f o r  two n e g a t i v e s make a n e g a t i v e when adding,  a p p l i e d t o m u l t i p l i c a t i o n or d i v i s i o n ) .  being  However, when low a c h i e v e r s  an e r r o r i t was u s u a l l y based on a r u l e o f t h e i r own c r e a t i o n  (e.g.  made when  t h e r e i s a n e g a t i v e s i g n , the answer i s n e g a t i v e ) and bore no resemblance to p r e v i o u s l y  learned  sign  rules.  Incomplete work seemed t o d i s t i n g u i s h achievement l e v e l s , a l t h o u g h t h i s was most n o t i c e a b l e achievers not  when comparing h i g h and low a c h i e v e r s .  would s i m p l y read the e q u a t i o n and then i n d i c a t e t h a t  u n d e r s t a n d t h e q u e s t i o n o r d i d not know how t o p r o c e e d .  achievers,  Many low  even a t t h e Grade 8 l e v e l ,  procedure, i n c l u d i n g systematic t r i a l  they d i d  High  were more w i l l i n g t o t r y some and e r r o r , t o get an answer.  d i d not,seem s a t i s f i e d with i n d i c a t i n g t h a t they c o u l d  They  not attempt an item;  even though, i n t h e case of t h e Grade 8s, they had never seen an e q u a t i o n  81  l i k e t h e two-step and m u l t i - s t e p e q u a t i o n s p r e s e n t e d . persevere  i n t h e s o l u t i o n o f an e q u a t i o n was d e f i n i t e l y one o f t h e major  d i f f e r e n c e s among achievement l e v e l s and supported Krutetskii The  This willingness to  t h e f i n d i n g s of  (1976).  e r r o r f r e q u e n c i e s on t h e instrument  i n d i v i d u a l d i a g n o s t i c concerns,  provided insight  as w e l l as u s e f u l i n f o r m a t i o n r e g a r d i n g t h e  n a t u r e o f t h e e r r o r s made a t d i f f e r e n t achievement l e v e l s . d i s c r e p a n c i e s i n t h e nature  into,  These  of t h e e r r o r s made by s t u d e n t s of d i f f e r e n t  achievement l e v e l s f o c u s e d on p o t e n t i a l areas o f c u r r i c u l a r concern i n algebra.  These d i s c r e p a n c i e s emphasized t h e importance o f t h e  understanding  o f b a s i c a l g e b r a i c concepts  and t h e f a c t t h a t f o r low  a c h i e v e r s l a c k of mastery of these concepts  seems t o p l a y a major r o l e i n  t h e i r c o n t i n u e d low achievement i n a l g e b r a . Identification  of  Systematic  and  Common  Errors  As mentioned p r e v i o u s l y i n t h e review o f t h e l i t e r a t u r e , (1982)  i d e n t i f i e d both s y s t e m a t i c and common a l g e b r a i c e r r o r s .  Anderson A  s y s t e m a t i c e r r o r o c c u r r e d when a s u b j e c t made a p a r t i c u l a r type o f e r r o r i n 50% o r more o f t h e a v a i l a b l e o p p o r t u n i t i e s .  A common e r r o r o c c u r r e d  across  s u b j e c t s when 5% o r more o f h e r 2 0 0 s u b j e c t s committed t h e same s y s t e m a t i c She found t h r e e e r r o r s which o c c u r r e d i n more than 20% o f her  error. sample: abx. be The  ax + (-bx ) = (b-a)x  2  where |b| > |a|, ax + x = ax , and ax + bx =  I t was o f i n t e r e s t t o t h e p r e s e n t  study t o see i f these e r r o r s would  r e p l i c a t e d and i f o t h e r s y s t e m a t i c o r common e r r o r s would be p r e s e n t . f o l l o w i n g q u e s t i o n addressed  this  concern:  2. Does t h e Diagnostic Checklist for Algebra a s s i s t i n the i d e n t i f i c a t i o n of sytematic e r r o r s f o r p a r t i c u l a r students and common e r r o r s among groups o f s t u d e n t s ? Nine o f t h e 36 s u b j e c t s used i n t h e f i n a l phase o f t h e study e x h i b i t e d s y s t e m a t i c e r r o r s on t h e Diagnostic n i n e were i n Grades 9 and 10.  Checklist  for Algebra.  There were f o u r g e n e r a l types o f s y s t e m a t i c  e r r o r s d i s p l a y e d by s t u d e n t s . Two were p r o c e d u r a l property of e q u a l i t y ) , subtraction,  sign:  Seven o f these  and two were r e s u l t a n t  ( l i k e terms,  (computational:  addition a d d i t i o n and  a d d i t i o n and s u b t r a c t i o n ) .  Three s t u d e n t s e x h i b i t e d s y s t e m a t i c e r r o r s with l i k e terms.  One  s u b j e c t c o n s i s t e n t l y made e r r o r s o f t h e type ax + b = (a+b)x, w h i l e  another  made e r r o r s o f t h e type ax + bx = abx, and t h e t h i r d made e r r o r s of t h e type ax + (-bx ) = (b-a)x, where  |b| > |a|. While  r e p r e s e n t e d a r e p l i c a t i o n o f two o f Anderson's  t h e l a t t e r two e r r o r s  (1982) s y s t e m a t i c e r r o r s ,  t h e y o n l y o c c u r r e d s y s t e m a t i c a l l y f o r t h e s e two s t u d e n t s . Four s t u d e n t s e x h i b i t e d s y s t e m a t i c e r r o r s with t h e a d d i t i o n p r o p e r t y o f equality.  One student c o n s i s t e n t l y added t h e same number t o both  r a t h e r than t h e o p p o s i t e . the student added  F o r example, i n t h e e q u a t i o n  sides,  17x + (-100) = -15,  (-100) t o both s i d e s o f t h e e q u a t i o n and o b t a i n e d the  r e s u l t t h a t 17x = -115, f a i l i n g t o n o t i c e t h e e r r o r t h a t were not o p p o s i t e s .  (-100) and (-100)  One student t r i e d t o a p p l y t h e a d d i t i o n p r o p e r t y o f  e q u a l i t y t o a l l s i t u a t i o n s o f combining  instances of the v a r i a b l e ,  when t h e y o c c u r r e d on t h e same s i d e o f t h e e q u a l i t y symbol. f o r t h e e q u a t i o n 8x + 9x = -17, t h i s student added  even  To i l l u s t r a t e ,  (-9x ) t o both s i d e s o f  the e q u a t i o n , c o r r e c t l y o b t a i n i n g 8x = (-9x ) + -17, but then became  confused,  and d i d not how  t o f i n i s h s o l v i n g the e q u a t i o n .  s t u d e n t s c o n s i s t e n t l y added the o p p o s i t e number t o one  The  other  two  s i d e of the  e q u a t i o n , but the same number t o the o t h e r s i d e of the e q u a t i o n .  For  example, i n the e q u a t i o n 17x +  t o the  left  side>  but  (-100) =  (-15), they would add  (-100) t o the r i g h t s i d e .  [17x =(-115)] as the f i r s t  100  T h i s gave them the same e q u a t i o n  student mentioned, yet t h e i r reasons f o r  a r r i v i n g a t t h i s e q u a t i o n were q u i t e d i f f e r e n t . Two One  students e x h i b i t e d  student e x h i b i t e d  computational  consistent  addition  and s u b t r a c t i o n  f a u l t y algorithms i n v o l v i n g  which seemed t o cause these e r r o r s .  The  errors.  regrouping,  o t h e r student c o n s i s t e n t l y chose  the wrong o p e r a t i o n . S i x of the n i n e s t u d e n t s making s y s t e m a t i c e r r o r s d i s p l a y e d sign e r r o r s with a d d i t i o n t h e s e e r r o r s due ( i e . two  and s u b t r a c t i o n .  Three s t u d e n t s seemed t o make  t o r u l e - b a s e d a p p l i c a t i o n of s i g n r u l e s f o r m u l t i p l i c a t i o n  n e g a t i v e s make a p o s i t i v e ) .  mechanical/perceptual  e r r o r s ; two  The  o t h e r t h r e e were due  from c o n s i s t e n t  from an i n a b i l i t y t o " n o t i c e " the n e g a t i v e  l i k e terms  sign errors  addition/  The  frequency  was:  (16%), a d d i t i o n p r o p e r t y of  (8%), and computational  equality  subtraction  errors  (5%) . Instrument  Validity  V a l i d i t y of a d i a g n o s t i c  one  signs.  w i t h which t h e s e common e r r o r s o c c u r r e d i n the sample addition/subtraction  to  c o p y i n g mistakes and  A l l of the s y s t e m a t i c e r r o r s were a l s o common e r r o r s .  (11%),  systematic  instrument has u s u a l l y been determined by  whether o r not i t a c t u a l l y enables i d e n t i f i c a t i o n o f the e r r o r s made by s t u d e n t s and the reasons  f o r making these e r r o r s .  has been g i v e n t o the u s u a l v a l i d i t y concerns o r c r i t e r i o n - r e f e r e n c e d t e s t development.  Very l i t t l e  attention  involved i n either  normative  However, a d i a g n o s t i c  instrument  which i n c o r p o r a t e s the s t r e n g t h s of each of these t e s t s w i t h i n the d i a g n o s t i c framework cannot  h e l p but be a more v a l i d  f o l l o w i n g r e s e a r c h q u e s t i o n was  instrument.  The  posed t o address the i s s u e o f instrument  validity: 3.  Is the Diagnostic  Checklist  for Algebra  valid?  Face and content v a l i d i t y were e s t a b l i s h e d through the methods of textbook a n a l y s i s , e r r o r - c a t e g o r i z a t i o n scheme development, and c o n s t r u c t i o n and s e l e c t i o n as o u t l i n e d i n Phases 1, 2, and 3. v a l i d i t y was  improved  i n Phase 4.  v a l i d i t y were undertaken  Measures o f c o n c u r r e n t and  equation Face  predictive  i n Phase 5 through the c o r r e l a t i o n a l a n a l y s i s of  outcomes on the d i a g n o s t i c instrument compared t o measures of achievement, IQ, and demographic d a t a .  While  mathematical  such "outcomes" on a  d i a g n o s t i c instrument s e r v e no d i a g n o s t i c purpose  (comparisons  " b e t t e r " i n a d i a g n o s t i c s e t t i n g i s absurd), such comparisons  of who  did  provide  measures o f v a l i d i t y which enhance the o v e r a l l c o n f i d e n c e t h a t t e a c h e r s would p l a c e i n the Diagnostic i n the a n a l y s e s . concerns  Checklist  for Algebra,  hence, t h e i r  D e t a i l e d s t a t i s t i c a l analyses r e l a t i n g to  are p r e s e n t e d i n Appendix D, P a r t 3.  However, due  sample s i z e and the f a c t t h a t i t s c o m p o s i t i o n was t h e s e c o r r e l a t i o n s must be viewed  with c a u t i o n .  from one  inclusion  validity t o the  limited  school setting,  85  Through comparisons of the mean p - v a l u e s  of each e q u a t i o n type with t h a t  e q u a t i o n t y p e ' s o r d i n a l r a n k i n g from the textbook  (See Table  1 9 8 9 ) was o b t a i n e d .  evidence  of s t r u c t u r a l v a l i d i t y  p-values  were o b t a i n e d by d e t e r m i n i n g the mean of the p - v a l u e s  equations B.4).  (Conry,  anlaysis  w i t h i n each e q u a t i o n type  (See Appendix B, P a r t  A Spearman's rho was c a l c u l a t e d at 0.90.  coefficient  p r o v i d e s f u r t h e r v a l i d i t y of the  p r e s e n t a t i o n o f the e q u a t i o n s With the e x c e p t i o n  The mean  4,  of the 4  Tables B . 2 -  This c o r r e l a t i o n  s t r u c t u r e and sequence of  w i t h i n the Diagnostic  of age and gender,  Checklist  for  -  IQ, and mathematics'  achievement  variables  from Grade 4 through 2 y e a r s a f t e r the d i a g n o s t i c  CSMS) were s i g n i f i c a n t .  Algebra.  a l l c o r r e l a t i o n s o b t a i n e d between  the d i a g n o s t i c , instrument "outcomes" and the demographic data grade),  7),  (age,  gender,  (mean mathematics' marks interviews,  Lankton,  While some o f the c o r r e l a t i o n s w e r e not h i g h  (refer  t o T a b l e D.5 o f Appendix D) they d i d p r o v i d e a measure of c o n c u r r e n t validity  (Lankton,  (mean mathematics'  CSMS,  IQ, grade l e v e l )  marks).  These c o r r e l a t i o n s p r o v i d e f u r t h e r evidence  the v a l i d i t y o f the Diagnostic Concepts,  Vocabulary,  C e r t a i n concepts reported to a f f e c t  as w e l l as p r e d i c t i v e v a l i d i t y  and  Checklist  for  Algebra.  Errors  and v o c a b u l a r y a s s o c i a t e d  the e q u a t i o n - s o l v i n g p r o c e s s  w i t h a l g e b r a have been t o some degree.  C o n t r a d i c t o r y f i n d i n g s have been r e p o r t e d f o r s t u d e n t s ' v a r i a b l e and i t s  influence  1982;  1978;  Kuchemann,  Rachlin,  1982;  for  u n d e r s t a n d i n g of  on the e q u a t i o n - s o l v i n g p r o c e s s  Wagner, 1977,  1981a;  Rosnick & Clement, 1980).  (Booth & H a r t ,  as opposed t o K i e r a n ,  An u n d e r s t a n d i n g of  1981b;  expressions  Table 7 R e l a t i o n s h i p * Between Text P o s i t i o n and Mean P -Values of E q u a t i o n Types  I  G e n e r a l Format of  Ordinal Position  E q u a t i o n Types  i n Textbooks  .  Mean P -Value**  One-Step  x+a=b*** x-a=b ax=b x/a=b II  Rank Order o f P -Values  .  2 5 6  68 53 0 50  5 6 7 8  3 4 8 7  57 53 36 40  9 10 11 12  10 9 11 12  22 23 15 14  1  7  Two-Step  ax+b=c ax+bx=c ax+b=cx a(x+b)= **** c  III.  1 2 3 4  Multi-Step  ax+b+cx+d=e ax+b=cx+d a(x+b)=cx+d a(x+b)=c(x+d)  *  Spearman's rho = 0.90  **  A l l p - v a l u e s l i s t e d are rounded t o two s i g n i f i c a n t and r e p r e s e n t hundredths.  ***  a,b,c,d,e,x a r e a l l i n t e g e r s , a*0. Based on t h e c o n s t r a i n t s of t h e study these e q u a t i o n s were l i m i t e d t o those c o n t a i n i n g i n t e g r a l r o o t s , as w e l l as i n t e g r a l c o e f f i c i e n t s .  ****  where c i s d i v i s i b l e by a.  digits  than i n d i v i d u a l d i a g n o s t i c p a t t e r n s .  Hence, t h e f o c u s o f p r e s e n t a t i o n w i l l  be s p e c i f i c examples o f t h e r e l a t i o n s h i p between a l g e b r a i c ' c o n c e p t s / v o c a b u l a r y and e r r o r s made i n t h e s o l v i n g o f e q u a t i o n s which might u s e f u l when t e a c h i n g a l g e b r a . a l g e b r a i c concepts  To determine  prove  how s t u d e n t s ' u n d e r s t a n d i n g o f  and v o c a b u l a r y r e l a t e d t o e r r o r s made i n s o l v i n g  e q u a t i o n s , t h e f o l l o w i n g r e s e a r c h q u e s t i o n was asked: 4.  A r e s t u d e n t s ' understandings o f a l g e b r a i c concepts and v o c a b u l a r y r e l a t e d t o t h e kinds o f e r r o r s they make i n the s o l u t i o n o f l i n e a r e q u a t i o n s i n one v a r i a b l e ?  Detailed s t a t i s t i c a l among concepts, to  a n a l y s e s o f t h e c o r r e l a t i o n s which were  vocabulary,  found  and e r r o r s a r e r e p o r t e d i n Appendix D, P a r t 4,  which t h e p r e v i o u s c a u t i o n a r y note r e g a r d i n g sample s i z e and composition  s h o u l d be a p p l i e d .  Hence, i n t h e p r e s e n t a t i o n o f r e s u l t s , no s t a t i s t i c a l  c l a i m s a r e made, but r a t h e r the i n t e r e s t i n g phenomena which o c c u r r e d d u r i n g diagnosis are presented. statistically  However, o n l y those i n s t a n c e s which were  s i g n i f i c a n t are discussed.  p a t t e r n s were j u s t t h a t , and of i n t e r e s t being  only t o the p a r t i c u l a r  singular  student  diagnosed.  An u n d e r s t a n d i n g  o f v a r i a b l e was expected t o be r e l a t e d t o t h e e r r o r s  which s t u d e n t s made i n combining to  I t i s assumed t h a t o t h e r  obtain correct f i n a l  u n l i k e terms, as w e l l as s t u d e n t s '  solutions f o r l i n e a r equations.  the u n d e r s t a n d i n g  of v a r i a b l e  number o f c o r r e c t  s o l u t i o n s a student o b t a i n e d .  Both a s p e c t s o f  ( i d e n t i f y and meaning) were r e l a t e d t o the  r e l a t i o n s h i p o f an u n d e r s t a n d i n g u n l i k e terms d i d not o c c u r .  ability  However, t h e expected  o f v a r i a b l e and e r r o r s made i n combining  T h i s l a c k o f r e l a t i o n s h i p may be an i n d i c a t i o n  that  l i k e terms i s  a f a r more c o m p l i c a t e d concept  i n d i c a t e d by p r e v i o u s  than may have  been  research.  The a b i l i t y t o use the e q u a l i t y symbol was r e l a t e d t o the a b i l i t y solve equations  correctly.  One t y p i c a l response  came from a Grade 8 boy  who r e q u i r e d r e m e d i a l work on h i s u n d e r s t a n d i n g of e q u a l i t y . "The answer's 609.  question," attempt  = 256,  with  256  =  x + (-344),  and a g a i n a s s e r t e d  r e f e r r i n g t o the f a c t  that  that |256|  l a r g e magnitude,  would be i n v e r s e l y  questions.  r a t h e r than a problem with numbers of = -75 and  T h i s was the case i n the  c r e a t e meaning" and "given meaning, checklist.  x (45) of the  create  T y p i c a l of s t u d e n t s '  This is  confusing."  out of e i g h t i n c o r r e c t  equations the  equation" responses when  of one Grade 9 When put  She e x h i b i t e d s i m i l a r  problems when s o l v i n g o n e - s t e p a d d i t i o n and s u b t r a c t i o n problems, in six  = -495.  "given an e q u a t i o n  She r e p l i e d : "Found 21 marbles and found some more. ...(pause)...  to  seemed t o be a f u n c t i o n of  asked t o c r e a t e meaning f o r x + 21 = 39 was the response  t o g e t h e r got back  order  s m a l l e r than the  r e l a t e d t o the u n d e r s t a n d i n g of one or more of  p o r t i o n s of the d i a g n o s t i c  girl.  the  He was unable  t h a t the number of i n c o r r e c t s o l u t i o n s  meaning-of-equation context,  changed  < |-344|.  as he c o r r e c t l y s o l v e d -63 + x  I t was e x p e c t e d  he  "The answer's  a s o l u t i o n of e i t h e r e q u a t i o n and t h i s  h i s view of the e q u a l i t y symbol,  without  He s t a t e d :  s m a l l e r than the q u e s t i o n . " when p r e s e n t e d with 800 + x =  When p r e s e n t e d  t o x + (-344)  to  resulting  solutions.  Many s u b j e c t s found c r e a t i n g an e q u a t i o n from a word problem t o be a very d i f f i c u l t  task;  When asked t o w r i t e an e q u a t i o n  for:  Penny n o t i c e d something i n t e r e s t i n g about her s a v i n g s account. I f you add $ 6 t o the amount you get the same r e s u l t as d o u b l i n g the amount and s u b t r a c t i n g $ 4 . How much money does Penny have i n her s a v i n g s account? s e v e r a l s u b j e c t s s h a r e d the view of one Grade 10 g i r l , "I can't do t h a t ! " e q u a l s 12 because was  When encouraged you double i t .  who  flatly  t o t r y , she d e c l a r e d :  "Do  Then minus 4 , e q u a l s 8 . "  stated:  6 times 2  Similarly,  she  unable t o s o l v e any o f the f o u r e q u a t i o n s w i t h i n s t a n c e s o f the  v a r i a b l e on both s i d e s , but not c o n t a i n i n g b r a c k e t s .  Students who  d i d not  u n d e r s t a n d the meaning of e q u a t i o n s i n one format, were a l s o unable t o s o l v e such e q u a t i o n s .  T h i s l a c k of u n d e r s t a n d i n g of the meaning o f  p a r t i c u l a r e q u a t i o n s seemed t o p r e c l u d e the f o r m u l a t i o n of procedures necessary f o r s o l v i n g s i m i l a r equations. An u n d e r s t a n d i n g of the e q u a l i t y symbol and the a b i l i t y t o c r e a t e an e q u a t i o n from a word problem were d i r e c t l y r e l a t e d .  When asked t o w r i t e an  a r i t h m e t i c sentence which had o p e r a t i o n s on both s i d e s , response was  t h a t of a Grade 8 boy,  who  wrote 1 + 2 = 3 + 2 = 5 .  one  typical  He e x p l a i n e d :  "You're a d d i n g 2 t o both, so t h e y ' r e e q u a l , and the answer i s 5 . " e x p l a i n e d , when answering the p r e v i o u s l y mentioned Penny's s a v i n g s account: to  get 8 . "  t h a t was of  "You  When q u e s t i o n e d why  add 6 and  He  later  word problem r e g a r d i n g  6 and get 1 2 , and then s u b t r a c t 4  he had added s i x t o i t s e l f , he e x p l a i n e d  what " d o u b l i n g " meant i n the q u e s t i o n .  This "operational"  sense  the e q u a l i t y symbol c a r r i e d over t o the a b i l i t y t o " t r a n s l a t e word  problems"  and seemed t o focus a t t e n t i o n oh the " o p e r a t i o n a l " words w i t h i n  t h e q u e s t i o n , r a t h e r than t h e i r meaning w i t h i n the c o n t e x t of the q u e s t i o n . Anderson  (1982)  had i n d i c a t e d t h a t s t u d e n t s d i d not make the same type  of  e r r o r s i n s i m p l i f y i n g a r i t h m e t i c e x p r e s s i o n s and s o l v i n g e q u a t i o n s  s i m i l a r formats. understanding  with  I t was o f i n t e r e s t t o i n v e s t i g a t e how t h e e r r o r s i n  e x p r e s s i o n s and u n d e r s t a n d i n g  equations  would be r e l a t e d .  A l l t h r e e o f t h e meaning o f e q u a t i o n q u e s t i o n s showed a p o s i t i v e r e l a t i o n s h i p t o a s t u d e n t ' s a b i l i t y t o e x p l a i n m u l t i p l i c a t i o n and d i v i s i o n i n v a r i a b l e expressions.  A t y p i c a l erroneous  c o n c e p t i o n i n t h i s area was  e x e m p l i f i e d by one Grade 9 g i r l who c o u l d not show t h e meaning o f 4n thought  t h a t n /4 o n l y r e p r e s e n t e d a f r a c t i o n .  account  problem mentioned p r e v i o u s l y , she s t a t e d :  F o r the same s a v i n g s "You take the 6 and  m u l t i p l y by 4, ... t h a t gets 24, then s u b t r a c t 4 and g e t 20." to  e x p l a i n , she r e p l i e d :  and a 4."  When asked  "Doubling means you m u l t i p l y , ... and t h e r e ' s a 6  This lack of a b i l i t y to e x p l a i n the operation of m u l t i p l i c a t i o n  seemed t o a l l o w " m u l t i p l y " t o mean " d o u b l i n g " f o r t h i s Three o f t h e common e r r o r s found equality,  and  i n t h i s study  student.  (addition property of  a d d i t i o n and s u b t r a c t i o n e r r o r s with computations and a d d i t i o n  and s u b t r a c t i o n e r r o r s with s i g n s ) showed t h e e x p e c t e d r e l a t i o n s h i p s w i t h each o t h e r .  positive  One-third of the students  c o n c u r r e n t l y e x h i b i t e d e r r o r s i n these t h r e e c a t e g o r i e s .  i n this  study  T y p i c a l o f these  was one Grade 9 boy, who would a p p l y t h e a d d i t i o n p r o p e r t y by adding t h e same number t o one s i d e and i t s o p p o s i t e t o t h e o t h e r s i d e of t h e e q u a t i o n . W i t h i n t h i s erroneous wrong c o m p u t a t i o n a l  a p p l i c a t i o n he would v a c i l l a t e between d o i n g t h e  operation  (e.g. he would w r i t e 25 + 19 and then say 6)  o r making a r u l e - b a s e d s i g n e r r o r (e.g. "You always s u b t r a c t t h e numbers on o p p o s i t e s i d e s , and i f t h e r e ' s a n e g a t i v e , then i t ' s n e g a t i v e . " ) .  The  91  i n t e r t w i n i n g o f t h e s e t h r e e common e r r o r s , which i n v o l v e such a s p e c t s o f t h e e q u a t i o n - s o l v i n g p r o c e s s , cannot be  fundamental  overlooked.  The many i n s t a n c e s o f p r o c e d u r a l and r e s u l t a n t e r r o r s which found o r i g i n s i n erroneous remediation.  c o n c e p t i o n s emphasize the need f o r e a r l y d i a g n o s i s and  Awareness of the t y p e s of erroneous  s t u d e n t s seem t o form may may  be  their  c o n c e p t i o n s which  a l e r t the t e a c h e r , so t h a t p r e v e n t a t i v e measures  undertaken.  Complexity/Structure/Magnitude  The work of Tatsuoka,  Effects  on  Errors  e t a l . (1980) i n d i c a t e d t h a t the magnitude of  s i g n e d numbers i n f l u e n c e d the r e s u l t s of i n t e g e r a r i t h m e t i c .  Booth  (1981)  found t h a t  l a r g e r numbers caused s t u d e n t s t o f o c u s on the p r o c e s s , w h i l e  Herscovics  (1979) c l a i m e d t h a t more e r r o r s o c c u r r e d as the numbers  i n c r e a s e d i n magnitude.  Kieran's  (1981b) r e s u l t s i n d i c a t e d t h a t magnitude  of numbers and c o m p l e x i t y of the e q u a t i o n i n t e r a c t e d .  Pettito  i n d i c a t e d t h a t the s t r u c t u r e o f the e q u a t i o n i n terms of the placement  variable  and the n u m e r i c a l magnitude had an i n t e r a c t i v e e f f e c t .  f i n d i n g s seemed t o i n d i c a t e the importance s t r u c t u r a l format,  of equation  These  complexity,  and n u m e r i c a l magnitude i n s o l v i n g e q u a t i o n s .  f o l l o w i n g r e s e a r c h q u e s t i o n was 5.  (1979)  posed t o i n v e s t i g a t e these  The  interactions:  Do the f a c t o r s of e q u a t i o n complexity, s t r u c t u r a l format, and n u m e r i c a l magnitude i n t e r a c t i n terms of the n a t u r e and f r e q u e n c y of the e r r o r s produced?  The e f f e c t s o f e q u a t i o n c o m p l e x i t y s t r u c t u r a l format  (e.g. ax =b,  x a=b,  ( I n | < 2 0 , 2 0 < I n | < 5 0 , 50<|n|<100, and  (one-, two-, b=ax, b=x  and  multi-step),  a ) , and n u m e r i c a l magnitude  100<|n|< 1 0 0 0 ) were e x p l o r e d i n terms  of the mean number of e r r o r s . which d e t a i l c o m p l e x i t y , the  structure,  expected that  an  increase  average number of e r r o r s made.  general,  the  e q u a t i o n s was e r r o r s made).  Part  and magnitude e r r o r means as w e l l  i n t e r a c t i o n e f f e c t s among these t h r e e a s p e c t s of e q u a t i o n I t was  the  T a b l e s are p r e s e n t e d i n Appendix D,  not  always the  as  construction.  i n e q u a t i o n c o m p l e x i t y would T h i s was  5  increase  case.  In  e f f e c t of e q u a t i o n c o m p l e x i t y on e r r o r s made i n s o l v i n g i n the  expected d i r e c t i o n  For the most p a r t ,  ( i e . the more complex, the more  equations containing  a lower mean e r r o r r a t e than those c o n t a i n i n g mean number of e r r o r s i n the an unexpected i n v e r s e  l a r g e r numbers.  f i n a l answers and  r e l a t i o n s h i p as  smaller  numbers  However-, the  rule-based sign errors  i n d i c a t e d i n the  had  following  had  table.  Table 8 The R e l a t i o n s h i p of Numerical Magnitude t o F i n a l Answers and Rule-Based Sign E r r o r s  N u m e r i c a l Magnitude  Nl N2 N3 N4  In each case e q u a t i o n s c o n t a i n i n g findings  Mean Number of E r r o r s f o r F i n a l Answers Rule-Based Signs  4.28  0 . 64 0 . 61 0.44 0.33  3 . 44 3.33  2.78  l a r g e r numbers had  seemed t o l e n d support t o Booth's  (1981) c o n t e n t i o n  numbers f o c u s e d a t t e n t i o n on the p r o c e s s .used, and made.  Another reason f o r t h i s r e d u c t i o n  fact that  fewer e r r o r s .  out  larger  hence fewer e r r o r s were  of e r r o r s may  s t u d e n t s were more l i k e l y t o w r i t e  that  These  be  found i n  t h e i r arithmetic  the  work when  faced in  low  w i t h numbers of l a r g e r magnitude. achievers,  who  o f t e n d i d not  T h i s was  write  p a r t i c u l a r l y noticeable  down t h e i r  arithmetic  calculations. There were two placement of the of the  main a s p e c t s of the  v a r i a b l e with r e s p e c t  v a r i a b l e with r e s p e c t  t o the  structure  e q u a l i t y symbol  e q u a l i t y symbol  (SI & S2)  (S3 & S4).  The  c o n s t a n t s i n the  equation.  variable(s)  with v a r i a b l e ( s )  on the  second aspect i n v o l v e d  d i f f e r e n c e s between the mean number of e r r o r s when the the  r i g h t of the  constants  (S2  r e s u l t s on  constants  & S4).  The  and  of v a r i a b l e ( s )  f i r s t aspect of s t r u c t u r e  computation a d d i t i o n / s u b t r a c t i o n  made more e r r o r s when the (SI & S2)  (SI & S3)  than when i t was  Analysis  of the  v a r i a b l e was on the  errors.  t o the  left  c o r r e l a t i o n m a t r i c e s of  answers.  (p < 0.001) between n u m e r i c a l magnitude and for  p r o c e d u r a l e r r o r s of l i k e terms  a d d i t i o n p r o p e r t y of e q u a l i t y resultant errors  first  on the  variable(s) t o the  was  left  of the  to  of  the  showed s i g n i f i c a n t Surprisingly, equality (S3  students symbol &  S4).  complexity/structure/numeric  16  Intercorrelations Within  i n t e r a c t i o n s were h i g h l y  Significant correlations s t r u c t u r a l format were found  (11 of the  (11 out  left  the  each e r r o r type were e x p l o r e d .  4-by-4 n u m e r i c / s t r u c t u r e g r i d , 13 of the (p < 0.001) f o r f i n a l  The  analyzing  r i g h t of the e q u a l i t y symbol  between each of t h e s e v a r i a b l e s and  significant  was  placement  r i g h t of  of the  i n t e r a c t i o n s d i s c l o s e d s e v e r a l noteworthy f i n d i n g s .  the  investigated:  t o the e q u a l i t y symbol and  a s p e c t compared mean number of e r r o r s when the of the  which were  16  i n v o l v i n g incomplete s o l u t i o n s  16  interactions),  the  interactions),  and  (12 out  16  of the  interactions).  No o t h e r such b l o c k s o f s i g n i f i c a n t  i n t e r c o r r e l a t i o n s were  found f o r e i t h e r c o m p l e x i t y / s t r u c t u r e or complexity/numeric  interactions.  These r e s u l t s s u p p o r t e d t h e c o n t i n u e d i n c l u s i o n o f t h e 4-by-4 n u m e r i c a l magnitude-by-structural-format complexity.  g r i d w i t h i n each l e v e l o f e q u a t i o n  F o r s t u d e n t s e x p e r i e n c i n g d i f f i c u l t i e s w i t h combining  terms, a p p l y i n g t h e a d d i t i o n p r o p e r t y o f e q u a l i t y , o r c o m p l e t i n g i n v e s t i g a t i o n o f e q u a t i o n s with v a r i e d s t r u c t u r a l formats  like  solutions,  and n u m e r i c a l  magnitudes may prove d i a g n o s t i c a l l y i n s t r u c t i v e and p r o v i d e a f i r m f o u n d a t i o n f o r t h e n a t u r e o f e q u a t i o n s which may be a t t h e r o o t o f these difficulties. Summary  The  of  Results  r e s u l t s o f t h e study i n d i c a t e d t h a t a number o f t h e e r r o r s which  were c h a r a c t e r i s t i c o f s t u d e n t s on the d i a g n o s t i c instrument  were r e l a t e d  t o achievement l e v e l .  These r e l a t i o n s h i p s were most e v i d e n t i n f i n a l  answers and incomplete  s o l u t i o n s on t h e e q u a t i o n p o r t i o n o f t h e d i a g n o s t i c  checklist  and i n many o f t h e c o n c e p t i o n s h e l d by s t u d e n t s r e g a r d i n g a l g e b r a  ( i d e n t i f y i n g v a r i a b l e , e x p l a i n i n g d i v i s i o n i n expressions, e x p l a i n i n g the meaning o f e q u a t i o n s , and t h e v o c a b u l a r y a s s o c i a t e d w i t h i n v e r s e s and i d e n t i t y elements).  The concepts  and v o c a b u l a r y p o r t i o n s of t h e instrument  e v i d e n c e d many i n t e r e s t i n g r e l a t i o n s h i p s with the performance on the e q u a t i o n s p o r t i o n o f t h e instrument. Krutetskii  These f i n d i n g s s u p p o r t e d t h e work o f  (1976) by i n d i c a t i n g t h a t h i g h a c h i e v e r s were a b l e t o e x p l a i n  the meaning o f e q u a t i o n s b e t t e r than low a c h i e v e r s , and t h i s  ability  r e l a t e d t o h i g h a c h i e v e r s ' performance on f i n a l answers i n s o l v i n g  e q u a t i o n s on t h e i n s t r u m e n t . Approximately  one q u a r t e r o f t h e s u b j e c t s e x h i b i t e d s y s t e m a t i c e r r o r s .  Most o f t h e s e were Grade 9 and Grade 10 s t u d e n t s . e r r o r t y p e s found equality,  (sign:  l i k e terms,  a d d i t i o n and s u b t r a c t i o n ,  and computation:  found t o be common e r r o r s .  The f o u r s y s t e m a t i c a d d i t i o n property of  a d d i t i o n and s u b t r a c t i o n ) were a l s o  These f i n d i n g s supported t h e work of Anderson  (1982) . I n v e s t i g a t i o n o f t h e r e l a t i o n s h i p o f t h e d i a g n o s t i c instrument t o measures of achievement i n mathematics r e v e a l e d some i n t e r e s t i n g  results.  S i x t y p e s o f e r r o r s which o c c u r r e d i n t h e e q u a t i o n s p o r t i o n o f t h e i n s t r u m e n t seemed r e l a t e d t o some o f the mathematics achievement  variables.  However, twelve o f t h e concepts and v o c a b u l a r y items showed p o s i t i v e r e l a t i o n s h i p s with a l l o f t h e mathematics achievement v a r i a b l e s . f i n d i n g seems t o support K r u t e t s k i i ' s a b i l i t y t o e x p l a i n mathematical  This  (1976) c o n t e n t i o n t h a t i t i s the  i d e a s without p e r f o r m i n g t h e procedures  which b e s t d i s t i n g u i s h e s h i g h a c h i e v e r s from low a c h i e v e r s i n mathematics. Equation complexity,  s t r u c t u r a l format,  s i g n i f i c a n t l y c o r r e l a t e d with e r r o r s . of Booth  (1981), K i e r a n  and n u m e r i c a l magnitude were  T h i s supported t h e p r e v i o u s r e s e a r c h  (1983) and P e t t i t o  (1979).  Significant  i n t e r c o r r e l a t i o n s were found between n u m e r i c a l magnitude and s t r u c t u r a l magnitude which supported t h e i r c o n t i n u e d i n c l u s i o n a t each l e v e l o f equation complexity i n the d i a g n o s t i c  instrument.  CHAPTER V DISCUSSION The d i s c u s s i o n o f t h e study reviews t h e p r o c e d u r a l d e s i g n used i n t h e development and v a l i d a t i o n o f t h e Diagnostic is  Checklist  f o l l o w e d by a r a t i o n a l e f o r use o f t h e c h e c k l i s t  assessment o f t h e c h e c k l i s t diagnostic u t i l i t y  This  i n d i a g n o s i s . An  i n terms o f r e l i a b i l i t y ,  i s then p r e s e n t e d .  for Algebra.  v a l i d i t y , and  The s i g n i f i c a n c e and l i m i t a t i o n s o f  the study a r e d i s c u s s e d and s u g g e s t i o n s a r e p r o v i d e d f o r t h e development o f f u t u r e d i a g n o s t i c i n s t r u m e n t s i n o t h e r areas o f mathematics.  Review  of Procedural  Design  The p r e s e n t study r e s u l t e d from t h e need f o r a d i a g n o s t i c instrument at the b e g i n n i n g stages o f s o l v i n g a l g e b r a i c e q u a t i o n s . s t r e s s e d t h e importance  o f i d e n t i f y i n g e a r l y erroneous  a l g e b r a b e f o r e t h e y became i n g r a i n e d procedures eliminate.  Some o f these erroneous  r u l e s l e a r n e d i n a r i t h m e t i c (Matz, and v a r i a b l e  (1976)  conceptions i n  f a r more d i f f i c u l t t o  conceptions involve o v e r g e n e r a l i z i n g 1980), erroneous  conceptions of e q u a l i t y  ( K i e r a n , 1983; Wagner, 1981a), simple a r i t h m e t i c e r r o r s  (Roberts, 1968), errors,  Krutetskii  sign errors  o r incomplete  (Tatsuoka e t a l . , 1980),  solutions  c o p y i n g and random  (Englehardt & Wiebe, 1981).  There was a  need f o r a d i a g n o s t i c instrument t h a t would i d e n t i f y t h e s e and o t h e r e r r o r s in  s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e . E r r o r s t e n d t o be based on "the s p e c i f i c content o f t h e concepts and  equations presented t o students.  I t was t h e r e f o r e c r i t i c a l t h a t t h e  c o n t e n t o f t h e d i a g n o s t i c instrument a c c u r a t e l y r e f l e c t e d t h e scope and  sequence o f b e g i n n i n g a l g e b r a c u r r i c u l a  .  In Phase 1 o f t h e study,  algebra  t e x t s were s c r u t i n i z e d and twelve e q u a t i o n types s e l e c t e d which r e p r e s e n t e d the o r d e r and content o f those t e x t s .  One hundred ninety-two  were c o n s t r u c t e d (12 e q u a t i o n types by 4 s t r u c t u r a l format magnitude) f o r p o s s i b l e i n c l u s i o n i n t h e instrument 1).  (Pettito,  by 4 n u m e r i c a l  (See Appendix B, P a r t  These e q u a t i o n s c o n t r o l l e d f o r e q u a t i o n c o m p l e x i t y ,  n u m e r i c a l magnitude, and s i g n placement  equations  structural  1979; Tatsuoka  format,  et a l . ,  1980) . Phase 2 o f t h e study i n v o l v e d t h e development o f an e r r o r c a t e g o r i z a t i o n scheme based Bright  (1982) .  on t h e t h e o r i e s o f Matz (1980)  T h i s e r r o r - c a t e g o r i z a t i o n scheme brought  and Bernard &  together the  s i g n i f i c a n t e r r o r c a t e g o r i e s i d e n t i f i e d by r e s e a r c h e r s i n t h e l e a r n i n g o f a l g e b r a over t h e p a s t s i x decades. c a t e g o r i e s o f Conceptual, Engelhardt  E r r o r s were s t r u c t u r e d i n t o the g e n e r a l  P r o c e d u r a l , and R e s u l t a n t  & Wiebe, 1981; Booth, 1981) .  (Carry e t a l . , 1976;  The e r r o r - c a t e g o r i z a t i o n  scheme  formed one o f t h e c o r n e r s t o n e s f o r t h e c o n s t r u c t i o n o f the d i a g n o s t i c instrument. In Phase 3, e i g h t y f i r s t - y e a r a l g e b r a s t u d e n t s were t e s t e d u s i n g the Lankton  t o determine  t h e i r achievement l e v e l i n a l g e b r a .  Each s u b j e c t  then r e c e i v e d one o f f o u r e q u a t i o n t e s t - f o r m s c o n s i s t i n g o f 48 e q u a t i o n s (See Appendix B, P a r t 2).  Each t e s t - f o r m c o n t a i n e d f o u r e q u a t i o n s  each o f t h e twelve e q u a t i o n t y p e s . n a t u r e o f t h e e r r o r committed.  from  Answers were coded t o i n d i c a t e t h e  E r r o r s were t r e a t e d as d i s t r a c t o r s f o r the  purpose o f d e t e r m i n i n g t h e d i f f i c u l t y  l e v e l o f each e q u a t i o n .  An  i t e m a n a l y s i s was performed  c o r r e l a t i o n with the t e s t ,  t o determine  each item's  p-value,  and c o r r e l a t i o n with t h e Lankton.  s e l e c t e d which had a p o s i t i v e c o r r e l a t i o n with t h e Lankton, c o r r e l a t i o n w i t h t h e t o t a l t e s t s c o r e , whose d i s t r a c t o r s  Items were a positive  ( e r r o r s ) had a  n e g a t i v e c o r r e l a t i o n with t h e t e s t and whose p - v a l u e was c l o s e s t t o 0.5. Each element o f t h e 4-by-4 n u m e r i c a l - m a g n i t u d e - b y - s t r u c t u r a l - f o r m a t was  i n c l u d e d a t each l e v e l o f e q u a t i o n complexity,  grid  due t o t h e s i g n i f i c a n t  i n t e r a c t i o n e f f e c t between n u m e r i c a l magnitude and s t r u c t u r a l format (See Appendix B, P a r t 3 ) . The f i n a l  48 e q u a t i o n s s e l e c t e d a r e l i s t e d i n  Appendix B, P a r t 4. Phase 4 o f t h e p r e s e n t study was d e s i g n e d t o ensure and t a s k s w i t h i n t h e d i a g n o s t i c instrument were c l e a r . format  that the directions The wording and  changes needed t o a c h i e v e t h i s purpose a r e i n c l u d e d i n Appendix C.  I t was a l s o o f i n t e r e s t t o i n v e s t i g a t e t h e d i f f e r e n c e s i n e r r o r p a t t e r n s which might o c c u r between high-and preliminary draft  o f t h e Diagnostic  low-achieving students. Checklist  for Algebra  The was used t o  i n v e s t i g a t e t h e e r r o r s o f two h i g h - a c h i e v i n g and two l o w - a c h i e v i n g Grade 9 students.  Low a c h i e v e r s made many more e r r o r s than h i g h a c h i e v e r s .  e r r o r s of high achievers p r i m a r i l y i n v o l v e d computational Low  and s i g n  a c h i e v e r s made more s i g n e r r o r s i n v o l v i n g c o p y i n g mistakes  high achievers. f o r them.  The errors.  than d i d  However, t h e two low a c h i e v e r s i n d i c a t e d t h i s was normal  An examination  o f p r o t o c o l s i n d i c a t e d t h a t t h e r e was no  a p p r e c i a b l e i n c r e a s e i n t h e number of c o p y i n g e r r o r s made d u r i n g t e s t i n g and t h a t c o p y i n g e r r o r s were e v e n l y d i s p e r s e d .  Low a c h i e v e r s made more  p r o c e d u r a l e r r o r s t h a n c o n c e p t u a l e r r o r s and made f a r more o f t h e s e t y p e s of  e r r o r s than d i d high a c h i e v e r s .  complete 60  Low a c h i e v e r s took over 90 minutes t o  t h e e n t i r e t e s t i n g , w h i l e h i g h a c h i e v e r s took s l i g h t l y  l e s s than  minutes. R e s u l t s f o r n u m e r i c a l magnitude i n d i c a t e d t h a t low a c h i e v e r s had  difficulty  at a l l l e v e l s , while high achievers experienced d i f f i c u l t y  on e q u a t i o n s o f l a r g e n u m e r i c a l magnitude (N4: 100<|n|<1000).  only  R e s u l t s of  s t r u c t u r a l format s u g g e s t e d t h a t low a c h i e v e r s e x p e r i e n c e d more d i f f i u l t y w i t h s t r u c t u r e s 1 and 3 ( v a r i a b l e t o t h e l e f t o f t h e c o n s t a n t s ) w h i l e h i g h achievers experienced d i f f i c u l t y  with s t r u c t u r e 4 ( v a r i a b l e t o the r i g h t of  b o t h t h e e q u a l i t y symbol and c o n s t a n t s ) .  The e f f e c t s o f e q u a t i o n  c o m p l e x i t y , n u m e r i c a l magnitude, and s t r u c t u r a l format overshadowed t h e c o n c e r n f o r t i m e l i m i t a t i o n s and n e c e s s i t a t e d t h e i n c l u s i o n o f 48 e q u a t i o n s i n the f i n a l d r a f t of the d i a g n o s t i c instrument. The purpose o f t h e f i n a l phase o f t h e s t u d y was f o u r f o l d : 1) i n v e s t i g a t i o n o f t h e e f f e c t s o f n u m e r i c a l magnitude and s t r u c t u r a l format;  2) e x p l o r a t i o n o f t h e n a t u r e and f r e q u e n c y o f e r r o r s made a t  d i f f e r e n t achievement l e v e l s ;  3) e x a m i n a t i o n o f i n t e r r e l a t i o n s h i p s o f  e r r o r s w i t h demographic v a r i a b l e s and concomitant of  measures;  4) comparison  t h e c o n c e p t u a l u n d e r s t a n d i n g s e x h i b i t e d and t h e e q u a t i o n - s o l v i n g e r r o r s  made on t h e d i a g n o s t i c i n s t r u m e n t . A l l 452 academic Grade 8, 9 and 10 s t u d e n t s i n a B r i t i s h j u n i o r h i g h s c h o o l were g i v e n t h e Lankton achievement l e v e l .  t o determine  Columbia  their algebraic  S u b j e c t s were t h e n a s s i g n e d t o one o f 18 c e l l s  (3  achievement l e v e l s by 3 grade cells cell  ranged  from 6 t o 51.  l e v e l s by 2 g e n d e r s ) .  Two s u b j e c t s were randomly s e l e c t e d from  for diagnostic interviews.  a d m i n i s t r a t i o n o f t h e CSMS  The s i z e o f these 18  These 36 s u b j e c t s r e c e i v e d a group  t o evaluate t h e i r understanding of v a r i a b l e .  Then each s u b j e c t was i n d i v i d u a l l y i n t e r v i e w e d and v i d e o - t a p e d . i n t e r v i e w s took a p p r o x i m a t e l y The  each  These  6 weeks t o complete.  r e s u l t s of t h e i n t e r v i e w s i n d i c a t e d t h a t t h e d i a g n o s t i c  checklist  p r o v i d e d i n s i g h t s i n t o t h e nature o f e r r o r s committed by b e g i n n i n g a l g e b r a students.  The reasons f o r u s i n g these p a r t i c u l a r procedures  i n the design  of t h e d i a g n o s t i c instrument and t h e d i a g n o s t i c i m p l i c a t i o n s o f the r e s u l t s o b t a i n e d i n t h e f i n a l phase o f t h e study a r e d i s c u s s e d i n d e t a i l i n the next  sections. Rationale  f o r t h e Diagnostic  Checklist  I t i s important t o d i s t i n g u i s h t h e Diagnostic from o t h e r d i a g n o s t i c t e s t s o r i n s t r u m e n t s .  for  Checklist  Algebra for  Algebra  The two p a r t s o f t h e c h e c k l i s t  would appear t o f u n c t i o n as f o r m a t i v e and summative a s p e c t s of b e g i n n i n g algebraic testing. checklist  Such i s not t h e case.  of the diagnostic  i s t o l o o k a t t h e e r r o r s s t u d e n t s make i n s o l v i n g  e q u a t i o n s i n one v a r i a b l e and t o determine erroneous  The purpose  linear  how these e r r o r s r e l a t e t o the  a l g e b r a i c c o n c e p t i o n s which a student h o l d s .  - a n a l y s e s have not ' c o n t r o l l e d the d i f f i c u l t y  Previous e r r o r  l e v e l o f q u e s t i o n s and have  used p o s t - f a c t o a n a l y s i s o f p a p e r - a n d - p e n c i l t e s t s o r p o s t - f a c t o i n t e r v i e w s t o determine Algebra  t h e nature o f the e r r o r s made.  The Diagnostic  Checklist  p r o v i d e s an i n n o v a t i v e form of i n s t r u m e n t a t i o n e n a b l i n g t h e  for  101  i n v e s t i g a t o r t o e x p l o r e e r r o r s as they o c c u r .  The c h e c k l i s t u t i l i z e s t h e  s t r e n g t h s o f both e r r o r - a n a l y s i s and s e m i - s t r u c t u r e d i n t e r v i e w s , thus p e r m i t t i n g i n - d e p t h q u e s t i o n i n g o f t h e student t o determine  t h e exact cause  of a p a r t i c u l a r e r r o r , as w e l l as p r o v i d i n g a v i s u a l m a t r i x o f e r r o r s f o r the purpose o f r e m e d i a l  instruction.  E r r o r s may be i n f l u e n c e d by t h e d i f f i c u l t y n o t e d by P e t t i t o  (1979),  items which were t o o d i f f i c u l t  s t u d e n t s t o choose a procedure & Hart  (1982)  As  d i d not a l l o w  f o r s o l v i n g the equation.  Similarly,  i n d i c a t e d t h a t , f o r items which were t o o easy,  r e v e r t e d t o "child-methods" procedures.  l e v e l of a question.  Booth  students  o r t r i a l - a n d - e r r o r and d i d not use a l g e b r a i c  S i n c e t h i s i n v e s t i g a t i o n f o c u s e s on t h e c o n c e p t u a l ,  p r o c e d u r a l , and r e s u l t a n t e r r o r s found by p r e v i o u s r e s e a r c h e r s , mid - d i f f i c u l t y l e v e l items seemed t o be most a p p r o p r i a t e f o r i n v e s t i g a t i n g procedural errors.  M i d - d i f f i c u l t y l e v e l items a l s o seemed t o be t h e best  way t o l i m i t e r r o r b i a s and t o p r o v i d e s t r u c t u r a l v a l i d i t y .  A Spearman's  rho o f 0 . 9 0 was o b t a i n e d when c o r r e l a t i n g the rank o f an e q u a t i o n type the t e x t b o o k s  w i t h t h e rank o b t a i n e d from t h e mean p - v a l u e  e q u a t i o n type  (Refer t o T a b l e 7 , p 8 6 ) .  from  f o r each  Such a h i g h c o r r e l a t i o n c o u l d not  have been o b t a i n e d from use o f o t h e r than m i d - d i f f i c u l t y l e v e l items.  This  h i g h c o r r e l a t i o n p r o v i d e s c o n v i n c i n g evidence t h a t t h e s t r u c t u r e o f the Diagnostic  Checklist  year algebra  for Algebra  i s compatible  with t h e s t r u c t u r e i n f i r s t -  textbooks.  Many o f t h e e r r o r s found on t h e d i a g n o s t i c c h e c k l i s t would have been missed  on o t h e r forms o f i n s t r u m e n t a t i o n .  Some o f t h e s e e r r o r s were found  102  due  t o t h e c a r e f u l c o n s t r u c t i o n o f t h e complexity,  s t r u c t u r e , and numerical  magnitudes o f t h e e q u a t i o n s ; o t h e r s were found due t o t h e use o f semistructured interviews.  The importance  e q u a t i o n s p r e s e n t e d t o students average  achievers.  of systematic v a r i a t i o n of the  was e x e m p l i f i e d by one o f t h e Grade 10  She o n l y made "borrowing"  (C3) which c o n t a i n e d l a r g e numbers  e r r o r s on complex  equations  (N4), but made no such mistakes  o t h e r c o m p l e x i t y o r n u m e r i c a l magnitude l e v e l .  at any  I f t h e s e e x p l i c i t types of  e q u a t i o n s had not been p r e s e n t i n t h e c h e c k l i s t ,  such d i a g n o s i s , and the  r e s u l t i n g r e m e d i a t i o n , may have not o c c u r r e d , l e a v i n g a gap i n e q u a t i o n s o l v i n g a b i l i t y f o r t h i s student.  L i k e w i s e , t h e importance  o f the use o f  i n t e r v i e w s was i l l u s t r a t e d by a h i g h a c h i e v i n g Grade 10 s t u d e n t ' s lack of conceptual understanding  o f l i k e terms.  When asked  apparent  i n the  c o n c e p t u a l s e c t i o n t o group l i k e terms, she seemed t o have no c o n c e p t i o n o f I t h e i r meaning.  However, when c o n f r o n t e d with e q u a t i o n s i n which she had t o  combine l i k e terms she d i d so r e a d i l y ,  and then commented t h a t she had no  i d e a why she had not been a b l e t o remember what they were p r e v i o u s l y . At the end o f t h e i n t e r v i e w , she asked t o go back and re-do t h e c o n c e p t u a l s e c t i o n on l i k e terms.  She demonstrated complete mastery, and then  announced t h a t i t was t h e procedures  which t r i g g e r e d t h e meaning f o r her,  r a t h e r than t h e name o f t h e concept.  Such i n s i g h t  into the learning  p r o c e s s o f a student would have been i m p o s s i b l e without  semi-structured  i n t e r v i e w t e c h n i q u e s and t h e i n c l u s i o n of p a r a l l e l c o n c e p t u a l and p r o c e d u r a l e r r o r s w i t h i n the instrument. Initial  r e s u l t s from t h e use of t h e Diagnostic  Checklist  for  Algebra  1 03  indicate that i t i s a useful tool for diagnosis.  I t p r o v i d e s many i n s i g h t s  i n t o t h e l e a r n i n g p r o c e s s e s o f a l g e b r a and c r e a t e s a f i r m f o u n d a t i o n upon which t o base r e m e d i a l i n s t r u c t i o n . for  initial  instruction,  Such i n s i g h t s a l s o have  implications  so t h a t the t e a c h e r i s aware o f t h e common e r r o r s  made by s t u d e n t s i n b e g i n n i n g a l g e b r a and can p r o v i d e examples i n which t h e s e e r r o r s can o c c u r and be c o r r e c t e d a t t h e i n i t i a l Assessment Statistical  of  the  Diagnostic  Checklist  stages o f l e a r n i n g . for  Algebra  Properties  Most p r e v i o u s a l g e b r a i c t e s t s used f o r d i a g n o s t i c purposes weaknesses i n v a l i d i t y  and r e l i a b i l i t y  (Petrosko, 1978).  p r o v i s i o n s were made f o r content and f a c e v a l i d i t y , predictive validity unreported.  were almost  The procedures  contained  While some  c o n c u r r e n t and  always o m i t t e d and r e l i a b i l i t y was o f t e n  used i n t h i s study were d e s i g n e d t o address  t h e s e weaknesses. V a l i d i t y  Content,  c o n c u r r e n t , f a c e , s t r u c t u r a l , and p r e d i c t i v e v a l i d i t y  e s t a b l i s h e d i n Phases 1, 2, 3, 4, and 5, r e s p e c t i v e l y . content v a l i d i t y  year s t u d e n t s .  Phase 2 improved f a c e  by p r e s e n t i n g e q u a t i o n s i n mean rank o r d e r o f f i r s t  t e x t s and by u t i l i z i n g  research-backed  e r r o r c a t e g o r i e s with  v a r i a t i o n o f n u m e r i c a l magnitude and s t r u c t u r a l format complexity. of  Phase 1 p r o v i d e d  o f t h e d i a g n o s t i c instrument by e n s u r i n g the c o n c e p t u a l  p o r t i o n s were a p p r o p r i a t e f o r f i r s t validity  were  Phase 3 p r o v i d e d c o n c u r r e n t v a l i d i t y  over  year algebra systematic equation  f o r the equation p o r t i o n  t h e instrument by comparing r e s u l t s w i t h t h e Lankton,  as w e l l as  structural validity  by c o r r e l a t i n g t h e t e x t and mean p - v a l u e  the twelve  equation types.  diagnostic  instrument.  Phase 4 improved f a c e v a l i d i t y  validity  through  f o r the " t o t a l  grades  was found  sufficiently  e r r o r s c o r e " on t h e d i a g n o s t i c instrument  (See Appendix D, Table D . 5 ) .  for  While t h i s  high to c l a i m p r e d i c t i v e v a l i d i t y  (r=0.35).  d i a g n o s t i c instrument failure,  measures d e s c r i b e d above.  (See Appendix D, P a r t 2 ) a c o r r e l a t i o n of  h i g h e r than t h e p r e d i c t i v e v a l i d i t y population  f o r the  as w e l l as some measure o f p r e d i c t i v e  comparisons w i t h t h e c r i t e r i o n  By c r e a t i n g "outcome s c o r e s " 0.48  and r e f i n e d the  Phase 5 p r o v i d e d c o n c u r r e n t v a l i d i t y  c o n c e p t u a l p o r t i o n o f t h e instrument,  r a n k i n g s of  was not  o f the instrument,  found f o r t h e Lankton  Because o f t h i s  and student  i t was  on t h e same  low p r e d i c t i v e v a l i d i t y t h e  s h o u l d not be used t o p r e d i c t  a l g e b r a i c success or  but would.best be used t o p r o v i d e a c o n s i s t e n t , s t r u c t u r e d method  i d e n t i f y i n g students'  errors.  R e l i a b i l i t y Hoyt E s t i m a t e s o f R e l i a b i l i t y Phase 3 f o r t h e e q u a t i o n s  used  i n the f i n a l  study.  c a l c u l a t e d on t h e f i n a l d i a g n o s t i c instrument (0.83),  0 . 8 2 t o 0 . 9 6 were o b t a i n e d i n  r a n g i n g from  the conceptual p o r t i o n ( 0 . 8 6 ) ,  Cronbach's Alpha  f o r the o v e r a l l  was  "outcome"  and t h e e q u a t i o n p o r t i o n  (0.75).  When t h e 2 7 c o n c e p t u a l items were s e p a r a t e l y compared w i t h t h e 2 1 e q u a t i o n e r r o r items 0.42  an a l p h a o f 0 . 6 5 was o b t a i n e d .  A split-half  was o b t a i n e d when comparing t h e 2 0 concept  items.  A split-half  reliability  coefficient  comparing p r o c e d u r a l e r r o r s t o r e s u l t a n t  r e l i a b i l i t y of  items t o t h e 7 v o c a b u l a r y  o f 0 . 8 3 was o b t a i n e d when  errors.  The  c o n c e p t u a l and e q u a t i o n p o r t i o n s o f t h e d i a g n o s t i c instrument  seemed t o be measuring d i f f e r e n t a s p e c t s o f a l g e b r a i c achievement, as d i d the v o c a b u l a r y and concept p o r t i o n s . These f i n d i n g s i n d i c a t e t h a t t h e r e i s a need f o r making d i s t i n c t i o n s among concepts, v o c a b u l a r y , solving i n diagnosis.  Reliability  Checklist  a c h i e v e d these  for Algebra  Diagnostic  In  and e q u a t i o n -  r e s u l t s i n d i c a t e that the  Diagnostic  distinctions.  Considerations  order f o r a c h e c k l i s t  diagnostic u t i l i t y ,  f o r l i n e a r e q u a t i o n s i n one v a r i a b l e t o have  i t must be a b l e t o i d e n t i f y s y s t e m a t i c and common  e r r o r s made by s t u d e n t s i n s o l v i n g e q u a t i o n s .  The k i n d s o f erroneous  c o n c e p t i o n s which s t u d e n t s have r e g a r d i n g a l g e b r a i c concepts must a l s o be identified. efficient  Most i m p o r t a n t l y t h e instrument  s h o u l d do t h i s t a s k i n an  manner.  Diagnostic  Categories  S e v e r a l o f t h e e r r o r c a t e g o r i e s used i n t h e d i a g n o s t i c instrument occurred infrequently.  Thus these e r r o r c a t e g o r i e s d i d not seem t o  c o n t r i b u t e s i g n i f i c a n t l y t o e i t h e r i n d i v i d u a l d i a g n o s i s o r t o group i n s t r u c t i o n a l concerns. the instrument  The e r r o r c a t e g o r i e s i n t h e c o n c e p t u a l p o r t i o n of  which o c c u r r e d with l i m i t e d frequency were t h e a b i l i t y t o  group e x p r e s s i o n s c o n t a i n i n g e i t h e r t h e a d d i t i o n , s u b t r a c t i o n o r d i v i s i o n symbols; t h e a b i l i t y t o e x p l a i n t h e meaning o f a d d i t i o n and s u b t r a c t i o n i n e x p r e s s i o n s c o n t a i n i n g b l a n k s and v a r i a b l e s , as w e l l as m u l t i p l i c a t i o n i n b l a n k e x p r e s s i o n s ; t h e a b i l i t y t o i d e n t i f y and e x p l a i n t h e meaning o f t h e e q u a l i t y symbol; and t h e v o c a b u l a r y a s s o c i a t e d with c o e f f i c i e n t s . A l l  o t h e r c o n c e p t u a l understandings  were r e q u i r e d f o r complete d i a g n o s i s .  S i m i l a r l y , t h e r e were e r r o r c a t e g o r i e s i n c l u d e d i n t h e s o l v i n g o f equations  which seemed o f l i m i t e d d i a g n o s t i c u t i l i t y ,  nature of t h e i r occurrence i n v o l v e d were annexation  during the interviews.  due t o t h e l i m i t e d  The p r o c e d u r a l e r r o r s  o f zero, i d e n t i t y c o n f u s i o n , p a r t i a l  distributive  o r d e r o f o p e r a t i o n s , m u l t i p l i c a t i o n p r o p e r t y o f e q u a l i t y and c o e f f i c i e n t s . The  r e s u l t a n t e r r o r s were  mechanical/perceptual,  computational  and random.  b a s i c f a c t s and f a u l t y  algorithms  A l l o t h e r p r o c e d u r a l and r e s u l t a n t  e r r o r c a t e g o r i e s were r e q u i r e d f o r complete d i a g n o s i s . I t would seem t h a t a l l o f t h e above c o n c e p t u a l , p r o c e d u r a l , and r e s u l t a n t e r r o r types o c c u r r e d so r a r e l y t h a t they c o u l d be removed from the d i a g n o s t i c instrument  with no l o s s t o t h e a c c u r a c y  of diagnosis.  e l i m i n a t i o n might prove u s e f u l i n r e d u c i n g a d m i n i s t r a t i v e time.  Such  However,  f u r t h e r c o n f i r m a t i o n from c o n t i n u e d use o f t h e d i a g n o s t i c instrument i s recommended b e f o r e such d e c i s i v e e l i m i n a t i o n o c c u r s . Systematic Systematic  and  Common  Errors  e r r o r s were d e f i n e d as e r r o r s which o c c u r r e d i n over 50% of  the a v a i l a b l e . o p p o r t u n i t i e s (Anderson,  1982).  In t h e p r e s e n t  study  s y s t e m a t i c e r r o r s i n c l u d e d l i k e terms, t h e a d d i t i o n p r o p e r t y of e q u a l i t y , and a d d i t i o n / s u b t r a c t i o n e r r o r s with both s i g n and computation. the 36 s u b j e c t s e x h i b i t e d s y s t e m a t i c e r r o r s . computational different  Nine of  Two o f t h e s t u d e n t s who made  a d d i t i o n and s u b t r a c t i o n e r r o r s made these e r r o r s f o r very  reasons.  One student had a s y s t e m a t i c f a u l t y a l g o r i t h m f o r both  o p e r a t i o n s which i n v o l v e d r e g r o u p i n g .  The o t h e r s y s t e m a t i c a l l y chose  107  adding, i n s t e a d of s u b t r a c t i n g . Anderson ( 1 9 8 2 ) (Hart, 1 9 8 3 ;  These f i n d i n g s s u p p o r t e d t h e work o f  and re-emphasized t h e importance o f i n t e r v i e w s i n d i a g n o s i s  Sleeman, 1984c).  The e r r o r s made by t h e s e two s t u d e n t s would  not have been i d e n t i f i e d by a p a p e r - a n d - p e n c i l t e s t . ' Common e r r o r s were d e f i n e d as s y s t e m a t i c more o f t h e sample (Anderson, 1982). were a l s o common e r r o r s .  e r r o r s which o c c u r r e d i n 5% o r  A l l of the four systematic  errors  I n terms o f t h e e x a c t e r r o r s found i n Anderson's  (1982) s t u d y , o n l y t h e e r r o r s o f ax -bx =  (b-a)x,  =abx  T h i s may have been due t o t h e  were r e p l i c a t e d i n t h e p r e s e n t s t u d y .  fact that quadratics the  had n o t been i n t r o d u c e d  |b|>|a|) and ax +bx  t o t h e s t u d e n t s a t t h e time o f  interviews. The  four-systematic  algebraic "addition."  and common e r r o r s found i n t h i s s t u d y  involved  A l l o f t h e s e e r r o r s , except l i k e terms, were  s i g n i f i c a n t l y c o r r e l a t e d w i t h students'  mean mathematics marks.  However,  t h e f a i l u r e o f any o f t h e s e e r r o r s t o c o r r e l a t e s i g n i f i c a n t l y w i t h IQ, t h e Lankton  o r t h e CSMS i n d i c a t e s t h a t t h e s e s y s t e m a t i c  and common e r r o r s may  be a f u n c t i o n o f i n s t r u c t i o n a l b i a s , r a t h e r t h a n m a t h e m a t i c a l a b i l i t y o r aptitude. Errors  The  Further  investigation i s required i n t h i s  Relating  to  Previous  area.  Research  e a r l y f i n d i n g s o f Wattawa (1927) and Pease (1929) as w e l l as t h e  more r e c e n t  work o f D a v i s & Cooney (1978) and Anderson - (1982) r e p o r t e d  up t o o n e - t h i r d o f t h e e r r o r s made i n a l g e b r a arithmetic errors.-  that  were a t t r i b u t a b l e t o  I n t h e p r e s e n t s t u d y , a p p r o x i m a t e l y 32% o f t h e e r r o r s  made were c o m p u t a t i o n a l .  Kieran's  (1983)  f i n d i n g t h a t n o v i c e s make t w i c e  108  as many number f a c t e r r o r s as e x p e r t s was  replicated. Addition/subtraction  s i g n e r r o r s o f t e n o c c u r r e d c o n c o m i t a n t l y with these computation e r r o r s , -emphasizing  re  the importance o f f a c i l i t y w i t h these two a r i t h m e t i c  o p e r a t i o n s f o r success i n a l g e b r a . The  r e s u l t s f u r t h e r i n d i c a t e d t h a t u n d e r s t a n d i n g of the concepts of  m u l t i p l i c a t i o n and d i v i s i o n and the a b i l i t y t o e x p l a i n these two  operations  i n c o n c r e t e terms were h i g h l y c o r r e l a t e d w i t h achievement i n a l g e b r a . These f i n d i n g s support the c o n t e n t i o n t h a t s t u d e n t s s h o u l d u t i l i z e  concrete  m a t e r i a l s t o h e l p them c o n c e p t u a l i z e . a r i t h m e t i c o p e r a t i o n s at the elementary  level  m a t e r i a l s may  (Sowder e t a l . , 1 9 8 6 )  still  and t h a t work with c o n c r e t e  be n e c e s s a r y at the secondary  The m a j o r i t y of s i g n i f i c a n t a l g e b r a i c concepts.  The  level.  f i n d i n g s i n t h i s study were r e l a t e d t o  significant  f i n d i n g s f o r the concepts o f v a r i a b l e  and e q u a l i t y s u p p o r t e d the p r e v i o u s r e s e a r c h f i n d i n g s o f K i e r a n Rosnick  & Clement  secondary  (1980).  While c o n c r e t e m a t e r i a l s may  l e v e l t o r e i n f o r c e some a r i t h m e t i c concepts,  suggested by the f i n d i n g s of t h i s study, i t may  (1983)  and  be needed at the as p r e v i o u s l y  be of even g r e a t e r  importance t h a t a d o l e s c e n t s use m a n i p u l a t i v e m a t e r i a l s t o g a i n a concept e q u a t i o n , v a r i a b l e , the e q u a l i t y symbol and the procedures s o l u t i o n of equations  ( H e r s c o v i c s , 1 9 7 9 ; Kinach,  f u r t h e r s u p p o r t e d by the s i g n i f i c a n t  1985).  of  i n v o l v e d i n the  This suggestion i s  i n t e r a c t i o n of the a b i l i t y t o e x p l a i n  the meaning o f e q u a t i o n s and the a b i l i t y t o o b t a i n c o r r e c t f i n a l answers i n solving equations. The  f i n d i n g s of the p r e s e n t study i n d i c a t e d t h a t s t u d e n t s d i d b e t t e r  1 09  between 50 and  w i t h numbers whose a b s o l u t e v a l u e was equations  (Cl).  T h i s may  textbooks  or i t may  on  one-step  t o the p r e v a l e n c e of these numbers i n  be t h a t these somewhat l a r g e r numbers cause s t u d e n t s t o  f o c u s on the p r o c e s s 1980).  be due  100  i f the p r o c e s s  i s easy  (Herscovics, 1979;  While t h e number of e r r o r s i n c r e a s e d w i t h c o m p l e x i t y  Booth, (Booth & Hart,  1 9 8 2 ) , the f a c t t h a t more e r r o r s o c c u r r e d w i t h l a r g e r numbers i n more complex e q u a t i o n s  (C2,  C3)  suggests  t h a t l a r g e numbers may  s t u d e n t s ' a b i l i t y t o f o r m u l a t e a procedure, complex i n n a t u r e  (Pettito,  1979).  interfere  when t h a t procedure  These two  with  i s more  f i n d i n g s may.help t o e x p l a i n  c o n t r a d i c t o r y f i n d i n g s r e p o r t e d by o t h e r r e s e a r c h e r s r e g a r d i n g the  effect  of n u m e r i c a l magnitude on the e q u a t i o n - s o l v i n g p r o c e s s . However, a d d i t i o n a l i n f o r m a t i o n r e g a r d i n g the e f f e c t  of  magnitude on the e q u a t i o n - s o l v i n g p r o c e s s , g a i n e d through interviews,  i n d i c a t e d t h a t the reasons  e a s i e r e q u a t i o n s may  have been due  t r u e f o c u s on p r o c e d u r e s further investigation,  the use  of  f o r the e r r o r s made when s o l v i n g  t o h a n d - c a l c u l a t i o n s , r a t h e r than 1982) .  (Anderson,  i t suggests  numerical  While t h i s  any  finding requires  t h a t s t u d e n t s s h o u l d be  required to  w r i t e out s o l u t i o n s t o easy q u e s t i o n s , r a t h e r than b e i n g a l l o w e d t o complete them m e n t a l l y . S t r o n g e r s t u d e n t s were a b l e t o e x p l a i n both the meaning of an and how  t o s o l v e i t , without  McKnight, 1 9 7 9 ; was  first  Booth & Hart,  n o t e d by K r u t e t s k i i  l e v e l i n the p r e s e n t  study.  a c t u a l l y f i n d i n g the s o l u t i o n 1982) . (1976) Student  equation  (Davis &  T h i s c h a r a c t e r i s t i c of h i g h a c h i e v e r s and h e l d a c r o s s both  age  and  achievement i n a l g e b r a might  grade be  11 0  improved by i n c l u s i o n o f such e x e r c i s e s achieving  i n r e g u l a r c l a s s w o r k , so t h a t  lower-  s t u d e n t s p r a c t i c e t h e s k i l l s o f e x p l a i n i n g t h e meaning o f  e q u a t i o n s and t h e p r o c e s s e s needed f o r s o l u t i o n , without a c t u a l l y s o l v i n g them. The  significant  f i n d i n g s f o r t h e v o c a b u l a r y a s s o c i a t e d with i d e n t i t y  elements and i n v e r s e s  seemed t o i n d i c a t e t h a t t e a c h e r s  include vocabulary i n t h e i r lessons i d e n t i t y elements and i n v e r s e s The  and s h o u l d  should  frequently  emphasize t h e importance of  i n the equation-solving  process.  f i n d i n g s t h a t o n l y f i n a l answers and incomplete s o l u t i o n s were  s i g n i f i c a n t l y c o r r e l a t e d with "outcomes" on t h e d i a g n o s t i c instrument were suprising.  Moreover, t h e f a c t t h a t both s y s t e m a t i c  found i n d i c a t e d t h a t p u r e l y sufficient  when d i a g n o s i n g  s t a t i s t i c a l analyses errors.  and common e r r o r s were  of information  The use o f i n t e r v i e w s  revealed  important d i a g n o s t i c d i f f e r e n c e s which were not s t a t i s t i c a l l y I n t e r v i e w s a r e recommended i n a l g e b r a i c d i a g n o s i s Summary The  of  Assessment  of  i d e n t i f i e d as b e i n g  both s y s t e m a t i c  and common.  on concepts i n v o l v e d i n a l g e b r a i c " a d d i t i o n . "  the d i a g n o s i s variable.  1983) .  study i n d i c a t e d t h a t t h e c h e c k l i s t had  and was v a l i d and r e l i a b l e .  a n a l y s i s with semi-structured  significant.  the Checklist  f i n d i n g s of the present  diagnostic usefulness  (Hart,  a r e not  interviews  Four e r r o r s were These f o u r e r r o r s  focused  The combination o f e r r o r -  provided  a u s e f u l methodology f o r  o f e r r o r s made by s t u d e n t s i n s o l v i n g l i n e a r e q u a t i o n s i n one  111  Significance  of  the  Study  The purpose o f t h i s study was t o develop a d i a g n o s t i c instrument which would i d e n t i f y t h e n a t u r e and frequency o f e r r o r s made by s t u d e n t s i n s o l v i n g l i n e a r e q u a t i o n s i n one v a r i a b l e and t o e x p l o r e t h e i n t e r r e l a t i o n s h i p s among t h e s e e r r o r s . it  was t h e f i r s t  T h i s study was s i g n i f i c a n t  because  study o f a l g e b r a i c e r r o r s which e s t a b l i s h e d content and  s t r u c t u r a l v a l i d i t y and t r i e d t o r e l a t e t h e c o n c e p t u a l understandings  held  by s t u d e n t s t o t h e a c t u a l p r o c e d u r a l and r e s u l t a n t e r r o r s made when s o l v i n g equations.  The r e s u l t s o f t h i s study p r o v i d e d d i a g n o s t i c i n s i g h t s i n t o the  l e a r n i n g p r o c e s s e s o f b e g i n n i n g students i n a l g e b r a , a n d . i d e n t i f i e d s y s t e m a t i c e r r o r s o f i n d i v i d u a l s t u d e n t s and common e r r o r s among s t u d e n t s . The  i d e n t i f i c a t i o n o f such e r r o r s p r o v i d e s important  diagnostic information  which may be u t i l i z e d i n r e m e d i a t i o n , as w e l l as c l a s s r o o m T h i s study was t h e f i r s t  instruction.  study t o combine e r r o r - a n a l y s i s with semi-  s t r u c t u r e d i n t e r v i e w t e c h n i q u e s t o i n v e s t i g a t e t h e d i f f e r e n c e s among t h e e r r o r s made by s t u d e n t s o f d i f f e r e n t achievement l e v e l s when s o l v i n g algebraic equations.  C o n t r o l o f e q u a t i o n t y p e s and format,  numerical  magnitude, and v a r i a b l e placement with r e s p e c t t o t h e e q u a l i t y symbol and c o n s t a n t s reduced t h e p o t e n t i a l f o r b i a s due t o e q u a t i o n c o n t e n t .  Many of  the p o s s i b l e s o u r c e s o f b i a s i n h e r e n t i n p r e v i o u s r e s e a r c h on a l g e b r a i c e r r o r s were a v o i d e d . The procedures  f o l l o w e d f o r instrument development r e s u l t e d i n t h e  c r e a t i o n o f a d i a g n o s t i c instrument which i s v a l i d ,  reliable,  Such t e c h n i q u e s which address concerns f o r f a c e , content,  and u s e f u l .  structural,  112  c o n c u r r e n t and p r e d i c t i v e v a l i d i t y must be i n c o r p o r a t e d diagnostic  instrument development.  t o address i s s u e s  of r e l i a b i l i t y ,  weakness i n d i a g n o s t i c An  Methods a r e a l s o p r o v i d e d i n t h i s study which have p r e v i o u s l y  been noted as a  instrument development.  attempt was made t o u t i l i z e  both t h e p r a c t i c a l and t h e o r e t i c a l work  on e r r o r s made by s t u d e n t s i n s o l v i n g The  i n future  l i n e a r e q u a t i o n s i n one v a r i a b l e .  combination of semi-structured interviews  with e r r o r a n a l y s i s  provided  a u s e f u l methodology f o r i d e n t i f y i n g common and s y s t e m a t i c e r r o r s i n (Sowder e t a l . , 1 9 8 6 ) .  algebra  to i n d i c a t e that  The r e s u l t s a c h i e v e d i n t h i s study seemed  the e r r o r - c a t e g o r i z a t i o n  scheme used i n t h e d i a g n o s t i c  i n s t r u m e n t was an e f f e c t i v e way o f a s s e s s i n g  errors  students i n algebra.  a l l o w e d easy i d e n t i f i c a t i o n of  The m a t r i x f o r m u l a t i o n  s y s t e m a t i c e r r o r s and t h e s e p a r a t i o n  committed by b e g i n n i n g  o f e q u a t i o n s by c o m p l e x i t y a l l o w e d f o r  easy i d e n t i f i c a t i o n o f e r r o r s a t each l e v e l o f e q u a t i o n The  complexity.  f o u r s y s t e m a t i c and common e r r o r s found i n t h e study appeared t o be  a function  of i n s t r u c t i o n  addition/subtraction errors). exposure  ( l i k e terms, a d d i t i o n p r o p e r t y o f e q u a l i t y ,  s i g n e r r o r s , and c o m p u t a t i o n a l  addition/subtraction  Some c o n c e p t u a l e r r o r s seem t o be r e l a t e d t o age and mathematical (explaining m u l t i p l i c a t i o n , using  c r e a t i n g e q u a t i o n s from words).  t h e e q u a l i t y symbol, and  However, t h e concept o f v a r i a b l e was not  r e l a t e d t o e i t h e r age o r mathematical exposure, s u p p o r t i n g t h e c o n t e n t i o n s of K i e r a n  (1983)  previous research were:  and Rosnick and Clement i n algebra  (1980) .  Other f i n d i n g s  which found support i n t h e p r e s e n t  t h e importance o f e x p r e s s i o n s i n t h e e q u a t i o n - s o l v i n g  from results  process  (Anderson, 1 9 8 2 ; Davis  & Cooney, 1 9 7 8 ) ,  1985;  (Kinach,  of  semi-structured  (Hart, 1 9 8 3 ; Booth & Hart,  i n t e r v i e w s f o r d i a g n o s t i c purposes et a l . , 1 9 8 4 ) ,  the u s e f u l n e s s  the importance of m a n i p u l a t i v e s Sowder e t a l . , 1 9 8 6 ) ,  1 9 8 2 ; Wagner  i n a l g e b r a and  and the f a c t t h a t h i g h  diagnosis  achievers  seemed t o have the a b i l i t y t o e x p l a i n e q u a t i o n - s o l v i n g p r o c e d u r e s executing  (1976)  them as n o t e d by K r u t e t s k i i  without  i n r e l a t i o n t o s o l v i n g word  problems. Limitations L i m i t a t i o n s of the study techniques,  content,  magnitude/structural The  use  of  Study  i n c l u d e d s e l e c t i o n of s u b j e c t s ,  numerical  magnitude o v e r l a p s ,  and  of v o l u n t e e r  numerical  s u b j e c t s l i m i t e d the g e n e r a l i z a b i l i t y of required.  have been more i n t e r e s t e d i n the e d u c a t i o n  Parents  G e n e r a l i z a b i l i t y may  u t i l i z i n g o n l y one  s c h o o l , two  t e a c h i n g of f i r s t - y e a r  a l s o be  l i m i t e d by the  male t e a c h e r s ,  who  g i v e consent  and one  student  s m a l l sample s i z e  textbook used i n the  t o the use  of t h i n k - a l o u d methods.  p r e v i o u s l y mentioned, i t i s p o s s i b l e t h a t t h i n k i n g a l o u d t e c h n i q u e s a c c u r a t e l y r e p l i c a t e thought p r o c e s s e s .  1980)  Students do not  use  once the procedure has become automatic  of automatic procedures may  do  As not  t e n d to v e r b a l i z e (Matz,  .  The  thus  algebra.  F u r t h e r l i m i t a t i o n s o c c u r r e d due  mathematical p r o c e s s e s  results,  of t h e i r c h i l d r e n , and  the sample might not be t r u l y r e p r e s e n t a t i v e of the g e n e r a l population.  interview  format i n t e r a c t i o n s .  p a r t i c u l a r l y s i n c e p a r e n t a l consent was may  the  have i n t e r e f e r e d with some  students  1  a b i l i t y t o a c c u r a t e l y d e s c r i b e what they were t h i n k i n g as they  solved equations  ( H e r s c o v i c s , 1979).  i n t e r f e r e d with r e t r i e v a l of concepts.  The use o f i n t e r v i e w s may a l s o have However, most s t u d e n t s i n the study  i n d i c a t e d t h a t h a v i n g t o " t h i n k - a l o u d " was not a problem.  They a l s o  i n d i c a t e d t h a t t h e y seemed t o make t h e same type o f mistakes i n t e r v i e w t h a t t h e y made on t h e i r homework.  Teacher  i n the  i n t e r v i e w s supported  t h i s contention. The c o n t e n t o f t h e developed d i a g n o s t i c instrument  r e f l e c t e d t h e scope  and sequence o f s e v e r a l c u r r e n t l y used f i r s t - y e a r a l g e b r a t e x t s . the u s e f u l n e s s o f t h i s d i a g n o s t i c c h e c k l i s t change. The  However,  i s l i m i t e d by c u r r i c u l u m  The content o f t h e c h e c k l i s t w i l l need t o be updated  periodically.  f i n d i n g s o f t h e p r e s e n t study were a l s o l i m i t e d by t h e f a c t  o n l y i n t e g e r c o e f f i c i e n t s and s o l u t i o n s were u t i l i z e d .  that  While t h e  l i m i t a t i o n o f i n t e g e r s i n t h e e q u a t i o n s was d e l i b e r a t e t o a l l o w f o r thorough  i n v e s t i g a t i o n o f s i g n and n u m e r i c a l magnitude e r r o r s i n e q u a t i o n -  s o l v i n g , many e r r o r s a s s o c i a t e d with f r a c t i o n s would not be i d e n t i f i e d and thus t h e e n t i r e scope o f f i r s t - y e a r a l g e b r a e q u a t i o n - s o l v i n g p r o c e s s e s was not e x p l o r e d . C u t - o f f v a l u e s f o r n u m e r i c a l magnitude were based on p r e v i o u s r e s e a r c h (Anderson,  1982;  Pettito,  1979).  Some o v e r l a p p i n g o f numbers o c c u r r e d  between t h e second and t h i r d l e v e l s o f n u m e r i c a l magnitude  (N2 and N3).  Such o v e r l a p caused some c o n f u s i o n s as t o the source o f some s t u d e n t s ' computational e r r o r s .  While t h e cause of e r r o r s was e s t a b l i s h e d  through  i n t e n s i v e q u e s t i o n i n g d u r i n g t h e i n t e r v i e w s , these o v e r l a p s d i d p r o v i d e a  l i m i t a t i o n i n t h e study. The  format  equations  o f t h e d i a g n o s t i c instrument s t r e s s e d t h e c o m p l e x i t y o f  (one-step, two-step,  multi-step).  e q u a t i o n s made i t easy t o determine  While t h i s  sequencing o f  where a student was i n terms o f  c u r r i c u l u m , t h i s sequencing made i t d i f f i c u l t  to f i n d patterns f o r either  n u m e r i c a l magnitude problems o r s t r u c t u r a l format problems. i n t e r a c t i o n e f f e c t o f these two f a c t o r s was s i g n i f i c a n t , p r o v i d e d another l i m i t a t i o n o f t h i s Suggestions  Since the  this  difficulty  study.  for  Future  Research  S e v e r a l s t u d i e s need t o be i n i t i a t e d which extend t h e parameters current research.  Firstly,  s c h o o l s t o address concerns  of the  t h e p r e s e n t study s h o u l d be r e p l i c a t e d i n o t h e r r e g a r d i n g t h e p o s s i b i l i t y t h a t s y s t e m a t i c and  common e r r o r s may be i n s t r u c t i o n - b a s e d .  T h i s would a l s o a s s i s t i n  d e t e r m i n i n g i f some e r r o r c a t e g o r i e s c o u l d be e l i m i n a t e d from t h e e r r o r c a t e g o r i z a t i o n scheme t o h e l p " s t r e a m l i n e " t h e i n s t r u m e n t . t e c h n i q u e s used t o e s t a b l i s h concepts and e q u a t i o n s w i t h c o e f f i c i e n t s and s o l u t i o n s s h o u l d be extended  t o concepts  Secondly, t h e  integral involving  f r a c t i o n s and e q u a t i o n s c o n t a i n i n g r a t i o n a l c o e f f i c i e n t s and s o l u t i o n s . The  importance  o f t h e c h o i c e o f m i d - d i f f i c u l t y l e v e l items f o r such an  i n s t r u m e n t cannot  be o v e r - s t r e s s e d , so t h a t t h e i d e n t i f i e d e r r o r s w i l l be a  f u n c t i o n of u n d e r s t a n d i n g and not of an i n a b i l i t y t o f o r m u l a t e a s o l u t i o n (Pettito,  1979) o r because a student c o u l d s o l v e t h e e q u a t i o n through non-  a l g e b r a i c means (Booth & Hart, 1982).  T h i r d l y , d i a g n o s t i c i n s t r u m e n t s need  t o be d e v e l o p e d which f o c u s on q u a d r a t i c e q u a t i o n s and systems o f e q u a t i o n s  116  to provide i n s i g h t  i n t o t h e e r r o r s which occur i n l a t e r work w i t h  algebraic  equations. In f u t u r e r e s e a r c h on a l g e b r a i c e r r o r s , t h e number o f incomplete s o l u t i o n s c o u l d be u t i l i z e d t o p r o v i d e a measure o f mastery f o r each equation type.  While t h e p r e s e n t study r e c o r d e d t h e s e e r r o r s ,  s u g g e s t e d t h a t once a student cannot  attempt  i t is  two o r more o f t h e q u e s t i o n s  f o r each e q u a t i o n type, t h e i n t e r v i e w be t e r m i n a t e d .  T h i s would p r o v i d e an  i n d i c a t i o n o f t h e mastery l e v e l o f t h e student, s h o r t e n t h e i n t e r v i e w time and i n most cases would not l i m i t While  the diagnosis.  t h e f i n d i n g s f o r t h e i n t e r a c t i o n o f n u m e r i c a l m a g n i t i t u d e and  s t r u c t u r a l format  were s i g n i f i c a n t ,  would l e a d t o p a r t i c u l a r  no d e f i n i t i v e p a t t e r n s were found which  remedial techniques.  I t i s suggested t h a t a  computer programme be developed t o enable t e a c h e r s t o e n t e r t h e n u m e r i c / s t r u c t u r e r e s u l t s from t h e d i a g n o s t i c instrument t o determine e f f e c t s f o r each t h i s task  student.  The p r e s e n t f o r m a t t i n g o f t h e instrument makes  onerous.  F u t u r e d i a g n o s t i c i n s t r u m e n t s s h o u l d attempt sets of numerical d i f f i c u l t y 100<|n|<1000) so. t h a t identified. each  ( i e . 0<|n|<20,  i n d i v i d u a l problem  t o use n o n - o v e r l a p p i n g  20<|n|<50,  50<|n|<100,  areas c o u l d be more r e a d i l y  Care w i l l need t o be taken t o o b t a i n m e a n i n g f u l  l e v e l o f n u m e r i c a l magnitude.  In t h e p r e s e n t study,  c a r e f u l i n t e r v i e w t e c h n i q u e s was r e q u i r e d t o determine  r e l i a n c e on  the d i s c r e p a n c i e s .  F o r example, a s t u d e n t made c o n s i s t e n t e r r o r s on e q u a t i o n s numbers between 20 and 5 0 .  equations at  involving  However, through c a r e f u l q u e s t i o n i n g , i t was  1 1 7  r e v e a l e d t h a t these p r i m a r i l y i n v o l v e d e r r o r s w i t h m u l t i p l i c a t i o n f a c t s f o r f o u r and  would have been b e t t e r d e s c r i b e d by numbers between 0 and  Mestre & Gerace rule-based,  (1986) p o i n t e d out t h a t students  r a t h e r than concept-based, d i s c i p l i n e .  t h i s same phenomenon i n the way  t h a t students  20.  viewed a l g e b r a as a Other r e s e a r c h e r s  noted  t r e a t a l g e b r a i c equations  as  something t o be manipulated, r a t h e r than r e p r e s e n t i n g something meaningful ( H e r s c o v i c s , 1979; t o support  these  Siford,  contentions  e q u a t i o n - s o l v i n g process underpinnings. students  1981). and  f i n d i n g s of the p r e s e n t  s t r e s s the importance of t e a c h i n g  As n o t e d by Sleeman  i s important,  When d e v e l o p i n g i s important  future studies should  a d i a g n o s t i c instrument  and  subject's a b i l i t y to solve equations.  equations The  key  i n one  aspects  reliability  and  instructional  conceptions.  which i n v o l v e s i n t e r v i e w s ,  Care needs t o be do not  taken t o ensure t h a t i n t e r f e r e with  G u i d e l i n e s s h o u l d be p r o v i d e d  a v o i d t e a c h i n g i n the d i a g n o s t i c s e t t i n g  processes  While,  on s u b j e c t s t o ensure the  interview questions  when u s i n g i n t e r v i e w s  focus on  of such erroneous  a d m i n i s t r a b i l i t y of the instrument.  The  (Herscovics,  to  1979).  used f o r f o r m u l a t i n g a d i a g n o s t i c instrument  for linear  of mathematics.  of t h i s f o r m u l a t i o n i n v o l v e d concerns f o r v a l i d i t y balance  the  (Wagner e t a l . , 1984), which o f t e n  v a r i a b l e c o u l d be a p p l i e d t o o t h e r areas  a  the  (1984a) the d i f f i c u l t i e s encountered by  t o t e s t the instrument  t h i n k - a l o u d techniques  seem  i n ways which emphasize i t s c o n c e p t u a l  methods which a v o i d the f o r m a t i o n  occurs  study  i n l e a r n i n g a l g e b r a have been g r e a t l y u n d e r - e s t i m a t e d .  diagnosis  it  The  between the p r a c t i c a l and  theoretical  and  u n d e r p i n n i n g s o f t e a c h i n g and r e s e a r c h were e x e m p l i f i e d i n the development of  the d i a g n o s t i c i n s t r u m e n t .  t h e Diagnostic of  Checklist  The c a r e f u l and s y s t e m a t i c approach  for Algebra  p r o v i d e s a model f o r the development  other diagnostic t o o l s using error a n a l y s i s .  such a s y s t e m a t i c approach  used i n  I t i s o n l y by  utilizing  t h a t p r o g r e s s can be made i n the d i a g n o s i s of  s t u d e n t s ' e r r o r s i n mathematics  (Bernard & B r i g h t ,  1982) .  As suggested by Tompkins, C o n n e l l y , and B e r n i e r (1981) : I f s u c c e s s f u l d i a g n o s t i c t e c h n i q u e s can be d e v e l o p e d and expanded and i f t h e s e t e c h n i q u e s can be s u c c e s s f u l l y t r a n s f e r r e d from the r e s e a r c h environment of the c l i n i c t o the l e a r n i n g environment of the classroom, then the e n t i r e massive area o f r e m e d i a t i o n can become the f o c u s of a t t e n t i o n (p. 62). The p r e s e n t study has been an attempt  t o develop such a d i a g n o s t i c  t e c h n i q u e f o r l i n e a r e q u a t i o n s i n one  variable.  119  REFERENCES A d i , H. (1978) . I n t e l l e c t u a l development equation s o l v i n g . Journal for Research 204-213.  and r e v e r s i b i l i t y o f thought i n in Mathematics Education, .9.(3),  Anderson, A. G. ( 1 9 8 2 ) . Error patterns in the simplification of polynomial expressions. U n p u b l i s h e d m a s t e r s t h e s i s , Memorial U n i v e r s i t y o f Newfoundland. S t . John's, Newfoundland. 1  Behr, M., Erlwanger S., & N i c h o l s , E. (1976). How children view equality sentences ( P r o j e c t of mathematical development o f c h i l d r e n ) . (ERIC Document R e p r o d u c t i o n S e r v i c e s No. ED144802). Bernard, J . E. & B r i g h t , G. W. (1982). Student performance in solving linear equations. Paper p r e s e n t e d a t t h e 4th Annual Meeting o f t h e I n t e r n a t i o n a l Group i n t h e Psychology o f Mathematics E d u c a t i o n , Antwerp, Belgium. (ERIC Document R e p r o d u c t i o n S e r v i c e s No. ED220267)'. Booth, L. (1981) . Strategies & errors in generalized arithmetic. P r o c e e d i n g s o f 5 t h Conference o f t h e I n t e r n a t i o n a l Group o f t h e P s y c h o l o g y o f Mathematics E d u c a t i o n (pl40-146). Grenoble, F r a n c e . Booth, L. (1982) . 4-6.  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Proceedings  of 5th  of Mathematics  An A n a l y t i c Framework f o r m a t h e m a t i c a l Conference  Education  of the  International  , (pp 165-170).  Grenoble,  variables.  Group for  Psychology  France.  Wagner, S., R a c h l i n , S., & Jensen, R. (1984). Algebra learning project: final report. U n i v e r s i t y of G e o r g i a and U n i v e r s i t y of C a l g a r y . (NIE C o n t r a c t No. 400-81-0028). Wattawa, V. The  (1927) .  Mathematics  Winer, B.  J.  (2nd e d ) .  (1971) .  New  A s t u d y of t h e e r r o r s i n n i n t h y e a r a l g e b r a  Teacher,  class.  2H(4) , 212-222.  Statistical  Principles  York, M c G r a w - H i l l .  in Experimental  Design  1 26  APPENDIX Phase Error  Categorization  A 2 Scheme  Development  1 27  The  purpose of t h i s appendix i s t o p r e s e n t a more d e t a i l e d d i s c u s s i o n of  how  the e r r o r - c a t e g o r i z a t i o n used i n the Diagnostic  was  developed.  Checklist  Algebra  A summary of r e s e a r c h f i n d i n g s f o r each e r r o r i n c l u d e d i n  the e r r o r - c a t e g o r i z a t i o n scheme i s p r e s e n t e d at the end tabular  for  of the d i s c u s s i o n i n  form.  Research i n a l g e b r a was  reviewed  and a l i s t  was  c r e a t e d of the  a u t h o r ( s ) , d a t e s , and the d i f f e r e n t e r r o r s i d e n t i f i e d w i t h i n the findings.  research  Many of the e r r o r s found had d i f f e r e n t names i n d i f f e r e n t  s t u d i e s , but were the same e r r o r , which supports the i d e a t h a t much of the r e s e a r c h i n a l g e b r a has been plagued by semantic, d i f f e r e n c e s as noted by F i r e s t o n e ( 1 9 8 7 ) .  r a t h e r than  conceptual,  Those e r r o r s which were not  l i s t e d by t h r e e o r more p r e v i o u s r e s e a r c h e r s were e l i m i n a t e d , because e r r o r s may  have been s p e c i f i c t o one p a r t i c u l a r p i e c e of r e s e a r c h , due  the n a t u r e of the e q u a t i o n s The  list  of remaining  used w i t h i n t h a t  e r r o r s was  account  (1982).  formulated  These r e s e a r c h e r s suggested  i n t o an e r r o r c a t e g o r i z a t i o n Booth  (1981),  and Bernard  t h a t d i a g n o s i s s h o u l d take  p r e v i o u s knowledge, p r o c e s s e s used, and r e s u l t s o b t a i n e d .  suggestions  to  study.  scheme based on the work of C a r r y e t a l . ( 1 9 8 0 ) , Bright  such  seemed t o imply t h a t a d i a g n o s t i c instrument  into  Such  f o r beginning  algebra should c l a s s i f y e r r o r s into three general types: p r o c e d u r a l , and  resultant.  a s s i g n e d t o one  of these t h r e e main c l a s s i f i c a t i o n s i n the  conceptual,  Each e r r o r found by p r e v i o u s r e s e a r c h e r s  was  error-  c a t e g o r i z a t i o n scheme. The  t h r e e main c a t e g o r i e s - c o n c e p t u a l , p r o c e d u r a l , r e s u l t a n t  - were  &  1  defined  i n t h e f o l l o w i n g manner.  Conceptual  errors are errors i n  u n d e r s t a n d i n g o f t h e concepts and v o c a b u l a r y a s s o c i a t e d algebra.  Procedural  involved  i n arithmetic  diagnostic insight  with  first-year  e r r o r s a r e e r r o r s made i n a t t e m p t i n g t o a p p l y  procedures t o the equation-solving  process.  computation, s i g n s ,  Resultant or copying.  algebraic  e r r o r s are e r r o r s By b a s i n g t h e  instrument on a c o n c e p t u a l - p r o c e d u r a l - r e s u l t a n t  model,  further  i n t o a s t u d e n t ' s a b i l i t y t o s o l v e l i n e a r e q u a t i o n s i n one v a r i a b l e  s h o u l d be p r o v i d e d i n these three  because t h e i n t e r r e l a t i o n s h i p s between e r r o r s  areas can be  to students' e r r o r s i n algebra  (Matz, 1980).  and e q u a t i o n s have  been shown t o i n t e r f e r e with s t u d e n t s ' a b i l i t y t o understand (Anderson, 1982; H e r s c o v i c s ,  Wagner, 1981a).  1980; K i e r a n ,  algebraic  1981a, 1983; S i f o r d ,  Hence, assessment o f b a s i c a l g e b r a i c concepts and  vocabulary i s necessary f o r Mis-application  contributes  M i s i n t e r p r e t a t i o n of the  meaning o f v a r i a b l e , t h e e q u a l i t y symbol, e x p r e s s i o n s ,  instruction  occurring  explored.  A l a c k o f u n d e r s t a n d i n g o f v o c a b u l a r y and a l g e b r a i c concepts  1981;  diagnosis.  o f a l g e b r a i c procedures may account f o r many o f the  e r r o r s which o c c u r i n a l g e b r a  (Matz, 1980) .  Faulty algorithms,  inefficient  p r o c e d u r e s , and f a i l u r e t o make t h e t r a n s i t i o n from mental a r i t h m e t i c t o algebraic operations (Davis  28  result  i n procedural  & Cooney, 1978; D r i s c o l l ,  e r r o r s i n l a t e r a l g e b r a i c work  1982; Hart, 1981; H e r s c o v i c s ,  Movshovitz-Hadar e t a l . , 1987; Sleeman,  1984a,  s c r u t i n y and i d e n t i f i c a t i o n o f t h e p r o c e d u r a l provide useful diagnostic  information  (Davis  1984b,  1984c).  1979; Careful  e r r o r s made by s t u d e n t s & Cooney, 1978).  As such,  1 29  means f o r i d e n t i f y i n g p r o c e d u r a l e r r o r s s h o u l d be i n c l u d e d i n a  diagnostic  instrument. Some e r r o r s found i n a l g e b r a are not based on e i t h e r or p r o c e d u r e s .  These e r r o r s are c a l l e d r e s u l t a n t  computational e r r o r s , e r r o r s may i n t e r f e r e  Sleeman,  is  resultant  1984c; Tatsuoka et errors is  ( E n g l e h a r d t , 1977;  al.,  1980).  essential  Hence,  E n g l e h a r d t & Wiebe,  a means  in a diagnostic  for  instrument.  p r o c e d u r a l , and  e r r o r s f o r complete d i a g n o s i s  Identical  an a l g e b r a i c c o n c e p t , a c c u r a t e copy,  different  Resultant  important t o d i s t i n g u i s h among c o n c e p t u a l ,  answers made by d i f f e r e n t  7n-l=48,  include  with the u n d e r s t a n d i n g of a l g e b r a i c concepts and the  identifying resultant It  concepts  e r r o r s , and they  s i g n e r r o r s , or h a b i t u a l c o p y i n g m i s t a k e s .  d e m o n s t r a t i o n of a l g e b r a i c procedures 1981;  algebraic  sign,  in algebra.  s t u d e n t s may be due t o an i n a b i l i t y t o understand  t o a p p l y an a l g e b r a i c p r o c e d u r e , o r computation r e s u l t .  One may make a l i k e - t e r m e r r o r ,  n=8;  w h i l e the o t h e r may i n d i c a t e t h a t  fact  error in d i v i s i o n occurred.  conceptual,  It  p r o c e d u r a l , and r e s u l t a n t  interrelationships.  is  research findings  8 f o r two v e r y 6n=48,  o n l y by d i s t i n g u i s h i n g among  errors that  I n c l u d e d with each e r r o r i s  summary of the  solving  7n=49, then s t a t e n=8 because a b a s i c  Such u n d e r s t a n d i n g i s  found t h a t e r r o r i n t h e i r r e s e a r c h .  of  in  i n d i c a t i n g that  one can understand t h e i r  c r i t i c a l f o r thorough  The e r r o r - c a t e g o r i z a t i o n scheme developed f o r t h i s Table A . l .  or t o o b t a i n an  F o r example,  two s t u d e n t s may o b t a i n the i n c o r r e c t r e s u l t reasons.  incorrect  study i s  a l i s t i n g of the  presented  investigators  This tabular presentation  i n a l g e b r a f o r the past  diagnosis.  who  represents  sixty-six  in  years.  a  1 30  The  columns i n t h a t  t a b l e p r o v i d e a summary of the  individual investigators.  The  which a p a r t i c u l a r e r r o r was The  table  reveals  (33 out  diagnosis  and  identified  rows p r o v i d e a summary of the  i d e n t i f i e d by  of 46).  investigators.  of the e r r o r s found are  For the p e r i o d  From  literature,  almost e q u a l  and  r e s u l t a n t e r r o r s can  43,  The  work from 1 9 8 3 . u n t i l the  seems t o be  focussing  resultant errors The  (16,  of 140).  more on c o n c e p t u a l and 8,  compared with 2/  developed e r r o r - c a t e g o r i z a t i o n  research  done i n a l g e b r a  b r i n g t o g e t h e r the future  out  research  diagnosis  may  to obtain  respectively,  another  seen  (49,  present  out  of  than  26).  scheme r e p r e s e n t s a s y n o p s i s of  i d e n t i f i e d e r r o r s under one  yet  be  procedural errors rather  over the p a s t seven decades.  f o c u s on d i a g n o s i s  in  of 1975-1982 when concern f o r  r e m e d i a t i o n became paramount i n the  respectively,  the  resultant  concern f o r c o n c e p t u a l , p r o c e d u r a l , 48,  by  frequency with  some i n t e r e s t i n g h i s t o r i c a l h i g h l i g h t s .  . p e r i o d of 1925-1975 the major t h r u s t nature  errors  leading  error-category.  the  I t i s an attempt  semantic u m b r e l l a ,  so  to remediation, rather  to  that than  Tfeble A . l Suimary of Errors in Algebra Investigators  Types of Errors  Variable Expression Equality Equation Coefficient L i t e Terms Inverses Identity Elements PROCEDURAL  X  x  X  X  mechanical/perceptual random inccnplete  X  X X X  X X  X  X  X X  X  X  X  x  X X X  X  X X  X. X  X  X X  X  X X X  X  X  x X  X x  X  X  X  X  X  X X  X X X  X X X  X X X  X  X X  X  X  X X  X  X X  X  X X X X  X X  X  X X  X  X  X  X X X  X X X X X X. X X  X  X  X  X  X  X X X  X  X  X X  X X  X  X  X  X x  }C  X  f< X X  X  X X  X  X X X  X X  X  X  X  X X X  X X  X X  X X  X  X X  X  X X X X  X  X  X  X  X  X X  X  X  X  X X X X X  X X X X  X  X X  X X  x X  X'  X  X  X  X  X X  X X  X X  X  X X X X  X  X  X X  X  X X  X  Zero Annexation Identity Confusion Like Terms (Conjoining) X Partial Distributive X X Order of Operations + property of = X x property of » numeric coefficients RESULTAOT-Camputational add/subtract X. X tnult iply /divide X X basic facts X faulty algorithm X wrong operation X RESULTANT-Sign add/subtract multiply/divide rule-based distributive RESULTANT-Other  x  X  X  X X  X  1 32  APPENDIX Phase Equation  Part  1:  Part Part Part  2: 3: 4:  Construction,  B 3  Testing,  and  Twelve Equation Types and for their Construction Four Test-Forms Interaction Effects Final Equations  the  Selection  Rules  1 33  The purpose o f t h i s appendix procedures  i s t o p r e s e n t a d e t a i l e d d i s c u s s i o n of the  and a n a l y s e s used i n t h e c o n s t r u c t i o n and s e l e c t i o n of the  e q u a t i o n s f o r t h e Diagnostic  Checklist  for Algebra.  final  P a r t 1 p r e s e n t s each  o f t h e t w e l v e e q u a t i o n t y p e s found i n Phase 1 of the s t u d y , t o g e t h e r w i t h a 4-by-4 n u m e r i c a l - m a g n i t u d e - b y - s t r u c t u r a l - f o r m a t  grid.  The  rules for  c o n s t r u c t i n g each e q u a t i o n w i t h i n the s i x t e e n c e l l s are p r e s e n t e d . c o n t a i n s c o p i e s of the f o u r t e s t - f o r m s used i n Phase 3.  Part 2  Part 3 d e t a i l s  the  i n t e r a c t i o n e f f e c t s among the c o m p l e x i t y of the e q u a t i o n s , t h e i r  structural  format, and the magnitude of the numbers used i n the e q u a t i o n s .  Part 4  p r e s e n t s a t a b u l a r l i s t i n g of t h e f i n a l e q u a t i o n s which were s e l e c t e d , t o g e t h e r w i t h s t a t i s t i c s which f a c i l i t a t e d t h e i r  selection.  Part Twelve and their  1  Equation the  Rules  Type for  Construction  EQUATION a  STRUCTURAL  20  N2  20<jn|<  50  N3  50<|n|< 100  N4  100<|n|<1000  15=  x+  29=  37  -27+x=23  -57=  i.1  - 5'Mx:---75  -1i»=x-t-  7+x-=-10  8  x + - - 3 5 2 - -701  -31=x+ 09=x+  L A T I M S Q U A R E AND S I G N / M A G N I T U D E First b c  c d  d  c  d  a  3 b  d  a  b  c  6  -UZ 59  2 5 S = x + - 3*4''+  BC0+x=6Q9  a b  S4  S3  xt  x+  FORMAT  S2  S1  |n|<  1  + x = b  NUMERICAL MAGNITUDE  N1  TYPE  R U L E S FOR EQUATION  Number  a=Positive b=Positive 'c=Negative d=Negati\/e  Large Small Larqe Small  Other  -5=  -9+x  <+5= 23+x £»1=-58+x -<*<*£» = 392+x  TYPE Number  +  -  CO  EQUATION TYPE  2  x - a = b  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S1  S3  S2  N1  | n|<  20  x-  13= -3  N2  2 0 < | n]<  50  x-  19= 25  N3  50< |n|<  100  x- -85= -6  -39-x=  58  01=x—59  1 0 0 < |n|<1000  x—121=230  339-x=-17't  tt6f.=x-536  N4  7-x=  S4  12  -1G=x- -3  -5=  _' 7_x= -27  -22=x- 39  26= - 2 1 - x  4  66=  7-x  96-x  -705=-380-x  LATIN SQUARE ANO SIGN/MAGNITUDE RULES FOR EQUATION TYPE  a  b c d  b c d  c d a  d a b  a  b  c  F i r s t Number n - P u s i t i v e Larqn b=Positivp Small (^Negative Lnrqe d=Neqative S m a l l  Other Number +  +  CO  EQUATION TYPE  3  ax = b  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE  S2  S1 N1  |n|<  20  x(  5)=  N2  20<!.nj<  50  x(  8)= -kB  N3  50<|n|< 100  x(  N4 100<|n|<1000  S3  2x= -18  20  -1HX=  kZ  S4  -16=x(  <•)  -kB=x(  -6)  - 18= -3x  kS= 7x '  63  -9x= - 5k  100=x( 1 0 ) *  56= -8x  x ( - <»5)=-•k35  19x= 3-30  976=x(- • 6 1 ) •  -1DOO=200x  -7) =  LATIN SQUARE AND SIGN/MAGNITUDE RULES FOR EQUATION TYPE a b  b  c"  c  d  d 3  c  d  a  b  d  a  b  c  F i r s t ; Number Small b ^ o s i t i v e Small =fteaative Larqe "^Neaative La rue  c  Other Number + _  +  -  * Due t o e x c l u s i o n or r a t i o n a l n u m b e r s , t h i s e q u a t i o n t y p e h a d trj be s w i t c h e d t o f i r s t n u m b e r l a r g e .  •  EQUATION TYPE U  x/a = b  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE  S2  S1  S3  S4  N1  n|<  20  x/  5=  U  13/x=-3  -7=x/  2  -1=  -7/x»  N2  20<|n|<  50  x/  8= - 6  -50/x= 5  -11=x/  -4  3=  <o/x*  N3  50<jnj< 100  9  -63/x=-7  9=x/  8  x/- 319=- -29  378/x=5^  N4 100<|n|<1000  x/ - 1 0 =  ^ • 0 = x / - •200  L A T I N 5QUARE AND SIGN/MAGNITUDE RULES FOR EQUATION TYPE  a  b  b  c d a  c d  c d a  d a  b  c  b  F i r s t Number a= P o s i t i v e L a r q e b=Positive Large c= N e q a t i v e L a r g e d=Negative Large  Other  Number +  -  +  -  * Dun t o e x c l u s i o n o f r a t i o n a l n u m b e r s , t h i s a q u a t i o n h a d t o b e s w i t c h e d t o f i r s t number s m a l l .  type  20=-100/x* -19=  361/x»  1  SIGN/MAGNITUDE ASSIGNMENT OF FOR EQUATION TYPES  Structural  Integer  Signs  EQUATION 5-12*  TYPES  Magnitude*  Level  a  b  c  d  e  4  +  +  +  +  -  Large  3  +  +  +  -  +  Small  2  +  +  -  -  -  Large  1  +  +  -  +  +  Small  3  +  -  +'  +  +  Large  4  +  -  +  -  -  Small  1  +  -  -  -  +  Large  2  +  -  -  +  -  Small  2  -  +  +  +  -  Large  1  -  +  +  -  +  Small  4  -  +  -  -  -  Large  3  -  +  -  +  +  Small  3  -  -  +  +  +  Large  4  -  -  +  -  -  Small  1  -  -  -  -  +  Large  +  -  Small  2  '  -  3  l a t i o n Types were a s s i g n e d t o t h e16 s i g n p o s s i b i l i t i e s f o r * c so t h a t s u b t r a c t i o n was used once i n each s t r u c t u r a l l e v e l and once at each n u m e r i c a l l e v e l . Assignment o f e q u a t i o n type was a l s o made so t h a t each s t r u c t u r a l l e v e l had even d i s t r i b u t i o n o f l a r g e / s m a l l f o r t h e s p e c i f i c comparisons between numbers. Each s t r u c t u r a l l e v e l a l s o had e q u a l d i s t r i b u t i o n o f s i g n s o f a, b, and c with more complex s i g n arrangements b e i n g a s s i g n e d t o s m a l l e r l e v e l s o f n u m e r i c a l magnitude. ** F o r each o f t h e E q u a t i o n Types 5 - 12, t h e r e i s an assignment l i n e at the bottom o f each g r i d . T h i s assignment l i n e i n d i c a t e s which o f t h e n u m e r i c a l c o e f f i c i e n t s ( e i t h e r a, b, c, d, o r e) have been a s s i g n e d t o t h e l a r g e / s m a l l magnitudes as i n d i c a t e d i n t h e c h a r t on t h i s page.  EQUATION TYPE  5*  ax + b = c  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S1  S3  S2  N1  |n|<  20  -3x-  N2  20<|n|<  50  -9x+ 23=  N3  50<|n|< 100  17x+--100= -15  -77- -x= -86  N4 100<|n|<1000  33x+ 445=--545  734+53x=--220  17= - a 50  15+-3x= -23-  0  x= -48  -13= -5x+  S4 2  0=--29x- 29  -3=  16+ -1x  28= -21-  7x  6x+ 33  61= -31+ 23x  259= 14x-637  100=1D00+ 30x  93=  The assignment of large/small magnitude refers to b compared to c within Equation Type 5.  EQUATION TYPE 6* ax + bx = c  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE  N1  |n|<  20  N2  20<|n|<  50  N3  50<|n|< 100  N4 100<|n|<1000  S3  S2  S1  S4  -x-  -5x= -12  18=-10x- -8x  -16=  -6x+ 2x  -15x-  x= 48  -48= -5x- 7x  37=  22x+ 15x  Bk  45x- -46x= -91  88= 56x- 67x  96= 81x- 87x  -370x+-•130x=-•1000  182x- 168x=-196  225='l20x-105x  720=-•3S0x+560x  8x+  9x= -17  33x+ -13x= -83x+  97x=  -50  i  *  The assignment of large/small magnitude refers to a compared to b within Equation Type 6.  EQUATION TYPE  7*  a x + b = cx  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S1  N1  |n|<  S2  S3  20  2x+ 20= - 3 x  -20-  3x= - 1 3 x  13x=  -x=  -6x  50x=  N2  20<|n|<  50  x + - 4 3 = - 2 3x  -35-  N3  50<|n|< 100  -79x+-75=-54x  -91-  N 4 100<|n|<1000  -15ux+782 =10'lx  :  *  The a s s i g n m e n t o f  53x= - 6 0 x  3 3 3 - 1 5 8 x = -- 1 2 1 x  7x-  S4 18  -7x=  <+1x- - 3 6  -28x=  -83x= - 7 5 x -  16+  x  21+-49x  56  9 3 x = - 8 7 + 74 x  124x=-- 2 7 6 x ~ - 8 0 0  971x=420+985x  l a r g e / s m a l l magnitude r e f e r s to a compared to c w i t h i n  E q u a t i o n Type 7 .  EQUATION TYPE a(x  + b) = c  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S1  N1  8*  |n|< 20  2(x-  1)= - 1 8  S2 2(  S4  S3  5 + x ) = -a  16=  4 ( x + 3)  20=-5( - 1 - x )  N2  20<|n|<  50  &Kx-  -5)= - 2 d  16(-7+x)=-48  46=  23(x+-^1)  26=13(  N3  50<|n|< 100  -7(x-  9)= 9 8  -6(-5+x)=-G0  -72=  -8(x+ 7)  57=19( - 2 - x )  -17(x--2D)=-204  -21(21+x)=399  N4 100<|n|<1000  *  The a s s i g n m e n t o f l a r g e / s m a l l m a g n i t u d e u i t h i n E q u a t i o n Type 8 .  288=-12(x+-24) r e f e r s to  the product  3-x)  -435=-5(12D-x)  o f ab as opposed t o c  EQUATION TYPE 9*  ax+b+cx+d=e  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE SI  N1  |n|< 20  N2  20<|nK_  50  N3  50<|n|< 100  N4 100<|n|<1000  -4x= -10  11= -kx-  3x+  3  13x=  0  30=-13x- 12- -6x+  7  -29=  68+ -43x= -63  %= 49x—39- 31x+  1  -62=-77-  x+  8+  -11x+ 14+ 14x+-29=33  29- -9x+  15+  42x+--28+ -54x+-18=38  -35—27x+  -4x+ -7+  -5x+ -5= 6  151x+ 95+-188x+162=14  -3-  S4  S3  S2  711-165x+--613+-•175x=-582  -8-  191=155x-146- 75x+-•303  0= -7—12x—17- 2x 9- 28x- -4~25x 2x—78-  x  -354=198-270x- 98-184x  ThE assignment of large/small magnitude refers to b compared to d within Equation Type 9. For the f i r s t two levels of structural difficulty a<c and for the last two levels of structural difficulty a>c.  EQUATION TYPE 1 0 * ax  + b = cx +  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S1  N1  |n|< 20  7x+  50 2 1 X +  N2  20<|n|<  N3  50<|n|< 100  N4 100<|n|<1000  d  -75x +  8 = -33= 69 =  S3  S2 3x=  9-  -x  -8  49- -27x =  16-  6x  -87  -15- -20x =  75-  _5x  -5x +  20  -4x + 3x +  - 5 1 3 X + - 156 = - 4 6 3 x + - -606  . - 7 -  7 1 5 - 123x=- 2 3 - - 118x  - 2 x - -0= - 1 5 x -  S4  5x-  6  1 1 -=4 0 x +  34  2 4 x - 52=  47x-  40  9 7 x - 1 8 3 = 1 2 3 x - - 337  4 + + .  89 +  -4x =  -5 +  5x  -35x =  -12 +  -3Dx  55x =  23 +  6 1x  - 3 0 0 + - 167x = - 1 0 0 U + - 237x  The a s s i g n m e n t o f l a r g e / s m a l l m a g n i t u d e refers to d as opposed to b Equation Type 1 0 . For the f i r s t two l e v e l s o f s t r u c t u r a l d i f f i c u l t y and f o r t h e l a s t two l e v e l s of s t r u c t u r a l d i f f i c u l t y a ^ c .  within a>c  EQUATION TYPE 1 1 * a(x  + b)  = cx + d  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S2  S1  S3  S4  N1  |n|<  20  -8(x-  -2)=-9x-  -1  2(  -5+x)=  9- -x  4x +  2=-3(x--4)  -4+ 12x=  N2  20<|n|<  50  13(x-  3)=-2x-  9  -9(  -4+x)=  4B—3x  17x+  8= 7 ( x - S )  -2+-18x=-  N3  50<|n|< 100  7(x — 14)=-3x-  -8  33(  3+x)=  -1--8x  52x+-16=15(x-  N 4 100<|n|<1000  -24G(x-  4)=54x- -84  -5( 161+x)=125-  2x  66x+  6)  44=26(x-26)  -9+  24x=  26+196x=  The a s s i g n m e n t o f l a r g e / s m a l l m a g n i t u d e r e f e r s t o b c o m p a r e d t o d w i t h i n E q u a t i o n Type 1 1 . F o r t h e f i r s t t w o l e v e l s o f s t r u c t u r a l d i f f i c u l t y a<-c f o r t h e l a s t two l e v e l s o f s t r u c t u r a l d i f f i c u l t y a > c .  -8(  -2 + x) 4+ x)  - y ( - 10 + x ) 45(  14 + x )  EQUATION TYPE 12* a(x + b) = c(x + d)  STRUCTURAL FORMAT  NUMERICAL MAGNITUDE S2  S1 N1  |n|< 20  N2  20<|n|<  50  N3  50<|n|< J O O  N4 100<|n|<1000  3(x+-2)= -2(x+ B ) -13(x+-3)=-11(x+-1)  -2( -9(  8-x)=  S3  1( 2+x)  - 1 - x ) = - 6 ( 6+x)  8(x+-9)  10 ( 10-x)= - 9 ( - 9 + x )  100(x+-8)=-60(x+ 0 )  25(- -24-x)=-15(48+x)  -9(x+  *  8)=  -3(x+  S4 4 ( x - 1)  - 3 ( - 2 --x)=  3 0- 4 - x )  10(x+--5)= 2 Q ( x - 2)  6(-6- -x)=  7( - 7 - x )  -9(x+  6)=  '+)=-12(x- 5)  1 3 9 ( x - 1 ) = 1 6 1 ( x - -5)  The assignment of large/small magnitude r e f e r s to d as opposed to b within Equation Type 12. For the f i r s t two l e v e l s of s t r u c t u r a l d i f f i c u l t y a>c and f o r the l a s t two l e v e l s of s t r u c t u r a l d i f f i c u l t y a<.c.  - 8 ( 1 2 --x)=-14(-- 3 - x ) 1Mt(  6--x)=160( 5 - x )  Part Four  2  Test-Forms  1 49  TEST-FORM NAME:  1  Read each equation carefully. Show a l l of your work in the space at the right of the equation. Place your answer in the correct answer space provided in the column at the right. You have the entire period to complete the 46 equations. When you are finished and have checked your work, raise your hand, and your paper will be collected. Good Luck! PART  A  1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)  x + 15 = 8 45 = 28 + x 89 = x + 59 800 + x = 609' -16 = x - -3 -47 - x = -27 x 85 = -6 -705 = -380 - x 2x = 18 -56 = 8x x(45) = -495 -48 = x(-6) -1 = -7,'x -11 = x/-4 -63/x = -7 X/-31 '= -28  PART  B  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32)  -3x - 17 = -8 28 = -21 - 7x 93 = 6x + 33 734 + 53x = -220 -x - -5x = -12 38x + -13x = -50 96 = 81x + -87x 225 = 120x -105x 2x + 20 = -3x -28x = 21 + -49x -83x = -75x - 56 333 - 158x = -121x 16 = 4(x + 3) 16(-7 + x) = -48 -7(x - 9) = 98 -435 = -5(120- x)  PART  C  33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43)  -3 - x +.8 + -4x = -10 - l l x +. 14 + 14x + -29 = 33 -64 = -77 - 2x 78 - x 191 = 155x - 146 - 75x + -303 4 + -4x - -5 + 5x -5x - 11 = -40x + 34 -15 20x - 75 - -5x -513x + -156 - -463x + 606 4x + 2 = -3(x - -4) -9 (-4 + x) = 48 - -3x 7(x 14) - -3x 8 26 + 196x » 45 (14 + x) -3 (-2 -x) - 3 ( - 4 -x) 10(x + -5) = 20(x - 2) 10(10 - x) - -9(-9 + x) 100(x + -8) - -60(x + 0)  44)  45) 46) 47) 48)  -  -  -  ONE-STEP  TWO-STEP  EQUATIONS  1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)  14) 15) 16)  EQUATIONS  MULTI-STEP  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32)  EQUATIONS  33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)  1 50 TEST-FORM 2 NAME: Read each equation carefully. Show a l l of your work in the space at the right of the equation. Place your answer in the correct answer space provided in the column at the right. You have the entire period to complete the 46 equations. When you are finished and have checked your work, raise your hand, and your paper will be collected. Good Luck! PART A - ONE-STEP EQUATIONS 1) 7 + x = -10 1) 2) x + 29 = 37 2) .. 3) 41 = -58 + x 3) 4) 256 = x + -344 4) 5) 5 = 7 -x 5) . . 6) -22 = x - 39 6) ..' 7) -39 - x = 58 7) 8) x - -121 =288 8) 9) x(-5) =20 9) 10) 17x = -51 10) 11) 100 = x(10) 11) 12) 1000 = 200x 12) 13) 7 = x/2 13) 14) 50/x = 5 14) 15) x/-10 =9 15) 16) -19 = 361/x 16) PART B - TWO STEP EQUATIONS 17) 5 + -3x =0 17) 18) -9x + 23 = 50 18) 19) 1 = -31 + 23x 19) 20) 259 = 14x - 637 20) 21) 18 = -lOx 8x 21) 22) -15x - x = 48 22) 23) -83x + 97x = 84 23) . 24) 720 = -380x + 560x 24) 25) -20 - 3x = -13x 25) 26) x + -48 = -23x 26) 27) 3x = -87 + 74x 27) 28) 124x = -276x 800 28) 29) 20 = -5(-1 - x) 29) 30) 46 = 23(x + -21) 30) 31) -6(-5 + x) - -60 31) 32) -17(x - -20) = -204 32) PART C - MULTI-STEP EQUATIONS 33) 11 = -4x 8 - 3x + 3 33) 34) 29 - -9x + 15 + 13x = 0 34) 35) 2x + -28 + -54x + -18 " 38 35) . 36) -354 = 198 - 270x - 98 - 184x 36) 37) 37) 7x + 8 - -5x + 20 38 + -35x = -12 + -30x 38) 38) 39) 24x - 52 = 47x - 40 39) 40) 715 - 123x - -23 - 118x 40) 41) -4+ I2x - -8(-2 + x) 41) 42) 17x + 8 - 7(x - 6) 42) 33(33 + x) - -1 - -8x 43) 43) 44) -246(x - 4) - 54x 84 44) 3(x + -2) = -2(x + 8) 45) 45) 6(-6 - x) = 7(-7 - x) 46) 46) 47) -9(x + 4) = -12(x - 5) 47) 25 (-24 - x) -15 (48 + x) 48) 48)  TEST-FORM 3 NAME: Read each e q u a t i o n c a r e f u l l y . Show a l l o f y o u r work i n t h e space a t t h e right of the equation. P l a c e your answer i n t h e c o r r e c t answer space p r o v i d e d i n t h e column a t t h e r i g h t . You have t h e e n t i r e p e r i o d t o complete t h e 46 e q u a t i o n s . When you a r e f i n i s h e d and have c h e c k e d y o u r work, r a i s e y o u r hand, and your paper w i l l be c o l l e c t e d . Good Luck! PART A - ONE STEP EQUATIONS  1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)  -14 = x + 6 -27 + x = 23 x + -57 - 41 -444 - 392 + x x - 13 = -3 26 = -21 - x 81 = x 59 389 - x = - 174 - 18 = -3x -9x = 54 x(8) = 48 -976 = x(61) 18/x = -3 x/8 = -6 20 = -100/x 400 = x/-20  PART  8  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32)  -13 = -5x + 2 -23 - x = -48 17x + -100 = -15 100 = 1000 + 30x -16 = -6x + 2x -48 = -5x - 7x 45x 46x = -91 -370x + -130x = -1000 13x = 7x - 18 -35 x = -6x -79x + -75 = -54x 971x = 420 + 985x 2 (x - 1) = -18 26 = 13(3 - x) -72 = -8 (x + 7) -21(21 + x) = 399  PART  C  33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44)  0 - -7 12x - -17 - 2x 30 - -13x - 12 6x + 7 -35 27x + 68 + -43x = -63 151x + 95 + -188x + 162 - 14 - 7 - 3x = 9 - -x 21x + -33 - -4x + -8 89 + 55x - 23 + 61x 97x - 183 - 123x - 337 -8(x 2) - -9x - -1 -2 + -18x - -11(4 + x) 52x + -16 - 15(x - 6) -5(161 + x) - 125 - 2x  45)  -2(8 - x) » 1(2 + x) -13(x + -3) - - l l ( x + -1) -8 (12 - x) - -14 (-3 - x) 138(x - 1) - 161(x - -5)  46) 47) 48)  -  -  TWO STEP  '  MULTI-STEP  1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)  EQUATIONS  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32)  EQUATIONS  33) .34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)  1 52 TEST-FORM NAME:  4  Read each equation carefully. Show a l l of your work in the space at the right Of the equation. Place your answer in the correct answer space provided in the column at the right. You have the entire period to complete the 46 equations. When you are finished and have checked your work, raise your hand, and your paper will be collected. Good Luck! PART  A  1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)  -5 = -9 + x -31 = x + -42 -64 + x = -75 x + -352 = -701 7 - x = 12 x - 19 = 25 66 = 96 - x 464 = x - 536 49 = -7x -19x = -380 x(-7) = -63 16 = x(4) -7 = x/2 -50/x = 5 x/-10 = 9 -19 = 361/x  PART  B - TWO STEP - 3 = 16 + -lx  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30)  -  ONE STEP  EQUATIONS  1) 2) 3) 4) 5) 6) 7) 8)  9) 10) 11) 12) 13) 14) . 15) 16)  EQUATIONS  0 = -29x - 29 -77 - -x = -36 33x + 445 = -545 8x + 9x = -17 37 = 22x + 15x 88 = 56x - 67x 182x - 168x = -196 -7x = 16 + x 50x = 41x 36 -91 - 53x = -60x -160x + 783 = lOlx 2 (5 + x) = -8 28(x 5) = -28 57 = l 9 ( - 2 - x)  31) 32)  288  PART 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)  C - MULTI-STEP EQUATIONS 4x + -7 + -5x + - 5 = 6 -29 = 9 - 28 - -4 - - 2 5 x 94 = 49x 39 - 31x + 1 711 - 165x + -613 + -175x - -582 -2x 8 = 5x - 6 49 - -27x = 16 - 24x - 7 6 x + 69 = 3x + -87 -300 + -167x = -1000 + ,-237x 2 ( - 5 + x) - 9 x 13(x - 3) - - 2 x - 9 - 9 + 24x - - 9 ( - 1 0 + x) 66x + 44 - 26(x - 26) - 3 ( x + 6) - 4(x - 1) - 9 ( - 1 - x) - - 6 ( 6 + x) - 9 ( x + 8) - 8<x + -9) 144(6 - x) - 160(5 - x)  = - 1 2 ( x + -24)  17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48)  Part Interaction  3 Effects  1 54  Phase 3 i n v o l v e d a d m i n i s t r a t i o n o f f o u r e q u a t i o n t e s t of  eighty subjects.  equations; each c e l l  Each t e s t  form c o n s i s t e d o f 48 o f the 192 c o n s t r u c t e d  f o u r from each o f t h e twelve o f t h e 4 by 4 n u m e r i c a l  e q u a t i o n t y p e s , making t h r e e  magnitude by s t r u c t u r a l format  An a n a l y s i s o f v a r i a n c e was performed on raw s c o r e s . complexity  forms t o a t o t a l  from  grid.  F a c t o r s were  (one-, two- and m u l t i - s t e p e q u a t i o n s ) , n u m e r i c a l  magnitude  < 2 0 , 20 < |n| < 50, 50 < |n| < 100, and 100 |n| < 1000), and  (In)  s t r u c t u r a l format presents  (e.g. n +a=b, a+n =b, b=n +a, and b=a+n ).  the equation  test-form  results.  I n d i v i d u a l l e v e l s of equation  complexity  ( C l , C2, C3) were compared  u s i n g S h e f f e ' s M u l t i p l e Range Test  (Glass & S t a n l e y ,  performance was h i g h e r on one-step  (Cl) than on two-step  (C3)  equations,  step  (Cl) and m u l t i - s t e p  Table B . l  1970).  While t h e mean  (C2) o r m u l t i - s t e p  r e s u l t s i n d i c a t e d t h a t o n l y t h e d i s c r e p a n c y between one(C3) equations  were s i g n i f i c a n t  [F(2,73)-2.50,  p < .05] . I n d i v i d u a l s t r u c t u r e s (SI, S2, S3, S4) were compared u s i n g M u l t i p l e Range T e s t . four subtests  These showed no s i g n i f i c a n t d i f f e r e n c e s a c r o s s a l l  [F(3,72)=2.87, p > .05].  M u l t i p l e c o n t r a s t s compared  placement o f t h e v a r i a b l e with r e g a r d t o the numbers as w e l l as t h e to  the l e f t  e q u a l i t y symbol  o f t h e numbers  (p < .05).  (SI & S3 vs S2 & S4)  (SI & S2 vs S3 & S 4 ) . P l a c i n g t h e v a r i a b l e  (SI & S3) seemed t o produce h i g h e r s c o r e s  p l a c i n g t h e v a r i a b l e t o the r i g h t o f the numbers was s i g n i f i c a n t  Sheffe's  than  (S2 & S 4 ) . T h i s c o n t r a s t  However, t h e p o s i t i o n o f t h e v a r i a b l e with  r e s p e c t t o t h e e q u a l i t y symbol made no a p p r e c i a b l e d i f f e r e n c e  1 55  (SI & S2 vs S3 & S 4 ) . There were s i g n i f i c a n t four subtests  f i n d i n g s f o r n u m e r i c a l magnitude on t h r e e of the  (See Table B . l ) .  d i f f e r e n c e s among e q u a t i o n s equations  These s i g n i f i c a n t d i f f e r e n c e s seemed due t o  withnumbers of s m a l l e r magnitude (Nl) and  w i t h numbers o f l a r g e r magnitude (N4) [F (3, 72) =2 .27,  However, t h e r e were no c o n s i s t e n t p a t t e r n s on d i f f e r e n t Equations  p < .05].  test-forms.  c o n t a i n i n g s m a l l e r numbers d i d not appear t o have s i g n i f i c a n t l y  h i g h e r s c o r e s than those Equations  c o n t a i n i n g l a r g e r numbers  (See Table B . l ) .  c o n t a i n i n g numbers whose a b s o l u t e v a l u e was g r e a t e r than  l e s s than o r e q u a l t o 100  50 and  (N3) had the same o r more c o r r e c t answers on  t h r e e o f t h e f o u r t e s t - f o r m s than e q u a t i o n s  c o n t a i n i n g numbers  whose  a b s o l u t e v a l u e was g r e a t e r than 20 and l e s s than o r e q u a l t o 50 (N2). On t h r e e of the f o u r t e s t - f o r m s t h e r e were i n t e r a c t i o n e f f e c t s between equation  complexity  and s t r u c t u r a l format,  and n u m e r i c a l magnitude. significant  as w e l l as e q u a t i o n  complexity  On a l l f o u r t e s t - f o r m s t h e r e were h i g h l y  i n t e r a c t i o n s between s t r u c t u r a l format and n u m e r i c a l magnitude.  These i n t e r a c t i o n s are p r e s e n t e d  i n F i g u r e s B . l , B.2, and B.3,  respectively. The  r e s u l t s presented  i n F i g u r e B . l i n d i c a t e d t h a t t h e r e was no  s t r u c t u r a l format which was e a s i e s t a t any g i v e n c o m p l e x i t y example, SI d i d not have the h i g h e s t p - v a l u e S i m i l a r r e s u l t s were found F i g u r e B.2). complexity  Again,  level,  f o r complexity  level.  For  on a l l f o u r t e s t - f o r m s .  and numeric i n t e r a c t i o n s (See  t h e r e was no d e f i n i t i v e p a t t e r n a t a p a r t i c u l a r  and w h i l e on Test-Form 3 t h e r e appeared t o be an  Table B . l Phase 3: E q u a t i o n T e s t - F o r m R e s u l t s f o r E q u a t i o n C o m p l e x i t y , S t r u c t u r a l Format, and Numerical Magnitude  Equation Test-Form 1 Factors Mean F Sig  Equation Cl C2 C3  Test-Form 2 Mean F Sig  Complexity .59 .27 .11 70.90***  Test-Form 3 Mean F Sig  .74 .50 .22  31.99***  4 . 47**  .34 .26 .32 .32  .52 ,42 .50 ,50  4 . 90*  1. 52  .33 .26 .41 .23  .54 .48 .49 .43  6.75**  .69 .32 .07 117.48***  .51 .32 .09  S t r u c t u r a l Format SI .30 S2 .28 S3 . 41 S4 .29 _77***  .39 ,37 .33 .34  Numerical Nl N2 N3 N4  .39 .37 .34 .35  Cl= C2= C3= Sl= S2= S3= S4=  Magnitude .35 .36 .36 .22 7.50***  Test-Form 4 Mean F Sig  34.11***  11.98***  One-step e q u a t i o n Two-step e q u a t i o n s M u l t i - s t e p equations v a r i a b l e s t o l e f t o f numbers & e q u a l i t y v a r i a b l e s t o r i g h t o f numbers & l e f t o f e q u a l i t y v a r i a b l e s t o l e f t o f numbers & r i g h t o f e q u a l i t y v a r i a b l e s t o r i g h t o f numbers & e q u a l i t y  Nl= |n| < 20 N2= 20< |n| < 50 N3= 50< 1n| < 100 N4= 100< | n | < 1000 Sig= L e v e l * = ** = *** =  of s i g n i f i c a n c e p < .05 p < .01 p < .001  Data f o r t h i s t a b l e were o b t a i n e d from BMDP8V - A n a l y s i s ( S P S S X - 1 9 8 2 ) .  of Variance  1 57  interaction  effect,  t h i s was  F i g u r e B . l and F i g u r e B.2, apparent. interaction  In examining effect  While main e f f e c t s  was  not s i g n i f i c a n t  the main e f f e c t  (p >.05).  f o r c o m p l e x i t y was  the data p r e s e n t e d i n F i g u r e B.3, noted f o r s t r u c t u r a l  for structural  format  e f f e c t between t h e s e two  factors  structural  a strong  and n u m e r i c a l magnitude.  due t o the s t r o n g  (See F i g u r e B.3). format,  The  interaction  interactions  among  and n u m e r i c a l magnitude  n e c e s s i t a t e d controls i n equation s e l e c t i o n . magnitude and s t r u c t u r a l  quite  format and numeric magnitude were found  ( r e f e r t o T a b l e B . l ) , these were not apparent  equation complexity,  However, i n both  A l l combinations  of n u m e r i c a l  format needed t o be r e p r e s e n t e d at each l e v e l of  e q u a t i o n c o m p l e x i t y i n the d i a g n o s t i c  instrument.  1 58  Figure B . l Phase 3:  I n t e r a c t i o n E f f e c t s of E q u a t i o n C o m p l e x i t y and S t r u c t u r a l Format  Test-Form 1  Test-Form 2 in  crj  OJ • r-l  QJ  ro :=> i  i  CL  a_ QJ. cn ro  QJ  ai ro PH QJ  OJ  > a:  >  1  1.0 . 9 .8 .7 .6 .5 . .4 .3 .2 .1 0  •A  1  2 . 3 Complexity  Test-Form 4 (TJ  cu 3  QJ  cn ro  \  u  QJ ct  1  2  3  Complexity  51 52 53 54  = = = =  3  Complexity  Test-Form 3  1.0 in .9 QJ .8 .7 .6 I .5 a. .4 QJ cn .3 ro rH .2 cu .1 > cc 0  2  ,0 9 8 7 6 5 4 3 2 1 0 1  2  3  Complexity  V a r i a b l e l e f t of e q u a l i t y , l e f t of constants V a r i a b l e l e f t of e q u a l i t y , r i g h t of constants V a r i a b l e r i g h t , of e q u a l i t y , l e f t o f c o n s t a n t s V a r i a b l e r i g h t of e q u a l i t y , r i g h t o f c o n s t a n t s  1 59  Figure  B.2  Phase 3:  I n t e r a c t i o n E f f e c t s of E q u a t i o n Complexity  and Numerical  Test-Form 1  Test-Form 2 cn  cu  QJ  •  r-t CO  i CL QJ  QJ QI  co u  QJ  ro u  cn  >  QJ  >  CI  CC  1  2  1.0 .9 .8 .7 .6 .5 .4  \  .3 .2 .1 0  3  1  2  Complexity  Test-Form 4  Test-Form 3 CD QJ rH  CO • > I  Q. QJ Ol CO Cn QJ  2  .0 ,9 ,8 ,7 6 5 4 3 2 1 0  3  Nl N2 ._ N3 ; N4  V  X  "NX  2 Complexity  -  j  V  1  Complexity  —  3  Complexity  1.0 cn .9 OJ u .8 •—i .7 CO > .6 I .5 CL .4 QJ .3 cn CO .2 fH .1 QJ > 0 1  Format  20 < 50 < 100 <  In I  <  In|  <  |n | < |n| <  20 50 100 1000  3  1 60  B.3  Figure  Phase 3 :  I n t e r a c t i o n E f f e c t s of S t r u c t u r a l Format and Numerical Magnitude  1  Test-Form  Test-Form 2  to  m  QJ 3  QJ  i  i  Q_  CL  QJ  QJ  H  n u  cn to QJ  > ct  2  ,0 9 8 7 6 .5 4 3 2 1 0  3  2  Structure  Structure  Test-Form 3  Test-Form  cn  UJ QJ  QJ  3  i  I  o_  CL  QJ  QJ OI  cn ro  ra In  OJ  QJ > CE  cr  2  3  ,0 ,9  .8 7 6 5 4 3 2 1 0  4  \  \  /  2  Structure  Nl N2 N3 N4  3  3  Structure  20 50 100  < < <  In | < |n| < |n| < Inl <  20. 50' 100 1000  4  161  Part Final  4  Equations  1 62  The  results  of t h e e q u a t i o n t e s t - f o r m s  study were u t i l i z e d i n s e l e c t i n g Diagnostic  Checklist  used, i n the t h i r d phase of  the e q u a t i o n s  for Algebra.  Due to the  for inclusion in  of  equation complexity  of  the  (one-,  interaction grid  T h i s meant t h a t  two-,  final  together (N/S),  48 equations  external c r i t e r i a Algebra  Test  were r e q u i r e d ( 3 e q u a t i o n c o m p l e x i t i e s  are p r e s e n t e d  (P),  (B-TT),  p-values  of the  p-values  the  was q u i t e  with  Lankton  with  the  First-Year  digits. used i n the  diagnostic  0 . 9 0 w i t h a mean p - v a l u e of  since  the  0.42.  The range  i n c l u s i o n of one i t e m from each c e l l  of  4 - b y - 4 n u m e r i c a l magnitude and s t r u c t u r a l format i n t e r a c t i o n g r i d was  r e q u i r e d at each l e v e l coefficients 0.04  large,  assignment  and c o r r e l a t i o n s are m u l t i p l i e d by  48 equations  final  i n s t r u m e n t ranged from 0 . 0 5 t o of  i n T a b l e s B . 2 , B . 3 , and B . 4 ,  raw s c o r e on the  The p - v a l u e s  x 4  interactions.  and b i s e r i a l c o r r e l a t i o n c o e f f i c i e n t  1 0 0 and rounded t o two s i g n i f i c a n t The  to c o n t r o l f o r  biserial correlation coefficient  (EC) - the t o t a l  (Lankton).  16 c e l l s  c o n t a i n e d the  w i t h t h e i r n u m e r i c a l magnitude and s t r u c t u r a l format  1 9 2 equations  total  levels  (4 n u m e r i c a l magnitude x 4 s t r u c t u r a l f o r m a t ) .  48 equations  difficulty level  effect  each of the t h r e e  and m u l t i - s t e p )  n u m e r i c a l magnitudes x 4 s t r u c t u r a l formats) The  the  strong i n t e r a c t i o n  between n u m e r i c a l magnitude and s t r u c t u r a l format,  the  to  Lankton.  of c o m p l e x i t y .  The b i s e r i a l c o r r e l a t i o n  of each e q u a t i o n used i n the d i a g n o s t i c  0 . 9 7 w i t h the t o t a l  1 9 2 equations  instrument ranged from  and from 0 . 1 1 t o  0 . 7 2 with  the  1 63  Table B.2 Statistics  EQUATION  TYPE:  Addition  f o r One-Step  ONE-STEP  N/S  x+57=41 -53+x=-75 800+x=609 256=x+-344  Equations  B-TT  EC*  N3S1 N3S2 N4S2 N4S3  74 82 50 67  87 46 04 39  70 37 48 28  NISI N1S3 N2S1 N2S4  53 50 77 32  51 51 60 04  52 51 66 26  N3S3 N3S4 N4S1 N4S4  68 80 42 90  43 60 37 82  26 38 49 12  N1S2 N1S4 N2S2 N2S3  53 40 55 50  62 37 04 54  06 46 41 64  Subtraction x-13=-3 -16=x—3 x-19=25 26=-21-x Multiplication 1. 100=x(10) 2. 56=-8x 3. x(-45)=-495 4. 1000=200x Division 1. 2. 3. 4.  18/x=-3 -l=-7/x -50/x=5 -ll=x/-4  *  N/S P B-TT EC  = = = =  Numerical Magnitude and S t r u c t u r a l D i f f i c u l t y L e v e l s p - v a l u e when e q u a t i o n s were t e s t e d i n Phase 3 b i s e r i a l c o r e l a t i o n of each e q u a t i o n with t h e t o t a l 192 e q u a t i o n s c o r r e l a t i o n of each e q u a t i o n with t h e Lankton  1 64  T a b l e B.3 Statistics  EQUATION  TYPE:  f o r Two-Step E q u a t i o n s  TWO-STEP  L i k e Numeric Terms  N/_S_  £  B-TT  EC*  1. 0=-2 9x-2 9 2. 17x+-100=-15 3. -77—x=-86 4. 61=-31+23x  N2S3 N3S1 N3S2 N3S4  59 53 50 67  87  25  56 26 38 32  NISI N2S4 N3S3 N4S1  64 59 41 47  66 65 43 99  58 40 45 47  N1S4 N2S1 N4S2 N4S4  45 52 20 26  82 57 75 97  26 36 32 41  N1S2 N1S3 N2S2 N4S3  40 40 40 40  63 78 63 78  51 36 51 36  69 88  L i k e V a r i a b l e Terms - 1 Side 1. 8x+9x=-17 2. 37=22x+15x 3. 88=56x-67x 4. -370x+-130=-1000 L i k e V a r i a b l e Terms - 2 Sides 1. -7x=16+x 2. x+-48=-23x 3. 333-158x=-121x 4. 971x=420+985x P a r e n t h e s e s / V a r i a b l e - 1 Side 1. 2. 3. 4.  2(5+x)=-8 16=4(x+3) 16(-7+x)=-48 288=-12(x+-24)  *  N/S = P = B-TT= EC =  Numerical Magnitude and S t r u c t u r a l D i f f i c u l t y L e v e l s p - v a l u e when e q u a t i o n s were t e s t e d i n Phase 3 b i s e r i a l c o r e l a t i o n o f each e q u a t i o n with t h e t o t a l 192 e q u a t i o n s c o r r e l a t i o n o f each e q u a t i o n with the Lankton  1 65  Table B.4 Statistics  EQUATION  TYPE:  MULTI-STEP  Variable/Numeric  Term - 1 Side  1. -4x+-7+-5x+-5=5 2. -3-x+8+-4x=-10 3. 0 = - 7 — 1 2 x — 1 7 - 2 x 4. 711-165x+-613+-175x=-582 Variable/Numeric 1. 2. 3. 4.  f o r M u l t i - S t e p Equations  N/S  B-TT  NISI N1S2 N1S4 N4S2  27 15 15 32  68 61 81 97  17 29 67 45  N1S3 N2S2 N2S4 N4S1  36 32 10 15  62 41 58 61  18 22 44 47  N2S1 N3S3 N3S4 N4S3  18 05 27 09  71 88 85 75  11 15 12 16  N2S3 N3S1 N3S2 N4S4  10 27 10 09  34 68 89 75  26 72 43 28  Term 2 S i d e s  -2x—8=5x-6 49—27x=16-6x 38+-35x=-12+-30x -513x+-156=-463+-606  L i k e V a r i a b l e Terms - 2 S i d e s 1. 13(x-3)=-2x-9 2. 52x+-16=15(x-6) 3. -9+24x=-9(-10+x) 4. 66x+44=26(x-26) P a r e n t h e s e s / V a r i a b l e - 1 Side 1. 10(x+-5)=20(x-2) 2. - 9 ( x + 8 ) = 8 ( x + - 9 ) 3. 1 0 ( 1 0 - x ) = - 9 ( - 9 + x ) 4. 144(6-x)=160(5-x)  *  N/S = P = B-TT= EC =  N u m e r i c a l Magnitude and S t r u c t u r a l D i f f i c u l t y L e v e l s p - v a l u e when e q u a t i o n s were t e s t e d i n Phase 3 b i s e r i a l c o r e l a t i o n o f each e q u a t i o n w i t h t h e t o t a l 192 e q u a t i o n s c o r r e l a t i o n o f each e q u a t i o n w i t h t h e Lankton  1 66  APPENDIX  PHASE  Wording  and  C  4  Format  Changes  1 67  The purpose of the i n t e r v i e w s of the f o u r s t u d e n t s used i n Phase 4 p r i m a r i l y t o r e f i n e the instrument concerns  i n terms of needed word changes  r e g a r d i n g time c o n s t r a i n t s .  The  refinements  The  types of erroneous  and  are p r e s e n t e d i n  terms of what changes were made t o the d i a g n o s t i c instrument, as i n d i v i d u a l i n t e r v i e w s .  r a t h e r than  understandings  which  o c c u r r e d are d i s c u s s e d , as w e l l as the steps taken t o r e c t i f y t h e s e Appendix E f o r f i n a l wording of the Diagnostic The  concept  a d d i t i o n of "How section.  Checklist  (See  Algebra). f o r the  do you know?" at the end of the meaning of v a r i a b l e  r e a l l y o n l y understood The  for  of v a r i a b l e q u e s t i o n s remained unchanged, except  By adding the q u e s t i o n , i t became apparent  of numbers.  was  v a r i a b l e s t o be  t h a t one  low a c h i e v e r  " t h i n g s " r a t h e r than r e p r e s e n t a t i o n s  o t h e r t h r e e students expanded t h e i r comments about  "the  v a r i a b l e s are the same" t o e x p l a i n t h a t they r e p r e s e n t e d the same number. When t o l d t o group e x p r e s s i o n cards by o p e r a t i o n , the h i g h e r a c h i e v i n g students experienced d i f f i c u l t y .  By changing  the d i r e c t i o n s t o "Group  t h e s e c a r d s by o p e r a t i o n s so ones r e p r e s e n t i n g the same e x p r e s s i o n s are t o g e t h e r , " a l l s t u d e n t s understood  the requirements  of the t a s k .  However,  both boys were unsure of the meaning of 4a, and both h i g h a c h i e v e r s were u n c e r t a i n i f 4+a meant 4/a The  or  a/4.  wording of the meaning of e x p r e s s i o n s s e c t i o n p r e s e n t e d g r e a t  difficulty  for a l l four subjects.  They were extremely  reluctant  and  c o n f u s e d when a t t e m p t i n g t o a t t a c h c o n c r e t e meaning t o e x p r e s s i o n s c o n t a i n i n g b l a n k s f o r the v a r i a b l e . t o "Use  the pennies  By changing  the wording from " E x p l a i n "  t o e x p l a i n the meaning of t h i s e x p r e s s i o n , " a l l  1 68  students  were f i n a l l y a b l e t o understand what t h e t a s k  many prompts were needed f o r some students the task.  required.  However,  t o enable them t o p r o c e e d  with  "Choose t h e number o f pennies t h a t t h e b l a n k r e p r e s e n t s " ,  the meaning so a Grade 1. student  "Show  c o u l d understand", and "Show me p h y s i c a l l y  what you do when you p e r f o r m t h e o p e r a t i o n o f ( a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n o r d i v i s i o n ) " were t h e a d d i t i o n a l prompts used t o c l a r i f y how t o e x p l a i n t h e meaning o f e x p r e s s i o n s expressions if  using concrete m a t e r i a l s .  For  c o n t a i n i n g l e t t e r s f o r t h e v a r i a b l e , "You may use t h e pennies,  you wish", was s u f f i c i e n t t o prompt t h e use o f c o n c r e t e  materials.  E x p l i c i t d i r e c t i o n s were r e q u i r e d i n t h e use o f e q u a l i t y s e c t i o n t o ensure t h a t h i g h a c h i e v e r s d i d not g i v e o n l y o r a l examples. was changed from "Give me an example" t o "Write your paper."  The wording  an a r i t h m e t i c sentence on  Two o f t h e s u b j e c t s , t h e h i g h - a c h i e v i n g  female and t h e low-  a c h i e v i n g male, s t a r t e d t o w r i t e more than one example t o enable use o f a l l the numbers i n t h e s e t .  The change i n t h e d i r e c t i o n s c l a r i f i e d t h e  vagueness o f t h e term "example." S e v e r a l concerns arose  f o r t h e meaning o f e q u a l i t y s e c t i o n .  prodded, s u b j e c t s c o n s i d e r e d  r e a d i n g t h e a r i t h m e t i c sentence and a g r e e i n g  t h a t i t was t r u e t o be a s u f f i c i e n t e x p l a n a t i o n . incomplete  i n d i c a t i o n o f a student's  e q u a l i t y was not s u f f i c i e n t responses g i v e n , t e a c h e r s •to "4+3=7".  understanding  However, such an o f t h e meaning o f  f o r d i a g n o s t i c purposes.  should  To improve t h e  l i s t e n f o r rewording o f t h e f i r s t  Such a rewording may i n d i c a t e t h a t t h e student  "answer" s h o u l d  Unless  question  t h i n k s an  f o l l o w t h e e q u a l i t y symbol and not "the problem"  (Behr e t  1 69  al., to  1976) .  "4+3=7."  Three of the f o u r s t u d e n t s r e v e r s e d t h e i r r e a d i n g of "7=4+3" However,  d i d not make the  when f a c e d with s i m i l i a r a l g e b r a i c e q u a t i o n s ,  reversal.  b e f o r e ? " were added t o the  "How do you know?" and "Have you seen 12x2=19+5 and 13=13 problems,  this  respectively.  student  equality,  l a t t e r p r o v i d e d an o p p o r t u n i t y f o r s t u d e n t s t o comment on  the meaningfulness achievers the  last  achievers  of the  reflexive  both i n d i c a t e d t h a t  p r o p e r t y of e q u a l i t y .  they had seen a number set  statement when v e r i f y i n g an e q u a t i o n ' s claimed that  An i n t e r e s t i n g  performed t o  The  former f o r c e d the and the  t o i n d i c a t e the o p e r a t i o n s  they  achieve  The h i g h equal to i t s e l f  solution.  they had never seen such statements  The low before.  c o n f u s i o n o c c u r r e d when students were asked t o  explain  the meaning of a g i v e n e q u a t i o n when the v a r i a b l e was supposed-, t o the amount of money they had i n t h e i r p o c k e t . took the  statement l i t e r a l l y t o mean t h a t  Three of the  represent  four subjects  the v a r i a b l e was the  amount of money they had i n t h e i r p o c k e t s ,  which was n o t h i n g !  actual By changing  the wording t o the s u b j u n c t i v e mood, "the amount of money y o u ' r e  supposed  t o have i n you pocket  to  recognize  (we're p r e t e n d i n g ) " the s t u d e n t s were a b l e  the n a t u r e of the t a s k .  explanations, Students  w h i l e the  The h i g h a c h i e v e r s  low a c h i e v e r s  as  were a b l e t o p r o v i d e  were n o t .  d i d not understand the next t a s k of c r e a t i n g meaning without  c o n t e x t u n t i l "problem" was changed t o of the e q u a t i o n s e c t i o n ,  "word p r o b l e m . "  the h i g h a c h i e v e r s  In the  last portion  r e q u i r e d the prompt t o make an  e q u a t i o n "on your paper" b e f o r e  they would w r i t e down the e q u a t i o n  were s a y i n g .  had d i f f i c u l t y r e a d i n g the q u e s t i o n s  The low a c h i e v e r s  they and  1 70  were unable t o c r e a t e e q u a t i o n s , a l t h o u g h they both t r i e d t o w r i t e down some o f the numbers i n the problems combined w i t h an "x" or an In the v o c a b u l a r y s e c t i o n , problems. letter.",  Even w i t h the h i n t :  coefficient  gave s t u d e n t s  "=".  particular  "Remember, i t ' s the number i n f r o n t o f the  some i n t e r e s t i n g r e s u l t s o c c u r r e d , g i v i n g f u r t h e r i n s i g h t  students' understanding, In "4+5n" one  not o n l y of c o e f f i c i e n t ,  but a l s o of l i k e  low a c h i e v e r i n d i c a t e d the c o e f f i c i e n t  coefficient  f o r "7-2n" was  coefficient  of "-n" was  5.  T h i s same s u b j e c t a l s o  negative.  was  c o r r e c t l y w i t h v a r i a b l e s whose c o e f f i c i e n t was  but was  terms.  9 and t h a t  the  indicated that  In l a t e r work w i t h e q u a t i o n s  s t u d e n t made s i m i l i a r e r r o r s with l i k e terms,  into  the  this  able to operate  n e g a t i v e one  (-1).  When g i v e n the d i r e c t i o n s i n the i n v e r s e s e c t i o n s "Write the o p p o s i t e ( r e c i p r o c a l ) of each of the f o l l o w i n g " , By d e l e t i n g the f i n a l phrase paper",  s t u d e n t s gave o n l y o r a l  and r e p l a c i n g i t with " e x p r e s s i o n on  a l l s u b j e c t s wrote answers on the paper p r o v i d e d .  i d e n t i t y element examples, i t was  n e c e s s a r y t o p o i n t t o the  o t h e r w i s e thought  P l a c e the answers on your paper."  use?"  students to Students  f o r the numbers  F o r example, when a s k i n g "Does i t matter  when r e f e r r i n g t o 3+n=3, i t was  t o p r o v i d e c l e a r d i r e c t i o n for. a l l To ensure t h a t  particular  t h a t both " l e t t e r s " and "numbers" r e f e r r e d t o the  v a r i a b l e r a t h e r than the l a t t e r r e f e r r i n g t o replacements i n the s e n t e n c e s .  you  When p r e s e n t i n g  p a r t s o f the e q u a t i o n s i n v o l v e d and t o s p e c i f i c a l l y d i r e c t " S o l v e each e q u a t i o n .  answers.  what numbers  n e c e s s a r y t o p o i n t t o both numbers  students.  s t u d e n t s r e c e i v e d the same d i r e c t i o n s f o r s o l v i n g  the  we  171  equations,  s p e c i f i c d i r e c t i o n s were g i v e n : "Solve each e q u a t i o n , e x p l a i n i n g  a l o u d what you are t h i n k i n g or d o i n g . " whether or not the f i n a l s o l u t i o n was  A l s o , the column f o r i n d i c a t i n g correct  (c) or i n c o r r e c t  (x)  was  added. The g e n e r a l format  of the d i a g n o s t i c c h e c k l i s t  seemed u s e f u l  a p p r o p r i a t e f o r d e t e r m i n i n g the l e v e l of u n d e r s t a n d i n g i n a l g e b r a i c concepts  and v o c a b u l a r y .  and  a t t a i n e d by  students  The e r r o r - c a t e g o r i z a t i o n scheme  d e p i c t e d student e r r o r s and p r o v i d e d a q u i c k v i s u a l r e c o r d of the s y s t e m a t i c e r r o r s of an i n d i v i d u a l s t u d e n t .  This v i s u a l c h e c k l i s t  of  e r r o r s a l s o p r o v i d e d a g r a p h i c r e p r e s e n t a t i o n of the common e r r o r s  across  s t u d e n t s when r e s u l t s were compared. A major concern  i n the use of s e m i - s t r u c t u u r e d i n t e r v i e w s i n the  d i a g n o s t i c instrument Completion  time  was  ranged  time c o n s t r a i n t s (Hart, 1983;  from 55  i n t e r v i e w e d i n Phase 4. t h i s time c o n s i d e r a b l y , .  minutes t o 105  1977).  minutes f o r the f o u r students  L i m i t i n g the number of e q u a t i o n s c o u l d s h o r t e n However, l i m i t i n g the number of e q u a t i o n s might  not produce an a c c u r a t e d e p i c t i o n of s t u d e n t s ' e r r o r s . p a r t i c u l a r concern  Opper,  i n view of the h i g h l y s i g n i f i c a n t  n u m e r i c a l magnitude and s t r u c t u r a l format  T h i s was  of  i n t e r a c t i o n between  which o c c u r r e d i n Phase  3.  1 72  APPENDIX Phase Statistical  Part Part  1: 2:  Part Part  3: 4:  Part  5:  Analyses  of  D 5 Error  Correlations  Forms Used for Subject Selection Correlations of Errors with Outcomes and Achievement Correlations of Errors with Other Measures Intercorrelations among Other Measures with Concepts and Errors Interaction Effects among Complexity, Structure, and Magnitude  173  The purpose of t h i s appendix i s t o d e t a i l t h e a n a l y s e s used i n t h e assessment o f t h e Diagnostic of  Checklist  for Algebra.  Part 1 provides  copies  t h e forms used f o r s u b j e c t s e l e c t i o n i n t h e f o u r t h and f i f t h phases o f  the study. instrument  P a r t 2 d e s c r i b e s t h e c r e a t i o n o f "outcomes" on the d i a g n o s t i c and t h e c o r r e l a t i o n s among these and t h e e r r o r s which were  i d e n t i f i e d by t h e instrument,  as w e l l as t a b l e s o f c o r r e l a t i o n s among  achievement l e v e l and t h e developed  errors.  Part 3 d e t a i l s the  i n t e r c o r r e l a t i o n s among demographic data, IQ, and mathematical achievement variables.  I t reports the s i g n i f i c a n t  c o r r e l a t i o n s among t h e s e measures  and t h e c o n c e p t u a l , p r o c e d u r a l , and r e s u l t a n t e r r o r s which o c c u r r e d i n the Diagnostic C h e c k l i s t f o r Algebra.  Part 4 a l s o presents a t a b l e of  i n t e r c o r r e l a t i o n s among c o n c e p t u a l , p r o c e d u r a l , and r e s u l t a n t e r r o r s . 5 shows t h e g r a p h i c i n t e r a c t i o n s among e q u a t i o n c o m p l e x i t y , format The one  and n u m e r i c a l  structural  magnitude.  r e s u l t s o f these a n a l y s e s a r e l i m i t e d because o f t h e use of o n l y  school s e t t i n g .  F u r t h e r assessment o f t h e instrument  s e t t i n g s would be n e c e s s a r y merit.  i n other  f o r these a n a l y s e s t o warrant f u l l  school  statistical  However, by t h e i n c l u s i o n o f such a n a l y s e s , i t i s hoped t h a t non-  meaningful  c l a i m s r e g a r d i n g t h e robustness  o f the instrument  w i l l be  a v o i d e d and t h a t t h e f i n d i n g s w i l l be o f d i a g n o s t i c and i n s t r u c t i o n a l value.  Part  Part  1:  Forms  used  in  Subject  Grade 9 s u b j e c t s p a r t i c i p a t e d of t h e i r c u r r i c u l a r a c t i v i t i e s . test-forms  Selection  i n the t h i r d phase of the Results  were g i v e n t o the t e a c h e r s  one  assessment purposes i n the c l a s s r o o m . Lankton and  the e q u a t i o n  test-forms  p a r e n t a l consent f o r student  interviews.  As t e a c h e r s  p a r t i c i p a t i o n was i n the  T h i s consent was  C o p i e s of the two  i n the  used the  r e s u l t s of  the  evaluation,  unnecessary f o r t h i s phase. individual  obtained  interviews,  p r i o r to  out the student  the questionnaire  interviews.  forms used f o r s u b j e c t  completeness of p r e s e n t a t i o n  equation  week a f t e r a d m i n i s t r a t i o n f o r  Students were a l s o asked to f i l l  p r i o r to p a r t i c i p a t i n g  the  as p a r t of t h e i r c l a s s r o o m  However, a l l s u b j e c t s p a r t i c i p a t i n g r e q u i r e d p a r e n t a l consent.  on the Lankton and  study as p a r t  selection  are i n c l u d e d f o r  of the procedures used i n Phase  5.  1 76  By c o m p l e t i n g the a t t a c h e d q u e s t i o n n a i r e , I am a g r e e i n g t o p a r t i c i p a t e i n the d i a g n o s t i c c h e c k l i s t r e s e a r c h on a l g e b r a i c e q u a t i o n s . I recognize t h a t I w i l l be i n t e r v i e w e d f o r about one hour while I s o l v e some a l g e b r a i c equations. I know t h a t I w i l l be asked t o " t h i n k a l o u d " as I s o l v e the e q u a t i o n s and t h a t the s e s s i o n w i l l be v i d e o - t a p e d . Only my f i r s t name w i l l be used i n r e p o r t s of my work (or I may i n d i c a t e a pseudonym, i f I so choose). I know t h a t I have the r i g h t t o withdraw from the r e s e a r c h at any time and t h a t t h i s w i l l i n no way a f f e c t my s c h o o l marks. QUESTIONNAIRE NAME: Sex :  M (Last),  F (First)  (circle  School: Teacher: Mathematics marks i n J u n i o r Hiah:  r  E d u c a t i o n of p a r e n t s (only check highest level) Father Mother Completed elementary Completed j u n i o r h i g h (grade 10) Completed h i g h s c h o o l (grade  12)  Vocational Training One  or more y e a r s  of c o l l e g e  Graduated from c o l l e g e Some graduate work Have a h i g h e r degree (masters/doctorate) Other  (specify)  one)  My e d u c a t i o n a l p l a n s (only check h i g h e s t l e v e l you p l a n to go) Self  177  Part  2:  Correlations  of  Errors  with  Outcomes  and  Achievement  The c r e a t i o n o f a d i a g n o s t i c i n s t r u m e n t s h o u l d be i n k e e p i n g w i t h measurement p r i n c i p l e s i n v o l v e d i n o t h e r " t e s t " c o n s t r u c t i o n , and s h o u l d ensure t h a t t h e instrument i s v a l i d . Diagnostic C h e c k l i s t f o r Algebra  The p r o c e d u r e s  attempted  used i n c r e a t i n g t h e  t o address  concerns  f o r both  v a l i d i t y and r e l i a b i l i t y w i t h i n a s e m i - s t r u c t u r e d , d i a g n o s t i c i n t e r v i e w setting.  One o f t h e apparent  dangers i n d i s c u s s i n g t h e r e s u l t s o f  i n t e r v i e w d a t a i s t h e tendency t o f o c u s on t h e minute d e t a i l s o f each i n d i v i d u a l d i a g n o s i s , and t o o f t e n i g n o r e some o f t h e g e n e r a l t r e n d s which may be o c c u r r i n g . for  W h i l e t h e s e minute d e t a i l s a r e i n t e r e s t i n g and i m p o r t a n t  t h e i n d i v i d u a l student being diagnosed,  emphasis on them when a s s e s s i n g  t h e d i a g n o s t i c i n s t r u m e n t may mask i m p o r t a n t f i n d i n g s which may have s i g n i f i c a n t implications f o r the i n s t r u c t i o n a l setting. a n a l y z i n g t h e d a t a from t h e Diagnostic  Checklist  Hence, i n  for Algebra  use o f  measurement p r i n c i p l e s and t e c h n i q u e s was a p p l i e d . Such a p p l i c a t i o n r e q u i r e d t h e c r e a t i o n o f "outcome s c o r e s " on t h e d i a g n o s t i c instrument.  Three such s c o r e s were c r e a t e d f o r each s u b j e c t on  the d i a g n o s t i c instrument. not m e a n i n g f u l  I t must be emphasized t h a t t h e s e " s c o r e s " were  f o r d i a g n o s t i c purposes r e l a t i n g t o i n d i v i d u a l s t u d e n t s .  However, t h e y d i d a l l o w f o r comparisons o f e r r o r s between s t u d e n t s and provided information regarding general trends of e r r o r s .  The f i r s t  score  r e p r e s e n t e d t h e t o t a l number o f e r r o r s a s t u d e n t c o u l d make on t h e d i a g n o s t i c instrument  (See T a b l e s D . l and D.2, T o t a l Number o f E r r o r s ) .  r e p r e s e n t e d t h e a r i t h m e t i c sum o f t h e second and t h i r d s c o r e s .  It  The second  s c o r e r e p r e s e n t e d t h e t o t a l number o f c o n c e p t u a l e r r o r s made by s t u d e n t s .  1 78  There were 2 7 d i f f e r e n t the d i a g n o s t i c instrument  items  i n t h e concepts  and v o c a b u l a r y p o r t i o n of  (See T a b l e s ' D . l and D . 2 , E r r o r s i n C o n c e p t s ) .  check under t h e mastery column counted  A  as no e r r o r ( 0 ) , under t h e p a r t i a l  column as one e r r o r ( 1 ) , and under t h e r e m e d i a l column as two e r r o r s ( 2 ) . T h i s meant t h a t t h e h i g h e s t p o s s i b l e s c o r e was 5 4 and r e p r e s e n t e d a measure of  conceptual errors,  r a t h e r than mastery o f c o n c e p t s .  represented 7 5 3 d i f f e r e n t  x 4 8 equations  1  (See T a b l e s D . l and D . 2 ,  T h i s was not t h e expected  1 0 0 8 ( 2 0 error categories  + 4 8 i n c o r r e c t f i n a l answers) due t o t h e nature o f t h e  c o n s t r u c t i o n of t h e e q u a t i o n s .  F o r example, many of t h e e q u a t i o n s  i n c l u d e b r a c k e t s , hence t h e o p p o r t u n i t y t o make a d i s t r i b u t i v e not p r e s e n t .  score  o p p o r t u n i t i e s f o r a s u b j e c t t o make p r o c e d u r a l or  r e s u l t a n t e r r o r s i n s o l v i n g the 4 8 equations E r r o r s i n Equations.) .  The t h i r d  d i d not  e r r o r was  Combining t h e number o f e r r o r s made on t h e second s c o r e ( 5 4 )  and t h e t h i r d s c o r e  ( 7 5 3 ) gave a t o t a l f o r t h e f i r s t  score of 8 0 7 .  This  r e p r e s e n t e d t h e t o t a l number o f e r r o r s t h a t any one student c o u l d p o s s i b l y make on t h e d i a g n o s t i c i n s t r u m e n t . The  reason  f o r c r e a t i n g t h e s e e r r o r s c o r e s was due t o t h e many  comparisons among e r r o r s i n concepts, o c c u r on t h e d i a g n o s t i c i n s t r u m e n t . o c c u r when m a k i n g - m u l t i p l e  procedures,  and r e s u l t s which c o u l d  An i n f l a t e d Type I (alpha) e r r o r c o u l d  comparisons, c a u s i n g i d e n t i f i c a t i o n o f  d i a g n o s t i c symptoms which were not s i g n i f i c a n t .  To l i m i t Type I e r r o r ,  o n l y those e r r o r s which s i g n i f i c a n t l y  (p<.05)  number o f e r r o r s ,  correlated  with t h e t o t a l  and then with e i t h e r t h e t o t a l c o n c e p t u a l e r r o r s o r t h e  t o t a l e r r o r s i n equations  were d i s c u s s e d  (See T a b l e s D . l and D . 2 ) .  when  Table D . l C o r r e l a t i o n s between Concept Scores and E r r o r Concepts  T o t a l Number of E r r o r s  Variable identify meaning Expressions identify + identify identify x identify / meaning blank + meaning blank meaning blank x meaning blank / meaning v a r i a b l e + meaning v a r i a b l e meaning v a r i a b l e x meaning v a r i a b l e / Equality identify use meaning Eguat ions meaning g i v e n c o n t e x t meaning w i t h o u t c o n t e x t g i v e n meaning c r e a t e Coefficients ^ identify Like-Terms identify group Inverses write opposites write reciprocals I d e n t i t y Elements zero one  All correlations and  rounded  ft P < .05 ** P £ .01 « « * P < .001  to  listed  have  Scores  Errors in Concepts  Errors in Equations  38* 37*  54*** 22  28 36*  34* 25 50** 16 -14 20 15 49** 04 06 36* 49**  31 29 61*** 47** 13 46** 46** 68*** 47** 45** 74*** 59***  31 20 40* 04 -21 09 04 37* -08 -06 20 40*  19 41* 12  41* 51** 47**  10 33* 00  37* 50** 74*ft«  63*** 66*** 63***  25 40* 68***  18  20  16  12 35*  01 27  14 33*  52** 60***  47** 65***  47** 51**  46** 46**  46** 59***  46** 36*  been  two s i g n i f i c a n t  m u l t i p l i e d by  digits.  100  1 79  Table D . l C o r r e l a t i o n s between Concept Scores and E r r o r Concepts '  T o t a l Number of E r r o r s  Variable identify meaning Expressions identify + identify identify x identify / meaning blank + meaning blank meaning blank x meaning blank / meaning v a r i a b l e + meaning v a r i a b l e meaning v a r i a b l e x meaning v a r i a b l e / Equality i d e n t i fy use meaning Equations meaning g i v e n c o n t e x t meaning without c o n t e x t g i v e n meaning c r e a t e Coefficients identify Like-Terms identify group Inverses write opposltes write reciprocals I d e n t i t y Elements zero one  All  correlations  and  rounded  * P < .05 ** P £ .01  ***  P <  .001  to  listed  have  Scores  E r r o r s In Concepts  Errors in Equations  38* 37*  54*** 22  28 36*  34* 25 50** 16 -14 20 15 49** 04 06 36* 49**  31 29 61*** 47** 13 46** 46** 68*** 47** 45** 74***  31 20 40* 04 -21 09 04 37* -08 -06 20 40*  59***  19 41* 12  41* 51** 47* *  37* 50** 74***  63*** 66*** 63***  25 40* 68***  18  20  16  12 35*  01 27  14 33*  52** 60***  4 7*« 65***  47** 51**  46** 46**  46** 59***  46** 36*  been  two s i g n i f i c a n t  m u l t i p l i e d by  digits.  10 ' 3 3 *  00  100  1 80  Table  D.2  C o r r e l a t i o n s between E r r o r Types and E r r o r Scores Error  Types  Procedural zero annexation identity confusion like-terms (conjoining) partial distributive order of o p e r a t i o n s addition property = m u l t i p l i c a t i o n property coefficient Resultant Computational addition/subtraction multiplication/division basic facts faulty algorithm wrong o p e r a t i o n Sign addition/subtraction multiplication/division rule-based distributive Other mechanical/perceptual random • incomplete s o l u t i o n s final  All and  answers  T o t a l Number of E r r o r s  =  00 22 38* -22 07 44** -07 -07  E r r o r s In Concepts  -16 15 24 -25 -27 08 -26 -33*  06 22 37* -19 17 49** -01 01  69*** 43** 23 21 56***  29 05 16 10 19  73*** 49** 23 22 61***  63*** 39* 58*** 38*  20 20 28 21  68*** 40* 60*** 38*  -11 -27 40*  19 -12 56***  13 -17 58*** 96***  c o r r e l a t i o n s l i s t e d have b e e n m u l t i p l i e d r o u n d e d t o two s i g n i f i c a n t digits.  * P < .05 * * P < .01 *** P < .001  Errors in Equations  52**  94***  181  the  i n t e r a c t i o n s between the c o n c e p t u a l and e q u a t i o n e r r o r s were analyzed,  o n l y t h o s e e r r o r s which s i g n i f i c a n t l y c o r r e l a t e d s c o r e s were d i s c u s s e d . correlations discussed  (p<.05) t o a l l t h r e e  error  These procedures h e l p e d t o ensure t h a t t h e were meaningful i n terms of measurement p r i n c i p l e s .  A summary of the f r e q u e n c i e s ,  means,  standard deviations,  and  s i g n i f i c a n t d i f f e r e n c e s between achievement and c o n c e p t u a l e r r o r s o r p r o c e d u r a l and r e s u l t a n t e r r o r s are p r e s e n t e d i n T a b l e s D.3 and D.4.  1 82  Table  Conceptual E r r o r and S i g n i f i c a n t  D.3  F r e q u e n c i e s , Means, Standard Differences  High A c h i e v e r Average A c h i e v e r Low Achiever Total X Total X Total X CONCEPTUAL Var i a b l e s Identify 5 0.42 meaning 1 0.08 Expressions Symb+ 0 0.00 Symb1 0.08 Symbx 3 0.25 Symb/ 5 0.42 Blnk + 4 0.33 Blnke 0.50 Blnkx 2 0.16 Blnk/ 15 1.25 Varb+ 6 0.50 4 0.33 Varb5 0.42 Varbx 1.08 Varb/ 13 Equality Identity 4 0.25 use 1 0.08 meaning 3 0.25 Equations with 14 1.16 1.00 without 12 create 9 0.75 Coefficients coef f 18 1.50 Like-Terms compare 19 1.58 group 14 1.16 Inverses Opposites 6 0. 50 Reciprocals 5 0.42 I d e n t i t y Elements zero 9 0.75 one 7 0.58  Levels  Combined Groups Moan S.D. F t e s t  11 2  0 .92 0 .17  16 4  1.33 0.33  0 .89 0 .19  0 .35 0 .05  2 0 5 4 7 5 7 14 6 6 13 13  0 .17 0 .00 0 .42 0 .33 0 .58 0 .42 0 .58 1 .16 0 .50 0 .50 1 .08 1 .08  2 2 4 8 8 11 10 22 10 10 10 19  0.17 0.17 0.33 0.67 0.67 0.92 0.83 1.83 0.83 0.83 0.83 1.58  0 .11 0 .17 0 .33 0 . 47 0 . 59 0 .61 0 .53 1 .42 0 .61 0 .56 0 .81 1 .25  0 .02 0 .04 0 .11 0 .21 0 .24 0 .31 0 .30 0 .78 0 .32 0 .28 0 .58 0 .86  6 3 4  0 .50 0 .25 0 .33  8 3 8  0.67 0.25 0.67  0 .50 0 .19 0 .42  0 .17 0 .05 0 .19  14 15 13  1 .16 1 .25 1 .42  17 1. 4 2 19 1.58 20 1.67  1 .25 1 . 28 1 .17  0 .88 0 :97 0 .99  15  1 .25  15  1.25  1 .33  0 .94  19 18  1 .58 1 .50  20 16  1.67 1.33  1 .61 1 .33  1 .23 1 .27  7 9  0 .58 0 .75  7 12  0.58 1.00  0 .56 0 .72  0 .27 0 .38  ft ft*  10 9  0 .83 0 .75  14 12  1.16 1.00  0 .92 0 .78  0 .45 0 .37  ft «  A b b r e v i a t i o n s of concepts l i s t e d i n t h i s t a b l e a r e i n t h e same order as those l i s t e d i n Table D . l . » p i .05 »* P i .01 «** p < .001  Deviations,  among Achievement  * *  •  *  « * **«  ftftft  ft*  183  Table  D.4  E r r o r Type F r e q u e n c i e s , Means, Standard D e v i a t i o n s , and S i g n i f i c a n t D i f f e r e n c e s among Achievement L e v e l s  High Achiever Total X  PROCEDURAL zeroannex idconfuse liketerms partldlst ordezoper +prop= xprop= coef £ tESULTANT Computation  +x/ bscfct fltyalg wrngoper Sign  +x/ rulebase dletrlb Other mechper random lncomplt tlnlans  Average Total  Achiever X  Low A c h i e v e r Total X  Combined Groups Mean S.D. K test  7 3 33 2 5 24 6 3  0. 0. 3. 0. 0. 2. 0. 0.  42 33 33 17 50 00 50 50  12 13 21 1 1 26 11 8  0, .67 0. .92 2. .33 0. .08 0. .08 2. , 17 0, .83 0. ,67  5 9 35 1 1 35 4 8  0 .42 0 .75 3 .17 0 .08 0 .17 2 .67 0 .42 0 .58  0. ,50 0. .67 2. .94 0. .11 0. . 2 5 2 . 28 0. .58 0 ..58  0 .66 0 .79 2 .93 0 .32 0 .50 2 .99 0 .97 0 .91  46 9 14 20 26  2. 92 1.92 1. 17 0 . 83 2 . 50  61 30 19 31 52  4 .58 , 2. .50 2. .17 1. .83 4 ,33 ,  74 43 25 21 58  4 .25 3 .33 2 .33 1 .67 4 .08  3. .81 2. .58 1. .89 1..44 3. .64  2 .55 1 .63 1 .51 1 . 16 2 .53  55 10 30 2  4. 00 1. 33 1. 75 0. 50  77 14 40 0  4, .58 2. .92 2, .42 0. ,00  79 28 37  5 .33 3 .08 1 .92 0 .67  5, .03 2 ..44 2. .03 0 ..53  2 .69 1 .87 1 .67 0 .81  1 .83 2 .33 5 .83 17 . 2 5  2, .06 1, .97 4 ., 36 1 3 . .83  2 .02 1 .96 4 .40 8 .69  39 18 112 181  2. 1. 4. 10.  42 75 08 25  39 20 148 262  2 ., 00 1. .83 3. .00 14 ,.00  1 46 20 160 290  A b b r e v i a t i o n s of e r r o r t y p e s l i s t e d In t h i s t a b l e a r e i n t h e same o r d e r a s t h o s e l i s t e d i n T a b l e D.2. * «* *»•  p i p i p i  .05 .01 .001  1 84  Part  Correlations  3:  of  Errors  with  Other  Measures  Because o f t h e p o s s i b l e number o f i n t e r c o r r e l a t i o n s among t h e demographic d a t a , IQ, and mathematics achievement v a r i a b l e s ideations  h e l d and e r r o r s made on t h e d i a g n o s t i c  and t h e  instrument only  those  v a r i a b l e s which showed s i g n i f i c a n t c o r r e l a t i o n s w i t h t h e t o t a l number of e r r o r s and with e i t h e r t o t a l number o f e r r o r s i n concepts o r t o t a l of e r r o r s  i n e q u a t i o n s were d i s c u s s e d .  Correlations  between these t h r e e  s c o r e s and t h e demographic, IQ, and achievement v a r i a b l e s Table  number  were p r e s e n t e d i n  D.6.  A l l t h r e e o f t h e mathematical achievement v a r i a b l e s  (Lankton, Mean Mark  i n Mathematics, CSMS) were s i g n i f i c a n t l y c o r r e l a t e d t o a l l instrument scores  ( r e f e r t o Table D.5).  Many s i g n i f i c a n t  c o r r e l a t i o n s o c c u r r e d which  r e l a t e d e r r o r s t o these mathematics achievement v a r i a b l e s . e r r o r s were r e l a t e d t o a l l t h r e e ,  while others r e l a t e d to only  of t h e mathematics achievement v a r i a b l e s . demographic d a t a and concepts o f a l g e b r a  The c r o s s t a b u l a t i o n s  and D . 7 .  one o r two of  and t h e e r r o r s made i n s o l v i n g  e q u a t i o n s r e s u l t e d i n a number o f s i g n i f i c a n t c o r r e l a t i o n s . p r e s e n t e d i n T a b l e s D.6  Some o f these  These were  Table D.5 C o r r e l a t i o n s between E r r o r Scores and Demographic IQ and Mathematical Achievement V a r i a b l e s  Variables Total  E r r o r Scores Conceptual  Equation  Grade  -48**  -28  -48**  Age  -26  -08  -28  08  -18  14  Gender IQ  -48**  -49**  -43**  Marks  -69***  -54***  -62***  Lankton  -60***  -68***  -50**  CSMS  -53***  -52**  -46**  * S i g n i f i c a n t at p<0.01 ** S i g n i f i c a n t a t p<0.001  186  Table D.6 S i g n i f i c a n t C o r r r e l a t i o n s f o r Concepts and Demographic IQ and M a t h e m a t i c a l Achievement V a r i a b l e s (p<.05)  Conceptual  Variable identify meaning Expressions Symb+ SymbSymbx Symb/ Blnk+ BlnkBlnkx Blnk/ Varb+ VarbVa rbx Varb/ Equality identify use meaning Equations with without create Coefficients coeff L i k e Terms compare group Inverses oppos recip Identities zero one  *  Grade  Age  Gender  32  IQ  Marks  Lankton  33  49 42  51  32 35  34 38  30  CSMS  47  36 -49  -34  •46 29  -37 31  48  36 38  37 35 28  29 52  45  35  29 44  36 29  35  44 50 53  33 51 67  34 46  40 39 53  29  33 -60 -43  37  39  50  40  50 60  41 31  46 35  44 44  46  42  A l l c o r r e l a t i o n s l i s t e d a r e rounded t o two s i g n i f i c a n t and r e p r e s e n t hundredths.  39  .35 31  digits  T a b l e D.7 S i g n i f i c a n t C o r r e l a t i o n s f o r E r r o r Types and Demographic, IQ, and M a t h e m a t i c a l Achievement V a r i a b l e s (p<.05)  E r r o r Types  Grade  Age  Gender  IQ  Marks  Lankton  CSMS  P rocedural  zeroannex IDconfuse liketerms partldist orderoper +prop= xprop= coeff  43* -33 -32 -35 35  32 32  Resultant Computat i o n a l  +x/ bscfct fltyalg wrngoper  -36 - 31  -35  -33  -31  32 29  34  Sign  +x/ rulebased distrib  38 -52  39  Other  mechper random incomplt Final  Answer  62 - 54  33 -37  44 -49 - 37  -67  A l l c o r r e l a t i o n s l i s t e d are rounded to two s i g n i f i c a n t and r e p r e s e n t hundredths.  -47 -  33  digits  -46  188  Part  Intercorrelations and Errors  4:  T a b l e D.8  among  Other  IQ showed h i g h l y  measures of mathematical r=0.55, p<0.001, CSMS:  (Marks/Lankton:  achievement  s i g n i f i c a n t c o r r e l a t i o n s with a l l (Marks:  r=0.54, p<0.001).  r=0.56, p<0.001;  The measures of  r=0.60, p<0.001; Marks/CSMS: r=0.65, p<0.001).  c o r r e l a t e d w i t h Grade l e v e l  The CSMS was  w i t h the d i a g n o s t i c  themselves  also s i g n i f i c a n t l y  This  seemed due  s i g n i f i c a n t l y correlated  not t o s t u d e n t s '  t o the e r r o r s made w h i l e s o l v i n g  equations  (See  Those e r r o r s which were s i g n i f i c a n t l y c o r r e l a t e d with grade  l e v e l were:  The  mathematical  (r=0.64, p<0.001).  checklist.  c o n c e p t i o n s , but r a t h e r  Lankton:  r=0.45, p<0.01;  Of the demographic data, o n l y grade l e v e l was  solutions  IQ,  W i t h i n the demographic data, o n l y  achievement were s i g n i f i c a n t l y c o r r e l a t e d amongst  T a b l e D.5).  Concepts  s i g n i f i c a n t i n t e r c o r r e l a t i o n of age and grade o c c u r r e d  (r=0.43, p<0.01).  Lankton/CSMS:  with  p r e s e n t s the i n t e r c o r r e l a t i o n s among the demographic data,  and mathematics' achievement v a r i a b l e s . the e x p e c t e d  Measures  rule-based sign errors  (r=0.39, p<0.05) and  incomplete  (r=0.63, p<0.001).  s i g n i f i c a n t i n t e r c o r r e l a t i o n s between concepts  are p r e s e n t e d T a b l e  D.9.  and e q u a t i o n  errors  T a b l e D.8 I n t e r c o r r e l a t i o n s o f Demographic Data, IQ, and M a t h e m a t i c a l Achievement  Grade  Grade  1.00  Age Gender  Age  Gender  IQ  Marks  Lankton  CSMS  0..43 *  0..00  0,.34  0 .41 .  0,.35  0,.64 **  1..00  0 .10 .  0 .26 ,  0 .25 ,  0 .26 ,  0 .17 .  1..00  0 .16 .  0,.08  0 .03 ,  - 0..02  1..00  0,.56 ** '0: .55 * * 0 .54 * *  IQ Marks  1..00  Lankton CSMS  0 ., 60 **  0,.45 *  1..00  0..65 ** 1,.00  * **  S i g n i f i c a n t a t p<.01 S i g n i f i c a n t a t p<.001  190  Table Significant  1  2  3  4  D.9  I n t e r c o r r e l a t i o n s between Concepts and E r r o r Types  5  6  7  8  9  10  11  12  13  14  15  16 17  18  19 20 21  22 23  35 35 1 1.00 1.00 36 34 36 39 2 -37 3 1.00 39 46 43 4B 39 33 -40 4 1.00 45 66 46 46 59 40 53 ' 35 -23 1.00 42 39 35 46 5 -47 1.00 56 56 43 45 -47 6 1.00 66 41 38 41 7 1.00 67 50 59 a 1.00 54 44 48 41 9 -39 -40 -36 -59 10 1.00 37 35 -35 -39 -46 1.00 42 -45 -37 -39 11 12 1.00 75 -39 -34 13 1.00 -36 14 1.00 36 15 1.00 53 41 16 1.00 58 81 61 40 37 17 1.00 63 57 37 18 1.00 45 35 37 19 1.00 51 38 20 1.00 21 1.00 50 22 1.00 23 1.00 24  *  ?4J» -39 -40 -38 -40 -46 -46 -43 -66 -55 -47 -41 -46 36 88 63 46 53 57 52 34 65 1.00  Correlations l i s t e d have been rounded to two s i g n i f i c a n t d i g i t s and represent hundredths. S i g n i f i c a n t correl at i on l e v e l s are indicated below: p<.05 :  r=0.3298  p<.01 :  r=0.4242  p<,001:  r=0.5256  ** These numbers represent the ideation and error va ria ble s.  l=Variable I d e n t i f i c a t i o n  Their d e f i n i t i o n s  S=6iven Meaning-Create Equation  are:  l7=Coaputational  Multiply/Divide  18=Coaputational Wrong Operation  2=Variabl'e Meaning  10=Additive Inverse  3=Identify M u l t i p l i c a t i o n Synbol  l l = M u l t i p l i c a t i v e Inverse  19=Sign Add/Subtract  4=Blank Division Expressions  12=Additive Identity  20=Sign Multiply/Divide  5=Variable M u l t i p l i c a t i o n Expressions  13=Hultiplicative  6=Variable Division Expressions  14=CoBibining Unlike Teras  22=Sign D i s t r i b u t i v e  7=Meaning of Equations Given Context  15=Addition Property of Equality  23=Inconplete Solutions  8=Meaning of Equations Without Context  16=Coaputational  24=Final Answers  Identity  Add/Subtract  21=Sign Rule-Based  Part  5:  Interaction Effects and Magnitude  F i g u r e s D . l , D.2,  Complexity,  Structure,  and D.3 p r e s e n t the i n t e r a c t i o n e f f e c t s among  c o m p l e x i t y and s t r u c t u r e , magnitude,  among  c o m p l e x i t y and magnitude, and s t r u c t u r e and  respectively.  T a b l e s D.10, D . l l , and D.12 g i v e t h e e r r o r means f o r c o m p l e x i t y , structure,  and magnitude,  respectively.  192  Figure D . l Phase 5:  I n t e r a c t i o n E f f e c t s o f E q u a t i o n Complexity  and S t r u c t u r a l Format  1.0  .9 O) Z3  #~—* > l  .8 .7 .6  CL  .5  CU CT)  .4  i_  >  .3 .2 .. 1 0 i  •S1 •52 53 54  = = = =  r~—J Complexity  V a r i a b l e l e f t of e q u a l i t y , l e f t of c o n s t a n t s V a r i a b l e l e f t of e q u a l i t y , r i g h t of c o n s t a n t s V a r i a b l e r i g h t of e q u a l i t y , l e f t o f c o n s t a n t s Variable right of e q u a l i t y , r i g h t of constants  1 93  Figure Phase 5:  D.2 I n t e r a c t i o n E f f e c t s of E q u a t i o n Complexity and Numerical Magnitude  1.0 .9  .8. GJ  > I  .5  <v CT  .3  «o  .2  >  ,1  0  1  —7.  J  Compiexliy  N1=Numerical N2=Numerical N3=Numerical •N4=Numericai  Magnitude 1 Magnitude 2 Magnitude 3 Magnitude 4  ( |n| < 20) ( 20 < |n| < 50) ( 50. < |n| < 100) ( 100 < |n| < 1000)  .194  D.3  Figure Phase 5:  I n t e r a c t i o n E f f e c t s of S t r u c t u r a l Format and Numerical Magnitude  1.0 .9 00  .8  CO Z3  .7  •—t  > I 0)  cn  a; > <  .6  .5 .4  .o  .2 .1 0 2  3  Structure — — —  N1=Numerical N2=Numerical N3=Numericai N4=Numerical  Magnitude Magnitude Magnitude Magnitude  1 2 3 4  |n| < 20 < |n| < 50 < |n| < ( 100 < |n| <  20) 50) 100) 1000)  Table D . 1 0 E q u a t i o n Complexity E r r o r Means  E r r o r Types  L e v e l s o f Complexity* Cl  C2  C3  0.00 0.03 0.03 0.00 0.03 0.25 0.03 0 . 00  0.03 0.03 0.61 0.03 0.00 0.39 0.00 0 .03  0.00 0.09 0 . 61 0.00 0.06 0.36 0.18 0 .15  0 .22 0.36 0.08 0.06 0 .53  0.25 0 .17 0.14 0.08 0.11  0.21 0 .15 0.06 0 .03 0.21  0 .72 0 .17 0 . 47 0.00  0.25 0.36 0.06 0 .22  0 .33 0.24 0 .12 0 .18  0.08 0.06 0.36 1.36  0.06 0.19 0.39 1.47  0.18 0 . 09 0 . 47 1 . 44  P rocedural  ZeroAnnexation IDconfusion Liketerms OrderOperations PartialDistributive +prop= xprop= Coefficient Resultant Computational  Add/Sub Mult/Div BasicFacts FaultyAlgorithm WrongOperation S ign  Add/Sub Mult/Div Rulebased Distributive Other  Mech/Perceptual Random Incomplete Final  Answer  Cl= Complexity L e v e l 1 C2= Complexity L e v e l 2 C3= Complexity L e v e l 3  one-step e q u a t i o n s two-step e q u a t i o n s m u l t i - s t e p equations  Table D . l l S t r u c t u r a l Format E r r o r Means  E r r o r Types SI  L e v e l s of S t r u c t u r a l Format* S2 S3 S4  Procedural  ZeroAnnexation IDconfusion Liketerms PartialDistributive OrderOperations +prop= xprop= Coefficient  0. .11 0. .08 0. .97 0. .03 0. .08 0. .56 0. .25 0. . 17  0 .03 0. .00 0. .50 0. .03 0. .06 0. .47 0. . 14 0. .06  0 ,03 . 0. .39 0. .53 0 .06 . 0 .08 . 0 .50 . 0 .08 . 0 .06 .  0. .33 0. .19 0. .94 0. .00 0. .08 0. .75 0. .11 0. .31  1..25 0 .50 . 0 .39 . 0 .39 , 0..89  1. .33 0 .58 . 0. .53 0. .50 0 .89 .  0 .78 . 0 .89 . 0 .47 . 0. .36 1. .17  0 . 44 0 . 61 0 .50 . 0. .19 0 ., 69  1..00 0,.31 0 .39 0 .22  1. .81 0. .86 0. .56 0, .25  1. .28 0. .72 0..64 0..03  0 .. 94 0 .56 , 0 . 44 0 .03  0 .72 0 .31 1 .48 3 .58  0. . 64 0 .58 , 0 .78 3 .33  0..42 0..47 0,.92 3..36  0..31 0 . 61 1 .19 3..56  lesultant lomputational  Add/Sub Mult/Div BasicFacts FaultyAlgorithm WrongOperation !ign  Add/Sub Mult/Div Rulebased Distribributive Ither  Mech/Perceptual Random Incomplete "inal  Answer  * Sl=Structure S2=Structure S3=Structure S4=Structure  1 2 3 4  (variable l e f t (variable l e f t (variable right (variable right  of e q u a l i t y , l e f t of number) of e q u a l i t y , r i g h t of number) of e q u a l i t y , l e f t of number) of e q u a l i t y , r i g h t of number  Table D.12 Numerical Magnitude Error Means  E r r o r Types Nl  L e v e l s o f Numerical Magnitude* N2 N3 N4  Procedural  ZeroAnnexation IDconfusion Liketerms PartialDistributive OrderOperations +prop= xprop= Coefficient.  0.03 0.14 1.1.9 0.03 0.08 0.97 0.19 0.83  0.03 0.28 0.67 0.00 0.03 0.44 0.06 0.22  0.03 0.11 0.56 0.06 0.11 0.53 0.17 0.14  0.42 0.14 0.53 0.03 0.03 0.33 0.17 0.06  0.67 0.67 0.28 0.17 0.83  0.92 0.36 0.31 0.22 0.92  0.97 0.78 0.61 0.08 0.97  1.25 0.78 0.69 0.97 0.92  1.2 8 0.75 0.64 0.39  1.25 0.39 0.61 0.00  1.56 0.56 0.44 0.06  0.94 0.75 0.33 0.08  0.31 0.33 1.22  0.79 0.28 1.36 3.44  0.64 0.58 1.14 3.33  0.44 0.78 0.64 2.78  Resultant Computational  Add/Sub Mult/Div BasicFacts FaultyAlgorithms WrongOperation S ign  Add/Sub Mult/Div Rulebased Distributive Other  Mech/Pereceptual Random Incomplete Final  Answer  Nl=Numerical N2=Numerical N3=Numerical N4=Numerical  4.28  Magnitude Magnitude Magnitude Magnitude  1 2 3 4  ( |n| < 20) ( 20 < |n| < 50) ( 50 < |n| < 100) ( 100 < |n| < 1000)  APPENDIX  DIAGNOSTIC (Linear  CHECKLIST  One  Variable  E  FOR  ALGEBRA  Equations)  DIAGNOSTIC CHECKLIST FOR ALGEBRA LINEAR ONE VARIABLE EQUATIONS Objectives Concepts A.1 Variable A.1.1 Identity Variable A.1.2 Meaning of Variable  Items What do you call the x in 5 + x? W  n  a  t  d  o  e  s  j t s  (  a  n  d  f o f ?  Tell me how the variables shown are related. 7w + 22 - 109 7n + 22 = 109  A.2 Expressions A.2.1 Identify Symbols  H  o  wd  0 y  o 7  u  Know?  Group these cards by operations so ones representing a +4 4+ a same expression 4a 4(a) 4xa 4 « a are together. _4 (Cards should be arranged ,  n  e  a  ln  a/4 a 4 4Ja" 44 A.2.2 Meaning of Expressions Containing •  in<  *• "  c a , 9 < 1orda  ')  Read each expression aloud and show its meaning with the pennies. (Additional prompts: (a) Explain it so a G'Bde 1 student would understand; (b) Show me how h works) (You may wish to use other concrete materials.) 6+D  Ox6 6 - D  D - 6 6  Containing tetters  Read each expression aloud and explain its meaning. You may use the pennies if you wish. (Again, student demonstration with concrete materials/pictorial representations may be appropriate.) 4x  A.3 EquaMty A.3.1 identity Symbol  X+4  4 - X J L X-=-4 4  What does this symbol (=») mean? Tell me another word for it.  A.3.2 Use of Equality  Write an arithmetic sentence on your paper using this symbol ( = ), any of these numbers: {1,2,3,4,5,6} and any arithmetic operations: { +, - ,x, •?}. Give me an example using any numbers you wish, where there are operations on both Bides of the equality symbol.  DIAGNOSTIC CHECKLIST FOR ALGEBRA LINEAR ONE VARIABLE EQUATIONS S a n  Items  Objectives A.3 Equality (continued) A.3.3 Meaning of Equality explain the meaning of each.  A.4 Equations A.4.1 Given equation with context create meaning  7=4+3  1 2 x 2 = 19 + 5  (listen lor reversal'. 4 + 3-7)  How do you know?  13 = 13 Have you seen this before?  Read each equation aloud Let's pretend x means the amount of money you're supposed to have in your pocket. Explain the meaning of each equation. x + 19 = 47  A.4.2 Given equation without context create meaning  3x + 4 = 10  2(x + 1 ) - 6  Read each equation aloud. Make up a word problem that someone might have been thinking of when they wrote 'he equation. x + 21 =39  A.4.3 Given meaning create equation  4x + 7 = 23  3{x + 2) = 15  Read each statement aloud and make an equation on your paper. Penny noticed something interesting about her savings account. If you add $6.00 to the amount you get the same result as doubling the amount and subtracting $4.00. How much money does Penny have in her savings account? John has a certain number of Michael Jackson records. He gives 4 to Mary and now has 11 records. How many records did he have at first? If you double the number of ABBA records that Mary has and add 7. you get the number of Michael Jackson records that John had at first. How many ABBA records does Mary have?  Comments  DIAGNOSTIC CHECKLIST FOR ALGEBRA LINEAR ONE VARIABLE EQUATIONS Objectives B. Vocabulary B.1 Coefficient  B.2 Like Terms B.2.1 Given the term for comparison  3  B.4.2 Multiplicative (One)  (point to the numbers and say) (point to the tetters and say)  17x 3a  5x  5a  x  5  -5  5n  Write the opposite (additive inverse) of each expression on your paper. x  19a  -25n  j j  4x-9  Write the reciprocal (multiplicative inverse) of each expression on your paper.  3  (point to the numbers and say) (point to the letters and say)  3x  -6x  5x  Solve each equation. Place the answer on your paper. 8+ n=8 n + 3=3 -3 + n=-3 What do you notice about the answers? Does it matter what numbers we use? Could we use variables (letters)? Do you know any other special properties of zero? 4 x n=4  nx-5=-5  E  I  Organize the cards into groups of like terms:  1. ^ 2 B.4 Identities B.4.1 Additive (Zero)  x  CJ  g  From the set cards, choose the like terms for 3x:  7 B.3.2 Multiplicative (Reciprocals)  E  For each expression, tell me the coefficient of n: (Hint: Remember coefficient means the number in front of the letter.) 3n 4 + Sn 7-2n -n (watch for 9n) (watch for 5n) (watch tor negative)  -5a B.3 Inverses B.3.1 Additive (Opposites)  ro  ; s  Items  -3x B.2.2 Choose from a group of expressions  OJ  7n = 7  What do you notice about the answers? Does it matter what numbers we use? Could we use variables (letters)? Do you know any other special properties of one?  I  Comments  o  < >  p g  (/> c  c  <  2  s •a  8  DO  S  «•  3  >2  a o a-o  a  0  oo s r™ *£•  6"  i  II  C i  5 II ro  s  2  03 x  ~  ro  -*  If; O  CO  8-1' =• £-1  <° •< B O  C  -n z O o 33  H  m >E" r O m O mO X c O QJ rn > z O H m 33 > o z  3  — |o  >  O  0 i -  &s  (/)  H  Correct Final S o l u t i o n TJ 30  zero annexation identity contusion  O O  like terms (conioming)  m o c  partial distributive order of operations + p r o p e n y ol equality  33  >  X properly ol equality coefficient errors  -v -  .IS  basic facts  4  faulty J g o r i t h m wrong operation  rule b a s e d distributive mechanical/perceptual^ random incomplete  202  r 33  m c r-  v> it/) - H -ic  z  DIAGNOSTIC CHECKLIST FOR ALGEBRA M> m •  B~  R M an*  VARIABLE EQUATIONS  C o m pu t a i i p n a l  Siqn  Olher it)  c ,9  LINEAR ONE  RESULTANT  PROCEDURAL  M  ~i H  S s 111111 s Di  > >  s  S  E i S 9 ^ 5 o v aj ra c = Q « > y E " 5 ° " 5 $ E S u £ >• * o 8 8 -S » * : ! « ^ g • . t o e ~ ai h - 7. 1 ... S -= 9 ' •!• ? 5 £ ? 5 ; ; ; ; ; " ° S . O ~ - . - - X O * * » T X o ••' * + K - 5 C 5 •  l\\llUl\  1  J  Objectives Items D.2 Two Variable Terms D 2.2 On both sides of equality -7x=16 + x x + - 48 = - 23x 333-I58x= - 1 2 1 x 971x = 420 + 985x D 3 Inclusion of Parentheses  •  2(5 t x) = - 8 1 6 = 4(X + 3) l f>( - T * x) - 48 288= - 12(x + -241  |  E. Multi Step E.I Combined Numeric and Variahle Terms - 4x + — 7 i - 5x + — 5 = 6 E l l On one side - 3 - x + 8 + ~ 4 x = - 10 of equality 0 = - 7 - - 1 2x - - 1 7 - 2x 711-165x+ "613+ ~175x= - 582  T  •  -  —  I 1  J  .  _  !  E i.2 On both sides - 2x - ~8 = 5x-6 of equality 4 9 - -27x = 16-6x 38+ -35x= " 12+ -30 - 5 1 3 K + - 1 5 6 = -463x + -606 E 2 Inclusion ot Parentheses E 2 1 On one side of equalitv  i  13(x-3)= 2x-9 52x + - 16 = 15(x - 6) - 9 + 24x= -9(-10 + x) 66x + 44 = 26(x - 26)  E.2 2 On both sides 10(x+ - 5 l = 20(x-2) of eouality "9(x + 8) = 8(x + -9) 10(10-xl = - 9 ( - 9 + x) 144(6 -x) = 160(5 - x)  Comments  j  \  .  1  i  ..  ..  

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