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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 STUDENTS' ERRORS MADE IN THE SOLUTION OF LINEAR EQUATIONS by CARRYL DIANE KOE B.Sc, Uni v e r s i t y of C a l i f o r n i a (Davis), 1967 M.A., Uni 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 Science Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1989 © C a r r y l Diane Koe, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department 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) ABSTRACT This study describes the development and v a l i d a t i o n of a diagnostic c h e c k l i s t intended to assess students' understandings of beginning algebraic concepts and i d e n t i f y the errors made on one-, two-, and multi-step l i n e a r equations i n one variable with i n t e g r a l c o e f f i c i e n t s and sol u t i o n s . The study occurred i n f i v e phases: (1) textbook analysis for scope of concepts and sequence of equations; (2) e r r o r - c a t e g o r i z a t i o n scheme development based on previous research; (3) construction, t e s t i n g , and s e l e c t i o n of equations based on systematic v a r i a t i o n of complexity, s t r u c t u r a l format and numerical sign and magnitude; (4) i n i t i a l t e s t i n g of the instrument f o r format and wording changes; (5) t e s t i n g of the f i n a l instrument and i n v e s t i g a t i o n of the nature, frequency, and i n t e r -r e l a t i o n s h i p s among the concepts held and the errors which occurred. The c h e c k l i s t used a semi-structured interview technique i n conjunction with e r r o r - a n a l y s i s . The r e s u l t s indicated that the c h e c k l i s t was a useful diagnostic t o o l and provided many in s i g h t s into the i n t e r r e l a t i o n s h i p s among students' concepts, errors, and achievement l e v e l s . Implications for i n s t r u c t i o n i n algebra were explored, and suggestions f o r future instrument development were made. TABLE OF CONTENTS i i i Page ABSTRACT i i LIST OF TABLES v i ACKNOWLEDGEMENT v i i CHAPTER I STATEMENT OF THE PROBLEM 1 Background of the Problem 1 Purpose and Nature of the Study 3 J u s t i f i c a t i o n of the Study 4 Research Questions 5 I I . REVIEW OF RELATED LITERATURE 7 Diagnostic Models 7 Algebraic Processing 7 Processing of Linear Equations 9 Summary of Diagnostic Models 11 Diagnostic Methods 11 Paper-and-Pencil Tests 11 Error-Analysis 12 Semi-Structured Interviews 18 Summary of Diagnostic Methods 20 Types of Errors i n Algebraic Diagnosis 21 Conceptual Errors 22 Variable 23 Expressions 2 6 Equality 28 Meaning of Equations 30 Identity Elements 32 Numerical Magnitude 33 Equation Structure and Complexity 34 Summary of Conceptual Errors 35 Procedural Errors 36 Order of Operations 3 6 Algebraic Properties 37 Summary of Procedural Errors 38 i v Resultant E r r o r s 38 Computational E r r o r s . 38 Sign E r r o r s 40 Mechanica l /Perceptual E r r o r s 41 Random E r r o r s 42 Incomplete Work 42 Summary and Recommendations 43 I I I DESIGN OF THE STUDY 47 Phase 1: Content Ana lys i s of F i r s t - Y e a r Algebra Textbooks . 47 Phase 2: Development of the E r r o r - Categor i za t ion Scheme . 49 Phase 3: Construct ion and Se lec t ion of Equations . . . . . 50 Construct ion of Equations 52 S e l e c t i o n of Equations 55 Subjects 55 Procedures . . . . . 55 Analyses 57 Phase 4: Refinement of the Instrument 59 Subjects 59 Procedures 5 9 Analyses . 60 Phase 5: Tes t ing of the F i n a l Instrument 60 Subjects 61 Procedures 61 Analyses : 62 Summary of Design 63 I V RESULTS 64 Summary of Phases 3 and 4 64 Phase 5: Instrument Tes t ing 68 Subjects 68 Procedures 72 Presentat ion of Results 73 Achievement Levels and E r r o r s 73 I d e n t i f i c a t i o n of Systematic and Common E r r o r s . . . 81 Instrument V a l i d i t y 83 Concepts, Vocabulary, and E r r o r s 85 Complexity /Structure/Magnitude E f f e c t s on E r r o r s . . 91 Summary of Results 94 V DISCUSSION 96 Review of Procedural Design 96 Rat ionale for the Diagnostic Checklist for Algebra . . . . 100 Assessment of the Diagnostic Checklist for Algebra . . . . 103 V S t a t i s t i c a l Properties 103 V a l i d i t y • 103 R e l i a b i l i t y 104 Diagnostic Considerations 105 Diagnostic Categories 105 Systematic and Common Errors 10 6 Errors Relating to Previous Research 107 Summary of Assessment of the Checklist 110 Sig n i f i c a n c e of the Study I l l Limitations of the Study 113 Suggestions f o r Future Research 115 REFERENCES 119 Appendix A: Phase 2: Error-Categorization Scheme Development 126 Appendix B: Phase 3: Equation Construction, Testing, and Selection 132 Appendix C: Phase 4: Wording and Format Changes 166 Appendix D: Phase 5: S t a t i s t i c a l Analyses of Error Correlations 172 Appendix E: Diagnostic Checklist for Algebra 198 v i LIST OF TABLES Table Page 1 Number of Researchers Finding Errors 46 2 Types of Linear Equations Presented i n 15 Textbooks . . . . . . 51 3 Phase 4: Comparison of High and Low Achievers' Error Frequencies on the Instrument 66 4 Individual Subjects' Values on Demographic, IQ, and Mathematical Achievement Variables 7 0 5 Phase 5: Comparison of High, Average, and Low Achievers' Conceptual Understandings on the Instrument 74 6 Phase 5: Comparison of High, Average, and Low Achievers' Error Frequencies on the Instrument 75 7 Relationship Between Text P o s i t i o n and Mean P-Values of Equation Types 8 6 8 The Relationship of Numerical Magnitude with F i n a l Answers and Rule-Based Sign Errors . . . . . . 92 V l l ACKNOWLEDGEMENTS I would l i k e to extend my appreciation to the many people who as s i s t e d me i n the completion of t h i s study. Dr. David R o b i t a i l l e served as my advisor and sounding board. His patient guidance and u n t i r i n g devotion to excellence i n prose and research has made the success of t h i s endeavour possible. Dr. James S h e r r i l l served as my research consultant, querying the developmental aspects; ensuring terseness and consistency of the research paradigm and the data presentation. Dr. Robert Conry acted as s t a t i s t i c a l consultant, providing guidance through the maze of analyses and the demands of 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 formulations. Special thanks go to Dr. Douglas Edge and Dr. Gaalen Erickson who consulted on diagnostic and conceptual apsects, r e s p e c t i v e l y . I am indebted to the s t a f f and students who p a r t i c i p a t e d i n the study, and made pos s i b l e t h i s i n v e s t i g a t i o n . L a s t l y , I wish to thank my family for t h e i r patience throughout many d i f f i c u l t days and weeks. Their f a i t h gave me the courage for the ever present re-writes. Special thanks to my husband, Gerry, for his encouragement and support. 1 CHAPTER I STATEMENT OF THE PROBLEM Diagnostic procedures need to be developed to help i d e n t i f y errors made i n beginning algebra (Davis & Cooney, 1978). As K r u t e t s k i i (1976) pointed out, a student's beginning, "external" errors must be i d e n t i f i e d and eradicated before they become " i n t e r n a l , mental operations." Students often create t h e i r own rules f or performing algebraic operations. S i m i l a r l y , they may a l t e r the new procedures they are learning to accommodate previous erroneous conceptions (Kieran, 1983). These erroneous understandings of ea r l y algebraic notions are very r e s i s t a n t to change at a l a t e r date (Rosnick & Clement, 1 9 8 0 ) . Furthermore, erroneous understandings formed i n beginning algebra may account f o r many errors found i n l a t e r mathematical work (Matz, 1980). Hence, i t i s important to i d e n t i f y these errors at t h e i r onset. Background of the Problem. The s o l u t i o n of l i n e a r equations i n one variable has a ce n t r a l role i n f i r s t - y e a r algebra courses. Within the algebra curriculum, t h i s topic forms the basis for the so l u t i o n of equations with two or more v a r i a b l e s . Unless students have a thorough foundation i n the so l u t i o n of l i n e a r equations, l a t e r progress i n algebra may be impeded. Therefore, i n v e s t i g a t i o n of the equation-solving process, i n terms of arithmetic and algebraic expressions, would seem to be an important step i n diagnosing errors which are made i n beginning algebra. In the 1920s, studies of f i r s t - y e a r algebra placed emphasis on product, or resultant, e r r o r s . A wide range of errors was i d e n t i f i e d ; including errors with arithmetic, signs, l i k e terms, and exponents, as well as incomplete solutions, careless errors, and u n c l a s s i f i e d errors (Pease, 1929; 2 Wattawa, 1927) . Unfortunately, for purposes of the present study, these product, or resultant, errors were i d e n t i f i e d without i n v e s t i g a t i o n of t h e i r underlying causes. More recent inv e s t i g a t o r s (Kieran, 1983; Wagner, Rachlin, & Jensen, 1984) have focussed on the processes used by students i n so l v i n g equations. They have noted that students who were adept at s o l v i n g one-step, and even two-step, equations of the form ax+b=c, where a, b, and c are whole numbers and a^O, were unable to solve equations of a more complex form. For example, students could solve 7x+l=50, but not 3x -5+4x+6=50. In solving the f i r s t equation, students seemed to be using mental arithmetic s k i l l s (e.g. "What plus 1 makes 5 0 ? " "What times 7 makes 49?") rather than algebraic operations (e.g. "Add -1 to both sides to obtain the equivalent equation 7x=49; then m u l t i p l y both sides by one-seventh to obtain the s o l u t i o n . " ) . The t r a n s i t i o n from mental arithmetic to algebraic operations apparently poses a major stumbling block for many otherwise academically able adolescents (Booth & Hart, 1982). Studies are needed which focus on the t r a n s i t i o n between mental arithmetic and algebraic procedures. Such focus may bring to l i g h t the reasons underlying p a r t i c u l a r procedural e r r o r s . During the past twelve years many studies have focussed on conceptual errors made by students i n so l v i n g l i n e a r equations i n one v a r i a b l e . These studies achieved the goals for which they were designed. However, for purposes of the present study, most of them f a i l e d to c o n t r o l one or more of the following: 1) factors which might influence e i t h e r the nature or frequency of the er r o r s ; 2) equation complexity: one-, two-, and multi-step equations were interspersed; 3) types of numbers: r a t i o n a l numbers were interspersed with integers; confounding errors i n algebra with errors i n f r a c t i o n s ; 4) the magnitude or signs 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 to e i t h e r the eq u a l i t y symbol or numeric constants. This f a i l u r e to con t r o l factors which may have influenced the type of conceptual errors made by students makes i t d i f f i c u l t to in t e r p r e t and apply the r e s u l t s of these recent studies to the present study. P u r p o s e a n d N a t u r e o f t h e S t u d y The purpose of the present study was to develop and va l i d a t e a diagnostic c h e c k l i s t (Diagnostic Checklist for Algebra) that would i d e n t i f y e rrors made by beginning algebra students when so l v i n g l i n e a r equations i n one v a r i a b l e . It would d i s t i n g u i s h among the conceptual, procedural, and resultant errors made by students when solv i n g such equations; thus, allowing the i n v e s t i g a t i o n of the nature and frequency of the e r r o r s . A semi-standardized interview technique was u t i l i z e d to f a c i l i t a t e such i n v e s t i g a t i o n . This technique was used within the confines of a diagnostic instrument which was i n part a c h e c k l i s t of mastery l e v e l s of concepts and i n part a c h e c k l i s t of errors which occurred when solv i n g l i n e a r equations of a s p e c i f i e d d i f f i c u l t y l e v e l . The f i r s t part of the instrument was used to investigate the conceptual framework of a f i r s t - y e a r algebra student and was i n the form of a c h e c k l i s t . It was used to i d e n t i f y the status of a p a r t i c u l a r algebraic concept within the student's cognitive framework. A mastery l e v e l for each concept was determined (mastery, p a r t i a l , remedial). The second part of the instrument was i n the form of a set of equations. These equations were used to i n v e s t i g a t e procedural and resultant e r r o r s . The set of equations was formulated to con t r o l complexity and structure, as well as numerical sign and magnitude. An er r o r - c a t e g o r i z a t i o n scheme was developed to f a c i l i t a t e i d e n t i f i c a t i o n of errors found by previous researchers i n algebra. The er r o r - c a t e g o r i z a t i o n scheme was used i n conjuction with the formulated set • 4 of equations. The analysis of the diagnostic data from Grade 8 , 9 , and 10 students was de s c r i p t i v e i n nature. The reason f o r t h i s was two-fold: diagnostic techniques employed and sampling techniques u t i l i z e d . Use of semi-structured interview techniques increased the time spent diagnosing i n d i v i d u a l students, which i n turn l i m i t e d the number of students whose errors were investigated. To incorporate a s t r a t i f i e d random sample within the l i m i t e d number of students used, the sample was drawn from a single school. The use of 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 selected data from the beginning algebra population was not f e a s i b l e . Hence, the data analysis used i n the study was d e s c r i p t i v e i n nature. V a l i d i t y issues were addressed throughout the formation, refinement, and f i n a l t e s t i n g of the instrument. The sequencing of equations was based on textbook analysis and s t a t i s t i c a l techniques were used to provide s t r u c t u r a l v a l i d i t y . Interaction e f f e c t s were analyzed to provide construct v a l i d i t y . Comparison of r e s u l t s with measures of algebraic achievement, IQ, and l a t e r grades provided measures of concurrent and p r e d i c t i v e v a l i d i t y . J u s t i f i c a t i o n of the Study Several researchers have noted the importance of ea r l y diagnosis of errors i n v o l v i n g the equation-solving processes used by students i n beginning algebra (Davis & Cooney, 1 9 7 8 ; Hart, 1 9 8 1 , 1 9 8 3 ; K r u t e t s k i i , 1 9 7 6 ; Rosnick & Clement, 1 9 8 0 ; Wagner et a l . , 1 9 8 4 ) . An instrument was needed to help explore students' understandings of algebraic concepts and vocabulary, to i d e n t i f y students' errors i n algebraic procedures, and to determine other errors associated with computation, signs, and t r a n s c r i b i n g . While many paper-and-pencil algebra t e s t s e x i s t , they do not meet the diagnostic needs stated above. ' Using paper-and-pencil t e s t s , correct 5 solutions 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 error types, nor i d e n t i f y errors which occur during the intermediate steps when solv i n g l i n e a r equations. However, by u t i l i z i n g semi-structured interview techniques within a diagnostic instrument, such diagnostic needs, created by the complex nature of algebra, can be met. .L i s t e n i n g to a student solve an equation aloud, and i n q u i r i n g about the reasoning processes used, may permit e a r l y i d e n t i f i c a t i o n of erroneous conceptions ( K i l p a t r i c k , 1 9 6 7 ) and f a c i l i t a t e understanding of the student's reasons f o r holding such conceptions (Kieran, 1 9 8 3 ; Hart, 1 9 8 3 ) . Thus, semi-structured interviews seem to meet the diagnostic needs to a greater degree than do paper-and-pencil t e s t s . Hart ( 1 9 8 3 ) expressed concern about the lack of a p p l i c a b i l i t y of q u a l i t a t i v e research to other students and teaching s i t u a t i o n s . However, when q u a l i t a t i v e research u t i l i z e s instruments which con t r o l the factors of equation complexity, s t r u c t u r a l format, and numerical magnitude, i d e n t i f i c a t i o n of the systematic errors made by a p a r t i c u l a r student and the common errors made among groups of students should be possible (Anderson, 1 9 8 2 ; Hart, 1 9 8 1 ) , regardless of changes i n s e t t i n g . Such i d e n t i f i c a t i o n should allow a p p l i c a b i l i t y of findings to other i n s t r u c t i o n a l s e t t i n g s . Furthermore, the i d e n t i f i c a t i o n of systematic and common errors i s necessary to provide a fir m foundation f o r both the teaching and the remediation of the equation-solving process. Investigation of the concepts, processes, and context i n which errors are made should not be overlooked i n algebraic diagnosis. R e s e a r c h Q u e s t i o n s Several c r i t e r i a were important i n the development and v a l i d a t i o n of the Diagnostic Checklist for Algebra. F i r s t , the diagnostic u t i l i t y of the instrument was considered to be of paramount importance. Second, the i d e n t i f i c a t i o n of the nature and frequency of errors occurring f o r a p a r t i c u l a r student was c r i t i c a l . Third, the i d e n t i f i c a t i o n of systematic and common errors made by students i n sol v i n g l i n e a r equations i n one var i a b l e was important. Fourth, the establishment of the v a l i d i t y of the instrument i n terms of other measures of achievement was necessary. F i f t h , the i n t e r a c t i o n e f f e c t s among error types and con t r o l factors required i n v e s t i g a t i o n . Hence, the following f i v e c r i t e r i a were considered necessary fo r the development of a sound diagnostic instrument i n algebra: diagnostic u t i l i t y , instrument v a l i d i t y , error i d e n t i f i c a t i o n , i n t e r a c t i o n e f f e c t s among errors, and i n t e r a c t i o n e f f e c t s among errors and co n t r o l f a c t o r s . The following research questions were formulated to determine i f the c r i t e r i a f o r instrument development had been met: 1. Does the 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 the nature and frequency of the errors made by a p a r t i c u l a r student? 2 . Does the 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 errors f o r p a r t i c u l a r students and common errors among groups of students? 3. Is the Diagnostic Checklist for Algebra v a l i d ? 4. Are students' understandings of algebraic concepts and vocabulary r e l a t e d to the kinds of errors they make i n the so l u t i o n of l i n e a r equations i n one variable? 5 . Do the factors of equation complexity, s t r u c t u r a l format, and numerical magnitude i n t e r a c t i n terms of the nature and frequency of the errors produced? C H A P T E R I I REVIEW OF R E L A T E D L I T E R A T U R E The review of the l i t e r a t u r e focuses on c r i t i c a l factors i n the development of a diagnostic instrument i n algebra. It i s divided into three sections: diagnostic models, diagnostic methods, and types of errors i n algebraic diagnosis. The f i r s t section presents two models f o r conceptualizing algebraic processing and diagnosing algebraic errors (Matz, 1 9 8 0 ; Bernard & Bright, 1 9 8 2 ) . The second section reviews paper-and-pencil t e s t s , e r r o r - a n a l y s i s , and semi-structured interviews. The t h i r d section examines the types of errors which have been reported by three or more researchers. The purposes of the review of the l i t e r a t u r e are to provide a research foundation for the formulation of a diagnostic instrument for l i n e a r equations i n one variable and to put the current study i n perspective. D i a g n o s t i c M o d e l s Several models have been proposed to explain errors made i n algebra (Bernard & Bright, 1 9 8 2 ; Davis, 1 9 7 9 , 1 9 8 0 ; Davis, McKnight, Parker & E l r i c k , 1 9 7 9 ; Booth, 1 9 8 1 ; Matz, 1 9 8 0 ) . However, only two of the models just c i t e d e x p l i c i t l y conceptualize errors involved i n the equation-solving process: algebraic processing (Matz, 1 9 8 0 ) and processing of l i n e a r equations (Bernard & Bright, 1 9 8 2 ) . These models provided a framework for cat e g o r i z i n g errors made i n solving algebraic equations. A l g e b r a i c P r o c e s s i n g Matz ( 1 9 8 0 ) indicated that students f a l l v i c t i m to one or more of three general types of e r r o r s : 1) unwarranted extrapolation of known, v a l i d rules to new problem s i t u a t i o n s ; 2) u n j u s t i f i e d extrapolation of one g e n e r a l i z a t i o n to another; and 3) i n a b i l i t y to change from one l e v e l of understanding of a concept to the next higher l e v e l . These three general e r r o r types form a t h e o r e t i c a l foundation for i n v e s t i g a t i n g errors made i n s o l v i n g one v a r i a b l e equations. Extrapolation errors can occur i n one of two ways: generalized d i s t r i b u t i v e or repeated a p p l i c a t i o n . The d i s t r i b u t i v e law of m u l t i p l i c a t i o n over addition i s an example of a v a l i d extrapolation [e.g. a(b+c)=ab+ac]. Treating the square root symbol as i f i t were something which could be d i s t r i b u t e d over addition provides an example of a common erroneous extrapolation (e.g. Va + b = Va+Vb). The incorrect use of the c a n c e l l a t i o n law provides an example of repeated applications [e.g. ax/a=x to (ax+by)/(a+b)=x+y], where the student applies c a n c e l l a t i o n to i n d i v i d u a l terms of the numerator and denominator and then adds the p a r t i a l r e s u l t s to obtain the answer. Matz noted that students sometimes a r r i v e at correct solutions to problems using techniques which are not generalizable (e.g. 16/64=1/4 done by c a n c e l l i n g the 6's). U n j u s t i f i e d extrapolation of one generalization to another often occurs i n i d e n t i t y confusions.. Students know A*0 has the value of 0 or A depending on whether * represents m u l t i p l i c a t i o n or addition. However, students sometimes confuse m u l t i p l i c a t i o n and addition. They f a i l to d i s t i n g u i s h between the two operations ( i . e . the "x" symbol vs the "+" symbol). Students' i n a b i l i t y to change from one l e v e l of understanding of a concept to the next higher l e v e l also seems to r e s u l t i n er r o r s . For instance i n arithmetic, when 7 and 8 are concatenated to form 78 there i s a concept of place value and i m p l i c i t addition. However, i n algebra, Ix denotes i m p l i c i t m u l t i p l i c a t i o n . The student i s required to keep and use both the o l d and new concepts i n order to understand expressions such as 7 8 x . Errors sometimes occur "as the r e s u l t (s) of reasonable, although unsuccessful, attempts to adapt previously acquired knowledge to a new s i t u a t i o n " (Matz, 1 9 8 0 , p. 1 5 4 ) . In order to diagnose these misapplications of knowledge, i t i s important to acquire an understanding of the erroneous conceptions involved i n students' errors (Herscovics, 1 9 7 9 ; Herscovics & Kieran, 1 9 8 0 ; Skemp, 1 9 8 0 ) . P r o c e s s i n g o f L i n e a r E q u a t i o n s Bernard & Bright ( 1 9 8 2 ) suggested that any model of equation-solving performance should: ... be a m u l t i - l e v e l account ranging across (a) perception and i n t e r p r e t a t i o n of algebraic symbolism, (b) conceptual understanding of the problem-solving task, (c) a p p l i c a t i o n of i n t e l l e c t u a l operations and processes, and (d) the development of s t r a t e g i e s and general methods f or solving any equation 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 for modeling equation-solving which contains s i x l e v e l s : task, method, strategy, process, operation, and t a c t i c . Each of these l e v e l s may be a source of algebraic e r r o r s . For example, when presented with the task of solving an equation such as 3(x+ 5 ) = 2 7 , a student may choose a step-by-step method, apply a strategy of removing brackets through the process of d i s t r i b u t i n g , and then use the operation of m u l t i p l i c a t i o n with the r e s u l t of writing "3x" (Bernard & Bright, adapted from Figure 3, p. 37). At each l e v e l of performance (task, method, strategy, process, operation, t a c t i c ) students have an opportunity to make e r r o r s . When a student i s presented with a task of sol v i n g an equation, p r i o r knowledge of the concepts of var i a b l e and equality (Kieran, 1981a, 1981b, 1 9 8 3 ; Wagner, 1981a) as well as arithmetic operations (Englehardt & Wiebe, 1981) act as input factors f o r the task. Many processing factors i n t e r a c t when choosing the methods, strategies, and processes used i n accomplishing the task. Sometimes confusions r e s u l t from the in c o r r e c t ordering of algebraic operations in v o l v i n g addition and m u l t i p l i c a t i o n properties of eq u a l i t y (Davis & Cooney, 1978), d i s t r i b u t i v e errors (Matz, 1980) and order of operations (Rosnick & Clement, 1980) . When performing the operations and t a c t i c s necessary f o r completion of the task, output errors may occur, such as v i s u a l and perceptual confusions (Matz, 1980) and careless mistakes (Englehardt & Wiebe, 1981) . Bernard & Bright (1982) indicated that students' understandings of algebraic concepts, symbolism, operations and strategies must be explored to provide comprehensive diagnosis. The i n t e r a c t i o n among these understandings may a f f e c t a student's perception of the task and the methods av a i l a b l e to accomplish that task. Too often studies on errors i n equation-solving emphasize the resultant errors with some focus on procedural errors and l i t t l e , i f any, concern f o r conceptual e r r o r s . Such emphasis leaves only an instrumental understanding of students' errors and "may focus attention on aspects of equation s o l v i n g that do not suggest remediation techniques" (Bernard & Bright, 1 9 8 2 , p. 2 1 ) . Hence, i t i s important to explore the i n t e r r e l a t i o n s h i p s among conceptual, procedural, and resultant errors to provide a thorough understanding of the nature of the errors which students make and to provide a basis f o r remediation. Summary o f D i a g n o s t i c M o d e l s Matz ( 1 9 8 0 ) and Bernard & Bright ( 1 9 8 2 ) described models of algebraic lea r n i n g that provide a framework within which algebraic errors might be explored. They emphasized the importance of p r i o r knowledge, ordering of concepts, processing a b i l i t y , and accuracy of r e t r i e v a l i n obtaining both p a r t i a l and f i n a l r e s u l t s . Their work underlines the importance of assessing input and process, as well as output, i n diagnosis. D i a g n o s t i c M e t h o d s Paper-and-pencil t e s t s , e r r o r - a n a l y s i s , and semi-structured interviews have been developed to investigate students' understanding of algebra. However, each of these methods has weaknesses, as well as strengths. P a p e r - a n d - P e n c i l T e s t s Petrosko ( 1 9 7 8 ) reviewed the strengths and weaknesses of 122 standardized algebra t e s t s a v a i l a b l e at the secondary l e v e l . These tests included achievement and diagnostic t e s t s , but not aptitude t e s t s , and were evaluated on 39 c r i t e r i a by at least two r a t e r s . Almost h a l f of the tests included a short statement regarding sources f o r items. • However, actual procedures used f o r s e l e c t i o n of items were presented i n only ten of the test manuals. Correlations with other t e s t s , experimental uses of the test, and t h e o r e t i c a l support were not provided i n 110 of the 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 studies were not mentioned i n any of the t e s t manuals. While 20 of the 122 tes t s provided acceptable i n t e r n a l consistency r e l i a b i l i t y c o e f f i c i e n t s (r>0.70), no other form of r e l i a b i l i t y was provided f o r any of the t e s t s . Petrosko (1978) noted that many tests were " d e f i c i e n t i n basic aspects of tes t q u a l i t y " (p. 146). Most standardized mathematics t e s t s do l i t t l e more than predict that a student w i l l "perform poorly," or has performed poorly, i n the p a r t i c u l a r areas t e s t e d (Swanson, Schwartz, Ginsberg & Kossan, 1981) . Paper-and-pencil t e s t s provide an accurate record of students' work, but f a i l to reveal t h e i r thought processes. "Paper-and-pencil tests ... are so often misinterpreted that they need to be supplemented with other means of evaluation" (Peck and Jencks, 1974, p. 54). Paper-and-pencil t e s t s provide minimal information regarding input errors, and almost no information regarding process errors and errors made when working through the intermediary steps ( p a r t i a l results) leading to the f i n a l s o l u t i o n . These d e f i c i e n c i e s l i m i t the diagnostic e ffectiveness of paper-and-pencil t e s t s . E r r o r - A n a l y s i s The purpose of err o r - a n a l y s i s i s thr e e f o l d : to determine the type, frequency, and source of errors made i n performing algebraic tasks. The ea r l y studies i n algebra focused on the nature and frequency of err o r s . More recent work has focused on the kinds of rules that students generate i n cre a t i n g these e r r o r s . Wattawa (1927) recorded the errors made on te s t s , written work, and o r a l responses i n a ninth-grade algebra class f o r a period of three months. He noted several mistakes r e l a t e d to s o l v i n g l i n e a r equations with i n t e g r a l c o e f f i c i e n t s (arithmetic: 4 1 . 4 % , algebraic laws: 1 7 . 2 % , sign e r r o r s : 1 3 . 8 % , and i n c o r r e c t operations: 6 . 9 % ) . Other errors occurred r e l a t i n g to f r a c t i o n s ( 1 7 . 2 % ) and quadratics ( 3 . 5 % ) . Many of the arithmetic and sign e r r o r s were a r e s u l t of f a u l t y reading or copying ( 1 0 . 8 % ) . Pease ( 1 9 2 9 ) investigated the errors of 3 5 0 f i r s t - y e a r algebra students i n four d i f f e r e n t schools using a t e s t of the four basic operations with numbers, monomials, and polynomials. He found that almost 25% of the subjects had d i f f i c u l t y with m u l t i p l y i n g by zero, e i t h e r with numbers or monomials. He also found that about 20% of the subjects experienced d i f f i c u l t i e s i n adding and subtracting monomials with no apparent numeric c o e f f i c i e n t (e.g. -aH—a) . Subtracting a p o s i t i v e monomial from a negative monomial also produced about a 20% error rate. Removing brackets preceded by a minus sign produced about a 25% error rate for both numbers and polynomials. Pease provided a summary of error types across operations for numbers, monomials and polynomials. These error types were sign ( 2 3 % ) , process ( 3 1 % ) , l i t e r a l number (e.g. a+a=a, 6%) , combination of terms ( 4 % ) , arithmetic ( 5 % ) , zero ( 4 % ) , careless errors ( 8 % ) , u n c l a s s i f i e d ( 1 1 % ) , and exponent ( 8 % ) . Randall ( 1 9 5 5 ) reported on the common errors made by 1 3 1 students from 56 high schools. Two or three students were chosen from each school by t h e i r mathematics teachers as being the best general mathematics students i n that p a r t i c u l a r school. The tes t administered to these students involved the four basic operations, l i n e a r equations, f a c t o r i n g , and word problems. Students had common errors of subtracting the larger number from the smaller number, combining unlike terms, and knowing procedures f o r f i n d i n g the unknown. A l l other common errors found r e l a t e d to f r a c t i o n s or decimals and were not applicable to the present study. Suggestions were made to improve the r e s u l t s by s t r e s s i n g arithmetic accuracy, by teaching some algebra to general mathematics students, and by emphasizing mathematical thinking through the use of word problems. Hil l i n g - S m i t h (1979) tested 105 th i r t e e n year-olds from the top 90% of the a b i l i t y range using ten questions devised to investigate t h e i r understanding of beginning algebra. He found the following types of errors: order of operations done l e f t to ri g h t (60%); confusing c o e f f i c i e n t s with exponents (5x with x 5, 24%; z 3 with 3z, 22%); adding unlike terms ( i e . 3+Q =3Q, 17%); confusion of in e q u a l i t y symbols (20%); and i n a b i l i t y to abstract and generalize (64%). Suggestions were made for teaching algebra using apparatus and demonstrations so that abstractions would be more concretely conceptualized, and for emphasizing the correct reading of mathematics f o r students. Tatsuoka, Bierenbaum, Ginsburg, & Kossan (1980) tested 127 Grade 8 students on addition of integers a f t e r three weeks of i n s t r u c t i o n . Two errors r e l a t i n g to the absolute value of the numbers were i d e n t i f i e d . Students e i t h e r added the absolute values qf the numbers or subtracted the number with smaller absolute value from the number with larger absolute value. Six errors were found which r e l a t e d to the sign of the answer. Students e i t h e r used the sign of the larger or smaller number, always used a p o s i t i v e or negative sign, or always used the sign of the f i r s t or second number. A "sign X absolute value error vector" was used to create items. Four items were required to determine a student's rule uniquely. The re s u l t s of the study ind i c a t e d that students most frequently used the sign df the number with the larger absolute value and that the rules students generated were based on the type of i n s t r u c t i o n they received. Rules i n v o l v i n g extrapolation of v a l i d rules using other operations, such as "two negatives make a p o s i t i v e , " were not included. The lack of i n c l u s i o n of such rules may have l i m i t e d the a p p l i c a b i l i t y of the r e s u l t s to classroom s i t u a t i o n s . Cote (1981) studied the operations of addition and subtraction of integers. The focus was on the confusions that a r i s e between the operation symbols and the sign of the numbers. F i f t y - f i v e students were taught addi t i o n and subtraction of integers. Three major types of errors occurred: d i r e c t subtraction (e.g. 17-38 meant 38-17), s u b s t i t u t i o n (e.g. -17+49 meant add 17 and 49 and make i t negative), and problems of int e g r a t i o n (e.g. -17 - 45 meant change 45 to -45 but then subtract 17 from 45 to give an answer of 28). Subjects were categorized by the number and types of errors they made. Results i n d i c a t e d that of the 30 subjects who made less than 10 erro r s , 18 were c l a s s i f i e d as d i r e c t subtraction, 9 as s u b s t i t u t i o n and 3 were u n c l a s s i f i e d . Of the 15 subjects who made more than ten errors, 6 were d i r e c t , 3 s u b s t i t u t i o n , and 6 int e g r a t i o n . The higher achievers d i d not make i n t e g r a t i o n e r r o r s . Ten subjects made no er r o r s . Cote's (1981) r e s u l t s suggested that the differences i n the error patterns of high and low achievers may provide in s i g h t into the d i f f i c u l t i e s which low achievers have with integers. The re s u l t s reinforced the importance of considering the sign, magnitude and order of presentation when using integers i n algebraic diagnosis. Anderson ( 1 9 8 2 ) used e r r o r - a n a l y s i s to compare the errors made on whole number, integer, and polynomial expressions. She found sign errors, incorrect-operation errors, d i s t r i b u t i o n errors, and exponent errors to be common among many students. However, students made these errors only i n the context of e i t h e r arithmetic or algebra, but not concomitantly. In four reports of a study on algebraic diagnosis, Sleeman (1984a, 1984b, 1984c, 1986) described the use of a computer-based Leeds Modelling System to diagnose errors of 14- and 15-year-old algebra students. A high percentage of the 14-year-olds' errors were undiagnosed by t h i s method. Interviews were subsequently used to determine the actual reasons for the er r o r s . There were four d i s t i n c t classes of errors found: 1) manipulative, i n which a correct rule e i t h e r has one stage omitted or replaced by an inappropriate operation; 2) parsing, i n which the student misgeneralizes the problem s i t u a t i o n (e.g. 2 + 3x = 5x); 3) c l e r i c a l , i n which the student misreads or miscopies; and 4) random. The f i r s t two of these error types support the t h e o r e t i c a l framework of algebraic errors proposed by Matz (1980). Sleeman (1984b) further suggested that "the d i f f i c u l t i e s i n lea r n i n g algebra have been greatly under-estimated" (p. 35). Movshovitz-Hadar, Zaslavsky, & Inbar ( 1 9 8 7 ) i n v e s t i g a t e d the errors made by high school students i n the Mathematics M a t r i c u l a t i o n Examination given i n I s r a e l to a l l students i n the n o n - s c i e n t i f i c stream at the end of the Grade 11 year. For two consecutive years the researchers analyzed 18 open-ended t e s t items c o n s i s t i n g of 13 general item types among which l i n e a r equations was only one item type. The category system which they developed was based on the empirical data gathered and not on previous research. The s i x e r r o r categories and t h e i r percentages for both years were: 1) misused data ( 22%, 2 0 % ) , 2) misinterpreted language (17%, 1 8 % ) , 3) l o g i c a l l y i n v a l i d inference (2%, 1%), 4) d i s t o r t e d theorem or d e f i n i t i o n (34%, 3 2 % ) , 5) u n v e r i f i e d s o l u t i o n (0%, 2 % ) , and 6) t e c h n i c a l e r r o r ( 25%, 2 7 % ) . The purpose of the model was to provide ins i g h t into students' errors to determine i f they occur across areas of mathematical topics or are associated with p a r t i c u l a r learning or teaching s t y l e s . The errors associated with l i n e a r equations i n one v a r i a b l e f e l l into the categories of misused data, misinterpreted language, d i s t o r t e d d e f i n i t i o n , and t e c h n i c a l e r r o r . Four coders were used for each subject. The coders categorized errors with an i n t e r - r a t e r r e l i a b i l i t y of 0 . 9 1 . In determining the category into which an error was c l a s s i f i e d , only the written work of the student, including "crossed-out, untidy parts" (Movshovitz-Hadar et a l . , 1 9 8 7 , p. 5) was analyzed. This meant that only the nature and frequency of the errors were categorized, without determining the underlying cause. The authors also found "correct solutions that arose from errors that cancelled one another" (p. 7 ) . There was no information 1 8 provided regarding the frequency of these types of correct solutions. The cause of the double errors was not investigated. Hence, a diagnosis of the reasons for students making these errors was not provided through the use of post-facto analysis of students' written work. Er r o r - a n a l y s i s provides a wealth of very precise information. However, the information i s often overwhelming. The focus tends to be on the frequency of i n d i v i d u a l errors, rather than on the i n t e r r e l a t i o n s h i p s of the errors and t h e i r underlying conceptual foundation. These underlying conceptual foundations 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 (Anderson, 1982) . S e m i - S t r u c t u r e d I n t e r v i e w s Diagnosticians have noted that interviews provide i n s i g h t into the conceptual foundations of errors which i s lacking i n other diagnostic approaches (Hart, 1981; Kieran, 1983; Wagner et a l . , 1984; Sleeman, 1986). However, interviews are time-consuming and r e l i a b i l i t y i s questionable (Carpenter, Blume, Herbert, Anick, & Pimm, 1982; Opper, 1977) . In addition, subjects' explanations of how they solved problems may not be a v a l i d r e f l e c t i o n of thought processes (Davis, 1980) . Procedures used by students when so l v i n g problems o r a l l y may be influenced by the mediation between thought and speech (Behr et a l . , 1976; Davis et a l . , 1979). V e r b a l i z a t i o n slows the response rate ( K i l p a t r i c k , 1967). Interaction with the interviewer may also influence r e s u l t s (Herscovics, 1979). Wording changes lower the r e l i a b i l i t y of "think-aloud" techniques (Opper, 1977) . However, despite these l i m i t a t i o n s , subjects' explanations of t h e i r own thought 1 9 processes are believed to provide more data and more r e l i a b l e data than paper-and-pencil t e s t s (Anderson, 1982; Hart, 1983). Hence, the issues of r e l i a b i l i t y and v a l i d i t y need to be addressed i n the development of a diagnostic interview technique. Structured interviews which present the same questions i n the same order seem to s t a b i l i z e students' responses, thereby improving r e l i a b i l i t y (Hart, 1 9 8 3 ; Wagner et a l . , 1984). Structured interviews reduce v a r i a b i l i t y i n cue words. However, the r i g i d i t y of completely structured interviews may l i m i t t h e i r usefulness. E f f e c t i v e diagnosis cannot ignore i n d i v i d u a l d ifferences. There needs to be a compromise between completely structured interviews and f r e e - a s s o c i a t i o n a l questioning. Semi-structured interviews seem to provide such a compromise. Semi-structured interviews present the same task to each student, but follow-up questions can be more f l e x i b l e (Wagner et a l . , 1984). Semi-structured interviews allow formulation 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, and v e r i f i c a t i o n of the systematic nature of errors (Swanson et a l . , 1981; Wagner, 1981b). Semi-structured interviews seem to increase 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 questioning and yet s t i l l provide useful diagnostic information about i n d i v i d u a l students. It should be noted that semi-structured interviews also have l i m i t a t i o n s . When using semi-structured interviews, Wagner et a l . (1984) reported several occurrences of confusion among the terms "v a r i a b l e , " "expression," and "equation." The "investigators were unsuccessful i n 20 formulating questions that would demonstrate a l l of these confusions i n c i s i v e l y " (p. 39). There i s a need for techniques to be incorporated within semi-structured interviews that w i l l i d e n t i f y conceptual confusions and provide i n s i g h t into 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 e r r o r s . One such technique which can be u t i l i z e d with semi-structured interviews i s d e t a i l e d e r r o r - a n a l y s i s , and t h i s approach has been used by several researchers (Anderson, 1982; Sleeman, 1984a, 1986; Tatsuoka et a l . , 1980; Wagner et a l . , 1984). However, except i n the case of the Wagner study, the use of semi-structured interviews occurred 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 . Students' post-facto v e r b a l i z a t i o n s of t h e i r thought processes may be influenced by memory-decay and the knowledge that they made an e r r o r . Hence, such v e r b a l i z a t i o n s may not be v a l i d representations of students' thought processes when i n i t i a l l y solving equations. It i s imperative that semi-structured interviews be conducted during the actual equation-solving process; while, ensuring that probing does not i n t e r f e r e with the c o n t i n u i t y of students' thought processes. By combining the techniques of e r r o r - a n a l y s i s , semi-structured interviews, and judicious probing within a diagnostic instrument, complementary and confirming information should be provided regarding the nature of the errors students make i n s o l v i n g l i n e a r equations i n one v a r i a b l e . Summary o f D i a g n o s t i c M e t h o d s Used alone, paper-and-pencil tests are not e f f e c t i v e as diagnostic instruments (Anderson, 1982; Peck & Jencks, 1974; Swanson et a l . , 1981). Semi-structured interviews seem to provide a procedure which i s e f f e c t i v e i n 21 diagnosing errors made i n sol v i n g equations (Wagner et a l . , 1984). But, again, interviews used alone do not provide s u f f i c i e n t s t r u c t u r i n g of the mathematical content. There i s a need to develop a diagnostic procedure which incorporates e r r o r - a n a l y s i s techniques with semi-structured interviews. Such a diagnostic procedure may be seen as u t i l i z i n g 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 methods to analyze students' understandings of and performance on algebraic tasks. As pointed out by Firestone (1987), "Used separately, q u a l i t a t i v e and quantitative studies provide d i f f e r e n t kinds of information. When focused on the same issue, q u a l i t a t i v e and qua n t i t a t i v e studies can triangu l a t e - that i s , use d i f f e r e n t methods to assess the robustness or s t a b i l i t y of fi n d i n g s . " The i n c l u s i o n of qua n t i t a t i v e and q u a l i t a t i v e methods of diagnosis within the same instrument would improve i t s diagnostic u t i l i t y . However, no diagnostic instruments were found which incorporated both quantitative and q u a l i t a t i v e analyses. Hence, there i s a need f or the development of such an instrument. T y p e s 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 The types of errors i n algebraic diagnosis are presented according to the c a t e g o r i z a t i o n scheme developed f o r use i n the Diagnostic Checklist for Algebra. This presentation provides a research basis f o r that development and an ordered reference f o r each category and sub-category. The d e f i n i t i o n s used i n the developed categorization scheme for errors are presented. Each error-type i s then presented with i t s review of the pertinent l i t e r a t u r e . Three main categories f o r errors i n algebra were formulated into 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 Carry et a l . (1980), Booth (1981), and Bernard & Bright (1982). As these researchers suggested, errors should be viewed i n terms of previous knowledge, processes used, and r e s u l t s obtained. The three main categories are c a l l e d Conceptual (knowledge of concepts and vocabulary associated with algebraic equations including v a r i a b l e , equality, expressions, equations, c o e f f i c i e n t s , l i k e terms, i d e n t i t i e s , and inverses), Procedural (errors made i n i n the s o l u t i o n of l i n e a r equations 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 addition and m u l t i p l i c a t i o n properties of equality, order of operations, and the d i s t r i b u t i v e property), and Resultant (computational and sign errors a r i s i n g from the four basic operations, mechanical/perceptual errors, random errors, and incomplete work). These errors could occur during the intermediate steps, as well as i n the f i n a l solutions of l i n e a r equations i n one v a r i a b l e . A d e t a i l e d discussion of the e r r o r - c a t e g o r i z a t i o n scheme and a tabular presentation of the researchers' findings of p a r t i c u l a r errors i s provided i n Appendix A. A summary of the research findings on errors i n algebra i s presented i n Table 1 at the conclusion of t h i s chapter. C o n c e p t u a l E r r o r s Englehardt & Wiebe (1981) defined conceptual errors as "errors a r i s i n g from incomplete, absent, or inc o r r e c t understanding of underlying mathematical concepts and p r i n c i p l e s " (p. 14). The choice of steps used i n s o l v i n g equations can be influenced by the l e v e l of understanding of the concepts of v a r i a b l e (Wagner, 1981a), l i k e terms (Anderson, 1982), and 23 e q u a l i t y symbol (Kieran, 1981a). The meaning of equations (Kieran, 1 9 8 3 ) and the use of i d e n t i t i e s and inverses (Matz, 1 9 8 0 ) also influenced some of the processes used by students i n so l v i n g equations. V a r i a b l e Students seem to have an extremely inconsistent understanding of the concept of v a r i a b l e . Some view a v a r i a b l e to be a l e t t e r of the alphabet, while others view i t as a replacement f o r a hidden number. Kiichemann (1978) argued that understanding of the concept of variable may be a function of age. Kieran (1983) and Wagner (1981a) reported understanding of the concept of v a r i a b l e to be r e l a t e d to the notation used. S i f o r d (1981) found that Grade 10 students were unable to accept a v a r i a b l e as a general unknown. They thought of variables only as representing one s p e c i f i c unknown number. It i s important i n diagnosis not only to determine a student's view of v a r i a b l e , but also to understand the s t a b i l i t y of t h i s view. A student's view of v a r i a b l e may influence the types of errors made when solv i n g equations. Kiichemann (1978) and Booth (1981) reported that the degree of understanding of the concept of variable seemed to i n t e r a c t with both the type and number of operations involved 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 so l v i n g a p a r t i c u l a r equation. Kiichemann (1978) found tha't the operations of subtraction and d i v i s i o n provided more insi g h t into the t e n a c i t y with which students would c l i n g to t h e i r p a r t i c u l a r view of v a r i a b l e . The number of operations and the occurrences of the variable within the equations caused s h i f t i n g views of the concept of variable (Booth, 1981). Davis et a l . (1979) also noted these phenomena and suggested that mathematically stronger students may be able to discuss equations without 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 to enable the higher achieving students to make the t r a n s i t i o n from arithmetic to algebra more r e a d i l y (Davis & McKnight, 1979). Rosnick & Clement (1980) and Rosnick (1981) found that even college students d i d not have a good understanding of the concept of v a r i a b l e . Both studies reported that over one-third of the college students were unable to solve the "Professor and Student" question c o r r e c t l y ( i . e . Write an equation f o r the statement: There are s i x times as many students as professors). These students seemed to view variable as the object i t s e l f (e.g. substitute S wherever you see the word students i n the sentence). This supports Kuchemann's (1978) concerns regarding students viewing variables as merely objects. However, i t c a l l s into question his contention that students' understanding of var i a b l e i s a function of age. Wagner (1981a) presented 15 middle school students, median age 13 years, and 15 high school students, median age 16.5 years, with a conservation task i n v o l v i n g a b i l i t y to d i s t i n g u i s h whether the alp h a b e t i c a l representation of a v a r i a b l e was a c r i t i c a l f a ctor i n the solu t i o n of simple l i n e a r equations: 7xW+22=109 7xN+22=109 Students were asked t h i s question: If you were to figu r e out what W should be to make t h i s statement true (pointing to the f i r s t equation) and what N should be to make t h i s statement true (pointing to the second equation), which would be larger, W or N? How can you t e l l ? (p. 109). 25 Most students seemed to react to t h i s question i n one of two ways: the change of l e t t e r was considered i r r e l e v a n t or the change of l e t t e r implied an e n t i r e l y new problem. Wagner found that the a b i l i t y to recognize that W and N were d i f f e r e n t names for the same number was s i g n i f i c a n t l y r e l a t e d to exposure to algebraic concepts, but not to the age of the student. This suggests that understanding of variable i s r e l a t e d to algebraic experience, rather than age. Wagner noted that the wording of her question was d e l i b e r a t e l y misleading. Some students may have f e l t that one l e t t e r was larger due to alphabetical order rather than mathematical properties. Kieran used Wagner's (1977, 1981a) equations, but changed the question format: "Are the solutions to these two equations the same or d i f f e r e n t ? " (p. 161). Kieran found that with t h i s wording change, 12-13 year olds who had no formal algebra knew that the value of the l e t t e r s was the same. Their usual explanation was through a 1-1 v e r t i c a l matching of symbols (See equations on p. 24). The fa c t that young, a l g e b r a i c a l l y inexperienced students knew that the l e t t e r s had the same value seemed contrary to both Wagner's (1981a) and Kuchemann's (1978) f i n d i n g s . Yet, t h e i r method of analysis supported Rosnick & Clement's (1980) contention that students view variables as objects. In view of these contradictory findings, further exploration of students' understanding of the concept of variable seems indicated. To summarize, studies i n v o l v i n g understanding of v a r i a b l e indicated that confusion seemed to centre around viewing the variable as representing an a c t u a l object or l e t t e r of the alphabet, rather than a magnitude. These 26 erroneous conceptions were not l i m i t e d to beginning algebra students, but p e r s i s t e d even at college l e v e l s , with age and algebraic background acting as p o s s i b l e modifying f a c t o r s . E x p r e s s i o n s The s i m p l i f i c a t i o n of expressions forms a major component of the equation-solving process. Several researchers have i d e n t i f i e d the a p p l i c a t i o n of the d i s t r i b u t i v e property and the combination of unlike terms to be the two most prominent s i m p l i f i c a t i o n procedures associated with errors found i n the solu t i o n of algebraic equations. As such, understanding of the concepts involved i n s i m p l i f i c a t i o n of expressions may influence the types of errors made i n l i n e a r equations i n one v a r i a b l e . Anderson (1982) studied the error patterns of 200 Grade 9 and 10 students on three t e s t s dealing with algebraic expressions, arithmetic expressions, and sing l e operations with integers. She found that very few students made both arithmetic and re l a t e d algebraic e r r o r s . For example, students c o r r e c t l y solved 697-(494+387), but i n c o r r e c t l y wrote a-(b+c) as a -b+c. This type of error seemed to suggest that students may perform algebraic manipulations without r e l a t i n g these manipulations to the underlying mathematical properties. A f t e r interviewing 16 of the 200 students used i n her study, Anderson (1982) found that most students just saw " l e t t e r s " as p r e c i s e l y that. They re f e r r e d to "x 2 " as "2x 's." This l e d them to make mistakes i n manipulating variables which they d i d not make i n manipulating numbers. Kuchemann (1978) and Kieran (1983) also suggested that the degree of understanding of these' concepts influenced the processes 27 used by students i n equat ion- so lv ing , and hence in f luenced r e s u l t s . Another explanat ion for the incons i s tency of Anderson's (1982) re su l t s between a lgebra i c and ar i thmet ic e rrors may be that ar i thmet i c and a lgebra ic items were not at the same l e v e l of d i f f i c u l t y . Ar i thmet ic items usua l ly invo lved l a r g e r numbers than d i d the corresponding a lgebra i c items. This use of l a r g e r numbers may have caused students to wri te out t h e i r work e x p l i c i t l y , rather than just answers, thus reducing the frequency of ar i thmet i c e r r o r s (e .g. x +x =x was a common a lgebra ic e r r o r , but 231+231=231 was not a common ar i thmet ic e r r o r ) . • Another p o s s i b i l i t y i s that the l a r g e r numbers helped students to focus on the process rather than simply t r y i n g to give an answer (Booth, 1981). Anderson (1982) claimed that systematic errors ex i s t i n a lgebra . A systematic e r r o r was def ined as an e r r o r made i n more than 50% of a v a i l a b l e oppor tun i t i e s for that p a r t i c u l a r e r r o r . A t o t a l of 30 d i f f e r e n t systematic e r r o r s were made by 7 9 of the 200 students i n the study. Of these only ten were a p p l i c a b l e to the present study. These invo lved s ign e r r o r s for m u l t i p l i c a t i o n [e .g . -5(2p -7)=-10p -35] and add i t i on (e .g . 5r H—21r =16r ) , 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 .g. -6(13a +8)=-78a +8 and -6(13a +8)=13a-48], and i n c o r r e c t operat ion errors [e .g . (17.x +2)-(12x +9)=29x -11] . S u r p r i s i n g l y , l i k e term e r r o r s [e .g. 27b -10=17h and 5r +-3(7r -2)=12r -2 ] , b a s i c fac t e r r o r s , and i n c o r r e c t w r i t i n g of operat ion symbols were not found to be systematic e r r o r s . In a d d i t i o n to systematic errors made by i n d i v i d u a l students, Anderson (1982) a l so i n v e s t i g a t e d common e r r o r s made among groups of students . A 28 common e r r o r was defined as a systematic error made by 10 or more students. Anderson found 15 such common errors, three of which occurred i n 20% or more of the sample. These three errors were ax +-bx =(b-a)x, (|b|>|a|); ax +x= ax 2; and ax +bx =abx. Of the twelve remaining common errors, only two involved polynomial expressions of degree one and were thus pertinent to the present i n v e s t i g a t i o n . They were ax +bx =(a+b)x 2 and a(bx +c)=abx +c. Despite the numerical magnitude inconsistencies among her t e s t s , Anderson's ( 1 9 8 2 ) study represented a unique attempt to explore 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 expressions artd s o l v i n g equations. Many common errors occurred i n the s i m p l i f i c a t i o n of expressions within the equations. This occurrence reinforces the need f o r i n v e s t i g a t i n g students', 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 process. E q u a l i t y Some students perceive the equality symbol to be somewhat l i k e an operation. For example, when di r e c t e d to read and explain a sentence such as 5 = 2 + 3 , students would respond that i t was backwards and re-write the sentence as 2+3=5 and then state that 2 plus 3 makes 5 . While t h i s may have been due to f a m i l i a r i t y with the second arithmetic sentence, the use of the word "makes" and the i n a b i l i t y to read the f i r s t sentence suggest that the students perceived the e q u a l i t y symbol to be "an i n d i c a t i o n to do something" (Behr, Erlwanger, & Nichols, 1 9 7 6 , p. 1 7 ) . The notion of e q u a l i t y as an operation was also supported by the work of Denmark, Barco, & Voran ( 1 9 7 6 ) . This notion of the e q u a l i t y symbol as an operator may not represent an " e r r o r , " but may a f f e c t the nature of the errors produced when equations are 29 presented with variables on both sides. Thus, the conception of the eq u a l i t y symbol as an operation requires i n v e s t i g a t i o n i n diagnosis of errors i n l i n e a r equations i n one v a r i a b l e . Behr et a l . ( 1 9 7 6 ) noted some discrepancy between written and o r a l presentations of questions in v o l v i n g the equality symbol (e.g. A = 2 + 3 , 3 = 3 , 3 + 2 = 2 + 3 ) . Some students i n Grades 1 - 6 were able to make sense of questions presented i n o r a l form. However, they were unable to understand the same questions i n written form. The discrepancy between students' understanding of the e q u a l i t y symbol i n written as opposed to o r a l presentation may bear further 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 at the secondary l e v e l . If beginning algebra students s t i l l view the e q u a l i t y symbol as an operation, they may be confused when working with equations i n algebra. This operational view of equality may cause confusion when they are required to use consecutive equal expressions i n sol v i n g written equations (Bernard & Bright, 1 9 8 2 ) . In addition, Matz ( 1 9 8 0 ) claims that there are two d i s t i n c t uses of the eq u a l i t y symbol i n algebra which may s u b s t a n t i a l l y extend the arithmetic notion of e q u a l i t y . He re f e r r e d to the two uses of eq u a l i t y as "tautology" and "constraint equation." In a "tautology," the student performs a chain of transformations on an expression to obtain i t s simplest form. Each subsequent expression i s the same as the previous one. In "constraint equations," the student performs transformations on both sides of the eq u a l i t y symbol, maintaining equivalence between the two sides of the equation, but not between each step of the transformation. The problem for 30 students i n beginning algebra i s that they must s i m p l i f y each side of the equation, keeping the l e f t side equal to i t s e l f and the righ t side equal to i t s e l f , thus u t i l i z i n g the t a u t o l o g i c a l sense of equality and then switch to the constraint use i n order to solve the equation, performing operations- so that the l e f t side i s equal to the righ t side at a l l times, but d e f i n i t e l y not equal to i t s e l f . While some may think these differences to be semantic i n nature, the d i f f i c u l t i e s experienced by students i n switching from s i m p l i f y i n g expressions to so l v i n g equations can be seen on a d a i l y basis i n beginning algebra classrooms. This d i s t i n c t i o n between the two possible views of the e q u a l i t y symbol may appear useful i n the classroom, however, i t does require empirical v a l i d a t i o n . As such the r e l a t i o n s h i p between the errors produced when using 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 "constraint equation" sense requires i n v e s t i g a t i o n to c l a r i f y the alleged need for these d i s t i n c t i o n s . In summary, research has shown that the operational use of e q u a l i t y i n arithmetic forms the c e n t r a l reason f o r many students' i n a b i l i t y to solve equations (Kieran, 1983) . Misunderstandings of e q u a l i t y as an operation, rather than a r e l a t i o n , seem to remain with students, even at the college l e v e l (Rosnick & Clement, 1980). Understanding of e q u a l i t y i s further confounded by equation complexity. As questions become more complex students may change t h e i r minds r e a d i l y regarding the e q u a l i t y of two expressions (Denmark et a l . , 197 6). Meaning o f E q u a t i o n s Linear equations a r i s e as mathematical models of problems i n the r e a l 31 world ( K r u t e t s k i i , 1 9 7 6 ) . Students tend to tr e a t equations as abstract, u n i n t e l l i g i b l e e n t i t i e s to be dealt with using a set of memorized procedures without conceptual underpinnings (Kieran, 1 9 8 0 ) . This lack of understanding of what an equation represents may s i g n i f i c a n t l y a f f e c t the equation-solving process (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 , seventh grade novices used one of three procedures to solve l i n e a r equations i n one v a r i a b l e : known number f a c t s , s u b s t i t u t i o n , or " i n v e r t " - leave the v a r i a b l e on the l e f t and transpose a l l numbers to the r i g h t side of the equation using inverse operations. A f t e r i n s t r u c t i o n which was "very strongly oriented toward the symmetric procedure of performing the same operation on both sides" (Kieran, 1 9 8 3 , p. 1 6 8 ) , the r e s u l t s f or novices appeared to be i n sharp contrast to those of the Grade 8 - 1 1 experts. While the four types of procedures were the same f o r both (known number fa c t s , s u b s t i t u t i o n , inversing, and same operation on both s i d e s ) , the frequency of use of each type of procedure i n d i c a t e d that experts strongly preferred inversing ( 1 5 2 instances compared to 19 instances of the other 3 procedures), while novices only marginally preferred the same inversing procedure (77 instances compared to 63 instances of the other procedures). Of greater s i g n i f i c a n c e was the fact that teaching of a p a r t i c u l a r method only influenced those students who already had a view of v a r i a b l e compatible with that method. While students whose view of v a r i a b l e was incompatible with t h e i r equation-solving procedures, changed t h e i r procedures to match t h e i r view of variable, even when such a change was i n d i r e c t c o ntradiction to the method being taught. 32 Kieran's (1983) findings 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 conceptions and procedures, as well as the necessity of determining, not only students' views of variable, but also t h e i r conceptions of the meaning of equations. I d e n t i t y E l e m e n t s Matz (1980) suggested that students were l i k e l y to make errors when a g e n e r a l i z a t i o n involved s p e c i f i c numerical values. He further contended that the i d e n t i t y elements for addition 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 learned that n * (special number) = n and that n * (special number) =0. It was the choice of the operator * together with the s p e c i a l number, 0 or 1 which determined the answer. Some students never seemed to learn that both the operation and the s p e c i a l number were c r i t i c a l . These students responded n to n * 1 and 0 to n * 0, unheeding of the fact that * might be + i n both instances. Others learned these i d e n t i t i e s together (ie. n x 1 = n, n x 0 = 0 , n + 0 = n) and randomly produced e i t h e r 0 or n for n * 0. Other errors possibly r e l a t e d to the i d e n t i t y elements occurred i n examples such as 2n - 2n = n, where the student explained that there was "zero n, but then wrote n, because 0 * n - n. In writing 3x + x = 3x, some students may explain: "There's nothing i n front of the other x, so i t ' s just zero." These examples suggest that generalizations i n v o l v i n g the i d e n t i t y elements may be a source of error i n algebra, but no studies reported analysis of the i d e n t i t y elements i n algebraic e r r o r s . 33 N u m e r i c a l M a g n i t u d e The magnitude of the numbers used i n an equation may also influence students' equation-solving a b i l i t y . Herscovics (1979) stressed: For small values of N, students have no d i f f i c u l t y i n s o l v i n g simple equations such as x+a=b, x-a=b, ax=b, x/a=b although a-x=b and a/x=b seem somewhat more d i f f i c u l t . It i s when the numbers used are large enough or when multiple operations appear i n the equation or when terms i n the unknown appear on both sides (ax+b=cx+d) that mental arithmetic usually ceases and algebraic methods come into t h e i r own (p. 110) .. The point where arithmetic algorithms end and algebraic procedures begin may be, i n part, a function of numerical magnitude. Booth (1981) also found that the use of large numbers (n>100) seemed to focus students' attention on the processes involved i n the s o l u t i o n of algebraic equations. She documented r e s u l t s which were consistent across subjects and across time, supporting Anderson (1982). Large numbers seemed to increase computational errors (Herscovics, 1979), yet reduced process e r r o r s . Control of numerical magnitude would appear to be a s i g n i f i c a n t concern i n diagnosis. Such control may encourage the use of algebraic, as opposed to arithmetic, methods for solving equations (Herscovics, 1979). The c o n t r o l of numerical magnitude i s further supported by the r e s u l t s of the Strategies and Errors i n Secondary Mathematics (SESM) project i n which Booth & Hart (1982) indicated that students 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 do not r e a d i l y extend to more complex questions or questions i n which the numbers are large or non-integer" (p. 4). These findings further emphasize the importance of numerical magnitude i n diagnosis. 34 The numbers used as c o e f f i c i e n t s may d i f f e r e n t i a l l y influence students' a b i l i t y to solve equations. Kieran (1981b) reported that although "... the large c o e f f i c i e n t s proved to be no hindrance at a l l f o r equations of the form ax = b and x + a = b , they tended to increase s l i g h t l y the complexity of the two-operation equations" (p. 164). An example of a two-operation equation i s 12x + 216 = 468. Error frequency seems to vary d i r e c t l y with the i n t e r a c t i o n between numerical magnitude and equation complexity. This i n t e r a c t i o n between numerical magnitude and equation complexity bears further i n v e s t i g a t i o n . It would seem that the magnitude of the numbers used i n equations influences the processes chosen by students i n solving equations. Numerical magnitude may also be i n t e r a c t i n g with the complexity of equations. As such, c o n t r o l of numerical magnitude may be a c r i t i c a l f a c t o r i n algebraic diagnosis. E q u a t i o n S t r u c t u r e a n d C o m p l e x i t y Equation structure and complexity may also influence students' a b i l i t y to solve equations. P e t t i t o (1979) defined 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 t r u c t u r a l format) i n terms of student f a m i l i a r i t y with equations i n r a t i o format. The f a m i l i a r equations (easiest s t r u c t u r a l d i f f i c u l t y level) had the v a r i a b l e appearing i n the righ t numerator (a/b=x/c), while the unfamiliar forms had the variable appearing i n the r i g h t denominator [a/b=c/(x+d)] and i n both denominators [a/x=b/(x+c)]. Nine ninth-grade students were able to solve the f a m i l i a r equations over three 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 could solve the f i r s t 35 (n<5) and second (n<ll) l e v e l s of unfamiliar numerically d i f f i c u l t equations could also solve the t h i r d l e v e l (n<57). P e t t i t o ( 1 9 7 9 ) concluded that "the i n a b i l i t y to solve the numerically d i f f i c u l t problem i s due, not to arithmetic e r r o r s , but to an i n a b i l i t y to organize a successful approach" (p. 74). In some cases the students were unable even to attempt the question. In other cases students were unable to modify the informal procedures involved i n obtaining equivalent f r a c t i o n s . The only l i m i t a t i o n to t h i s study was the lack of the use of the second l e v e l of numerical d i f f i c u l t y f o r the second 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 . Despite the apparent impact of both s t r u c t u r a l format and numerical magnitude on the s o l u t i o n of equations, the majority of studies reviewed f a i l e d to c o n t r o l or manipulate these two f a c t o r s . Unfortunately, P e t t i t o ' s (1979) findings are l i m i t e d by lack of adequate co n t r o l and a possible numeric by structure i n t e r a c t i o n . However, her findings supported Kieran's (1981b) contention that magnitude and complexity may i n t e r a c t . Equation complexity, s t r u c t u r a l format and numerical magnitude need to be c o n t r o l l e d i n analyzing errors i n the s o l u t i o n of algebraic equations. Summary o f C o n c e p t u a l E r r o r s The influence of conceptual errors on the equation-solving process requires further i n v e s t i g a t i o n . The contradictory findings surrounding students' understanding of variable and i n t e r a c t i o n of that understanding with age and mathematical background require c l a r i f i c a t i o n . The e f f e c t of students' views of the equality symbol on the procedures used i n solving equations needs to be considered (Kieran, 1 9 8 3 ) along with the role of i d e n t i t i e s and inverses (Matz, 1980). Complexity, s t r u c t u r a l format, and numerical magnitude need to be c o n t r o l l e d within the equations used i n diagnosis. Studies need to be developed which systematically vary each of these factors comparing conceptions held, procedures used, and re s u l t s obtained. It i s c r i t i c a l f o r diagnosticians to understand students' conceptions, uninfluenced by i n v e s t i g a t i o n of t h e i r reasons f o r s e l e c t i n g p a r t i c u l a r procedures or obtaining p a r t i c u l a r r e s u l t s when solving l i n e a r ' equations i n one v a r i a b l e . P r o c e d u r a l E r r o r s Procedural errors i n arithmetic are "errors derived from systematic and inappropriate algorithmic procedures" (Engelhardt & Wiebe, 1981, p. 14). I algebra, procedural errors encompass the errors occurring i n s i m p l i f y i n g arithmetic expressions and misusing the rules f o r order of operations, and extend to errors, i n s i m p l i f y i n g algebraic expressions, in c l u d i n g p a r t i a l d i s t r i b u t i v e errors, sign errors with the d i s t r i b u t i v e property and combining unlike terms. O r d e r o f O p e r a t i o n s The tendency of students to perform arithmetic operations i n order from l e f t to ri g h t may influence the types of errors they make i n solving equations. Many students view arithmetic operations as a s t r i n g of operations to be performed i n a l e f t to right sequence (Hilling-Smith, 197 9 Kieran, 1979). Kieran (1979) found that i n creating arithmetic i d e n t i t i e s , subjects would write "operation by operation, as they were thinking them, • and were keeping a running t o t a l . a s they went along" (p. 2). For example, 37 students would write 4x2-3=5+10+3 as an i d e n t i t y . When given the suggestion that brackets be in s e r t e d around 5+10, many students thought t h i s was e n t i r e l y unnecessary. They thought that the bracketed numbers had to appear f i r s t , i n order to' correspond to t h e i r l e f t to r i g h t sequencing. Systematic v a r i a t i o n of l e f t to r i g h t sequencing may be required to determine understanding of algebraic operations i n e r r o r - a n a l y s i s . A l g e b r a i c P r o p e r t i e s M i s a p p l i c a t i o n of the additive and m u l t i p l i c a t i v e properties of equality may r e s u l t i n errors i n v o l v i n g c o e f f i c i e n t s , inverses, or combining unlike terms. Two of the common errors found among both f i r s t - and second-year algebra students were adding opposites inappropriately (-a+ax =x ) and combining m u l t i p l i c a t i v e inverse and additive inverse (the r e c i p r o c a l of a i s -1/a) (Davis & Cooney, 1978) . Even college students made s i m i l i a r errors when so l v i n g algebraic equations (Carry et a l . , 1980; Lewis, 1980). The tendency to make algebraic errors may be an outcome of over-g e n e r a l i z a t i o n of r u l e s . Booth & Hart (1982) interviewed approximately 70 students from the Concepts i n Secondary Mathematics and Science (CSMS) project who had made errors which were also made by a large number of students on the CSMS Algebra t e s t . When presented with an item involving the t o t a l number of soccer goals and the fact that one team made x goals and the other y goals, some students were able to write down x+y, but then continued t h e i r answer to x+y=z, s t a t i n g : " I f you add two numbers you get another number so I suppose adding two l e t t e r s gives you another l e t t e r " (p. 9). Students not only seek closure when adding l e t t e r s , but also have a 38 tendency to add 4 and 3n and to obtain e i t h e r 7 or In. Matz (1980) noted that students who make mistakes often t r y to generalize the rule to f i t the problem (e.g. since 3 + 4 = 7 then 3 + 4r> = 7n) . Summary o f P r o c e d u r a l E r r o r s Procedural errors seem to a r i s e from the tendency of students to generalize arithmetic rules and inappropriately apply them to algebraic situations-. It would appear that the rules most often generalized pertain to the d i s t r i b u t i v e property. It i s c r i t i c a l to understand students' procedures i n so l v i n g algebraic equations so that misapplications of rules can be detected. R e s u l t a n t E r r o r s Resultant errors are those errors which occur i n wr i t i n g or s t a t i n g a r e s u l t e i t h e r i n the p a r t i a l s o l u t i o n of a problem or i n i t s f i n a l answer. They a r i s e from computational, sign, mechanical, perceptual, or random err o r s , as well as from an i n a b i l i t y to complete the so l u t i o n of equations. Resultant errors should be explored at each stage of the equation-solving process, as students may make errors and s t i l l obtain a correct f i n a l answer (Anderson, 1982). C o m p u t a t i o n a l E r r o r s Computational errors involve any of the four basic operations (addition, subtraction, m u l t i p l i c a t i o n , d i v i s i o n ) . For each operation, errors may be made i n basic f a c t s , f a u l t y algorithms, or incor r e c t choice of that operation. Considerable information e x i s t s regarding computational errors, due to the frequency of t h e i r i n v e s t i g a t i o n (Brown & Burton, 1978; CSMS, 39 descr ibed i n Hart , 1981; SESM, descr ibed i n Booth, 1984) . Therefore, only a l i m i t e d d i s c u s s i o n of the more s a l i e n t r e s u l t s i s prov ided . Roberts (1968), us ing 148 Grade 3 students, e s t a b l i s h e d four major e r r o r ca tegor ie s : wrong operat ion , obvious computational e r r o r , de fec t ive a lgor i thm, and random response. Using Robert 's (1968) framework as a s t a r t i n g p o i n t , Engelhardt (1977) tes ted 198 Grade 3 and Grade 6 students on an 84-item computational t e s t . He analyzed the errors and i d e n t i f i e d eight types: b a s i c fac t (38%), grouping (22%), inappropr ia te i n v e r s i o n (21%), i n c o r r e c t operat ion (4%) , de fec t ive a lgor i thm (18%), incomplete a lgori thm (7%), i d e n t i t y (1%), and zero (6%). Basic fact e rrors appeared to be the most common e r r o r s made by students . These data evidenced that over h a l f the e r r o r s made i n ar i thmet i c were computational i n nature . Later studies by Englehardt & Wiebe (1981) and Englehardt (1982) confirmed these e r r o r c l a s s i f i c a t i o n s . Graeber & Wallace (1977) i d e n t i f i e d systematic a d d i t i o n and subtrac t ion e r r o r s at the Grade 3, 4, and 7 l e v e l s us ing I n d i v i d u a l l y Prescr ibed I n s t r u c t i o n (IPI) p r e t e s t s . They reported that both systematic errors increased with grade l e v e l . An e r r o r was c l a s s i f i e d as systematic i f "three or more of the i n d i v i d u a l ' s t es t items evidenced the same e r r o r pattern" (p. 8) . Several systematic e r r o r s were i d e n t i f i e d across grade l e v e l s . These inc luded omit t ing c a r r y i n g i n add i t i on or borrowing i n subtrac t ion , adding a l l d i g i t s i r r e s p e c t i v e of place value , and i g n o r i n g the tens column. In a d d i t i o n to systematic e r r o r s , random errors and incomplete work were a l so reported . 40 Computational errors have been reported as a major source of d i f f i c u l t y i n s o l v i n g l i n e a r equations i n algebra (Davis & Cooney, 1978; Pease, 1929; Wattawa, 1927). Their i d e n t i f i c a t i o n forms a c r i t i c a l aspect of diagnosis i n algebra to provide d i s t i n c t i o n s between algebraic and arithmetic e r r o r s . However, no systematic exploration of computational errors f o r l i n e a r • equations i n one var i a b l e was found i n the l i t e r a t u r e . S i g n E r r o r s Sign errors frequently occur i n the solu t i o n of l i n e a r equations. They may involve any of the four basic operations, misapplication of sign rules (e.g. two negatives make a p o s i t i v e when adding) or misuse of the d i s t r i b u t i v e property [e.g. -3(x -5)= -3x -15]. Tatsuoka et a l . (1980) reported that i n s t r u c t i o n a l approaches correlated with rule-based sign e r r o r s . That i s to say, i f a teacher stated a rule for one operation, then students would use i t inappropriately with another operation. Davis et a l . (1979), using an 8-item modification of an e a r l i e r 14-item version of Tatsuoka's (1977) integer addition t e s t , found that approximately one-third of the answers could be explained by the "symmetric subtraction" e r r o r . For example a student might state: Three minus f i v e equals two. The order does not matter. If you see a you just subtract the smaller number from the larger number." Davis et a l . (1979) suggested that students may make sign errors because of i n c o r r e c t a p p l i c a t i o n of previously learned r u l e s . Theoretical and empirical evidence of t h i s have been provided by Matz (1980) and Tatsuoka et a l . (1980), r e s p e c t i v e l y . Cote (1981) reported s i m i l a r findings and further noted, that some students viewed integer arithmetic as merely a c o l l e c t i o n of unrelated r u l e s . When diagnosing a student's errors i n so l v i n g equations, the p a r t i c u l a r combination of signs and magnitudes of the numbers involved i n the selected equations may not allow the student to make a mistake. This i s p a r t i c u l a r l y true f o r some of the more complex student-generated rules (Tatsuoka et a l . , 1980). Hence, the systematic v a r i a t i o n of the signs and magnitude of numbers i s necessary to f a c i l i t a t e the i d e n t i f i c a t i o n of the misapplied or student-generated r u l e s . Without such formulation of diagnostic equations, many rules, which randomly produce correct and in c o r r e c t answers, depending on the numerical signs and magnitudes involved, may go unnoticed. 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 Students c o n s i s t e n t l y seem to make a small percentage of mechanical/perceptual errors when solv i n g equations. Mechanical/perceptual errors are "systematic and non-systematic errors appearing to r e s u l t from v i s u a l or motor-related d i f f i c u l t i e s such as i l l e g i b l e number formation, misalignment of numbers, or miscopying/misreading d i g i t s or symbols" (Engelhardt & Wiebe, 1981, p. 14). Wattawa (1927) found that 26% of the errors made by a ninth-year algebra class were due to copying or reading mistakes. More recently, Davis &• Cooney (1978) reported a 9% error rate among f i r s t - y e a r and a 13% error rate among second-year algebra students i n v o l v i n g miscopying or misreading items. Such errors need to be considered when diagnosing errors occurring i n algebra. R a n d o m E r r o r s Within the l i t e r a t u r e , many researchers refer to "careless" e r r o r s . It i s an i l l - d e f i n e d category and, hence, the more well-defined category of "random" errors has been used.in the present study. Random errors may be described as "non-systematic errors appearing to r e s u l t from momentary i n a t t e n t i o n to relevant s t i m u l i " (Engelhardt & Wiebe, 1981, p. 15). They can be di s t i n g u i s h e d from other errors by t h e i r non-systematic nature as well as the responses students give when questioned as to how a p a r t i c u l a r answer was obtained. Usually when t h i s type of erro r i s pointed out to the students, they respond: "How s i l l y ! I know better," and proceed to correct the er r o r immediately. Wagner et a l . (1984) explained that. It i s not c l e a r what causes-students to make low-level errors on problems that present hi 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 accidental, perhaps i t re s u l t s from a temporary overload i n short-term memory, or perhaps i t r e f l e c t s a reversion to an e a r l i e r "naive" rule i n the face of a higher l e v e l problem that, for the moment, exacts f u l l concentration (p. 31) . While random errors seem to represent a low l e v e l type' of error, they need to be considered when diagnosing errors, i n algebra. I n c o m p l e t e Work Incomplete work appears as an error category i n many studies involving algebra.' Wattawa (1927) found that 5% of. students f a i l e d to complete the solutions of equations at the beginning of the year. The analogous figure increased to 9% by the end of the year. Davis & Cooney (197 8) reported a 12% rate of incomplete solutions. However, s l i g h t l y over 90% of t h i s 43 incomplete work occurred i n equations containing f r a c t i o n s . It i s unclear i f such a high percentage of errors would occur i n the so l u t i o n of l i n e a r equations i n one va r i a b l e containing only integers. Investigation of students' incomplete work may be c r i t i c a l to understanding errors made i n so l v i n g l i n e a r equations i n one va r i a b l e . Summary a n d R e c o m m e n d a t i o n s Several factors seemed to influence the nature and frequency of errors reported by researchers i n beginning algebra. Students' views of a variable as an object and of equ a l i t y as an operation seem c e n t r a l to students' i n a b i l i t y to solve l i n e a r equations. However, there are contradictory f i n d i n g s regarding the r e l a t i o n s h i p of these two concepts to age and mathematical background. These contradictions are further compounded by the fact that both concepts i n t e r a c t with equation complexity. Such research findings necessitate further i n v e s t i g a t i o n . As indicated by several researchers, s i m i l a r erroneous understandings of the meaning of expressions and equations have l e d students to make procedural errors i n v o l v i n g order of operations, the d i s t r i b u t i v e property, combining of unlike terms, and the additi o n and m u l t i p l i c a t i o n properties of equa l i t y . Magnitude of numbers used i n equations, placement of signed numbers within equations, and the complexity of the equation i n terms of the number of occurrences of the var i a b l e and the number of operations have a l l been reported to be s i g n i f i c a n t factors a f f e c t i n g student performance i n solving equations. While i t was apparent from most studies that numerical magnitude influenced r e s u l t s , there seemed to be two very d i f f e r e n t views of why t h i s 44 occurred. Some studies suggested that larger numbers af f e c t e d students' a b i l i t y to r e t r i e v e correct processes f o r solving equations. Other studies i n d i c a t e d that larger numbers forced students to focus on the process, rather than guessing and s u b s t i t u t i n g values into the equation. These issues are further confounded by the tendency of students to perform operations i n a l e f t to r i g h t order. The i n t e r r e l a t i o n s h i p s among numerical magnitude and the equations' structure and complexity have been l a r g e l y unexplored. Computational errors seem to be a s i g n i f i c a n t source of d i f f i c u l t y for some students when solv i n g l i n e a r equations. The r e l a t i o n s h i p between computational errors and algebraic errors requires i n v e s t i g a t i o n . Furthermore, there seems to be a p o s i t i v e c o r r e l a t i o n (r=0.52, Englehardt & Wiebe, 1981) between students' understanding of arithmetic concepts and the computational errors they make. There have been no inve s t i g a t i o n s to determine i f such a c o r r e l a t i o n e x i s t s between understanding of algebraic concepts and the procedural or resultant errors made i n solv i n g equations. Mechanical, perceptual, and random errors may be a function of in a t t e n t i o n to the task at hand. In the case of the f i r s t two errors, students usually miscopy or misread the question or some aspect of t h e i r work. In the case of random errors, there i s some evidence to suggest that students may be attending to higher l e v e l concepts and procedures involved i n s o l v i n g equations and subsequently make a mistake. The d i f f e r e n t causes of i n a t t e n t i o n require c a r e f u l consideration i n diagnosis. Radatz (1979) suggested that mechanical errors may be due to "lack of i n t e r e s t or to 45 d i v e r s i o n " (p. 1 8 ) . While overlap between mechanical/perceptual and random errors may occur, f o r the purposes of the present study the d i s t i n c t i o n between these e r r o r types was maintained. A summary of the number of investigators who found a p a r t i c u l a r error -within 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 use i n the present study i s presented i n Table 1 . The findings were separated into two groupings, from 1 9 2 2 - 1 9 7 4 and from 1 9 7 5 - 1 9 8 8 . As can be seen, the focus i n f the f i r s t group was on resultant errors (33 instances over 12 categories) compared to conceptual errors (5 instances over 8 categories) and procedural errors (8 instances over 8 categories). While the second group s t i l l found that the highest frequency of errors were resultant errors (48 instances over 12 categories), errors were more evenly d i s t r i b u t e d among the types reported i n Table 1 ( i e . conceptual - 48 instances over 8 categories; procedural - 39 instances over 8 categories). 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 use i n t h i s study provided an opportunity to view the changes i n research focus over the past seven decades. Detailed findings of errors f o r i n d i v i d u a l investigators may be found i n Appendix A. Bernard & Bright ( 1 9 8 2 ) i n d i c a t e d that "the s i m i l a r i t y of error categories across a v a r i e t y of studies suggests that convergence may be occurring regarding an appropriate categorization of these e r r o r s " (p. 14) . The e r r o r - c a t e g o r i z a t i o n scheme used i n the present study attempts t h i s convergence and provides a framework i n which to view previous research regarding the conceptual, procedural, and resultant errors that students make when solv i n g l i n e a r equations i n one va r i a b l e . 46 Table 1 Number of Researchers Finding Error-Categorization Scheme Errors Type of Error Number of Researchers : 1925-1974 1975-1988 CONCEPTUAL Variable 0 9 Expression 3 9 Equ a l i t y 0 5 Equation 1 6 C o e f f i c i e n t 0 5 Like Terms 1 8 Inverses 0 3 Identity Elements 0 3 PROCEDURAL Zero Annexation 1 3 Ident i t y Confusion 0 3 Like Terms (conjoining) 1 7 P a r t i a l D i s t r i b u t i v e 3 9 Order of Operations 0 4 + Property of = 2 4 x Property of = 1 4 C o e f f i c i e n t Errors 0 5 RESULTANT C o m p u t a t i o n a l add/subtract 3 6 mu l t i p l y / d i v i d e 3 5 basic f a c t s 2 5 f a u l t y algorithm 3 4 wrong operation 3 9 S i g n add/subtract 2 3 mu l t i p l y / d i v i d e 3 2 rule-based 0 3 d i s t r i b u t i v e 2 3 O t h e r mechanical/perceptual 5 4 random 4 1 incomplete work 3 3 CHAPTER I I I DESIGN OF THE STUDY The purpose of the study was to develop a d iagnos t i c c h e c k l i s t to 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 algebra students i n s o l v i n g l i n e a r equations i n one v a r i a b l e . Development and v a l i d a t i o n of the c h e c k l i s t occurred i n f i v e phases: 1. ana lys i s of f i r s t - y e a r algebra textbooks for content and sequence, 2. 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 previous research, 3. cons truc t ion and s e l e c t i o n of equations, 4. refinement of the instrument, 5. t e s t i n g of the f i n a l instrument. Each phase was designed to address fac tors i d e n t i f i e d by previous researchers as i n f l u e n c i n g students' e rrors i n s o l v i n g l i n e a r equations. Because the design of the c h e c k l i s t was contingent upon the r e s u l t s of Phase 1 and Phase 2, both the design aspects and the f indings for these two phases are reported i n t h i s chapter p r i o r to presentat ion of the rest of the des ign . P h a s e 1: C o n t e n t A n a l y s i s o f F i r s t - Y e a r A l g e b r a T e x t b o o k s An ana lys i s of 15 f i r s t - y e a r algebra textbooks was undertaken to se lect content for the d iagnos t i c instrument. The textbooks se l ec ted were publ i shed between 1978 and 1984 and represented the major f i r s t - y e a r algebra textbooks used i n North America. Four aspects were analyzed: concepts, vocabulary, types of equations, and i n s t r u c t i o n a l sequence. The r e s u l t s of t h i s a n a l y s i s , together with f indings from previous research, formed the bas i s for the content and sequence of the d iagnos t i c instrument. The textbooks d iscussed the concepts of v a r i a b l e s , express ions , equality, and meaning of equations (See Appendix E, Section A of the Diagnostic Checklist for Algebra for the items associated with these concepts). The concepts of variables and expressions were usually presented through d e f i n i t i o n s , examples, and sometimes counter-examples. The concept of e q u a l i t y was most often presented i m p l i c i t l y as that of a balance between the l e f t - s i d e and r i g h t - s i d e of an equation. Only two of the textbooks e x p l i c i t l y introduced equality as an equivalence r e l a t i o n with the properties of symmetry, r e f l e x i v i t y , and t r a n s i t i v i t y . The meaning of equations was usually done i m p l i c t l y through word problems, although many textbooks d i d include introductions through cue words and some work with phrases and sentences. A few textbooks gave students equations and asked f o r an explanation of t h e i r meaning. The vocabulary associated with c o e f f i c i e n t s and l i k e terms was most often introduced at the same time as expressions. Again, t h i s was done through the use of bold type, examples, and counter-examples. The vocabulary associated with additive and m u l t i p l i c a t i v e inverses was most often introduced with the addition and m u l t i p l i c a t i o n properties of e q u a l i t y . For i d e n t i t y elements most of the textbooks gave non-rigorous presentations through examples of the properties of one and zero. However, the same two textbooks which introduced equality as an equivalence r e l a t i o n also introduced inverses and i d e n t i t y elements as part of the properties of the r e a l number system (See Appendix E, Section B of the Diagnostic Checklist for Algebra f o r the items used to investigate understanding of t h i s vocabulary). In general, the textbooks introduced 12 types of equations to students p r i o r to the introduction of 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 or sol u t i o n s . A l l twelve types of equations were included i n the Diagnostic Checklist for Algebra because algebraic errors seem to be s p e c i f i c to the p a r t i c u l a r equations used (Davis & Cooney, 197 8). As suggested by C o l l i s (1975), the equations occur i n the diagnostic instrument i n the mean rank p o s i t i o n as determined from the textbooks. These twelve equation types together with t h e i r o r d i n a l p o s i t i o n are presented i n Table 2. P h a s e 2: D e v e l o p m e n t 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 The e r r o r - c a t e g o r i z a t i o n scheme was developed to f a c i l i t a t e the i d e n t i f i c a t i o n of the s p e c i f i c errors which a student might make when sol v i n g l i n e a r equations i n one variable, and was based on the l i t e r a t u r e reviewed i n Chapter 2. To reduce the p r o b a b i l i t y of i n c l u s i o n of errors which were s p e c i f i c to a p a r t i c u l a r study, only those errors found by three or more researchers were included. Errors were grouped into three major categories: conceptual, procedural, and resultant. The conceptual category included the four concepts and four vocabulary items mentioned e a r l i e r i n the textbook a n a l y s i s . Errors i d e n t i f i e d within the procedural category included annexation of zero, i d e n t i t y confusion, incorrect 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 errors, errors made i n applying the rules f o r order of operations, the addition and m u l t i p l i c a t i o n properties of equality, and errors i n obtaining the c o e f f i c i e n t of the v a r i a b l e . The resultant category was p a r t i t i o n e d into three sub-categories: computational, sign, and other e r r o r s . Computational errors included errors i n any of the four basic operations, basic f a c t s , defective algorithms, or the use of wrong operations. Sign errors were errors i n p o s i t i v e or negative signs with any of the four basic operations, rule-based errors i n determining the sign, or sign e r r o r s using the d i s t r i b u t i v e property. Other errors included mechanical and perceptual problems, random errors, or incomplete work. Ov e r a l l , 20 procedural and resultant errors were included i n the e r r o r -categorizationscheme. These 20 errors represented a consolidation of the type of err o r s found by previous researchers. The name chosen for each of these 20 errors represented the name most often used i n previous research. Adetailed discussion of the err o r - c a t e g o r i z a t i o n scheme i s presented i n Appendix A, and includes a tabular presentation of the errors found by each researcher which occurred i n the er r o r - c a t e g o r i z a t i o n scheme. P h a s e 3: C o n s t r u c t i o n a n d S e l e c t i o n o f E q u a t i o n s The purpose of the t h i r d phase of the study was to construct a set of equations and subsequently select a portion of these f o r i n c l u s i o n i n the diagnostic instrument. Previous researchers had u t i l i z e d equations 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 chapters of a current textbook (Davis & Cooney, 1 9 7 8 ) or by "maximizing the number of manipulations made i n solving the equations" (Anderson, 1 9 8 2 ) . Such s e l e c t i o n procedures may produce q u e s t i o n - s p e c i f i c errors, as many aspects of the composition of an equation have been reported to influence the nature and frequency of the errors made by students i n solv i n g l i n e a r equations i n one va r i a b l e . These aspects include: equation complexity (Booth & Hart, 1 9 8 2 ) , numerical magnitude Table 2 Types of Linear Equations Presented i n 15 Textbooks Equation General Ordinal P o s i t i o n Types Format i n Textbooks I . O n e - S t e p Addition x+a=b* 1 Subtraction x-a=b 2 M u l t i p l i c a t i o n ax=b 3 D i v i s i o n x/a=b 4 I I . T w o - S t e p Like Terms Numerical ax+b=c 5 Variable One Side of ax+bx=c 6 Both Sides of = ax+b=cx 7 Parentheses a(x+b)=c** 8 I l l . M u l t i - S t e p Combined Numeric/Variable One Side of = ax+b+cx+d=e 9 Both Sides of = ax+b=cx+d 10 Parentheses One Side of = a(x+b)=cx+d 11 Both Sides of = a(x+b)=c(x+d) 12 a,b,c,d,e,x are a l l integers, a*0. Based on the constraints of the study these equations were l i m i t e d to those containing i n t e g r a l roots, as well 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 variable (Kieran, 1983), structure of the equation (P e t t i t o , 1979), and order of sign presentation (Tatsuoka et a l . , 1980). To co n t r o l f or such influences, the systematic v a r i a t i o n of these aspects of the equations constructed f o r use i n the diagnostic instrument ensured that errors were representative of students' erroneous understandings, and not simply a r e f l e c t i o n of the type of equation presented. C o n s t r u c t i o n o f E q u a t i o n s The construction of equations for use i n the Diagnostic Checklist for Algebra employed systematic v a r i a t i o n of the signs of the numbers, the absolute value of the numbers i n r e l a t i o n to each other, and the r e l a t i o n s h i p of the v a r i a b l e to both the numbers and the e q u a l i t y symbol. Such v a r i a t i o n was achieved through the creation of an equation-type-by-numerical-magnitude-by-structural-format g r i d of equations. The 12 equation types i d e n t i f i e d i n the textbook analysis were u t i l i z e d . Sixteen equations were developed for each equation type c o n s i s t i n g of four l e v e l s of numerical magnitude and four categories of s t r u c t u r a l format. Within these 4-by-4 grids, order of numerical sign and relative.magnitude of numbers were systematically varied as suggested by Tatsuoka et a l . (1980). This r e s u l t e d i n the construction of 192 equations: 12 equation types x 4 s t r u c t u r a l format categories x 4 numerical magnitude l e v e l s . The 4-by-4 grids f o r each of the twelve types of equations and the accompanying magnitude and sign rules used within each g r i d are presented i n Appendix B, Part 1. 53 Four l e v e l s of numerical magnitude were defined, using absolute value. Cut-off values were based on the work of P e t t i t o (1979) and Anderson (1982) . Equations which s a t i s f i e d the numerical magnitude constraints and the sign and numerical 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 design rules s p e c i f i e d i n Appendix A were not always possible to construct. For example, consider an equation of the form ax +b=cx i n which |c|>|a| and the signs of a and c were opposite, and yet each of a, b, and c were of second numerical magnitude (20<|n|< 50). For example, while 24x +50=~26x was possible ( i e . |"26|>|24| and |~26 + "24|<50), a l l equations at t h i s l e v e l would require solutions of -1 or 1 i n order to simultaneously meet the magnitude, as well as sign and i n t e g r a l c o n s t r a i n t s . Hence, for a l l equations where t h i s type of problem occurred, as many numerical values as possible were kept within the designated l i m i t s , or the c o e f f i c i e n t s were within the correct l i m i t s once brackets were removed [e.g. ~ 9 (x +8)=8(x +~9) was an N_3_ equation when the brackets were removed] . S t r u c t u r a l format of equations also influences performance i n equation-s o l v i n g ( P e t t i t o , 1979). Some students experience d i f f i c u l t y when the v a r i a b l e appears e i t h e r a f t e r the number (Herscovics & Kieran, 1980)' or to the r i g h t of the e q u a l i t y symbol (Behr et a l . , 1976; Wagner, 1981a) . Numerical Magnitude Integer Values In| < 20 (Nl) (N2) (N3) (N4) 20< |n| < ' 50 50< |n| < 100 100< |n| < 1000 54 Therefore, systematic v a r i a t i o n of the placement of variables with respect to both the eq u a l i t y symbol and the constants within an equation i s necessary. These v a r i a t i o n s required the creation of four categories of s t r u c t u r a l format: S t r u c t u r a l Format 1 (SI): Equations i n which the va r i a b l e f i r s t occurred on the l e f t side of the equality symbol, followed by numbers (eg. x +15=8). S t r u c t u r a l Format 2 (S2): Equations i n which the va r i a b l e f i r s t occurred on the l e f t side.of the equality symbol, but occurred on the right side of the numbers (eg. 15+x = 8 ) . S t r u c t u r a l Format 3 (S3>: Equations i n which the va r i a b l e f i r s t occurred on the r i g h t side of the equality symbol, followed by numbers (eg. 8=x +15) . St r u c t u r a l Format 4 (S4): Equations i n which the var i a b l e f i r s t occurred on the r i g h t side of the eq u a l i t y symbol, and occurred on the right side of the numbers (eg. 8=15+x ). In those equations where there were two occurrences of the var i a b l e (Equation Types 6, 7, and 9 - 12, See Table 2, page 51), the intermediate step, where the variables were combined into one variable term, was used to determine the structure of the equation. An example of such a Level 1 structure would be: 13 (x - 3) = -2x - 9 13x - 39 = -2x - 9 15x = 30 In the f i r s t l i n e , the variables on both sides of the equation were followed by numbers, and i t was assumed that students would t r y to make the c o e f f i c i e n t of the combined variable terms p o s i t i v e . That i s to say, a f t e r removing parentheses i n the above example, students would add 2x to both sides of the equation to make 15x and then add 39 to both sides of the equation to make 30. 55 S e l e c t i o n o f E q u a t i o n s There were four factors involved i n the s e l e c t i o n of the equations for use i n the Diagnostic Checklist for Algebra. The f i r s t was to determine the e f f e c t of systematic v a r i a t i o n of the complexity of an equation, i t s structure, and the magnitude of the numbers used i n the equation on the errors produced by students. The second was to provide a measure of v a l i d a t i o n of the developed e r r o r - c a t e g o r i z a t i o n scheme. The t h i r d was to reduce the number of equations from 192 to a more p r a c t i c a l l e v e l . The fourth was to ensure that the equations selected would d i s t i n g u i s h among l e v e l s of achievement i n algebra. In order to control f or these factors, i t was necessary to a s c e r t a i n subjects' achievement i n algebra, to administer the constructed equations, and to determine the nature of the errors produced i n s o l v i n g them. S u b j e c t s A l l four academic Grade 9 mathematics classes i n one junior high school were used i n t h i s phase. The subjects represented the e n t i r e population of f i r s t - y e a r algebra students i n the school. Absenteeism res u l t e d i n a loss of s i x of the 86 subjects. P r o c e d u r e s In the Spring of 1 9 8 3 , a l l subjects were administered the Lankton First-Year Algebra Test (Lankton, 1 9 6 5 , Form E ) to determine t h e i r achievement l e v e l s i n algebra. High achievers were defined as subjects scoring one or more standard deviations above the mean on the Lankton. Low achievers were defined as those scoring one or more standard deviations below the mean. The Lankton consists of 50 items, requires 40 minutes f o r administration, and was designed* as a mid-year, Algebra One assessment t o o l . The content of the t e s t includes d e f i n i t i o n of terms and meaning of signs and symbols (8 items), fundamental algebraic operations, f a c t o r i n g and e x t r a c t i n g roots (12 items), equations and i n e q u a l i t i e s ( 1 3 items), algebraic expressions and formulas, functions, v a r i a t i o n and problem so l v i n g (8 items), and graphic representation (9 items). S p l i t -h a l f r e l i a b i l i t y c o e f f i c i e n t s f o r Form E ranged from 0 . 8 2 to 0 . 8 6 (Lankton, 1 9 6 5 ) . The 1 9 2 equations needed to be administered to subjects to determine t h e i r d i f f i c u l t y l e v e l . A balanced L a t i n Square design was used for assignment of the 1 9 2 equations to one of four test-forms (See Winer, 1 9 7 1 , Chapter 9 ) . Each test-form consisted of 48 items, four from each of the twelve equation types (See Appendix B, Part 2 ) . The balanced assignment of equations to test-forms also ensured the balanced d i s t r i b u t i o n of numerical magnitude and s t r u c t u r a l format throughout the 48 equations. Each c e l l of the numerical-magnitude-by-structural-format g r i d was represented three times throughout each test-form, although for d i f f e r e n t equation types. The balanced assignment of equations to test-forms also enabled analysis of the e f f e c t of item presentation and c o n t r o l l e d f or i n d i v i d u a l differences among the four algebra classes. A l l subjects were given one of the four test-forms during the mathematics cla s s following administration of the Lankton. This immediate administration of the equations helped to eliminate any e f f e c t that i n s t r u c t i o n might have had on the solving of the equations. No feedback regarding r e s u l t s on the Lankton was given to subjects p r i o r to administration of the equation test-forms. The four test-forms were equally d i s t r i b u t e d within the four classes involved i n the study. A n a l y s e s An item analysis was made of each test-form using LERTAP (Nelson, 1972). As post-facto analysis of the errors was made, the twenty error categories used i n the diagnostic instrument were reduced to ten to f a c i l i t a t e the i n v e s t i g a t i o n of the general nature of the errors occurring. The errors of zero annexation and i d e n t i t y confusion were not included i n the analysis due to the d i f f i c u l t y of t h e i r i d e n t i f i c a t i o n without interviewing the student. One of the ten categories was reserved f o r correct answers, as the occurrence of correct answers was to be checked i n the f i n a l instrument. Total correct answers for each subject could also be co r r e l a t e d with the i n d i v i d u a l r e s u l t s on the Lankton to ensure s i m i l a r i t y of achievement r e s u l t s . Each error made on a p a r t i c u l a r equation was coded as follows: a = Right Answer b = Other (mechanical/perceptual, random) c = Incomplete solutions d = Computational errors ( a l l f i v e errors combined) e = Sign errors ( a l l except d i s t r i b u t i v e ) f = Sign er r o r on d i s t r i b u t i v e property g = Misuse of + or x property of equality h = C o e f f i c i e n t errors (for +, -, x, or + and order of operations) i = Combining unlike terms j = P a r t i a l d i s t r i b u t i v e The comments i n parentheses are the names given to the errors i n the error-c a t e g o r i z a t i o n scheme presented i n Appendix A which were included i n each category of the LERTAP analysis. The LERTAP analysis treated each equation as i f i t were part of a multiple-choice t e s t , with er r o r categories serving as d i s t r a c t o r s . Other than the two errors mentioned above, a l l error categories i d e n t i f i e d f o r use i n the diagnostic instrument occurred as errors f o r one or more of the subjects tested on the equations. This lent some measure of face v a l i d i t y to the er 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 diagnostic instrument were chosen from the 192 equations using the r e s u l t s from the LERTAP analysis together with analysis 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, numerical magnitude, and s t r u c t u r a l format. A d e t a i l e d presentation of the r e s u l t s of these 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, Part 3. Based on the l a t t e r a n alysis, each c e l l of the numerical-magnitude-by-structural-format g r i d was included for. each complexity l e v e l . Such i n c l u s i o n meant there would be 16 items included f o r each l e v e l of complexity, making a t o t a l of 48 equations to be .included i n the diagnostic instrument. Within these constraints, items were selected whose correct answer had a p o s i t i v e c o r r e l a t i o n with the t o t a l score on the test-form, and whose in c o r r e c t answers had a negative c o r r e l a t i o n with the t o t a l score on the test-form. To ensure that the errors i d e n t i f i e d were not influenced by item d i f f i c u l t y l e v e l ( i e. easy items would only r e s u l t i n random errors and few procedural errors would occur; d i f f i c u l t items would mainly r e s u l t i n incomplete work, and, again, very few procedural errors would occur) , items were selected from the above group of equations whose p-values were closest to 0 . 5 . From these selected equations, only those were used whose c o r r e l a t i o n s with the Lankton were also p o s i t i v e . The f i n a l equations, together with t h e i r s t a t i s t i c a l correlatons, are presented i n Appendix B, Part 4 . When comparisons were made between the equation sections of the Lankton and the f i n a l 48 equations selected f o r use i n the Diagnostic Checklist for Algebra the c o r r e l a t i o n s ranged from 0 . 6 1 to 0 . 8 6 . These c o r r e l a t i o n s suggested acceptable concurrent v a l i d i t y with the equation portions of the Lankton. P h a s e 4: R e f i n e m e n t o f t h e I n s t r u m e n t The fourth phase of the study was used to r e f i n e the wording and format of the diagnostic instrument. This refinement included changes i n wording to c l a r i f y items, i n c l u s i o n of hints f or diagnosticians of possible cues which might s t i l l need to be made for further c l a r i f i c a t i o n , and changes to the format which would allow f or diagnostic comments and tabulations. It also served to investigate the time needed for completion of the diagnostic instrument. S u b j e c t s Four subjects, one male and one female from each of the high and low achievement groups, were randomly selected from the eighty subjects used i n the Phase 3 of the study. P r o c e d u r e s The four subjects were administered the Diagnostic Checklist for Algebra i n a c l i n i c a l - i n t e r v i e w s e t t i n g . They were also questioned to determine i f the wording of the instrument was c l e a r . This was p a r t i c u l a r l y important i n the Conceptual-expressions portion of the diagnostic instrument, where students had to show t h e i r understanding of arithmetic and algebraic expressions through the manipulation of concrete materials. A f t e r each response, subjects were asked to explain the thought processes they used i n di s p l a y i n g understanding of concepts and vocabulary i n algebra ( K i l p a t r i c k , 1967). A n a l y s e s Video-tapes of the diagnostic interviews were viewed to determine areas of common confusion surrounding any aspect of the diagnostic instrument. Each subject viewed h i s or her own video-tape with the interviewer. Each was questioned regarding responses, errors, and apparent nervousness. On the basis of observations of these four students, changes were made to the wording of the Diagnostic Checklist for Algebra to c l a r i f y meaning and reduce t e s t i n g time, as well as changes to the placement of video-equipment to minimize d i s t r a c t i o n s f o r the subjects. For a complete d e s c r i p t i o n of the changes which occurred i n the c h e c k l i s t during Phase 4, re f e r to Appendix C. P h a s e 5: T e s t i n g o f t h e F i n a l I n s t r u m e n t The purpose of the f i n a l phase of the study was to address the research questions posed i n Chapter 1. The f i f t h phase of the study involved administration of the re f i n e d diagnostic instrument to a larger group of students. Each was given the Diagnostic Checklist for Algebra i n an i n d i v i d u a l , c l i n i c a l - i n t e r v i e w . The nature and frequency of errors made on the diagnostic instrument were investigated. Comparisons were made between high, average, and low achievers. The re l a t i o n s h i p s among concepts, vocabulary, e r r o r patterns, and equation-solving procedures were explored. S u b j e c t s A s t r a t i f i e d random sample of 36 subjects was selected from the academic mathematics classes i n the same junior high school used i n Phases 3 and 4 . Subjects who had been retained and thus exposed to e i t h e r Phase 3 or Phase 4 were omitted from the sample. The s t r a t a were: grade l e v e l ( 8 , 9 , 1 0 ) , algebraic achievement l e v e l (high, average, low) and gender (male, female). P r o c e d u r e s A l l academic mathematics students were given the Lankton First-Year Algebra Test (Lankton, 1 9 6 5 , Form E ) i n a group session to determine t h e i r algebraic achievement l e v e l . Students were then assigned to one of eighteen groups according to grade l e v e l ( 8 / 9 / 1 0 ) , achievement l e v e l (high/average/low), and gender (male/female). Two subjects were randomly selected from each of 18 groups. Parental consent forms were given to each of the 36 subjects. (See Appendix D, Part 1 ) . If parental consent to p a r t i c i p a t e i n the diagnostic interviews was not obtained, another subject was randomly selected from the appropriate group. A l l parental consent forms were returned p r i o r to the diagnostic interviews. The Concepts in Secondary Mathematics and Science (CSMS) tes t (Booth & Hart, 1982) was administered to a l l subjects i n a separate group session p r i o r to any interviews. The purpose of the i n c l u s i o n of t h i s t e s t i n the study was to determine whether or not the two questions on understanding of va r i a b l e used i n the Diagnostic Checklist for Algebra (based on Wagner, 1981a) adequately described students' understanding of v a r i a b l e . The CSMS te s t consists of 51 items and was based on the work of C o l l i s (1975) and Kuchemann (-1978). Results on the CSMS would be compared to student's understanding of the concept of variable and the errors made r e l a t i n g to vari a b l e s i n so l v i n g equations (e.g. combining unlike terms). A Latin-Square design was used to select the date and time of interviews f o r each subject to control f o r the e f f e c t s of grade, achievement, and gender. The interviews extended over a six-week period because of 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 well as absenteeism. A l l subjects were i n d i v i d u a l l y interviewed, and interviews were video-taped. Preliminary diagnostic r e s u l t s were discussed with each student at the conclusion of the interview. A n a l y s e s Comparisons were made with appropriate items from the Lankton and with students' mathematics marks to e s t a b l i s h concurrent v a l i d i t y . Frequency d i s t r i b u t i o n s were used to examine the e f f e c t s of equation complexity, s t r u c t u r a l format, and numerical magnitude. Frequency d i s t r i b u t i o n s were also c a l c u l a t e d f o r a l l error categories. Errors were co r r e l a t e d with IQ, age, mathematical achievement - Lankton, Marks, CSMS - and other errors. 63 The two teachers of the 36 subjects were interviewed regarding the errors made by the students. The purpose of the teacher interviews was to determine i f e r r o r s made on the diagnostic instrument were s i m i l i a r to errors which students made i n the classroom s e t t i n g . Summary o f S t u d y D e s i g n Previous research has indic a t e d a need for diagnostic t o o l s at the secondary l e v e l to explore errors i n algebra; the present study was designed to meet t h i s need using the framework of the "input-process-output" model of learning (Booth, 1981). Scrutiny of f i r s t - y e a r algebra textbooks i d e n t i f i e d twelve types of equations commonly taught i n f i r s t -year algebra courses, as well as concepts and vocabulary which might influence procedures used and errors made i n solv i n g l i n e a r equations i n one v a r i a b l e . Equations were constructed which r e f l e c t e d research concerns fo r equation complexity, numerical magnitude and sign, and s t r u c t u r a l format. An er r o r - c a t e g o r i z a t i o n scheme was developed and used to i d e n t i f y e rrors made i n solv i n g equations. The diagnostic instrument combined the r i g o r of 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 of semi-structured interviews, minimizing the negative aspects of each technique. The procedures used to develop the diagnostic instrument were selected to increase the p r o b a b i l i t y that errors i d e n t i f i e d would be representative of the types of errors made by students i n f i r s t - y e a r algebra courses. CHAPTER IV RESULTS The r e s u l t s presented are the pertinent findings from the t h i r d , fourth, and f i f t h phases of the study. The outcomes of the item analysis and i n t e r a c t i o n e f f e c t s involved i n the s e l e c t i o n of equations are summarized from Phase 3 of the study. The r e s u l t s of the refinement procedures used i n Phase 4 of the study are discussed. The nature and frequency of the errors made by students using the Diagnostic Checklist for Algebra i n Phase 5 are presented. However, t h e i r presentation i s organized around the research questions, rather than s p e c i f i c interview r e s u l t s . Examples from i n d i v i d u a l interviews are d e t a i l e d to c l a r i f y f i n d i n g s . Summary of Phases 3 and 4 Phase 3 involved s e l e c t i o n of equations from the 192 constructed equations (12 equation types by 4 s t r u c t u r a l formats by 4 numerical magnitudes) f o r use i n the diagnostic instrument. The possible i n t e r a c t i o n e f f e c t s among complexity of the equations, s t r u c t u r a l format, and numerical magnitude was a concern from previous research which required empirical c l a r i f i c a t i o n . Analysis of the re s u l t s of the equation t e s t forms i n d i c a t e d that there was a strong i n t e r a c t i o n e f f e c t between s t r u c t u r a l format and numerical magnitude (See Appendix B, Part 3). It was decided to include a l l of the' r e s u l t i n g c e l l s of t h i s 4 x 4 g r i d at each l e v e l of equation complexity f o r complete diagnosis. This necessitated the s e l e c t i o n of 48 equations f o r i n c l u s i o n i n the Diagnostic Checklist for Algebra. The fourth phase of the study r e f i n e d the wording and format of the instrument, and investigated i t s usefulness i n i d e n t i f y i n g systematic errors of i n d i v i d u a l s and common errors among groups of students. The re s u l t s of the wording changes are presented i n Appendix C. The nature and frequency of the errors produced by the four subjects involved i n Phase 4 are presented here. The two low achievers seemed to have almost equal d i f f i c u l t y across a l l l e v e l s of structure and magnitude. The one exception was with equations which were structured with the .variable to the l e f t of the equality and constants (SI). Twelve of the 28 equations on which low achievers made errors were SI equations. These equations were twice as d i f f i c u l t for'the two low achievers as were any other l e v e l of s t r u c t u r a l format. Five of the 11 equations on which high achievers made errors were equations of numerical magnitude four (N4: 100<In|<1000). Looking at the s t r u c t u r a l focus of these eleven equations, i t seemed that equations where the var i a b l e was to the right of the constants (S2 & S4) caused the most d i f f i c u l t y (7 of the 11 equations) f o r high achievers. These differences i n the patterns of errors seemed to indicate the need for i n c l u s i o n of the 4-by-4 magnitude-by-structure g r i d at each equation l e v e l . Thus, the f i n a l instrument contained 48 equations. Subjects showed differences i n the nature and frequency of t h e i r conceptual, procedural, and resultant e r r o r s . They were rated for t h e i r understanding of concepts and vocabulary (See Table 3, Conceptual). A T a b l e 3 P h a s e 4: C o m p a r i s o n o f H i g h a n d 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 I n s t r u m e n t E r r o r T y p e s A c h i e v e m e n t L e v e l s H i g h Low C o n c e p t u a l V a r i a b l e E x p r e s s i o n s E q u a l i t y E q u a t i o n s C o e f f i c i e n t s L i k e T e r m s I n v e r s e s I d e n t i t i e s P r o c e d u r a l Z e r o A n n e x a t i o n 2 0 I d e n t i t y C o n f u s i o n 0 1 L i k e T e r m s 0 6 P a r t i a l D i s t r i b u t i v e 0 0 O r d e r o f O p e r a t i o n s 0 0 + P r o p e r t y o f = 0 9 x P r o p e r t y o f = 4 4 C o e f f i c i e n t 0 0 M * P M / P M / P M M M / P R R R P / R R M / P P M P / R Resultant C o m p u t a t i o n a l A d d / S u b t r a c t 1 2 M u l t i p l y / D i v i d e 3 5 B a s i c f a c t s 1 5 F a u l t y a l g o r i t h m 2 2 W r o n g o p e r a t i o n 0 2 S i g n A d d / S u b t r a c t 2 2 M u l t i p l y / D i v i d e 3 7 R u l e b a s e d 3 3 D i s t r i b u t i v e 0 1 O t h e r M e c h a n i c a l / p e r c e p t 0 4 R a n d o m 0 0 I n c o m p l e t e 0 1 F i n a l Answer 14 2 9 * M = M a s t e r y P = P a r t i a l R = R e m e d i a l r a t i n g of "mastery" (M) meant that both subjects could f u l l y explain the item, i n c l u d i n g an a b i l i t y to demonstrate t h e i r knowledge. A r a t i n g of "mastery/partial" (M/P) meant that one subject had mastered the concept and that the other had only p a r t i a l l y mastered the concept. A r a t i n g of " p a r t i a l " (P) meant that both subjects only p a r t l y explained or demonstrated the given concept (e.g. they may have known that a l e t t e r represented any number as a placeholder, but had forgotten the word " v a r i a b l e " ) . A r a t i n g of "parti a l / r e m e d i a l " (P/R) meant that one subject had p a r t i a l l y mastered the concept and that the other required remedial assistance f o r the given concept. A r a t i n g of "remedial" (R) meant that both subjects d i d not have any basic understanding of the given concept and required remedial assistance. Only two concepts were below a p a r t i a l mastery l e v e l f or the two high achievers, while there were s i x concepts below p a r t i a l mastery for the low achievers (See Table 3, Conceptual). Each of the high achievers made one type of procedural e r r o r . The boy had one instance of a zero annexation error, and the g i r l had four instances of errors with the m u l t i p l i c a t i o n property of eq u a l i t y . However, the procedural errors made by the low achievers were almost evenly divided among the two subjects, except that the boy was the one to make an i d e n t i t y element confusion between zero and one (See Table 3, Procedural). The high achievers made no resultant-other e r r o r s , and t h e i r errors were almost evenly divided between computation and sign i n the resultant category. For low achievers, computation and sign errors seemed to involve mainly m u l t i p l i c a t i o n and d i v i s i o n . They also made several copying mistakes inv o l v i n g signs (See Table 3 under mechanical/perceptual). In general, high achievers made almost s i x times as many resultant errors (29 ) as procedural errors ( 5 ) . Low achievers, on the other hand, made only s l i g h t l y over three times as many resultant errors (63 ) as procedural errors ( 2 0 ) . These differences i n both the frequency, as well as the nature', of the errors that were made by students of 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 implications and may pinpoint some of the common errors found among students. There were i n d i c a t i o n s that the diagnostic instrument d i d i d e n t i f y systematic errors within subjects and common errors across subjects. Preliminary analyses indicated that the instrument had p o t e n t i a l for diagnosing students' errors i n algebra. P h a s e 5: I n s t r u m e n t T e s t i n g In the f i f t h phase of the study, the errors made by students were analyzed and the re l a t i o n s h i p s among these errors were investigated. These analyses involved comparison between a student's achievement l e v e l i n algebra and the erro r patterns found among concepts, procedures, and r e s u l t s . They also involved the comparison of errors found at d i f f e r e n t l e v e l s of equation complexity, s t r u c t u r a l format, and numerical magnitude. S u b j e c t s The e n t i r e academic Grade 8 , Grade 9 , and Grade 10 populations (n = 1 4 3 , 1 4 1 , 8 8 , respectively) i n the junior high school used i n the t h i r d and fourth phases were administered the Lankton First-Year Algebra 69 t e s t . For each grade l e v e l , subjects who scored one or more standard deviations above the mean were c l a s s i f i e d as high achievers, those within one standard deviation of the mean as average achievers, and those who scored one or more standard deviations below the mean as low achievers. Subjects were assigned to c e l l s based on gender (male/female), grade l e v e l ( 8 / 9 / 1 0 ) and achievement l e v e l (high/average/low). Two subjects were randomly selected from each of these eighteen c e l l s f o r i n c l u s i o n i n the diagnostic interviews. Any subjects retained i n the school who had been involved i n the t h i r d or fourth phases of the study were not included i n the s e l e c t i o n process. Table 4 presents the demographic data, IQ, and mean mathematics marks for each subject included i n the f i n a l interviews. This information was obtained from i n d i v i d u a l P u p i l Record Cards. Within the table, the f i r s t four subjects at each grade l e v e l were high achievers, the next four were average achievers, and the l a s t four were low achievers. There were s i x students whose IQs seemed contra-indicated by t h e i r achievement l e v e l placement on the Lankton (See Table 4 ) . This may have been due i n part to the fact that the only IQ scores a v a i l a b l e for students was the Otis-Lennon (Form M ), which i s a group administered IQ t e s t given to subjects upon entry at the Grade 8 l e v e l i n the selected junior high school, and as such t h e i r IQ scores may not have been representative of t h e i r true c a p a b i l i t i e s i n the school s e t t i n g . At the Grade 10 l e v e l , one male average achiever had a comparatively high IQ ( 131 ) and one had a comparatively low IQ ( 9 0 ) . However, each made 2 9 errors on the equation portion of the diagnostic T a b l e 4 P h a s e 5 : S e l e c t e d D a t a f o r S u b j e c t s G r a d e G e n d e r Acre 10 M a r k L a n k t o n CSMS 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 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 A t e a c h g r a d e l e v e l , t h e f i r s t f o u r s u b j e c t s a r e h i g h a c h i e v e r s , t h e n e x t f o u r s u b j e c t s a r e a v e r a g e a c h i e v e r s , a n d t h e l a s t f o u r s u b j e c t s a r e l o w a c h i e v e r s a s d e f i n e d b y t h e L a n k t o n . N=36 instrument, which was near the average of 30 errors f o r that grade and achievement l e v e l . At the Grade 8 l e v e l , the two male high achievers had r e l a t i v e l y low IQ scores (97, 98) and low average marks, yet the f i r s t made 65 errors and the second made 57 errors on the diagnostic instrument, which was near the average of 60 errors f o r that grade and achievement l e v e l . S i m i l a r l y , one of the Grade 8 female average achievers had a r e l a t i v e l y low IQ (93), and a somewhat higher mark than expected (C+), but made 71 errors on the equation portion of the diagnostic instrument compared to an average of 70 errors f o r that grade and achievement l e v e l . Conversely, one female low achiever at the Grade 8 l e v e l had a r e l a t i v e l y high IQ (125). However, t h i s subject made 95 errors on the equation portion of the diagnostic instrument, which was p r e c i s e l y the average of 95 errors f o r her grade and achievement l e v e l . '. . . These discrepancies between IQ, achievement l e v e l , marks, and performance on diagnostic c h e c k l i s t resulted i n some other curious r e l a t i o n s h i p s . For example, the average IQ for the low Grade 8s (106) was higher than the average IQ for e i t h e r the high or average Grade 8 groups (102, 101). The marks f or a l l groups seemed low, and at the Grade 8 l e v e l were barely d i s t i n g u i s h a b l e between achievement l e v e l s . The CSMS res u l t s d i d seem to distingush between both achievement l e v e l and grade l e v e l with the exception of the high achieving 9s and 10s which were i n d i s t i n g u i s h -able. These discrepancies within the sample may have had some influence on the r e s u l t s i n the f i n a l phase of the study. A f t e r s e l e c t i o n , p o t e n t i a l subjects met as a group during the lunch hour following administration of the Lankton. The purpose of the study and the nature of student p a r t i c i p a t i o n were c a r e f u l l y explained. Subjects were given a parental consent form (See Appendix D, Part 1) and were asked to return t h i s p r i o r to t h e i r interview. Three subjects were unable to p a r t i c i p a t e , two because of a lack of parental consent, and one because of long-term i l l n e s s . New subjects from the same c e l l s were randomly selected as replacements. P r o c e d u r e s Approximately two weeks p r i o r to the f i r s t interview i n ea r l y May, subjects were given a group administration of the Concepts in Secondary Mathematics and Science (CSMS) t e s t . This test was designed to measure l e v e l of understanding of the concept of v a r i a b l e . Any absentees were administered the CSMS i n the next two days, so that a l l subjects had completed both the Lankton and the CSMS p r i o r to the.diagnostic interviews. A Latin-Square Design was used to select the date and time of interviews to cont r o l f o r gender, grade l e v e l , and achievement l e v e l . However, some interview times had to be re-scheduled because of i n s t r u c t i o n a l demands i n the classroom, e x t r a - c u r r i c u l a r sports a c t i v i t y c o n f l i c t s , and absenteeism. Interviews occurred i n a small, audio-visual storage room. A l l interviews were video-taped. At the beginning of each interview, the purpose of the interview and video-taping was re-explained to each student and the student questionnaire form was reviewed. At the end of the interviews subjects were given feedback regarding the nature of t h e i r errors on the diagnostic instrument. Questions regarding the study were answered. Aft e r school hours, remediation was provided upon request. P r e s e n t a t i o n o f R e s u l t s The r e s u l t s of the interviews are presented i n terms of the research questions asked. The errors made by students of d i f f e r e n t achievement l e v e l s were analyzed and systematic and common errors were i d e n t i f i e d . The re l a t i o n s h i p s of errors to demographic data, IQ, and mathematical achievement variables were explored. The nature and frequency of errors made at d i f f e r e n t l e v e l s of equation complexity, s t r u c t u r a l format, and numerical magnitude were compared. A c h i e v e m e n t L e v e l s a n d E r r o r s K r u t e t s k i i (1976) noted differences among students i n t h e i r a b i l i t y to solve algebraic problems. "Good problem solvers" were able to explain what they were doing i n terms of both concepts and procedures. "Poor problem so l v e r s " seemed more procedure-oriented and were unable to verba l i z e reasons f o r what they were doing or thinking. While K r u t e t s k i i ' s work r e f e r r e d to word problems, s i m i l i a r findings could occur i n the solving of l i n e a r equations i n one va r i a b l e . The following research question was asked: 1. Does the 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 the nature and frequency of the errors made by a p a r t i c u l a r student? In keeping with the investigations done by K r u t e t s k i i (1976), the students were grouped according to achievement l e v e l to f a c i l i t a t e i n v e s t i g a t i o n of the nature and frequency of the errors made. An average l e v e l of understanding for each concept and vocabulary item from the conceptual portion of the c h e c k l i s t are presented i n Table 5 within each achievement l e v e l at each grade l e v e l (Refer to page 67 for the r a t i n g of understanding d e f i n i t i o n s ) . Error frequency r e s u l t s from the equation portion of the c h e c k l i s t are presented i n Table 6 f o r each err o r category within each achievement l e v e l at each grade l e v e l . This separation of grade l e v e l and achievement l e v e l seemed necessary as the diagnostic instrument inv e s t i g a t e d content covered p r i o r to Christmas at the Grade 9 l e v e l , but t h i s content had not been attempted by May i n the Grade 8 c l a s s e s . As i n the fourth phase of the study, the errors among high, average and low achievers d i f f e r e d i n quantitative and q u a l i t a t i v e aspects across a l l three e r r o r types [conceptual (See Table 5 ) , procedural, and resultant (See Table 6)]. To compute the error rates f or conceptual understandings (See Table 5) , remedial was counted as 2., p a r t i a l as 1 , and mastery as Q_. High achievers had a 28% e r r o r rate, average achievers had a 32% error rate, and low achievers had a 47% error rate. The reasons for these differences i n e r r o r rate were investigated and r e s u l t s i n d i c a t e d that c e r t a i n aspects of both the concepts and vocabulary sections contributed to the d i f f e r e n c e s . There were four d i f f e r e n t aspects of the concepts section of the •diagnostic c h e c k l i s t which r e l a t e d to students' achievement l e v e l s . These were the a b i l i t y of students to: 1) i d e n t i f y the variable and explain i t s meaning, 2 ) explain concretely the meaning of d i v i s i o n i n expressions Table 5 Phase 5: Comparison of High, Average, and Low Achievers' Conceptual Understandings on the Instrument Understandings* Achievement Levels** High Average Low . 8 9 10 8 9 10 8 9 10 C o n c e p t u a l Variable P M M P/R M/P M/P R M/P P/l Expressions P P M/P P/R P P P/R P/R R Equality- P M M P/R M/P M P/R P P Equation R P P R P/R P R R R C o e f f i c i e n t R P/R P/R R P P R P R Like Terms R M/P R R R P R R R Inverses P M/P M R P M P/R M/P P Identity Elements P M/P M P/R M/P M P/R P P Grand N = 3 6 The conceptual portion of the diagnostic c h e c k l i s t i s presented i n terms of understanding of the concepts and vocabulary associated with so l v i n g l i n e a r equations. (Refer to page 67 for 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 of High, Average, and Low Achievers' E r r o r Frequencies on the Instrument E r r o r Types E r r o r Frequencies f o r Achievement L e v e l s * High Average Low 8 9 10 8 9 10 8 9 10 P r o c e d u r a l Zero Annexation 2 3 2 4 4 4 0 2 3 I d e n t i t y Confusion 0 2 1 6 4 3 4 1 4 Li k e Terms 9 9 15 13 8 0 8 15 12 P a r t i a l D i s t r i b u t i v e 0 1 1 0 0 1 0 1 0 Order of Operations 3 1 1 0 1 0 0 1 0 + Property, of = • 7 13 4 12 12 2 5 . 14 16 x Property of = 0 2 4 0 6 5 0 4 0 C o e f f i c i e n t 0 2 1 1 2 5 1 6 1 R e s u l t a n t C o m p u t a t i o n a l Add/Subtract 20 13 13 24 27 10 15 28 31 M u l t i p l y / D i v i d e 4 2 3 9 10 11 18 8 7 Basi c Facts 6 0 8 7 6 6 4 13 8 F a u l t y A l g o r i t h m 5 9 6 2 22 7 6 8 7 Wrong Operation ' 11 7 8 22 19 11 14 ' 21 23 S i g n Add/Subtract 33 10 12 22 38 17 13 35 31 M u l t i p l y / D i v i d e 4 4 2 7 2 5 13 11 4 Rule Based 21 6 3 20 15 5 12 11 14 D i s t r i b u t i v e 2 0 0 0 0 0 1 0 0 O t h e r Mechanical/Percept 9 20 10 5 25 9 4 15 27 Random 3 9 6 3 7 10 1 11 9 Incomplete 104 5 3 101 37 0 119 18 23 F i n a l A n s w e r 135 23 23 135 59 32 156 95 75 * As ranked on the Lankton First-Year Algebra Test Grand N = 36 containing blanks and variables, 3) explain concretely the meaning of m u l t i p l i c a t i o n i n expressions containing variables, and 4) create meaning of equations with and without context as well as given the meaning-create an equation. High-achieving students were able to explain these things almost without exception and regardless of grade l e v e l ; while the opposite was true f o r low-achieving students. For example, when explaining the meaning of d i v i s i o n i n expressions containing variables (x/4), one high-achieving Grade 8 explained: "Well, x could be 12, then 12 divi d e d by 4 means 3 groups of 4." During t h i s explanation she put groups of 4 pennies together and noted 3 groups. For the same question, a low-achieving Grade 8 stated: "I can make up any number for x. It means 3 div i d e d by 4." During t h i s explanation she put out 3 pennies and then added one more penny to these 3. Some average achievers showed mastery at the Grade 10 l e v e l , while those of lower grade l e v e l s d i d not tend to show such mastery. When asked the " i d e n t i f y v a r i a b l e " question: "What do you c a l l the x. i n 5+x? What does i t stand for?", an average-achieving Grade 10 c o r r e c t l y stated: " I t ' s a l e t t e r - umm va r i a b l e . It could mean any number." However, f o r the same question, an average-achieving Grade 8 commented: "The 'pi'? An answer?" Some portions of the vocabulary section of the diagnostic instrument also produced major differences among l e v e l s of achievement. As with the concepts section, there were four vocabulary'items involved. These were the vocabulary associated with: 1) additive inverses, 2) m u l t i p l i c a t i v e inverses, 3) the additive i d e n t i t y element, and 4) the 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 of the i n t e r e s t i n g findings for the Grade 10s was the fact that the four average achieving students had p a r t i a l mastery of naming and c o l l e c t i n g l i k e terms, while the high-achieving students d i d not (See Table 5). However, i n l a t e r work i n equation-solving, the high-achieving students were able to carry out algebraic procedures combining l i k e terras, and two of them even commented that they had no idea why they had e a r l i e r missed naming and grouping " l i k e terms." The high achievers used the word spontaneously i n t h e i r equation-solving procedures, and i t seemed to be the procedures which t r i g g e r e d the meaning of l i k e terms f o r them. The high achievers s t i l l made numerous errors using l i k e terms i n sol v i n g equations (33 compared to average achievers 21), but 19 of these were s e l f - c o r r e c t e d during s o l u t i o n , while for average achievers these mistakes with l i k e terms went by unnoticed. Procedural errors i n v o l v i n g the addition property of e q u a l i t y were much higher for the Grade 9 high achievers and average achievers than for the corresponding Grade 10s (Refer to Table 6). However, t h i s was not the case f o r low achievers, who made a large number of errors at both the Grade 9 and Grade 10 l e v e l s . An example of a t y p i c a l error made by Grade 9 low achievers 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 side of the equation was done c o r r e c t l y . However, when he s a i d "You subtract Ix from both sides," he wrote the "-" f i r s t on the l e f t side and then the "7x " i n front of i t . When questioned about why he wrote what he did, he ind i c a t e d again that you have to subtract the same thing from both sides. He seemed obl i v i o u s to the lack of commutativity of the operation of subtraction and was unconcerned about the correctness of the answer. Some findings f o r the procedural and resultant errors seemed opposite to expectations. For example, at the Grade 8 l e v e l the low group had the fewest errors f o r zero annexation, addition/ subtraction, basic f a c t s , r u l e -based, mechanical/perceptual and random. However, analysis of the protocols of these low achievers showed that they tended not to depend on memory f o r any work, but rather wrote out e x p l i c i t l y any work dealing with zero annexation or basic f a c t s , even to the point of making a column of f i v e 6s to add them rather than r e c a l l i n g the basic f a c t , f i v e times s i x . This tendency of low-achieving Grade 8s to write out t h e i r computational work, d i d not seem to continue at the Grade 9 and 10 l e v e l s f o r low achievers, and a c t u a l l y became a way to d i s t i n g u i s h between low and high achievers. It i s the high achievers at the Grades 9 and 10 l e v e l s who tended to write out t h e i r computational and procedural work and the low achievers who d i d not. It should also be noted that some errors d i d not occur for the low-achieving Grade 8s because they d i d not attempt some questions at a l l , while other Grade 8s attempted them, made some errors, and then f a i l e d to complete the s o l u t i o n . Similar examples are found for the low Grade 9s for the same reason. As might be expected, f i n a l answers being e i t h e r correct or incorrect provided c l e a r d i s t i n c t i o n s among high, average, and low achievers i n terms of errors made i n solv i n g equations. High achievers at the Grade 8, 9, and 10 l e v e l s , c o r r e c t l y answered 3 0 % , 88%, and 88%, re s p e c t i v e l y . Average Grade 8, 9, and 10 students c o r r e c t l y answered 3 0 % , 69%, and 83%, re s p e c t i v e l y . Low achievers at the Grade 8, 9, and 10 l e v e l s c o r r e c t l y answered 19%, 51%, and 61%, res p e c t i v e l y . Frequency of resultant sign errors 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 to d i s t i n g u i s h among achievement l e v e l s . When a high-achieving student made an erro r i t was often a confusion with a correct rule f o r another operation (e.g. two negatives make a negative when adding, being applied to m u l t i p l i c a t i o n or d i v i s i o n ) . However, when low achievers made an e r r o r i t was usually based on a rule of t h e i r own creation (e.g. when there i s a negative sign, the answer i s negative) and bore no resemblance to previously learned sign r u l e s . Incomplete work seemed to d i s t i n g u i s h achievement l e v e l s , although t h i s was most noticeable when comparing high and low achievers. Many low achievers would simply read the equation and then indi c a t e that they d i d not understand the question or d i d not know how to proceed. High achievers, even at the Grade 8 l e v e l , were more w i l l i n g to t r y some procedure, i n c l u d i n g systematic t r i a l and error, to get an answer. They 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 that they could not attempt an item; even though, i n the case of the Grade 8s, they had never seen an equation 81 l i k e the two-step and multi-step equations presented. This willingness to persevere i n the so l u t i o n of an equation was d e f i n i t e l y one of the major diff e r e n c e s among achievement l e v e l s and supported the findings of K r u t e t s k i i ( 1 9 7 6 ) . The e r r o r frequencies on the instrument provided i n s i g h t into, i n d i v i d u a l diagnostic concerns, as well as useful information regarding the nature of the errors made at d i f f e r e n t achievement l e v e l s . These discrepancies i n the nature of the errors made by students of d i f f e r e n t achievement l e v e l s focused on p o t e n t i a l areas of c u r r i c u l a r concern i n algebra. These discrepancies emphasized the importance of the understanding of basic algebraic concepts and the fact that f o r low achievers lack of mastery of these concepts seems to play a major role i n t h e i r continued low achievement i n algebra. 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 a n d Common E r r o r s As mentioned previously i n the review of the l i t e r a t u r e , Anderson ( 1 9 8 2 ) i d e n t i f i e d both systematic and common algebraic e r r o r s . A systematic e r r o r occurred when a subject made a p a r t i c u l a r type of error i n 50% or more of the av a i l a b l e opportunities. A common erro r occurred across subjects when 5% or more of her 2 0 0 subjects committed the same systematic e r r o r . She found three errors which occurred i n more than 20% of her sample: ax + (-bx ) = (b-a)x where |b| > |a|, ax + x = ax 2, and ax + bx = abx. It was of i n t e r e s t to the present study to see i f these errors would be r e p l i c a t e d and i f other systematic or common errors would be present. The following question addressed t h i s concern: 2. Does the 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 errors f o r p a r t i c u l a r students and common errors among groups of students? Nine of the 36 subjects used i n the f i n a l phase of the study exhibited systematic errors on the Diagnostic Checklist for Algebra. Seven of these nine were i n Grades 9 and 10. There were four general types of systematic errors displayed by students. Two were procedural ( l i k e terms, addition property of e q u a l i t y ) , and two were resultant (computational: addition and subtraction, sign: addition and subtraction). Three students exhibited systematic errors with l i k e terms. One subject c o n s i s t e n t l y made errors of the type ax + b = (a+b)x, while another made err o r s of the type ax + bx = abx, and the t h i r d made errors of the type ax + (-bx ) = (b-a)x, where |b| > |a|. While the l a t t e r two errors represented a r e p l i c a t i o n of two of Anderson's (1982) systematic errors, they only occurred systematically f o r these two students. Four students exhibited systematic errors with the addition property of equ a l i t y . One student c o n s i s t e n t l y added the same number to both sides, rather than the opposite. For example, i n the equation 17x + (-100) = -15, the student added (-100) to both sides of the equation and obtained the r e s u l t that 17x = -115, f a i l i n g to notice the error that (-100) and (-100) were not opposites. One student t r i e d to apply the addition property of eq u a l i t y to a l l s i t u a t i o n s of combining instances of the vari a b l e , even when they occurred on the same side of the equality symbol. To i l l u s t r a t e , f o r the equation 8x + 9x = -17, t h i s student added (-9x ) to both sides of the equation, c o r r e c t l y obtaining 8x = (-9x ) + -17, but then became confused, and d i d not how to f i n i s h s o l v i n g the equation. The other two students c o n s i s t e n t l y added the opposite number to one side of the equation, but the same number to the other side of the equation. For example, i n the equation 17x + (-100) = (-15), they would add 100 to the l e f t side> but (-100) to the rig h t side. This gave them the same equation [17x =(-115)] as the f i r s t student mentioned, yet t h e i r reasons for a r r i v i n g at t h i s equation were quite d i f f e r e n t . Two students exhibited computational addition and subtraction errors. One student exhibited consistent f a u l t y algorithms i n v o l v i n g regrouping, which seemed to cause these e r r o r s . The other student c o n s i s t e n t l y chose the wrong operation. Six of the nine students making systematic errors displayed systematic sign errors with addition and subtraction. Three students seemed to make these errors due to rule-based a p p l i c a t i o n of sign rules f or m u l t i p l i c a t i o n ( i e . two negatives make a p o s i t i v e ) . The other three were due to mechanical/perceptual errors; two from consistent copying mistakes and one from an i n a b i l i t y to "notice" the negative signs. A l l of the systematic errors were also common er r o r s . The frequency with which these common errors occurred i n the sample was: addition/subtraction sign errors (16%), addition property of equality (11%), l i k e terms (8%), and computational addition/ subtraction errors (5%) . I n s t r u m e n t V a l i d i t y V a l i d i t y of a diagnostic instrument has usually been determined by whether or 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 of the errors made by students and the reasons for making these e r r o r s . Very l i t t l e attention has been given to the usual v a l i d i t y concerns involved i n e i t h e r normative or c r i t e r i o n - r e f e r e n c e d t e s t development. However, a diagnostic instrument which incorporates the strengths of each of these t e s t s within the diagnostic framework cannot help but be a more v a l i d instrument. The following research question was posed to address the issue of instrument v a l i d i t y : 3. Is the Diagnostic Checklist for Algebra valid? Face and content v a l i d i t y were established through the methods of textbook analysis, e r r o r - c a t e g o r i z a t i o n scheme development, and equation construction and s e l e c t i o n as outlined i n Phases 1, 2, and 3. Face v a l i d i t y was improved i n Phase 4. Measures of concurrent and p r e d i c t i v e v a l i d i t y were undertaken i n Phase 5 through the c o r r e l a t i o n a l analysis of outcomes on the diagnostic instrument compared to measures of mathematical achievement, IQ, and demographic data. While such "outcomes" on a diagnostic instrument serve no diagnostic purpose (comparisons of who d i d "better" i n a diagnostic s e t t i n g i s absurd), such comparisons provide measures of v a l i d i t y which enhance the o v e r a l l confidence that teachers would place i n the Diagnostic Checklist for Algebra, hence, t h e i r i n c l u s i o n i n the analyses. Detailed s t a t i s t i c a l analyses r e l a t i n g to v a l i d i t y concerns are presented i n Appendix D, Part 3. However, due to the l i m i t e d sample s i z e and the fact that i t s composition was from one school setting, these c o r r e l a t i o n s must be viewed with caution. 85 Through comparisons of the mean p-values of each equation type with that equation type 's o r d i n a l ranking from the textbook an lays i s (See Table 7), evidence of s t r u c t u r a l v a l i d i t y (Conry, 1 9 8 9 ) was obtained. The mean p-values were obtained by determining the mean of the p-values of the 4 equations wi th in each equation type (See Appendix B, Part 4, Tables B . 2 -B . 4 ) . A Spearman's rho was c a l c u l a t e d at 0.90. This c o r r e l a t i o n c o e f f i c i e n t provides fur ther v a l i d i t y of the s t ruc ture and sequence of presenta t ion of the equations wi th in the Diagnostic Checklist for Algebra. With the exception of age and gender, a l l c o r r e l a t i o n s obtained between the d iagnost ic , instrument "outcomes" and the demographic data (age, gender, grade) , IQ, and mathematics' achievement v a r i a b l e s (mean mathematics' marks - from Grade 4 through 2 years a f t er the d iagnos t i c in terv iews , Lankton, CSMS) were s i g n i f i c a n t . While some of the corre lat ionswere not high (refer to Table D.5 of Appendix D) they d i d provide a measure of concurrent v a l i d i t y (Lankton, CSMS, IQ, grade l eve l ) as wel l as p r e d i c t i v e v a l i d i t y (mean mathematics' marks). These c o r r e l a t i o n s provide fur ther evidence for the v a l i d i t y of the Diagnostic Checklist for Algebra. 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 e r t a i n concepts and vocabulary assoc iated with algebra have been reported to a f f ec t the equat ion- so lv ing process to some degree. C o n t r a d i c t o r y f indings have been reported for students' understanding of v a r i a b l e and i t s inf luence on the equat ion- so lv ing process (Booth & Hart , 1982; Kuchemann, 1978; Wagner, 1977, 1981a; as opposed to K i e r a n , 1981b; R a c h l i n , 1982; Rosnick & Clement, 1980). An understanding of expressions Table 7 Relationship* Between Text P o s i t i o n and Mean P -Values of Equation Types General Format of Ordinal P o s i t i o n Rank Order Mean Equation Types i n Textbooks of P -Values P -Value** I . O n e - S t e p x+a=b*** x-a=b ax=b x/a=b I I . T w o - S t e p ax+b=c ax+bx=c ax+b=cx a(x+b)= c**** I I I . M u l t i - S t e p 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-values l i s t e d are rounded to two s i g n i f i c a n t d i g i t s and represent hundredths. *** a,b,c,d,e,x are a l l integers, a*0. Based on the constraints of the study these equations were l i m i t e d to those containing i n t e g r a l roots, as well 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. 1 2 68 2 5 53 3 1 7 0 4 6 50 5 3 57 6 4 53 7 8 36 8 7 40 9 10 22 10 9 23 11 11 15 12 12 14 than i n d i v i d u a l diagnostic patterns. Hence, the focus of presentation w i l l be s p e c i f i c examples of the r e l a t i o n s h i p between algebraic'concepts/ vocabulary and errors made i n the solving of equations which might prove u s e f u l when teaching algebra. To determine how students' understanding of algebraic concepts and vocabulary r e l a t e d to errors made i n solv i n g equations, the following research question was asked: 4. Are students' understandings of algebraic concepts and vocabulary r e l a t e d to the kinds of errors they make i n the s o l u t i o n of l i n e a r equations i n one variable? D e t a i l e d s t a t i s t i c a l analyses of the co r r e l a t i o n s which were found among concepts, vocabulary, and errors are reported i n Appendix D, Part 4, to which the previous cautionary note regarding sample siz e and composition should be applied. Hence, i n the presentation of r e s u l t s , no s t a t i s t i c a l claims are made, but rather the i n t e r e s t i n g phenomena which occurred during diagnosis are presented. However, only those instances which were s t a t i s t i c a l l y s i g n i f i c a n t are discussed. It i s assumed that other singular patterns were just that, and of i n t e r e s t only to the p a r t i c u l a r student being diagnosed. An understanding of variable was expected to be r e l a t e d to the errors which students made i n combining unlike terms, as well as students' a b i l i t y to obtain correct f i n a l solutions f o r l i n e a r equations. Both aspects of the understanding of var i a b l e ( i d e n t i f y and meaning) were r e l a t e d to the number of correct solutions a student obtained. However, the expected r e l a t i o n s h i p of an understanding of variable and errors made i n combining unlike terms d i d not occur. This lack of 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 ar more complicated concept than may have been i n d i c a t e d by previous research. The a b i l i t y to use the e q u a l i t y symbol was r e l a t e d to the a b i l i t y to solve equations c o r r e c t l y . One t y p i c a l response came from a Grade 8 boy who requ ired remedial work on h i s understanding of e q u a l i t y . He s tated: "The answer's smal ler than the ques t ion ." when presented with 800 + x = 609. When presented with 256 = x + (-344), he changed the order to x + (-344) = 256, and again asserted that "The answer's smal ler than the ques t ion ," r e f e r r i n g to the fact that |256| < | -344 | . He was unable to attempt a s o l u t i o n of e i t h e r equation and t h i s seemed to be a funct ion of h i s view of the e q u a l i t y symbol, rather than a problem with numbers of large magnitude, as he c o r r e c t l y solved -63 + x = -75 and x (45) = -495. It was expected that the number of i n c o r r e c t so lu t ions of the equations would be i n v e r s e l y r e l a t e d to the understanding of one or more of the meaning-of-equation quest ions . This was the case i n the "given an equation without context , create meaning" and "given meaning, create equation" port ions of the d iagnos t i c c h e c k l i s t . T y p i c a l of students' responses when asked to create meaning for x + 21 = 39 was the response of one Grade 9 g i r l . She r e p l i e d : "Found 21 marbles and found some more. When put together got back . . . ( p a u s e ) . . . This i s confus ing ." 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 one-step add i t i on and subtrac t ion problems, r e s u l t i n g i n s i x out of e ight i n c o r r e c t s o l u t i o n s . Many subjects found c r e a t i n g an equation from a word problem to be a very d i f f i c u l t task; When asked to write an equation f o r : Penny noticed something i n t e r e s t i n g about her savings account. If you add $6 to the amount you get the same r e s u l t as doubling the amount and subtracting $ 4 . How much money does Penny have i n her savings account? several subjects shared the view of one Grade 10 g i r l , who f l a t l y stated: "I can't do that!" When encouraged to t r y , she declared: "Do 6 times 2 equals 12 because you double i t . Then minus 4 , equals 8 . " S i m i l a r l y , she was unable to solve any of the four equations with instances of the v a r i a b l e on both sides, but not containing brackets. Students who d i d not understand the meaning of equations i n one format, were also unable to solve such equations. This lack of understanding of the meaning of p a r t i c u l a r equations seemed to preclude the formulation of procedures necessary f o r s o l v i n g s i m i l a r equations. An understanding of the equality symbol and the a b i l i t y to create an equation from a word problem were d i r e c t l y r e lated. When asked to write an arithmetic sentence which had operations on both sides, one t y p i c a l response was that of a Grade 8 boy, who wrote 1+2=3+2=5. He explained: "You're adding 2 to both, so they're equal, and the answer i s 5 . " He l a t e r explained, when answering the previously mentioned word problem regarding Penny's savings account: "You add 6 and 6 and get 1 2 , and then subtract 4 to get 8 . " When questioned why he had added s i x to i t s e l f , he explained that was what "doubling" meant i n the question. This "operational" sense of the e q u a l i t y symbol c a r r i e d over to the a b i l i t y to "translate word problems" and seemed to focus attention oh the "operational" words within the question, rather than t h e i r meaning within the context of the question. Anderson ( 1 9 8 2 ) had ind i c a t e d that students d i d not make the same type of e rrors i n s i m p l i f y i n g arithmetic expressions and sol v i n g equations with s i m i l a r formats. It was of i n t e r e s t to investigate how the errors i n understanding expressions and understanding equations would be related. A l l three of the meaning of equation questions showed a p o s i t i v e r e l a t i o n s h i p to a student's a b i l i t y to explain 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 conception i n t h i s area was exemplified by one Grade 9 g i r l who could not show the meaning of 4n and thought that n /4 only represented a f r a c t i o n . For the same savings account problem mentioned previously, she stated: "You take the 6 and mul t i p l y by 4, ... that gets 24, then subtract 4 and get 20." When asked to explain, she r e p l i e d : "Doubling means you multiply, ... and there's a 6 and a 4." This lack of a b i l i t y to explain the operation of m u l t i p l i c a t i o n seemed to allow "multiply" to mean "doubling" f o r t h i s student. Three of the common errors found i n t h i s study (addition property of equality, a d d i t i o n and subtraction errors with computations and addition and subtraction errors with signs) showed the expected p o s i t i v e r e l a t i o n s h i p s with each other. One-third of the students i n t h i s study concurrently exhibited errors i n these three categories. T y p i c a l of these was one Grade 9 boy, who would apply the addition property by adding the same number to one side and i t s opposite to the other side of the equation. Within t h i s erroneous a p p l i c a t i o n he would v a c i l l a t e between doing the wrong computational operation (e.g. he would write 25 + 19 and then say 6) or making a rule-based sign error (e.g. "You always subtract the numbers on opposite sides, and i f there's a negative, then i t ' s negative."). The 91 intertwining of these three common errors, which involve such fundamental aspects of the equation-solving process, cannot be overlooked. The many instances of procedural and resultant errors which found t h e i r o r i g i n s i n erroneous conceptions emphasize the need f o r e a r l y diagnosis and remediation. Awareness of the types of erroneous conceptions which students seem to form may a l e r t the teacher, so that preventative measures may be undertaken. 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 o n E r r o r s The work of Tatsuoka, et a l . (1980) indicated that the magnitude of signed numbers influenced the r e s u l t s of integer arithmetic. Booth (1981) found that larger numbers caused students to focus on the process, while Herscovics (1979) claimed that more errors occurred as the numbers increased i n magnitude. Kieran's (1981b) r e s u l t s i n d i c a t e d that magnitude of numbers and complexity of the equation interacted. P e t t i t o (1979) ind i c a t e d that the structure of the equation i n terms of the v a r i a b l e placement and the numerical magnitude had an i n t e r a c t i v e e f f e c t . These findings seemed to ind i c a t e the importance of equation complexity, s t r u c t u r a l format, and numerical magnitude i n so l v i n g equations. The following research question was posed to investigate these i n t e r a c t i o n s : 5 . Do the factors of equation complexity, s t r u c t u r a l format, and numerical magnitude i n t e r a c t i n terms of the nature and frequency of the errors produced? The e f f e c t s of equation complexity (one-, two-, and multi-step), s t r u c t u r a l format (e.g. ax =b, x a=b, b=ax, b=x a), and numerical magnitude ( I n | < 2 0 , 2 0 < I n | < 5 0 , 50<|n|<100, and 100<|n|< 1 0 0 0 ) were explored i n terms of the mean number of e r r o r s . Tables are presented i n Appendix D, Part 5 which d e t a i l complexity, structure, and magnitude err o r means as well as the i n t e r a c t i o n e f f e c t s among these three aspects of equation construction. It was expected that an increase i n equation complexity would increase the average number of errors made. This was not always the case. In general, the e f f e c t of equation complexity on errors made i n s o l v i n g equations was i n the expected d i r e c t i o n ( i e . the more complex, the more errors made). For the most part, equations containing smaller numbers had a lower mean erro r rate than those containing larger numbers. However-, the mean number of errors i n the f i n a l answers and rule-based sign errors had an unexpected inverse r e l a t i o n s h i p as indicated i n the following table. Table 8 The Relationship of Numerical Magnitude to F i n a l Answers and Rule-Based Sign Errors Numerical Magnitude Mean Number of Errors for F i n a l Answers Rule-Based Signs Nl N2 N3 N4 4.28 3 . 44 3 . 3 3 2.78 0 . 64 0 . 61 0.44 0.33 In each case equations containing larger numbers had fewer e r r o r s . These findings seemed to lend support to Booth's (1981) contention that larger numbers focused attention on the process .used, and hence fewer errors were made. Another reason for t h i s reduction of errors may be found i n the f a c t that students were more l i k e l y to write out t h e i r arithmetic work when faced with numbers of larger magnitude. This was p a r t i c u l a r l y noticeable i n low achievers, who often d i d not write down t h e i r arithmetic c a l c u l a t i o n s . There were two main aspects of the structure which were investigated: placement of the v a r i a b l e with respect to the e q u a l i t y symbol and placement of the v a r i a b l e with respect to the constants i n the equation. The f i r s t aspect compared mean number of errors when the variable(s) was on the l e f t of the e q u a l i t y symbol (SI & S2) with variable(s) on the r i g h t of the e q u a l i t y symbol (S3 & S4). The second aspect involved analyzing d i f f e r e n c e s between the mean number of errors when the variable(s) was to the r i g h t of the constants (SI & S3) and of variable(s) to the l e f t of the constants (S2 & S4). The f i r s t aspect of structure showed s i g n i f i c a n t r e s u l t s on computation addition/subtraction e r r o r s . S u r p r i s i n g l y , students made more errors when the variable was to the l e f t of the e q u a l i t y symbol (SI & S2) than when i t was on the r i g h t of the e q u a l i t y symbol (S3 & S4). Analysis of the c o r r e l a t i o n matrices of complexity/structure/numeric i n t e r a c t i o n s d i s c l o s e d several noteworthy fin d i n g s . I n t e r c o r r e l a t i o n s between each of these variables and each error type were explored. Within the 4-by-4 numeric/structure g r i d , 13 of the 16 i n t e r a c t i o n s were highly s i g n i f i c a n t (p < 0.001) for f i n a l answers. S i g n i f i c a n t c o r r e l a t i o n s (p < 0.001) between numerical magnitude and s t r u c t u r a l format were found fo r procedural errors of l i k e terms (11 of the 16 i n t e r a c t i o n s ) , the a d d i t i o n property of equality (11 out of the 16 i n t e r a c t i o n s ) , and resultant errors i n v o l v i n g incomplete solutions (12 out of the 16 i n t e r a c t i o n s ) . No other such blocks of 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 for e i t h e r complexity/structure or complexity/numeric i n t e r a c t i o n s . These r e s u l t s supported the continued i n c l u s i o n of the 4-by-4 numerical-magnitude-by-structural-format g r i d within each l e v e l of equation complexity. For students experiencing d i f f i c u l t i e s with combining l i k e terms, applying the addition property of equality, or completing solutions, i n v e s t i g a t i o n of equations with varied s t r u c t u r a l formats and numerical 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 provide a f i r m foundation f o r the nature of equations which may be at the root of these d i f f i c u l t i e s . Summary o f R e s u l t s The r e s u l t s of the study ind i c a t e d that a number of the errors which were c h a r a c t e r i s t i c of students on the diagnostic instrument were rel a t e d to achievement l e v e l . These re l a t i o n s h i p s were most evident i n f i n a l answers and incomplete solutions on the equation portion of the diagnostic c h e c k l i s t and i n many of the conceptions held by students regarding algebra ( i d e n t i f y i n g v a r i a b l e , explaining d i v i s i o n i n expressions, explaining the meaning of equations, and the vocabulary associated with inverses and i d e n t i t y elements). The concepts and vocabulary portions of the instrument evidenced 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 equations portion of the instrument. These findings supported the work of K r u t e t s k i i (1976) by i n d i c a t i n g that high achievers were able to explain the meaning of equations better than low achievers, and t h i s a b i l i t y r e l a t e d to high achievers' performance on f i n a l answers i n solv i n g equations on the instrument. Approximately one quarter of the subjects exhibited systematic e r r o r s . Most of these were Grade 9 and Grade 10 students. The four systematic e r r o r types found (sign: addition and subtraction, addition property of equal i t y , l i k e terms, and computation: addition and subtraction) were also found to be common er r o r s . These findings supported the work of Anderson (1982) . Investigation of the r e l a t i o n s h i p of the diagnostic instrument to measures of achievement i n mathematics revealed some i n t e r e s t i n g r e s u l t s . Six types of errors which occurred i n the equations portion of the instrument seemed r e l a t e d to some of the mathematics achievement vari a b l e s . However, twelve of the concepts and vocabulary 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 of the mathematics achievement v a r i a b l e s . This f i n d i n g seems to support K r u t e t s k i i ' s (1976) contention that i t i s the a b i l i t y to explain mathematical ideas without performing the procedures which best distinguishes high achievers from low achievers i n mathematics. Equation complexity, s t r u c t u r a l format, and numerical magnitude 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 e r r o r s . This supported the previous research of Booth (1981), Kieran (1983) and P e t t i t o (1979). 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 between numerical magnitude and s t r u c t u r a l magnitude which supported t h e i r continued i n c l u s i o n at each l e v e l of equation complexity i n the diagnostic instrument. CHAPTER V DISCUSSION The discussion of the study reviews the procedural design used i n the development and v a l i d a t i o n of the Diagnostic Checklist for Algebra. This i s followed by a ra t i o n a l e f o r use of the c h e c k l i s t i n diagnosis. An assessment of the c h e c k l i s t i n terms of r e l i a b i l i t y , v a l i d i t y , and diagnostic u t i l i t y i s then presented. 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 of the study are discussed and suggestions are provided f o r the development of future diagnostic instruments i n other areas of mathematics. Review of Procedural Design The present study res u l t e d from the need f or a diagnostic instrument at the beginning stages of solving algebraic equations. K r u t e t s k i i (1976) stressed the importance of i d e n t i f y i n g e a r l y erroneous conceptions i n algebra before they became ingrained procedures f a r more d i f f i c u l t to eliminate. Some of these erroneous conceptions involve overgeneralizing rules learned i n arithmetic (Matz, 1980), erroneous conceptions of equality and v a r i a b l e (Kieran, 1983; Wagner, 1981a), simple arithmetic errors (Roberts, 1968), sign errors (Tatsuoka et a l . , 1980), copying and random errors, or incomplete solutions (Englehardt & Wiebe, 1981). There was a need f o r a diagnostic instrument that would i d e n t i f y these and other errors i n s o l v i n g l i n e a r equations i n one va r i a b l e . Errors tend to be based on "the s p e c i f i c content of the concepts and equations presented to students. It was therefore c r i t i c a l that the content of the diagnostic instrument accurately r e f l e c t e d the scope and sequence of beginning algebra c u r r i c u l a . In Phase 1 of the study, algebra texts were s c r u t i n i z e d and twelve equation types selected which represented the order and content of those texts. One hundred ninety-two equations were constructed (12 equation types by 4 s t r u c t u r a l format by 4 numerical magnitude) for possible i n c l u s i o n i n the instrument (See Appendix B, Part 1). These equations c o n t r o l l e d f o r equation complexity, s t r u c t u r a l format, numerical magnitude, and sign placement (Pettito, 1979; Tatsuoka et a l . , 1980) . Phase 2 of the study involved the development of an e r r o r -c a tegorization scheme based on the theories of Matz (1980) and Bernard & Bright (1982) . This e r r o r - c a t e g o r i z a t i o n scheme brought together the s i g n i f i c a n t e r r o r categories i d e n t i f i e d by researchers i n the learning of algebra over the past s i x decades. Errors were structured into the general categories of Conceptual, Procedural, and Resultant (Carry et a l . , 1976; Engelhardt & Wiebe, 1981; Booth, 1981) . The err o r - c a t e g o r i z a t i o n scheme formed one of the cornerstones f o r the construction of the diagnostic instrument. In Phase 3, eighty f i r s t - y e a r algebra students were tested using the Lankton to determine t h e i r achievement l e v e l i n algebra. Each subject then received one of four equation test-forms c o n s i s t i n g of 48 equations (See Appendix B, Part 2). Each test-form contained four equations from each of the twelve equation types. Answers were coded to indica t e the nature of the erro r committed. Errors were treated as d i s t r a c t o r s f o r the purpose of determining the d i f f i c u l t y l e v e l of each equation. An item analysis was performed to determine each item's p-value, c o r r e l a t i o n with the t e s t , and c o r r e l a t i o n with the Lankton. Items were selected which had a p o s i t i v e c o r r e l a t i o n with the Lankton, a p o s i t i v e c o r r e l a t i o n with the t o t a l t e s t score, whose d i s t r a c t o r s (errors) had a negative c o r r e l a t i o n with the tes t and whose p-value was cl o s e s t to 0.5. Each element of the 4-by-4 numerical-magnitude-by-structural-format g r i d was included at each l e v e l of equation complexity, due to the 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 numerical magnitude and s t r u c t u r a l format (See Appendix B, Part 3). The f i n a l 48 equations selected are l i s t e d i n Appendix B, Part 4. Phase 4 of the present study was designed to ensure that the di r e c t i o n s and tasks within the diagnostic instrument were c l e a r . The wording and format changes needed to achieve t h i s purpose are included i n Appendix C. It was also of i n t e r e s t to investigate the differences i n error patterns which might occur between high-and low-achieving students. The preliminary d r a f t of the Diagnostic Checklist for Algebra was used to invest i g a t e the errors of two high-achieving and two low-achieving Grade 9 students. Low achievers made many more errors than high achievers. The errors of high achievers p r i m a r i l y involved computational and sign errors. Low achievers made more sign errors i n v o l v i n g copying mistakes than d i d high achievers. However, the two low achievers ind i c a t e d t h i s was normal f o r them. An examination of protocols ind i c a t e d that there was no appreciable increase i n the number of copying errors made during t e s t i n g and that copying errors were evenly dispersed. Low achievers made more procedural e r r o r s than conceptual e r r o r s and made f a r more of these types of e r r o r s than d i d high achievers. Low achievers took over 90 minutes to complete the e n t i r e t e s t i n g , while high achievers took s l i g h t l y l e s s than 60 minutes. R e s u l t s f o r numerical magnitude i n d i c a t e d that low achievers had d i f f i c u l t y at a l l l e v e l s , while high achievers experienced d i f f i c u l t y only on equations of l a r g e numerical magnitude (N4: 100<|n|<1000). Results of s t r u c t u r a l format suggested that low achievers experienced 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 to the l e f t of the constants) while high 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 to the r i g h t of both the e q u a l i t y symbol and co n s t a n t s ) . The e f f e c t s of equation complexity, numerical magnitude, and s t r u c t u r a l format overshadowed the concern f o r time l i m i t a t i o n s and n e c e s s i t a t e d the i n c l u s i o n of 48 equations 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 of the f i n a l phase of the study was f o u r f o l d : 1) i n v e s t i g a t i o n of the e f f e c t s of numerical 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 of the nature and frequency of e r r o r s made at d i f f e r e n t achievement l e v e l s ; 3) examination of i n t e r r e l a t i o n s h i p s of e r r o r s w i t h demographic v a r i a b l e s and concomitant measures; 4) comparison of the conceptual understandings e x h i b i t e d and the e q u a t i o n - s o l v i n g e r r o r s made on the d i a g n o s t i c instrument. A l l 452 academic Grade 8, 9 and 10 students i n a B r i t i s h Columbia j u n i o r high school were given the Lankton to determine t h e i r a l g e b r a i c achievement l e v e l . Subjects were then assigned to one of 18 c e l l s (3 achievement l e v e l s by 3 grade l e v e l s by 2 genders). The s i z e of these 18 c e l l s ranged from 6 to 51. Two subjects were randomly selected from each c e l l f o r diagnostic interviews. These 36 subjects received a group administration of the CSMS to evaluate t h e i r understanding of va r i a b l e . Then each subject was i n d i v i d u a l l y interviewed and video-taped. These interviews took approximately 6 weeks to complete. The r e s u l t s of the interviews ind i c a t e d that the diagnostic c h e c k l i s t provided i n s i g h t s into the nature of errors committed by beginning algebra students. The reasons f o r using these p a r t i c u l a r procedures i n the design of the diagnostic instrument and the diagnostic implications of the res u l t s obtained i n the f i n a l phase of the study are discussed i n d e t a i l i n the next sections. R a t i o n a l e f o r t h e Diagnostic Checklist for Algebra I t i s important to d i s t i n g u i s h the Diagnostic Checklist for Algebra from other diagnostic t e s t s or instruments. The two parts of the c h e c k l i s t would appear to function as formative and summative aspects of beginning algebraic t e s t i n g . Such i s not the case. The purpose of the diagnostic c h e c k l i s t i s to look at the errors students make i n solv i n g l i n e a r equations i n one var i a b l e and to determine how these errors r e l a t e to the erroneous algebraic conceptions which a student holds. Previous error -analyses have not 'controlled the d i f f i c u l t y l e v e l of questions and have used post-facto analysis of paper-and-pencil t e s t s or post-facto interviews to determine the nature of the errors made. The Diagnostic Checklist for Algebra provides an innovative form of instrumentation enabling the 101 i n v e s t i g a t o r to explore errors as they occur. The c h e c k l i s t u t i l i z e s the strengths of both e r r o r - a n a l y s i s and semi-structured interviews, thus permitting in-depth questioning of the student to determine the exact cause of a p a r t i c u l a r error, as well as providing a v i s u a l matrix of errors for the purpose of remedial i n s t r u c t i o n . Errors may be influenced by the d i f f i c u l t y l e v e l of a question. As noted by P e t t i t o ( 1 9 7 9 ) , items which were too d i f f i c u l t d i d not allow students to choose a procedure f o r solv i n g the equation. S i m i l a r l y , Booth & Hart ( 1 9 8 2 ) i n d i c a t e d that, for items which were too easy, students reverted to "child-methods" or t r i a l - a n d - e r r o r and d i d not use algebraic procedures. Since t h i s i n v e s t i g a t i o n focuses on the conceptual, procedural, and resultant errors found by previous researchers, mid - d i f f i c u l t y l e v e l items seemed to be most appropriate f o r i n v e s t i g a t i n g procedural e r r o r s . M i d - d i f f i c u l t y l e v e l items also seemed to be the best way to l i m i t e r r o r bias and to provide s t r u c t u r a l v a l i d i t y . A Spearman's rho of 0 . 9 0 was obtained when c o r r e l a t i n g the rank of an equation type from the textbooks with the rank obtained from the mean p-value f o r each equation type (Refer to Table 7 , p 8 6 ). Such a high c o r r e l a t i o n could not have been obtained from use of other than m i d - d i f f i c u l t y l e v e l items. This high c o r r e l a t i o n provides convincing evidence that the structure of the Diagnostic Checklist for Algebra i s compatible with the structure i n f i r s t -year algebra textbooks. Many of the errors found on the diagnostic c h e c k l i s t would have been missed on other forms of instrumentation. Some of these errors were found 102 due to the c a r e f u l construction of the complexity, structure, and numerical magnitudes of the equations; others were found due to the use of semi-structured interviews. The importance of systematic v a r i a t i o n of the equations presented to students was exemplified by one of the Grade 10 average achievers. She only made "borrowing" errors on complex equations (C3) which contained large numbers (N4), but made no such mistakes at any other complexity or numerical magnitude l e v e l . If these e x p l i c i t types of equations had not been present i n the c h e c k l i s t , such diagnosis, and the r e s u l t i n g remediation, may have not occurred, leaving a gap i n equation-s o l v i n g a b i l i t y f o r t h i s student. Likewise, the importance of the use of interviews was i l l u s t r a t e d by a high achieving Grade 10 student's apparent lack of conceptual understanding of l i k e terms. When asked i n the conceptual section to group l i k e terms, she seemed to have no conception of I t h e i r meaning. However, when confronted with equations i n which she had to combine l i k e terms she d i d so re a d i l y , and then commented that she had no idea why she had not been able to remember what they were previously. At the end of the interview, she asked to go back and re-do the conceptual section on l i k e terms. She demonstrated complete mastery, and then announced that i t was the procedures which t r i g g e r e d the meaning for her, rather than the name of the concept. Such insi g h t into the learning process of a student would have been impossible without semi-structured interview techniques and the i n c l u s i o n of p a r a l l e l conceptual and procedural errors within the instrument. I n i t i a l r e s u l t s from the use of the Diagnostic Checklist for Algebra 1 03 i n d i c a t e that i t i s a useful t o o l f o r diagnosis. It provides many insights into the learning processes of algebra and creates a fir m foundation upon which to base remedial i n s t r u c t i o n . Such i n s i g h t s also have implications f o r i n i t i a l i n s t r u c t i o n , so that the teacher i s aware of the common errors made by students i n beginning algebra and can provide examples i n which these errors can occur and be corrected at the i n i t i a l stages of learning. A s s e s s m e n t o f t h e Diagnostic Checklist for Algebra S t a t i s t i c a l P r o p e r t i e s Most previous algebraic t e s t s used for diagnostic purposes contained 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). While some provisions were made for content and face v a l i d i t y , concurrent and p r e d i c t i v e v a l i d i t y were almost always omitted and r e l i a b i l i t y was often unreported. The procedures used i n t h i s study were designed to address these weaknesses. V a l i d i t y Content, concurrent, face, 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 were esta b l i s h e d i n Phases 1, 2, 3, 4, and 5, res p e c t i v e l y . Phase 1 provided content v a l i d i t y of the diagnostic instrument by ensuring the conceptual portions were appropriate f o r f i r s t year students. Phase 2 improved face v a l i d i t y by presenting equations i n mean rank order of f i r s t year algebra texts and by u t i l i z i n g research-backed error categories with systematic v a r i a t i o n of numerical magnitude and s t r u c t u r a l format over equation complexity. Phase 3 provided concurrent v a l i d i t y f o r the equation portion of the instrument by comparing r e s u l t s with the Lankton, as well as s t r u c t u r a l v a l i d i t y by c o r r e l a t i n g the text and mean p-value rankings of the twelve equation types. Phase 4 improved face v a l i d i t y and re f i n e d the diagnostic instrument. Phase 5 provided concurrent v a l i d i t y f o r the conceptual portion of the instrument, as well as some measure of pr e d i c t i v e v a l i d i t y through comparisons with the c r i t e r i o n measures described above. By c r e a t i n g "outcome scores" (See Appendix D, Part 2) a c o r r e l a t i o n of 0 . 4 8 f o r the " t o t a l e r r o r score" on the diagnostic instrument and student grades was found (See Appendix D, Table D . 5 ) . While t h i s was not s u f f i c i e n t l y high to claim p r e d i c t i v e v a l i d i t y of the instrument, i t was higher than the p r e d i c t i v e v a l i d i t y found f o r the Lankton on the same population ( r = 0 . 3 5 ) . Because of t h i s low p r e d i c t i v e v a l i d i t y the diagnostic instrument should not be used to predict algebraic success or f a i l u r e , but would.best be used to provide a consistent, structured method fo r i d e n t i f y i n g students' e r r o r s . R e l i a b i l i t y Hoyt Estimates of R e l i a b i l i t y ranging from 0 . 8 2 to 0 . 9 6 were obtained i n Phase 3 f o r the equations used i n the f i n a l study. Cronbach's Alpha was ca l c u l a t e d on the f i n a l diagnostic instrument f o r the o v e r a l l "outcome" ( 0 . 8 3 ) , the conceptual portion ( 0 . 8 6 ) , and the equation portion ( 0 . 7 5 ) . When the 27 conceptual items were separately compared with the 2 1 equation err o r items an alpha of 0 . 6 5 was obtained. A s p l i t - h a l f r e l i a b i l i t y of 0 . 4 2 was obtained when comparing the 20 concept items to the 7 vocabulary items. A s p l i t - h a l f r e l i a b i l i t y c o e f f i c i e n t of 0 . 8 3 was obtained when comparing procedural errors to resultant e r r o r s . The conceptual and equation portions of the diagnostic instrument seemed to be measuring d i f f e r e n t aspects of algebraic achievement, as d i d the vocabulary and concept portions. These findings i n d i c a t e that there i s a need f o r making d i s t i n c t i o n s among concepts, vocabulary, and equation-s o l v i n g i n diagnosis. R e l i a b i l i t y r e s u l t s indicate that the Diagnostic Checklist for Algebra achieved these d i s t i n c t i o n s . D i a g n o s t i c C o n s i d e r a t i o n s In order f o r a c h e c k l i s t f o r l i n e a r equations i n one var i a b l e to have diagnostic u t i l i t y , i t must be able to i d e n t i f y systematic and common errors made by students i n solv i n g equations. The kinds of erroneous conceptions which students have regarding algebraic concepts must also be i d e n t i f i e d . Most importantly the instrument should do t h i s task i n an e f f i c i e n t manner. D i a g n o s t i c C a t e g o r i e s Several of the error categories used i n the diagnostic instrument occurred infrequently. Thus these error categories d i d not seem to contribute s i g n i f i c a n t l y to e i t h e r i n d i v i d u a l diagnosis or to group i n s t r u c t i o n a l concerns. The error categories i n the conceptual portion of the instrument which occurred with l i m i t e d frequency were the a b i l i t y to group expressions containing e i t h e r the addition, subtraction or d i v i s i o n symbols; the a b i l i t y to explain the meaning of addition and subtraction i n expressions containing blanks and variable s , as well as m u l t i p l i c a t i o n i n blank expressions; the a b i l i t y to i d e n t i f y and explain the meaning of the equ a l i t y symbol; and the vocabulary associated with c o e f f i c i e n t s . A l l other conceptual understandings were required f o r complete diagnosis. S i m i l a r l y , there were error categories included i n the solv i n g of equations which seemed of l i m i t e d diagnostic u t i l i t y , due to the l i m i t e d nature of t h e i r occurrence during the interviews. The procedural errors involved were annexation of zero, i d e n t i t y confusion, p a r t i a l d i s t r i b u t i v e order of operations, m u l t i p l i c a t i o n property of e q u a l i t y and c o e f f i c i e n t s . The resultant errors were computational basic facts and f a u l t y algorithms mechanical/perceptual, and random. A l l other procedural and resultant e r r o r categories were required f o r complete diagnosis. It would seem that a l l of the above conceptual, procedural, and resultant e r r o r types occurred so r a r e l y that they could be removed from the diagnostic instrument with no loss to the accuracy of diagnosis. Such el i m i n a t i o n might prove useful i n reducing administrative time. However, further confirmation from continued use of the diagnostic instrument i s recommended before such d e c i s i v e elimination occurs. S y s t e m a t i c and Common E r r o r s Systematic errors were defined as errors which occurred 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 the present study systematic e r r o r s included l i k e terms, the addition property of equality, and addition/subtraction errors with both sign and computation. Nine of the 36 subjects exhibited systematic e r r o r s . Two of the students who made computational addition and subtraction errors made these errors f o r very d i f f e r e n t reasons. One student had a systematic f a u l t y algorithm f o r both operations which involved regrouping. The other systematically chose 107 adding, i n s t e a d of s u b t r a c t i n g . These f i n d i n g s supported the work of Anderson ( 1 9 8 2 ) and re-emphasized the importance of i n t e r v i e w s i n diagnosis (Hart, 1 9 8 3 ; Sleeman, 1984c). The e r r o r s made by these two students would not have been i d e n t i f i e d by a paper-and-pencil t e s t . ' Common e r r o r s were d e f i n e d as systematic e r r o r s which occurred i n 5% or more of the sample (Anderson, 1982). A l l of the four systematic e r r o r s were a l s o common e r r o r s . In terms of the exact e r r o r s found i n Anderson's (1982) study, only the e r r o r s of ax -bx = (b-a)x, |b|>|a|) and ax +bx =abx were r e p l i c a t e d i n the present study. This may have been due to the f a c t t h a t q u a d r a t i c s had not been introduced to the students at the time of the i n t e r v i e w s . The four-systematic and common e r r o r s found i n t h i s study i n v o l v e d a l g e b r a i c " a d d i t i o n . " A l l of these 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, the f a i l u r e of any of these e r r o r s to c o r r e l a t e s i g n i f i c a n t l y with IQ, the Lankton or the CSMS i n d i c a t e s that these systematic and common e r r o r s may be a f u n c t i o n of i n s t r u c t i o n a l b i a s , r a t h e r than mathematical a b i l i t y or a p t i t u d e . Further i n v e s t i g a t i o n i s re q u i r e d i n t h i s area. E r r o r s R e l a t i n g t o P r e v i o u s R e s e a r c h The e a r l y f i n d i n g s of Wattawa (1927) and Pease (1929) as w e l l as the more recent work of Davis & Cooney (1978) and Anderson - (1982) reported that up t o o n e - t h i r d of the e r r o r s made i n algebra were a t t r i b u t a b l e to a r i t h m e t i c e r r o r s . - In the present study, approximately 32% of the e r r o r s made were computational. Kieran's ( 1 9 8 3 ) f i n d i n g that novices make twice 108 as many number fact errors as experts was r e p l i c a t e d . Addition/subtraction sign errors often occurred concomitantly with these computation errors, re -emphasizing the importance of f a c i l i t y with these two arithmetic operations f o r success i n algebra. The r e s u l t s further indicated that understanding 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 to explain these two operations i n concrete terms were highly c o r r e l a t e d with achievement i n algebra. These findings support the contention that students should u t i l i z e concrete materials to help them conceptualize.arithmetic operations at the elementary l e v e l (Sowder et a l . , 1 9 8 6 ) and that work with concrete materials may s t i l l be necessary at the secondary l e v e l . The majority of s i g n i f i c a n t findings i n t h i s study were r e l a t e d to algebraic concepts. The s i g n i f i c a n t findings f or the concepts of variable and e q u a l i t y supported the previous research findings of Kieran ( 1 9 8 3 ) and Rosnick & Clement ( 1 9 8 0 ) . While concrete materials may be needed at the secondary l e v e l to rein f o r c e some arithmetic concepts, as previously suggested by the findings of t h i s study, i t may be of even greater importance that adolescents use manipulative materials to gain a concept of equation, v a r i a b l e , the e q u a l i t y symbol and the procedures involved i n the s o l u t i o n of equations (Herscovics, 1 9 7 9 ; Kinach, 1 9 8 5 ) . This suggestion i s further supported by the s i g n i f i c a n t i n t e r a c t i o n of the a b i l i t y to explain the meaning of equations and the a b i l i t y to obtain correct f i n a l answers in s o l v i n g equations. The findings of the present study indicated that students d i d better 1 09 with numbers whose absolute value was between 50 and 100 on one-step equations ( C l ) . This may be due to the prevalence of these numbers i n textbooks or i t may be that these somewhat larger numbers cause students to focus on the process i f the process i s easy (Herscovics, 1979; Booth, 1980). While the number of errors increased with complexity (Booth & Hart, 1982), the fa c t that more errors occurred with larger numbers i n more complex equations (C2, C3) suggests that large numbers may i n t e r f e r e with students' a b i l i t y to formulate a procedure, when that procedure i s more complex i n nature (Pettito, 1979 ) . These two findings may.help to explain contradictory findings reported by other researchers regarding the e f f e c t of numerical magnitude on the equation-solving process. However, a d d i t i o n a l information regarding the e f f e c t of numerical magnitude on the equation-solving process, gained through the use of interviews, i n d i c a t e d that the reasons for the errors made when solv i n g e a s i e r equations may have been due to hand-calculations, rather than any true focus on procedures (Anderson, 1982) . While t h i s f i n d i n g requires further i n v e s t i g a t i o n , i t suggests that students should be required to write out solutions to easy questions, rather than being allowed to complete them mentally. Stronger students were able to explain both the meaning of an equation and how to solve i t , without a c t u a l l y f i n d i n g the s o l u t i o n (Davis & McKnight, 1979; Booth & Hart, 1982) . This c h a r a c t e r i s t i c of high achievers was f i r s t noted by K r u t e t s k i i (1976) and held across both age and grade l e v e l i n the present study. Student achievement i n algebra might be 1 1 0 improved by i n c l u s i o n of such exercises i n regular classwork, so that lower-achieving students p r a c t i c e the s k i l l s of explaining the meaning of equations and the processes needed for solu t i o n , without a c t u a l l y solving them. The s i g n i f i c a n t findings f o r the vocabulary associated with i d e n t i t y elements and inverses seemed to in d i c a t e that teachers should frequently include vocabulary i n t h e i r lessons and should emphasize the importance of i d e n t i t y elements and inverses i n the equation-solving process. The findings that only f i n a l answers and incomplete solutions 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 the diagnostic instrument were su p r i s i n g . Moreover, the fact that both systematic and common errors were found i n d i c a t e d that purely s t a t i s t i c a l analyses of information are not s u f f i c i e n t when diagnosing e r r o r s . The use of interviews revealed important diagnostic differences which were not s t a t i s t i c a l l y s i g n i f i c a n t . Interviews are recommended i n algebraic diagnosis (Hart, 1983) . Summary o f A s s e s s m e n t o f t h e C h e c k l i s t The findings of the present study indicated that the c h e c k l i s t had diagnostic usefulness and was v a l i d and r e l i a b l e . Four errors were i d e n t i f i e d as being both systematic and common. These four errors focused on concepts involved i n algebraic "addition." The combination of e r r o r -analysis with semi-structured interviews provided a us e f u l methodology for the diagnosis of errors made by students i n solving l i n e a r equations i n one va r i a b l e . 111 S i g n i f i c a n c e o f the Study The purpose of t h i s study was to develop a diagnostic instrument which would i d e n t i f y the nature and frequency of errors made by students i n so l v i n g l i n e a r equations i n one variable and to explore the i n t e r r e l a t i o n s h i p s among these e r r o r s . This study was s i g n i f i c a n t because i t was the f i r s t study of algebraic errors which es 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 to r e l a t e the conceptual understandings held by students to the actual procedural and resultant errors made when solving equations. The r e s u l t s of t h i s study provided diagnostic i n s i g h t s into the learning processes of beginning students i n algebra, a n d . i d e n t i f i e d systematic errors of i n d i v i d u a l students and common errors among students. The i d e n t i f i c a t i o n of such errors provides important diagnostic information which may be u t i l i z e d i n remediation, as well as classroom i n s t r u c t i o n . This study was the f i r s t study to combine e r r o r - a n a l y s i s with semi-structured interview techniques to investigate the diff e r e n c e s among the errors made by students of d i f f e r e n t achievement l e v e l s when solv i n g algebraic equations. Control of equation types and format, numerical magnitude, and var i a b l e placement with respect to the eq u a l i t y symbol and constants reduced the p o t e n t i a l f o r bias due to equation content. Many of the possible sources of bias inherent i n previous research on algebraic errors were avoided. The procedures followed f o r instrument development resu l t e d i n the crea t i o n of a diagnostic instrument which i s v a l i d , r e l i a b l e , and useful. Such techniques which address concerns f o r face, content, s t r u c t u r a l , 1 1 2 concurrent and p r e d i c t i v e v a l i d i t y must be incorporated i n future diagnostic instrument development. Methods are also provided i n t h i s study to address issues of r e l i a b i l i t y , which have previously been noted as a weakness i n diagnostic instrument development. An attempt was made to u t i l i z e both the p r a c t i c a l and t h e o r e t i c a l work on errors made by students i n solv i n g l i n e a r equations i n one va r i a b l e . The combination of semi-structured interviews with e r r o r analysis provided a useful methodology f o r i d e n t i f y i n g common and systematic errors i n algebra (Sowder et a l . , 1 9 8 6 ) . The r e s u l t s achieved i n t h i s study seemed to i n d i c a t e that the err o r - c a t e g o r i z a t i o n scheme used i n the diagnostic instrument was an e f f e c t i v e way of assessing errors committed by beginning students i n algebra. The matrix formulation allowed easy i d e n t i f i c a t i o n of systematic errors and the separation of equations by complexity allowed for easy i d e n t i f i c a t i o n of errors at each l e v e l of equation complexity. The four systematic and common errors found i n the study appeared to be a function of i n s t r u c t i o n ( l i k e terms, addition property of equality, addition/subtraction sign errors, and computational addition/subtraction e r r o r s ) . Some conceptual errors seem to be re l a t e d to age and mathematical exposure (explaining m u l t i p l i c a t i o n , using the equ a l i t y symbol, and cre a t i n g equations from words). However, the concept of va r i a b l e was not re l a t e d to e i t h e r age or mathematical exposure, supporting the contentions of Kieran ( 1 9 8 3 ) and Rosnick and Clement ( 1 9 8 0 ) . Other findings from previous research i n algebra which found support i n the present r e s u l t s were: the importance of expressions i n the equation-solving process (Anderson, 1 9 8 2 ; Davis & Cooney, 1 9 7 8 ) , the usefulness of semi-structured interviews f o r diagnostic purposes (Hart, 1 9 8 3 ; Booth & Hart, 1 9 8 2 ; Wagner et a l . , 1 9 8 4 ) , the importance of manipulatives i n algebra and diagnosis (Kinach, 1 9 8 5 ; Sowder et a l . , 1 9 8 6 ) , and the f a c t that high achievers seemed to have the a b i l i t y to explain equation-solving procedures without executing them as noted by K r u t e t s k i i ( 1 9 7 6 ) i n r e l a t i o n to s o l v i n g word problems. L i m i t a t i o n s o f t h e S t u d y Limitations of the study included s e l e c t i o n of subjects, interview techniques, content, numerical magnitude overlaps, and numerical magnitude/structural format i n t e r a c t i o n s . The use of volunteer subjects 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 r e s u l t s , p a r t i c u l a r l y since parental consent was required. Parents who give consent may have been more in t e r e s t e d i n the education of t h e i r children, and thus the sample might not be t r u l y representative of the general student population. G e n e r a l i z a b i l i t y may also be l i m i t e d by the small sample size u t i l i z i n g only one school, two male teachers, and one textbook used i n the teaching of f i r s t - y e a r algebra. Further l i m i t a t i o n s occurred due to the use of think-aloud methods. As previously mentioned, i t i s possible that thinking aloud techniques do not accurately r e p l i c a t e thought processes. Students do not tend to verbalize mathematical processes once the procedure has become automatic (Matz, 1 9 8 0 ) . The use of automatic procedures may have i n t e r e f e r e d with some students 1 a b i l i t y to accurately describe what they were thinking as they solved equations (Herscovics, 1979). The use of interviews may also have i n t e r f e r e d with r e t r i e v a l of concepts. However, most students i n the study i n d i c a t e d that having to "think-aloud" was not a problem. They also i n d i c a t e d that they seemed to make the same type of mistakes i n the interview that they made on t h e i r homework. Teacher interviews supported t h i s contention. The content of the developed diagnostic instrument r e f l e c t e d the scope and sequence of several currently used f i r s t - y e a r algebra tex t s . However, the usefulness of t h i s diagnostic c h e c k l i s t i s l i m i t e d by curriculum change. The content of the c h e c k l i s t w i l l need to be updated p e r i o d i c a l l y . The findings of the present study were also l i m i t e d by the fact that only integer c o e f f i c i e n t s and solutions were u t i l i z e d . While the l i m i t a t i o n of integers i n the equations was deliberate to allow f o r thorough i n v e s t i g a t i o n of sign and numerical magnitude errors i n equation-solving, many errors associated 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 the e n t i r e scope of f i r s t - y e a r algebra equation-solving processes was not explored. Cut-off values f o r numerical magnitude were based on previous research (Anderson, 1982; P e t t i t o , 1979). Some overlapping of numbers occurred between the second and t h i r d l e v e l s of numerical magnitude (N2 and N3). Such overlap caused some confusions as to the source of some students' computational e r r o r s . While the cause of errors was established through intensive questioning during the interviews, these overlaps d i d provide a l i m i t a t i o n i n the study. The format of the diagnostic instrument stressed the complexity of equations (one-step, two-step, multi-step). While t h i s sequencing of equations made i t easy to determine where a student was i n terms of curriculum, 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 e i t h e r numerical magnitude problems or s t r u c t u r a l format problems. Since the i n t e r a c t i o n e f f e c t of these two factors was s i g n i f i c a n t , t h i s d i f f i c u l t y provided another l i m i t a t i o n of t h i s study. S u g g e s t i o n s f o r F u t u r e R e s e a r c h Several studies need to be i n i t i a t e d which extend the parameters of the current research. F i r s t l y , the present study should be r e p l i c a t e d i n other schools to address concerns regarding the p o s s i b i l i t y that systematic and common errors may be instruction-based. This would also a s s i s t i n determining i f some erro r categories could be eliminated from the e r r o r -c a t e g o r i z a t i o n scheme to help "streamline" the instrument. Secondly, the techniques used to e s t a b l i s h concepts and equations with i n t e g r a l c o e f f i c i e n t s and solutions should be extended to concepts i n v o l v i n g f r a c t i o n s and equations containing r a t i o n a l c o e f f i c i e n t s and solutions. The importance of the choice of m i d - d i f f i c u l t y l e v e l items for such an instrument cannot be over-stressed, so that the i d e n t i f i e d errors w i l l be a function of understanding and not of an i n a b i l i t y to formulate a solu t i o n ( P e t t i t o , 1979) or because a student could solve the equation through non-algebraic means (Booth & Hart, 1982). Thirdly, diagnostic instruments need to be developed which focus on quadratic equations and systems of equations 1 1 6 to provide i n s i g h t into the errors which occur i n l a t e r work with algebraic equations. In future research on algebraic errors, the number of incomplete solutions could be u t i l i z e d to provide a measure of mastery for each equation type. While the present study recorded these errors, i t i s suggested that once a student cannot attempt two or more of the questions f o r each equation type, the interview be terminated. This would provide an i n d i c a t i o n of the mastery l e v e l of the student, shorten the interview time and i n most cases would not l i m i t the diagnosis. While the findings f or the i n t e r a c t i o n of numerical magnititude and s t r u c t u r a l format were s i g n i f i c a n t , no d e f i n i t i v e patterns were found which would lead to p a r t i c u l a r remedial techniques. It i s suggested that a computer programme be developed to enable teachers to enter the numeric/structure r e s u l t s from the diagnostic instrument to determine e f f e c t s f or each student. The present formatting of the instrument makes t h i s task onerous. Future diagnostic instruments should attempt to use non-overlapping sets of numerical d i f f i c u l t y ( i e . 0<|n|<20, 20<|n|<50, 50<|n|<100, 100<|n|<1000) so. that i n d i v i d u a l problem areas could be more r e a d i l y i d e n t i f i e d . Care w i l l need to be taken to obtain meaningful equations at each l e v e l of numerical magnitude. In the present study, r e l i a n c e on c a r e f u l interview techniques was required to determine the discrepancies. For example, a student made consistent errors on equations i n v o l v i n g numbers between 20 and 50 . However, through c a r e f u l questioning, i t was 1 1 7 revealed that these p r i m a r i l y involved errors with m u l t i p l i c a t i o n facts for four and would have been better described by numbers between 0 and 20. Mestre & Gerace (1986) pointed out that students viewed algebra as a rule-based, rather than concept-based, d i s c i p l i n e . Other researchers noted t h i s same phenomenon i n the way that students tr e a t algebraic equations as something to be manipulated, rather than representing something meaningful (Herscovics, 1979; S i f o r d , 1981). The findings of the present study seem to support these contentions and stress the importance of teaching the equation-solving process i n ways which emphasize i t s conceptual underpinnings. As noted by Sleeman (1984a) the d i f f i c u l t i e s encountered by students i n learning algebra have been greatly under-estimated. While, diagnosis i s important, future studies should focus on i n s t r u c t i o n a l methods which avoid the formation of such erroneous conceptions. When developing a diagnostic instrument which involves interviews, i t i s important to t e s t the instrument on subjects to ensure the a d m i n i s t r a b i l i t y of the instrument. Care needs to be taken to ensure that think-aloud techniques and interview questions do not i n t e r f e r e with the subject's a b i l i t y to solve equations. Guidelines should be provided to avoid teaching i n the diagnostic s e t t i n g (Wagner et a l . , 1984), which often occurs when using interviews (Herscovics, 1979). The processes used f o r formulating a diagnostic instrument for l i n e a r equations i n one v a r i a b l e could be applied to other areas of mathematics. The key aspects of t h i s formulation involved concerns for v a l i d i t y and r e l i a b i l i t y and a balance between the p r a c t i c a l and t h e o r e t i c a l underpinnings of teaching and research were exemplified i n the development of the diagnostic instrument. The c a r e f u l and systematic approach used i n the Diagnostic Checklist for Algebra provides a model f o r the development of other diagnostic t o o l s using error a n a l y s i s . It i s only by u t i l i z i n g such a systematic approach that progress can be made i n the diagnosis of students' errors i n mathematics (Bernard & Bright, 1982) . 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New York, McGraw-Hill. 1 26 A P P E N D I X A P h a s e 2 E r r o r C a t e g o r i z a t i o n Scheme D e v e l o p m e n t 1 27 The purpose of t h i s appendix i s to present a more d e t a i l e d discussion 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 Checklist for Algebra was developed. A summary of research findings for each er r o r included 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 presented at the end of the discussion i n tabular form. Research i n algebra was reviewed and a l i s t was created of the author(s), dates, and the d i f f e r e n t errors i d e n t i f i e d within the research f i n d i n g s . Many of the errors found had d i f f e r e n t names i n d i f f e r e n t studies, but were the same error, which supports the idea that much of the research i n algebra has been plagued by semantic, rather than conceptual, di f f e r e n c e s as noted by Firestone ( 1 9 8 7 ) . Those errors which were not l i s t e d by three or more previous researchers were eliminated, because such errors may have been s p e c i f i c to one p a r t i c u l a r piece of research, due to the nature of the equations used within that study. The l i s t of remaining errors was formulated into an e r r o r categorization scheme based on the work of Carry et a l . ( 1 9 8 0 ) , Booth ( 1 9 8 1 ) , and Bernard & Bright ( 1 9 8 2 ) . These researchers suggested that diagnosis should take into account previous knowledge, processes used, and r e s u l t s obtained. Such suggestions seemed to imply that a diagnostic instrument for beginning algebra should c l a s s i f y errors into three general types: conceptual, procedural, and r e s u l t a n t . Each error found by previous researchers was assigned to one of these three main c l a s s i f i c a t i o n s i n the e r r o r -c a t e g o r i z a t i o n scheme. The three main categories - conceptual, procedural, resultant - were 1 28 defined i n the following manner. Conceptual errors are errors i n understanding of the concepts and vocabulary associated with f i r s t - y e a r algebra. Procedural errors are errors made i n attempting to apply algebraic procedures to the equation-solving process. Resultant errors are errors involved i n arithmetic computation, signs, or copying. By basing the diagnostic instrument on a conceptual-procedural-resultant model, further i n s i g h t into a student's a b i l i t y to solve l i n e a r equations i n one variable should be provided because the i n t e r r e l a t i o n s h i p s between errors occurring i n these three areas can be explored. A lack of understanding of vocabulary and algebraic concepts contributes to students' errors i n algebra (Matz, 1980). M i s i n t e r p r e t a t i o n of the meaning of vari a b l e , the eq u a l i t y symbol, expressions, and equations have been shown to i n t e r f e r e with students' a b i l i t y to understand algebraic i n s t r u c t i o n (Anderson, 1982; Herscovics, 1980; Kieran, 1981a, 1983; Si f o r d , 1981; Wagner, 1981a). Hence, assessment of basic algebraic concepts and vocabulary i s necessary f o r diagnosis. M i s - a p p l i c a t i o n of algebraic procedures may account f o r many of the errors which occur i n algebra (Matz, 1980) . Faulty algorithms, i n e f f i c i e n t procedures, and f a i l u r e to make the t r a n s i t i o n from mental arithmetic to algebraic operations r e s u l t i n procedural errors i n l a t e r algebraic work (Davis & Cooney, 1978; D r i s c o l l , 1982; Hart, 1981; Herscovics, 1979; Movshovitz-Hadar et a l . , 1987; Sleeman, 1 9 8 4 a , 1984b, 1 9 8 4 c ) . Careful s c r u t i n y and i d e n t i f i c a t i o n of the procedural errors made by students provide useful diagnostic information (Davis & Cooney, 1978). As such, 1 29 means for i d e n t i f y i n g procedural e rrors should be inc luded i n a d iagnost i c instrument. Some e r r o r s found i n algebra are not based on e i t h e r a lgebra ic concepts or procedures . These e r r o r s are c a l l e d resu l tant e r r o r s , and they inc lude computational e r r o r s , s ign e r r o r s , or h a b i t u a l copying mistakes . Resultant e r r o r s may i n t e r f e r e with the understanding of a lgebra ic concepts and the demonstration of a lgebra ic procedures (Englehardt, 1977; Englehardt & Wiebe, 1981; Sleeman, 1984c; Tatsuoka et a l . , 1980). Hence, a means for i d e n t i f y i n g re su l tant e r r o r s i s e s s e n t i a l i n a d iagnos t i c instrument. It i s important to d i s t i n g u i s h among conceptual , procedura l , and r e s u l t a n t e r r o r s for complete diagnosis i n a lgebra . I d e n t i c a l i n c o r r e c t answers made by d i f f e r e n t students may be due to an i n a b i l i t y to understand an a l g e b r a i c concept, to apply an a lgebra ic procedure, or to obtain an accurate copy, s ign , or computation r e s u l t . For example, i n s o l v i n g 7n-l=48, two students may obtain the i n c o r r e c t r e s u l t of 8 for two very d i f f e r e n t reasons. One may make a l i k e - t e r m e r r o r , i n d i c a t i n g that 6n=48, n=8; while the other may i n d i c a t e that 7n=49, then state n=8 because a bas ic fac t e r r o r i n d i v i s i o n occurred . It i s only by d i s t i n g u i s h i n g among conceptual , p r o c e d u r a l , and resu l tant errors that one can understand t h e i r i n t e r r e l a t i o n s h i p s . Such understanding i s c r i t i c a l f or thorough d iagnos i s . The e r r o r - c a t e g o r i z a t i o n scheme developed for t h i s study i s presented i n Table A . l . Included with each e r r o r i s a l i s t i n g of the i n v e s t i g a t o r s who found that e r r o r i n t h e i r research . This tabu lar presentat ion represents a summary of the research f indings i n algebra for the past s i x t y - s i x years . 1 30 The columns i n that table provide a summary of the errors i d e n t i f i e d by i n d i v i d u a l i n v e s t i g a t o r s . The rows provide a summary of the frequency with which a p a r t i c u l a r e r r o r was i d e n t i f i e d by i n v e s t i g a t o r s . The table reveals 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 . From the .period of 1925-1975 the major thrust of the errors found are resultant i n nature (33 out of 46). For the period of 1975-1982 when concern for diagnosis and remediation became paramount i n the l i t e r a t u r e , almost equal concern f o r conceptual, procedural, and resultant errors can be seen (49, 43, 48, r e s p e c t i v e l y , out of 140). The work from 1983.until the present seems to be focussing more on conceptual and procedural errors rather than resultant errors (16, 8, compared with 2/ respectively, out of 26). The developed e r r o r - c a t e g o r i z a t i o n scheme represents a synopsis of the research done i n algebra over the past seven decades. It i s an attempt to bring together the i d e n t i f i e d errors under one semantic umbrella, so that future research may focus on diagnosis leading to remediation, rather than diagnosis to obtain yet another error-category. Tfeble A.l Suimary of Errors in Algebra Types of Errors Investigators Variable X X X X X X X X X X X X Expression X x X X x X X X X X X X X X X X Equality X X X X X Equation X X X X X X X X X X. X Coefficient x X X X X X Lite Terms X X X X X X X X X X X X Inverses X X X Identity Elements X X X X PROCEDURAL Zero Annexation X X X X Identity Confusion X X X Like Terms (Conjoining) X X X x X X X X X X X X X Partial Distributive X X X X X X X X X X X X X X Order of Operations X X X X X + property of = X x X X X X X x property of » X X X X X X numeric coefficients X' x X X X X X RESULTAOT-Camputational add/subtract X. X X X X X X X tnult iply /divide X X X X X X X X X basic facts X X X X X X faulty algorithm X X X X X X wrong operation X X X X X X X X X X X X RESULTANT-Sign add/subtract X X X x }C multiply/divide X. X X X f< rule-based X X X X distributive X X X X X X RESULTANT-Other mechanical/perceptual X X X X X X X X X random X X X X X X inccnplete X X X X X 1 32 A P P E N D I X B P h a s e 3 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 , a n d S e l e c t i o n P a r t 1: T w e l v e E q u a t i o n T y p e s a n d t h e R u l e s f o r t h e i r C o n s t r u c t i o n P a r t 2: F o u r T e s t - F o r m s P a r t 3: I n t e r a c t i o n E f f e c t s P a r t 4: F i n a l E q u a t i o n s 1 33 The purpose of t h i s appendix i s to present a d e t a i l e d d i s c u s s i o n of the procedures and analyses used i n the c o n s t r u c t i o n and s e l e c t i o n of the f i n a l equations f o r the Diagnostic Checklist for Algebra. Part 1 presents each of the twelve equation types found i n Phase 1 of the study, together with a 4-by-4 numerical-magnitude-by-structural-format g r i d . The r u l e s f o r c o n s t r u c t i n g each equation w i t h i n the s i x t e e n c e l l s are presented. Part 2 contains copies of the four test-forms used i n Phase 3. 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 complexity of the equations, t h e i r s t r u c t u r a l format, and the magnitude of the numbers used i n the equations. Part 4 presents a t a b u l a r l i s t i n g of the f i n a l equations which were s e l e c t e d , together with 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 s e l e c t i o n . P a r t 1 T w e l v e E q u a t i o n T y p e a n d t h e R u l e s f o r t h e i r C o n s t r u c t i o n E Q U A T I O N T Y P E 1 a + x = b NUMERICAL MAGNITUDE S1 S T R U C T U R A L S2 FORMAT S3 S4 N1 |n|< 20 x t 15= 8 7+x-=-10 -1i»=x-t- 6 -5= - 9 +x N2 20<jn|< 50 x+ 2 9 = 37 -27+x=23 -31=x+ -UZ <+5= 23+x N3 50<|n|< 100 x+ -57= i.1 - 5 'Mx:-- -75 09=x+ 59 £»1=-58+x N4 100<|n|<1000 x+- -352- -701 BC0+x=6Q9 25S=x+- 3*4'+ -<*<*£» = 392+x L A T I M SQUARE AND S I G N / M A G N I T U D E R U L E S FOR E Q U A T I O N T Y P E F i r s t N u m b e r O t h e r N u m b e r a b c d a = P o s i t i v e L a r g e + b c d 3 b = P o s i t i v e S m a l l -c d a b ' c = N e g a t i v e L a r q e d a b c d = N e g a t i \ / e S m a l l -CO EQUATION TYPE 2 x - a = b NUMERICAL MAGNITUDE STRUCTURAL FORMAT S1 S2 S3 S4 N1 | n|< 20 x- 13= -3 7-x= 12 -1G=x- -3 -5= 7-x N 2 20<| n]< 50 x - 19= 25 _'47_x= -27 -22=x- 39 26= -21 - x N 3 50< |n|< 100 x- -85= -6 -39-x= 58 01=x—59 66= 96-x N 4 100< |n|<1000 x—121=230 339-x=-17't tt6f.=x-536 -705=-380-x LATIN SQUARE ANO SIGN/MAGNITUDE RULES FOR EQUATION TYPE F i r s t Number Other Number a b c d n - P u s i t i v e Larqn -b c d a b=Positivp Small + c d a b (^Negative Lnrqe -d a b c d=Neqative Small + CO EQUATION TYPE 3 ax = b NUMERICAL MAGNITUDE S1 STRUCTURAL S2 FORMAT S3 S4 N1 |n|< 20 x ( 5)= 20 2x= -18 -16=x( <•) - 18= -3x N2 20<!.nj< 50 x ( 8)= -kB -1HX= kZ -kB=x( -6) kS= 7x ' N3 50<|n|< 100 x ( -7) = 63 -9x= - 5k 100=x( 10) * 56= -8x • N4 100<|n|<1000 x ( - <»5)=-•k35 19x= 3-30 976=x(-•61) • -1DOO=200x LATIN SQUARE AND SIGN/MAGNITUDE RULES FOR EQUATION TYPE Fi r s t ; Number Other Number a b c" d Smal l + b c d 3 b ^ o s i t i v e Smal l _ c d a b c=fteaative Larqe + d a b c "^Neaative La rue -* Due to 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 had trj be s w i t c h e d to f i r s t number l a r g e . EQUATION TYPE U x / a = b NUMERICAL MAGNITUDE S1 S2 STRUCTURAL FORMAT S3 S4 N1 n|< 20 x/ 5= U 1 3 / x = - 3 - 7 = x / 2 -1= -7/x» N2 20<|n|< 50 x/ 8= - 6 -50/x= 5 - 1 1 = x / - 4 3= <o/x* N3 50<jnj< 100 x/ -10= 9 - 6 3 / x = - 7 9 = x / 8 20=-100/x* N4 100<|n|<1000 x/- 319=- -29 378 /x=5^ ^ • 0 = x / - •200 -19= 361/x» LATIN 5QUARE AND SIGN/MAGNITUDE RULES FOR EQUATION TYPE F i r s t Number O t h e r Number a b c d a= P o s i t i v e L a r q e + b c d a b = P o s i t i v e L a r g e -c d a b c= N e q a t i v e L a r g e + d a b c d = N e g a t i v e L a r g e -* Dun to 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 , this a q u a t i o n t y p e had t o be s w i t c h e d t o f i r s t number s m a l l . 1 3 SIGN/MAGNITUDE ASSIGNMENT OF EQUATION TYPES FOR EQUATION TYPES 5-12* S t r u c t u r a l L e v e l I n t e g e r S i g n s a b c d e M a g n i t u d e * 4 + + + + - L a r g e 3 + + + - + S m a l l 2 + + - - - L a r g e 1 + + - + + S m a l l 3 + - +' + + L a r g e 4 + - + - - S m a l l 1 + - - - + L a r g e 2 + - - + - S m a l l 2 - + + + - L a r g e 1 - + + - + S m a l l 4 - + - - - L a r g e 3 - + - + + S m a l l 3 - - + + + L a r g e 4 - - + - - S m a l l 1 - - - - + L a r g e 2 ' - + - S m a l l lation Types were assigned to the 16 sign p o s s i b i l i t i e s f o r * c so that subtraction 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 numerical l e v e l . Assignment of equation type was also made so that 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 of large/small f o r the 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 also had equal d i s t r i b u t i o n of signs of a, b, and c with more complex sign arrangements being assigned to smaller l e v e l s of numerical magnitude. ** For each of the Equation Types 5 - 12, there i s an assignment l i n e at the bottom of each g r i d . This assignment l i n e indicates which of the numerical c o e f f i c i e n t s (either a, b, c, d, or e) have been assigned to the large/small magnitudes as indicated i n the chart on t h i s page. EQUATION TYPE 5* ax + b = c NUMERICAL MAGNITUDE S1 S2 STRUCTURAL FORMAT S3 S4 N1 |n|< 20 -3x- 17= - a 15+-3x= 0 -13= -5x+ 2 -3= 16+ -1x N2 20<|n|< 50 -9x+ 23= 50 -23- x= -48 0=--29x- 29 28= -21- 7x N3 50<|n|< 100 17x+--100= -15 -77- -x= -86 93= 6x+ 33 61= -31+ 23x N4 100<|n|<1000 33x+ 445=--545 734+53x=--220 259= 14x-637 100=1D00+ 30x The assignment of large/small magnitude refers to b compared to c within Equation Type 5. EQUATION TYPE 6* ax + bx = c NUMERICAL MAGNITUDE S1 STRUCTURAL S2 FORMAT S3 S4 N1 |n|< 20 8x+ 9x= -17 -x- -5x= -12 18=-10x- -8x -16= -6x+ 2x N2 20<|n|< 50 33x+ -13x= -50 -15x- x= 48 -48= -5x- 7x 37= 22x+ 15x N3 50<|n|< 100 -83x+ 97x= Bk 45x- -46x= -91 88= 56x- 67x 96= 81x- 87x N4 100<|n|<1000 -370x+-•130x=-•1000 182x-i 168x=-196 225='l20x-105x 720=-•3S0x+560x * The assignment of large/small magnitude refers to a compared to b within Equation Type 6. EQUATION TYPE 7 * a x + b = c x NUMERICAL MAGNITUDE S1 S2 STRUCTURAL FORMAT S3 S4 N1 |n|< 20 2x+ 20= - 3 x - 2 0 - 3x= - 1 3 x 13x= 7 x - 18 - 7x= 16+ x N2 20<|n|< 50 x+-43=-2 3x - 3 5 - -x= - 6 x 50x= <+1x- - 3 6 - 2 8 x = 21+-49x N 3 50<|n|< 100 - 7 9 x + - 7 5 = - 5 4 x - 9 1 - 53x= - 6 0 x -83x= - 7 5 x - 56 93x=-87+ 74 x N4 100<|n|<1000 -15ux+782 : =10' lx 333-158x=--121x 124x=-- 2 7 6 x ~ -800 971x=420+985x * 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 a c o m p a r e d t o c w i t h i n E q u a t i o n Type 7 . EQUATION TYPE 8 * a ( x + b ) = c NUMERICAL MAGNITUDE STRUCTURAL FORMAT S1 S2 S3 S4 N1 |n|< 20 2 ( x - 1)= - 1 8 2( 5+x)= -a 16= 4 ( x + 3) 2 0 = - 5 ( - 1 - x ) N2 20<|n|< 50 & K x - - 5 ) = - 2 d 16 ( -7+x )= -48 46= 2 3 ( x + - ^ 1 ) 26=13( 3 - x ) N3 50<|n|< 100 - 7 ( x - 9)= 9 8 - 6 ( - 5 + x ) = - G 0 -72= - 8 ( x + 7 ) 57=19( - 2 - x ) N4 100<|n|<1000 - 1 7 ( x - - 2 D ) = - 2 0 4 -21 (21+x )=399 2 8 8 = - 1 2 ( x + - 2 4 ) - 4 3 5 = - 5 ( 1 2 D - x ) * 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 t h e p r o d u c t o f ab a s o p p o s e d t o c u i t h i n E q u a t i o n Type 8 . EQUATION TYPE 9* a x + b + c x + d = e NUMERICAL MAGNITUDE SI S2 STRUCTURAL FORMAT S3 S4 N1 |n|< 20 -4x+ -7+ -5x+ -5= 6 -3- x+ 8+ -4x= -10 11= -kx- -8- 3x+ 3 0= -7—12x—17- 2x N2 20<|nK_ 50 -11x+ 14+ 14x+-29=33 29- -9x+ 15+ 13x= 0 30=-13x- 12- -6x+ 7 -29= 9- 28x- -4~25x N3 50<|n|< 100 42x+--28+ -54x+-18=38 -35—27x+ 68+ -43x= -63 %= 49x—39- 31x+ 1 -62=-77- 2x—78- x N4 100<|n|<1000 151x+ 95+-188x+162=14 711-165x+--613+-•175x=-582 191=155x-146- 75x+-•303 -354=198-270x- 98-184x ThE assignment of large/small magnitude refers to b compared to d within Equation Type 9. For the first two levels of structural difficulty a<c and for the last two levels of structural difficulty a>c. EQUATION TYPE 1 0 * a x + b = c x + d NUMERICAL MAGNITUDE STRUCTURAL FORMAT S1 S2 S3 S4 N1 |n|< 20 7 x + 8 = - 5 x + 2 0 . - 7 - 3x= 9 - - x - 2 x - - 0 = 5 x - 6 4 + - 4 x = - 5 + 5 x N2 20<|n|< 50 2 1 X + - 3 3 = - 4 x + - 8 4 9 - - 2 7 x = 1 6 - 6x - 1 5 x - 11= - 4 0 x + 34 + . - 3 5 x = - 1 2 + - 3 D x N3 50<|n|< 100 - 7 5 x + 6 9 = 3 x + - 8 7 - 1 5 - - 2 0 x = 7 5 - _5x 2 4 x - 5 2 = 4 7 x - 4 0 8 9 + 5 5 x = 2 3 + 6 1x N4 100<|n|<1000 - 5 1 3 X + - 1 5 6 = - 4 6 3 x + - - 6 0 6 7 1 5 - 1 2 3 x = - 2 3 - - 1 1 8 x 9 7 x - 1 8 3 = 1 2 3 x - - 3 3 7 - 3 0 0 + - 1 6 7 x = - 1 0 0 U + - 2 3 7 x 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 d a s o p p o s e d t o b w i t h i n E q u a t i o n T y p e 1 0 . 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 a n d f o r t h e l a 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 . EQUATION TYPE 1 1 * a ( x + b ) = c x + d NUMERICAL MAGNITUDE S1 STRUCTURAL S2 FORMAT S3 S4 N1 |n|< 20 - 8 ( x - - 2 ) = - 9 x - - 1 2 ( - 5 + x ) = 9 - - x 4 x + 2 = - 3 ( x - - 4 ) - 4 + 1 2 x = - 8 ( - 2 + x ) N2 20<|n|< 50 1 3 ( x - 3 ) = - 2 x - 9 - 9 ( - 4 + x ) = 4 B — 3 x 1 7 x + 8= 7 ( x - S ) - 2 + - 1 8 x = - 4 + x ) N 3 50<|n|< 100 7 ( x — 1 4 ) = - 3 x - - 8 3 3 ( 3 + x ) = - 1 - - 8 x 5 2 x + - 1 6 = 1 5 ( x - 6 ) - 9 + 2 4 x = - y ( - 10 + x ) N 4 100<|n|<1000 - 2 4 G ( x - 4 ) = 5 4 x - - 8 4 - 5 ( 1 6 1 + x ) = 1 2 5 - 2 x 6 6 x + 4 4 = 2 6 ( x - 2 6 ) 2 6 + 1 9 6 x = 4 5 ( 14 + x ) T h e 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 T y p e 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 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 . EQUATION TYPE 12* a(x + b) = c(x + d) NUMERICAL MAGNITUDE S1 STRUCTURAL S2 FORMAT S3 S4 N1 |n|< 20 3(x+-2)= -2(x+ B) -2( 8-x)= 1( 2+x) -3(x+ 6)= 4 ( x - 1) -3 ( -2 --x)= 30 - 4 - x ) N2 20<|n|< 50 -13(x+-3)=-11(x+-1) -9( -1-x)= -6( 6+x) 10(x+--5)= 2Q(x- 2) 6(-6- -x)= 7( - 7 - x ) N3 50<|n|< JOO -9(x+ 8)= 8(x+-9) 10 ( 10-x)= -9(-9+x) -9(x+ '+)=-12(x- 5) -8(12--x)=-14(-- 3 - x ) N4 100<|n|<1000 100(x+-8)=-60(x+ 0) 25(- -24-x)=-15(48+x) 139(x- 1)=161(x--5) 1Mt( 6--x)=160( 5 -x ) * The assignment of large/small magnitude refers to d as opposed to b within Equation Type 12. For the f i r s t two levels of st r u c t u r a l d i f f i c u l t y a>c and for the l a s t two levels of structural d i f f i c u l t y a<.c. P a r t 2 F o u r T e s t - F o r m s 1 4 9 T E S T - F O R M 1 N A M E : 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! P A R T A - O N E - S T E P E Q U A T I O N S 1) x + 15 = 8 1) 2) 45 = 28 + x 2) 3) 89 = x + 59 3) 4) 800 + x = 609' 4) 5) -16 = x - -3 5) 6) -47 - x = -27 6) 7) x 85 = -6 7) 8) -705 = -380 - x 8) 9) 2x = 18 9) 10) -56 = 8x 10) 11) x(45) = -495 11) 12) -48 = x(-6) 12) 13) -1 = -7,'x 13) 14) -11 = x/-4 14) 15) -63/x = -7 15) 16) X/-31 '= -28 16) PART B - TWO-STEP EQUATIONS 17) -3x - 17 = -8 17) 18) 28 = -21 - 7x 18) 19) 93 = 6x + 33 19) 20) 734 + 53x = -220 20) 21) -x - -5x = -12 21) 22) 38x + -13x = -50 22) 23) 96 = 81x + -87x 23) 24) 225 = 120x -105x 24) 25) 2x + 20 = -3x 25) 26) -28x = 21 + -49x 26) 27) -83x = -75x - 56 27) 28) 333 - 158x = -121x 28) 29) 16 = 4(x + 3) 29) 30) 16(-7 + x) = -48 30) 31) -7(x - 9) = 98 31) 32) -435 = -5(120- x) 32) P A R T C - M U L T I - S T E P E Q U A T I O N S 33) -3 - x +.8 + -4x = -10 33) 34) -llx +. 14 + 14x + -29 = 33 34) 35) -64 = -77 - 2x 78 - x 35) 36) 191 = 155x - 146 - 75x + -303 36) 37) 4 + -4x - -5 + 5x 37) 38) -5x - 11 = -40x + 34 38) 39) -15 20x - 75 - -5x 39) 40) -513x + -156 - -463x + 606 40) 41) 4x + 2 = -3(x - -4) 41) 42) -9 (-4 + x) = 48 - -3x 42) 43) 7(x 14) - -3x 8 43) 44) 26 + 196x » 45 (14 + x) 44) 45) -3 (-2 -x) - 3 ( - 4 -x) 45) 46) 10(x + -5) = 20(x - 2) 46) 47) 10(10 - x) - -9(-9 + x) 47) 48) 100(x + -8) - -60(x + 0) 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) 7x + 8 - -5x + 20 37) 38) 38 + -35x = -12 + -30x 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) 43) 33(33 + x) - -1 - -8x 43) 44) -246(x - 4) - 54x 84 44) 45) 3(x + -2) = -2(x + 8) 45) 46) 6(-6 - x) = 7(-7 - x) 46) 47) -9(x + 4) = -12(x - 5) 47) 48) 25 (-24 - x) -15 (48 + x) 48) TEST-FORM 3 NAME: Read each equation c a r e f u l l y . Show a l l of your work i n the space at the ri g h t of the equation. Place your answer i n the c o r r e c t answer space provided i n the column at the r i g h t . You have the e n t i r e p e r i o d to complete the 46 equations. When you are f i n i s h e d and have checked your work, r a i s e your 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) -14 = x + 6 1) 2) -27 + x = 23 2) 3) x + -57 - 41 3) 4) -444 - 392 + x 4) 5) x - 13 = -3 5) 6) 26 = -21 - x ' 6) 7) 81 = x 59 7) 8) 389 - x = - 174 8) 9) - 18 = -3x 9) 10) -9x = 54 10) 11) x(8) = 48 11) 12) -976 = x(61) 12) 13) 18/x = -3 13) 14) x/8 = -6 14) 15) 20 = -100/x 15) 16) 400 = x/-20 16) PART 8 - TWO STEP EQUATIONS 17) -13 = -5x + 2 17) 18) -23 - x = -48 18) 19) 17x + -100 = -15 19) 20) 100 = 1000 + 30x 20) 21) -16 = -6x + 2x 21) 22) -48 = -5x - 7x 22) 23) 45x 46x = -91 23) 24) -370x + -130x = -1000 24) 25) 13x = 7x - 18 25) 26) -35 x = -6x 26) 27) -79x + -75 = -54x 27) 28) 971x = 420 + 985x 28) 29) 2 (x - 1) = -18 29) 30) 26 = 13(3 - x) 30) 31) -72 = -8 (x + 7) 31) 32) -21(21 + x) = 399 32) PART C - MULTI-STEP EQUATIONS 33) 0 - -7 12x - -17 - 2x 33) 34) 30 - -13x - 12 6x + 7 .34) 35) -35 27x + 68 + -43x = -63 35) 36) 151x + 95 + -188x + 162 - 14 36) 37) - 7 - 3x = 9 - -x 37) 38) 21x + -33 - -4x + -8 38) 39) 89 + 55x - 23 + 61x 39) 40) 97x - 183 - 123x - 337 40) 41) -8(x 2) - -9x - -1 41) 42) -2 + -18x - -11(4 + x) 42) 43) 52x + -16 - 15(x - 6) 43) 44) -5(161 + x) - 125 - 2x 44) 45) -2(8 - x) » 1(2 + x) 45) 46) -13(x + -3) - - l l ( x + -1) 46) 47) -8 (12 - x) - -14 (-3 - x) 47) 48) 138(x - 1) - 161(x - -5) 48) 1 52 TEST-FORM 4 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) -5 = -9 + x 1) 2) -31 = x + -42 2) 3) -64 + x = -75 3) 4) x + -352 = -701 4) 5) 7 - x = 12 5) 6) x - 19 = 25 6) 7) 66 = 96 - x 7) 8) 464 = x - 536 8) 9) 49 = -7x 9) 10) -19x = -380 10) 11) x(-7) = -63 11) 12) 16 = x(4) 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) -3 = 16 + -lx 17) 18) 0 = -29x - 29 18) 19) -77 - -x = -36 19) 20) 33x + 445 = -545 20) 21) 8x + 9x = -17 21) 22) 37 = 22x + 15x 22) 23) 88 = 56x - 67x 23) 24) 182x - 168x = -196 24) 25) -7x = 16 + x 25) 26) 50x = 41x 36 26) 27) -91 - 53x = -60x 27) 28) -160x + 783 = lOlx 28) 29) 2 (5 + x) = -8 29) 30) 28(x 5) = -28 30) 31) 57 = l9 ( -2 - x) 31) 32) 288 = -12(x + -24) 32) PART C - MULTI-STEP EQUATIONS 33) 4x + -7 + -5x + - 5 = 6 33) 34) -29 = 9 - 28 - -4 - -25x 34) 35) 94 = 49x 39 - 31x + 1 35) 36) 711 - 165x + -613 + -175x - -582 36) 37) -2x 8 = 5x - 6 37) 38) 49 - -27x = 16 - 24x 38) 39) -76x + 69 = 3x + -87 39) 40) -300 + -167x = -1000 + ,-237x 40) 41) 2 ( -5 + x) - 9 x 41) 42) 13(x - 3) - -2x - 9 42) 43) -9 + 24x - -9( -10 + x) 43) 44) 66x + 44 - 26(x - 26) 44) 45) -3 (x + 6) - 4(x - 1) 45) 46) -9 ( -1 - x) - -6(6 + x) 46) 47) - 9 (x + 8) - 8<x + -9) 47) 48) 144(6 - x) - 160(5 - x) 48) P a r t 3 I n t e r a c t i o n E f f e c t s 1 54 Phase 3 involved administration of four equation t e s t forms to a t o t a l of eighty subjects. Each te s t form consisted of 48 of the 192 constructed equations; four from each of the twelve equation types, making three from each c e l l of the 4 by 4 numerical magnitude by s t r u c t u r a l format g r i d . An analysis of variance was performed on raw scores. Factors were complexity (one-, two- and multi-step equations), numerical magnitude (In) < 2 0 , 20 < |n| < 50, 50 < |n| < 100, and 100 |n| < 1000), and s t r u c t u r a l format (e.g. n +a=b, a+n =b, b=n +a, and b=a+n ). Table B.l presents the equation test-form r e s u l t s . I n dividual l e v e l s of equation complexity (Cl, C2, C3) were compared using Sheffe's M u l t i p l e Range Test (Glass & Stanley, 1970). While the mean performance was higher on one-step (Cl) than on two-step (C2) or multi-step (C3) equations, r e s u l t s indicated that only the discrepancy between one-step (Cl) and multi-step (C3) equations were s i g n i f i c a n t [F(2,73) -2 .50, p < .05] . Individual structures (SI, S2, S3, S4) were compared using Sheffe's M u l t i p l e Range Test. These showed no s i g n i f i c a n t d ifferences across a l l four subtests [F(3,72)=2.87, p > .05]. Multiple contrasts compared placement of the va r i a b l e with regard to the numbers (SI & S3 vs S2 & S4) as well as the eq u a l i t y symbol (SI & S2 vs S3 & S4). Placing the variable to the l e f t of the numbers (SI & S3) seemed to produce higher scores than p l a c i n g the var i a b l e to the right of the numbers (S2 & S4). This contrast was s i g n i f i c a n t (p < .05). However, the p o s i t i o n of the variable with respect to the eq u a l i t y symbol made no appreciable d i f f e r e n c e 1 55 (SI & S2 vs S3 & S4). There were s i g n i f i c a n t findings f o r numerical magnitude on three of the four subtests (See Table B . l ) . 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 to diffe r e n c e s among equations withnumbers of smaller magnitude (Nl) and equations with numbers of larger magnitude (N4) [F (3, 72) =2 .27, p < .05]. However, there were no consistent patterns on d i f f e r e n t test-forms. Equations containing smaller numbers d i d not appear to have s i g n i f i c a n t l y higher scores than those containing larger numbers (See Table B . l ) . Equations containing numbers whose absolute value was greater than 50 and less than or equal to 100 (N3) had the same or more correct answers on three of the four test-forms than equations containing numbers whose absolute value was greater than 20 and less than or equal to 50 (N2). On three of the four test-forms there 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, as well as equation complexity and numerical magnitude. On a l l four test-forms there were highly s i g n i f i c a n t 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 numerical magnitude. These i n t e r a c t i o n s are presented i n Figures B . l , B.2, and B.3, re s p e c t i v e l y . The r e s u l t s presented i n Figure B.l indicated that there was no s t r u c t u r a l format which was easiest at any given complexity l e v e l . For example, SI d i d not have the highest p-value on a l l four test-forms. Similar r e s u l t s were found f o r complexity and numeric i n t e r a c t i o n s (See Figure B.2). Again, there was no d e f i n i t i v e pattern at a p a r t i c u l a r complexity l e v e l , and while on Test-Form 3 there appeared to be an Table B . l Phase 3: Equation Test-Form Results for Equation Complexity, S t r u c t u r a l Format, and Numerical Magnitude Equation Test-Form 1 Factors Mean F S ig Test-Form 2 Mean F Sig Test-Form 3 Mean F Sig Test-Form 4 Mean F Sig Equation Complexity C l .59 C2 .27 C3 .11 70.90*** .69 .51 .32 .32 .07 117.48*** .09 .74 .50 34.11*** .22 31.99*** S t r u c t u r a l Format SI S2 S3 S4 .30 .28 . 41 .29 _77*** .39 ,37 .33 .34 4 . 47** .34 .26 .32 .32 .52 ,42 .50 ,50 4 . 90* Numerical Magnitude Nl N2 N3 N4 .35 .36 .36 .22 7.50*** .39 .37 .34 .35 1. 52 .33 .26 .41 .23 11.98*** .54 .48 .49 .43 6.75** Cl= One-step equation C2= Two-step equations C3= Multi-step equations Sl= v a r i a b l e s to l e f t of numbers & equality S2= variables to r i g h t of numbers & l e f t of equality S3= variables to l e f t of numbers & right of equality S4= variables to r i g h t of numbers & equality Nl= | n | < 20 N2= 20< |n| < 50 N3= 50< 1n| < 100 N4= 100< | n | < 1000 Sig= Leve 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 table were obtained from BMDP8V - Analysis of Variance ( S P S S X - 1 9 8 2 ) . 1 57 i n t e r a c t i o n e f f e c t , t h i s was not s i g n i f i c a n t (p >.05). However, i n both Figure B . l and Figure B.2, the main e f f e c t f o r complexity was quite apparent. In examining the data presented i n Figure B.3, a strong i n t e r a c t i o n e f f e c t was noted for s t r u c t u r a l format and numerical magnitude. While main e f f e c t s f o r s t r u c t u r a l format and numeric magnitude were found (refer to Table B . l ) , these were not apparent due to the strong i n t e r a c t i o n e f f e c t between these two factors (See Figure B.3). The i n t e r a c t i o n s among equation complexity, s t r u c t u r a l format, and numerical magnitude necessitated controls i n equation s e l e c t i o n . A l l combinations of numerical magnitude and s t r u c t u r a l format needed to be represented at each l e v e l of equation complexity i n the diagnostic instrument. 1 5 8 Figure B . l Phase 3: I n t e r a c t i o n E f f e c t s of Equation Complexity and S t r u c t u r a l Format crj QJ i a_ QJ. cn ro OJ > a: Test-Form 1 in OJ • r-l ro :=> i CL QJ ai ro PH QJ > 1.0 . 9 .8 . 7 . 6 .5 . .4 .3 .2 . 1 0 Test-Form 2 • A 1 2 . 3 Complexity 1 2 3 Complexity in QJ I a. QJ c n ro rH cu > c c 1.0 . 9 .8 . 7 . 6 .5 . 4 .3 . 2 . 1 0 Test-Form 3 \ 1 2 3 Complexity (TJ cu 3 QJ cn ro u QJ ct ,0 9 8 7 6 5 4 3 2 1 0 Test-Form 4 1 2 3 Complexity 51 = 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 52 = 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 53 = V a r i a b l e right, of e q u a l i t y , l e f t of constants 54 = 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 of constants 1 59 Figure B.2 Phase 3: Interaction E f f e c t s of Equation Complexity and Numerical Format cu i CL QJ cn co u QJ > CC Test-Form 1 cn QJ • r-t CO QJ QI ro u QJ > C I 1.0 . 9 .8 . 7 .6 .5 . 4 .3 .2 . 1 0 Test-Form 2 \ 1 2 3 Complexity 1 2 3 Complexity cn OJ u •—i CO > I CL QJ cn CO fH QJ > 1.0 . 9 . 8 . 7 . 6 .5 . 4 .3 .2 . 1 0 Test-Form 3 1 2 3 Complexity CD QJ r H CO • > I Q. QJ Ol CO Cn QJ .0 , 9 ,8 ,7 6 5 4 3 2 1 0 Test-Form 4 V V X "NX 1 2 3 Complexity - Nl — j N2 . _ N3 ; N4 I n I < 20 < I n | < 50 < |n | < 20 50 100 100 < |n| < 1000 1 60 Figure B . 3 Phase 3 : Interaction E f f e c t s of S t r u c t u r a l Format and Numerical Magnitude to QJ 3 i Q_ QJ cn to H QJ > ct Test-Form 1 m QJ i CL QJ n u ,0 9 8 7 6 .5 4 3 2 1 0 Test-Form 2 2 3 Structure 2 3 Structure cn QJ i o_ QJ cn ro OJ cr Test-Form 3 UJ QJ 3 I CL QJ OI ra In QJ > CE ,0 , 9 .8 7 6 5 4 3 2 1 0 Test-Form 4 \ \ / 2 3 4 Structure 2 3 Structure Nl N2 N3 N4 20 < 50 < 100 < I n | < 20. | n | < 50' | n | < 100 Inl < 1000 161 P a r t 4 F i n a l E q u a t i o n s 1 62 The r e s u l t s of the equation test - forms used, i n the t h i r d phase of the study were u t i l i z e d i n s e l e c t i n g the equations for i n c l u s i o n i n the Diagnostic Checklist for Algebra. Due to the strong i n t e r a c t i o n e f fec t between numerical magnitude and s t r u c t u r a l format, each of the three l eve l s of equation complexity (one-, two-, and mult i - s tep) contained the 16 c e l l s of the i n t e r a c t i o n g r i d (4 numerical magnitude x 4 s t r u c t u r a l format) . This meant that 48 equations were requ ired (3 equation complexi t ies x 4 numerical magnitudes x 4 s t r u c t u r a l formats) to c o n t r o l for i n t e r a c t i o n s . The f i n a l 48 equations are presented i n Tables B . 2 , B . 3 , and B . 4 , together with t h e i r numerical magnitude and s t r u c t u r a l format assignment (N/S), d i f f i c u l t y l e v e l (P), 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 with the t o t a l 192 equations (B-TT), 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 with the ex terna l c r i t e r i a (EC) - the t o t a l raw score on the Lankton First-Year Algebra Test (Lankton). The p-values 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 1 0 0 and rounded to two s i g n i f i c a n t d i g i t s . The p-values of the f i n a l 48 equations used i n the d iagnos t i c instrument ranged from 0 . 0 5 to 0 . 9 0 with a mean p-value of 0 . 4 2 . The range of p-values was qui te large , s ince i n c l u s i o n of one item from each c e l l of the 4 - b y - 4 numerical 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 requ ired at each l e v e l of complexity. The b i s e r i a l c o r r e l a t i o n c o e f f i c i e n t s of each equation used i n the d iagnost i c instrument ranged from 0 . 0 4 to 0 . 9 7 with the t o t a l 1 9 2 equations and from 0 . 1 1 to 0 . 7 2 with the Lankton. 1 63 Table B.2 S t a t i s t i c s f o r One-Step Equations EQUATION TYPE: ONE-STEP Addition N/S B-TT EC* x+57=41 -53+x=-75 800+x=609 256=x+-344 N3S1 N3S2 N4S2 N4S3 74 82 50 67 87 46 04 39 70 37 48 28 Subtraction x-13=-3 -16=x—3 x-19=25 26=-21-x NISI N1S3 N2S1 N2S4 53 50 77 32 51 51 60 04 52 51 66 26 M u l t i p l i c a t i o n 1. 100=x(10) 2. 56=-8x 3. x(-45)=-495 4. 1000=200x N3S3 N3S4 N4S1 N4S4 68 80 42 90 43 60 37 82 26 38 49 12 D i v i s i o n 1. 2 . 3. 4. 18/x=-3 -l=-7/x -50/x=5 -ll=x/-4 N1S2 N1S4 N2S2 N2S3 53 40 55 50 62 37 04 54 06 46 41 64 * N/S = Numerical Magnitude and Str u c t u r a l D i f f i c u l t y Levels P = p-value when equations were tested i n Phase 3 B-TT = b i s e r i a l c o r e l a t i o n of each equation with the t o t a l 192 equations EC = c o r r e l a t i o n of each equation with the Lankton 1 64 Table B.3 S t a t i s t i c s for Two-Step Equations EQUATION TYPE: TWO-STEP Like 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 69 88 25 56 26 38 32 Like Variable Terms - 1 Side 1. 8x+9x=-17 2. 37=22x+15x 3. 88=56x-67x 4. -370x+-130=-1000 NISI N2S4 N3S3 N4S1 64 59 41 47 66 65 43 99 58 40 45 47 Like Variable Terms - 2 Sides 1. -7x=16+x 2. x+-48=-23x 3. 333-158x=-121x 4. 971x=420+985x N1S4 N2S1 N4S2 N4S4 45 52 20 26 82 57 75 97 26 36 32 41 Parentheses/Variable - 1 Side 1. 2(5+x)=-8 2. 16=4(x+3) 3. 16(-7+x)=-48 4. 288=-12(x+-24) N1S2 N1S3 N2S2 N4S3 40 40 40 40 63 78 63 78 51 36 51 36 * N/S = Numerical Magnitude and Str u c t u r a l D i f f i c u l t y Levels P = p-value when equations were tested i n Phase 3 B-TT= b i s e r i a l c o r e l a t i o n of each equation with the t o t a l 192 equations EC = c o r r e l a t i o n of each equation with the Lankton 1 65 Table B.4 S t a t i s t i c s f o r Multi-Step Equations EQUATION TYPE: MULTI-STEP Variable/Numeric Term - 1 Side N/S B-TT 1. -4x+-7+-5x+-5=5 NISI 27 2. -3-x+8+-4x=-10 N1S2 15 3. 0=-7—12x—17-2x N1S4 15 4. 711-165x+-613+-175x=-582 N4S2 32 68 61 81 97 17 29 67 45 Variable/Numeric Term 2 Sides 1. -2x—8=5x-6 N1S3 2. 49—27x=16-6x N2S2 3. 38+-35x=-12+-30x N2S4 4. -513x+-156=-463+-606 N4S1 36 32 10 15 62 41 58 61 18 22 44 47 Like Variable Terms - 2 Sides 1. 13(x-3)=-2x-9 N2S1 2. 52x+-16=15(x-6) N3S3 3. -9+24x=-9(-10+x) N3S4 4. 66x+44=26(x-26) N4S3 Parentheses/Variable - 1 Side 1. 10(x+-5)=20(x-2) N2S3 2. -9(x+8)=8(x+-9) N3S1 3. 10(10-x )=-9( -9+x) N3S2 4. 144(6-x)=160(5-x) N4S4 18 05 27 09 10 27 10 09 71 88 85 75 34 68 89 75 11 15 12 16 26 72 43 28 * N/S = Numerical Magnitude and St r u c t u r a l D i f f i c u l t y Levels P = p-value when equations were tested i n Phase 3 B-TT= b i s e r i a l c o r e l a t i o n of each equation with the t o t a l 192 equations EC = c o r r e l a t i o n of each equation with the Lankton 1 66 APPENDIX C PHASE 4 Wording and Format Changes 1 67 The purpose of the interviews of the four students used i n Phase 4 was p r i m a r i l y to r e f i n e the instrument i n terms of needed word changes and concerns regarding time constraints. The refinements are presented i n terms of what changes were made to the diagnostic instrument, rather than as i n d i v i d u a l interviews. The types of erroneous understandings which occurred are discussed, as well as the steps taken to r e c t i f y these (See Appendix E for f i n a l wording of the Diagnostic Checklist for Algebra). The concept of v a r i a b l e questions remained unchanged, except for the addition of "How do you know?" at the end of the meaning of variable section. By adding the question, i t became apparent that one low achiever r e a l l y only understood variables to be "things" rather than representations of numbers. The other three students expanded t h e i r comments about "the variables are the same" to explain that they represented the same number. When t o l d to group expression cards by operation, the higher achieving 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 to "Group these cards by operations so ones representing the same expressions are together," a l l students understood the requirements of the task. However, both boys were unsure of the meaning of 4a, and both high achievers were uncertain i f 4+a meant 4/a or a/4. The wording of the meaning of expressions section presented great d i f f i c u l t y f o r a l l four subjects. They were extremely reluctant and confused when attempting to attach concrete meaning to expressions containing blanks for the v a r i a b l e . By changing the wording from "Explain" to "Use the pennies to explain the meaning of t h i s expression," a l l 1 68 students were f i n a l l y able to understand what the task required. However, many prompts were needed f o r some students to enable them to proceed with the task. "Choose the number of pennies that the blank represents", "Show the meaning so a Grade 1. student could understand", and "Show me p h y s i c a l l y what you do when you perform the operation of (addition, subtraction, m u l t i p l i c a t i o n or d i v i s i o n ) " were the a d d i t i o n a l prompts used to c l a r i f y how to explain the meaning of expressions using concrete materials. For expressions containing l e t t e r s f o r the variable, "You may use the pennies, i f you wish", was s u f f i c i e n t to prompt the use of concrete materials. E x p l i c i t d i r e c t i o n s were required i n the use of eq u a l i t y section to ensure that high achievers d i d not give only o r a l examples. The wording was changed from "Give me an example" to "Write an arithmetic sentence on your paper." Two of the subjects, the high-achieving female and the low-achieving male, st a r t e d to write more than one example to enable use of a l l the numbers i n the set. The change i n the d i r e c t i o n s c l a r i f i e d the vagueness of the term "example." Several concerns arose for the meaning of equality section. Unless prodded, subjects considered reading the arithmetic sentence and agreeing that i t was true to be a s u f f i c i e n t explanation. However, such an incomplete i n d i c a t i o n of a student's understanding of the meaning of eq u a l i t y was not s u f f i c i e n t f o r diagnostic purposes. To improve the responses given, teachers should l i s t e n f o r rewording of the f i r s t question •to "4+3=7". Such a rewording may indicate that the student thinks an "answer" should follow the equality symbol and not "the problem" (Behr et 1 69 a l . , 1976) . Three of the four students reversed t h e i r reading of "7=4+3" to "4+3=7." However, when faced with s i m i l i a r a lgebra i c equations, they d i d not make the r e v e r s a l . "How do you know?" and "Have you seen t h i s before?" were added to the 12x2=19+5 and 13=13 problems, r e s p e c t i v e l y . The former forced the student to i n d i c a t e the operations performed to achieve e q u a l i t y , and the l a t t e r provided an opportunity for students to comment on the meaningfulness of the r e f l e x i v e property of e q u a l i t y . The high achievers both i n d i c a t e d that they had seen a number set equal to i t s e l f as the l a s t statement when v e r i f y i n g an equat ion's s o l u t i o n . The low achievers claimed that they had never seen such statements before . An i n t e r e s t i n g confusion occurred when students were asked to exp la in the meaning of a given equation when the v a r i a b l e was supposed-, to represent the amount of money they had i n t h e i r pocket. Three of the four subjects took the statement l i t e r a l l y to mean that the v a r i a b l e was the ac tua l amount of money they had i n t h e i r pockets, which was nothing! By changing the wording to the subjunct ive mood, "the amount of money you're supposed to have i n you pocket (we're pretending)" the students were able to recognize the nature of the task . The high achievers were able to provide explanat ions , while the low achievers were not . Students d i d not understand the next task of c r e a t i n g meaning without context u n t i l "problem" was changed to "word problem." In the l a s t por t ion of the equation s e c t i o n , the high achievers requ ired the prompt to make an equation "on your paper" before they would write down the equation they were say ing . The low achievers had d i f f i c u l t y reading the questions and 1 70 were unable to create equations, although they both t r i e d to write down some of the numbers i n the problems combined with an "x" or an "=". In the vocabulary section, coefficient gave students p a r t i c u l a r problems. Even with the h i n t : "Remember, i t ' s the number i n front of the l e t t e r . " , some i n t e r e s t i n g r e s u l t s occurred, g i v i n g further i n s i g h t into students' understanding, not only of c o e f f i c i e n t , but also of l i k e terms. In "4+5n" one low achiever indicated the c o e f f i c i e n t was 9 and that the c o e f f i c i e n t f o r "7-2n" was 5. This same subject also indicated that the c o e f f i c i e n t of "-n" was negative. In l a t e r work with equations t h i s student made s i m i l i a r errors with l i k e terms, but was able to operate c o r r e c t l y with variables whose c o e f f i c i e n t was negative one (-1). When given the d i r e c t i o n s i n the inverse sections "Write the opposite (reciprocal) of each of the following", students gave only o r a l answers. By d e l e t i n g the f i n a l phrase and replacing i t with "expression on you paper", a l l subjects wrote answers on the paper provided. When presenting i d e n t i t y element examples, i t was necessary to point to the p a r t i c u l a r parts of the equations involved and to s p e c i f i c a l l y d i r e c t students to "Solve each equation. Place the answers on your paper." Students otherwise thought that both " l e t t e r s " and "numbers" r e f e r r e d to the v a r i a b l e rather than the l a t t e r r e f e r r i n g to replacements for the numbers i n the sentences. For example, when asking "Does i t matter what numbers we use?" when r e f e r r i n g to 3+n=3, i t was necessary to point to both numbers to provide c l e a r d i r e c t i o n for. a l l students. To ensure that students received the same d i r e c t i o n s f or solving the 171 equations, s p e c i f i c d i r e c t i o n s were given: "Solve each equation, explaining aloud what you are thinking or doing." Also, the column for i n d i c a t i n g whether or not the f i n a l s o l u t i o n was correct (c) or inc o r r e c t (x) was added. The general format of the diagnostic c h e c k l i s t seemed useful and appropriate f o r determining the l e v e l of understanding attained by students i n algebraic concepts and vocabulary. The e r r o r - c a t e g o r i z a t i o n scheme depicted student errors and provided a quick v i s u a l record of the systematic errors of an i n d i v i d u a l student. This v i s u a l c h e c k l i s t of errors also provided a graphic representation of the common errors across students when r e s u l t s were compared. A major concern i n the use of semi-structuured interviews i n the diagnostic instrument was time constraints (Hart, 1983; Opper, 1977). Completion time ranged from 55 minutes to 105 minutes f o r the four students interviewed i n Phase 4. Limiting the number of equations could shorten t h i s time considerably, . However, l i m i t i n g the number of equations might not produce an accurate depiction of students' e r r o r s . This was of p a r t i c u l a r concern i n view of the highly s i g n i f i c a n t i n t e r a c t i o n between numerical magnitude and s t r u c t u r a l format which occurred i n Phase 3. 1 72 A P P E N D I X D P h a s e 5 S t a t i s t i c a l A n a l y s e s o f E r r o r C o r r e l a t i o n s P a r t 1: F o r m s U s e d f o r S u b j e c t S e l e c t i o n P a r t 2 : C o r r e l a t i o n s o f E r r o r s w i t h O u t c o m e s a n d A c h i e v e m e n t P a r t 3: C o r r e l a t i o n s o f E r r o r s w i t h O t h e r M e a s u r e s P a r t 4: I n t e r c o r r e l a t i o n s among O t h e r M e a s u r e s w i t h C o n c e p t s a n d E r r o r s P a r t 5: 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 , S t r u c t u r e , a n d M a g n i t u d e 173 The purpose of t h i s appendix i s to d e t a i l the analyses used i n the assessment of the Diagnostic Checklist for Algebra. Part 1 provides copies of the forms used f o r subject s e l e c t i o n i n the fourth and f i f t h phases of the study. Part 2 describes the creation of "outcomes" on the diagnostic instrument and the co r r e l a t i o n s among these and the errors which were i d e n t i f i e d by the instrument, as well as tables of c o r r e l a t i o n s among achievement l e v e l and the developed e r r o r s . 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 v a r i a b l e s . It 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 these measures and the conceptual, procedural, and resultant errors which occurred i n the Diagnostic Checklist f o r Algebra. Part 4 also presents a table of i n t e r c o r r e l a t i o n s among conceptual, procedural, and resultant e r r o r s . Part 5 shows the graphic i n t e r a c t i o n s among equation complexity, s t r u c t u r a l format and numerical magnitude. The r e s u l t s of these analyses are l i m i t e d because of the use of only one school s e t t i n g . Further assessment of the instrument i n other school settings would be necessary f o r these analyses to warrant f u l l s t a t i s t i c a l merit. However, by the i n c l u s i o n of such analyses, i t i s hoped that non-meaningful claims regarding the robustness of the instrument w i l l be avoided and that the findings w i l l be of diagnostic and i n s t r u c t i o n a l value. P a r t 1: F o r m s u s e d i n S u b j e c t S e l e c t i o n Grade 9 subjects p a r t i c i p a t e d i n the t h i r d phase of the study as part 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 . Results on the Lankton and the equation test-forms were given to the teachers one week a f t e r administration for assessment purposes i n the classroom. As teachers used the r e s u l t s of the Lankton and the equation test-forms as part of t h e i r classroom evaluation, parental consent f o r student p a r t i c i p a t i o n was unnecessary for t h i s phase. However, a l l subjects p a r t i c i p a t i n g i n the i n d i v i d u a l interviews, required parental consent. This consent was obtained p r i o r to the interviews. Students were also asked to f i l l out the student questionnaire p r i o r to p a r t i c i p a t i n g i n the interviews. Copies of the two forms used for subject s e l e c t i o n are included for completeness of presentation of the procedures used i n Phase 5. 1 76 By completing the attached questionnaire, I am agreeing to p a r t i c i p a t e i n the diagnostic c h e c k l i s t research on algebraic equations. I recognize that I w i l l be interviewed for about one hour while I solve some algebraic equations. I know that I w i l l be asked to "think aloud" as I solve the equations and that the session w i l l be video-taped. Only my f i r s t name w i l l be used i n reports of my work (or I may i n d i c a t e a pseudonym, i f I so choose). I know that I have the righ t to withdraw from the research at any time and that t h i s w i l l i n no way a f f e c t my school marks. QUESTIONNAIRE NAME: Sex : M F (Last), (First) ( c i r c l e one) School: Teacher: Mathematics marks i n Junior Hiah: r Education of parents My educational plans (only check (only check highest highest level) l e v e l you plan to go) Father Mother Self Completed elementary Completed junior high (grade 10) Completed high school (grade 12) Vocational T r a i n i n g One or more years of college Graduated from college Some graduate work Have a higher degree (masters/doctorate) Other (specify) 177 P a r t 2: C o r r e l a t i o n s o f E r r o r s w i t h O u t c o m e s a n d A c h i e v e m e n t The c r e a t i o n of a d i a g n o s t i c instrument should be i n keeping with measurement p r i n c i p l e s i n v o l v e d i n other " t e s t " c o n s t r u c t i o n , and should ensure that the instrument i s v a l i d . The procedures used i n c r e a t i n g the Di a g n o s t i c C h e c k l i s t f o r Algebra attempted 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 semi-structured, d i a g n o s t i c i n t e r v i e w s e t t i n g . One of the apparent dangers i n d i s c u s s i n g the r e s u l t s of i n t e r v i e w data i s the tendency to focus on the minute d e t a i l s of each i n d i v i d u a l d i a g n o s i s , and to oft e n ignore some of the general trends which may be o c c u r r i n g . While these minute d e t a i l s are i n t e r e s t i n g and important f o r the i n d i v i d u a l student being diagnosed, emphasis on them when assessing the d i a g n o s t i c instrument may mask important f i n d i n g s which may have s i g n i f i c a n t i m p l i c a t i o n s f o r the i n s t r u c t i o n a l s e t t i n g . Hence, i n a n a l y z i n g the data from the Diagnostic Checklist for Algebra use of measurement p r i n c i p l e s and techniques 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 the c r e a t i o n of "outcome scores" on the d i a g n o s t i c instrument. Three such scores were created f o r each subject on the d i a g n o s t i c instrument. I t must be emphasized that these "scores" were not meaningful f o r d i a g n o s t i c purposes r e l a t i n g to i n d i v i d u a l students. However, they d i d a l l o w f o r comparisons of e r r o r s between students and provided i n f o r m a t i o n regarding general trends of e r r o r s . The f i r s t score represented the t o t a l number of e r r o r s a student c o u l d make on the d i a g n o s t i c instrument (See Tables D.l and D.2, T o t a l Number of E r r o r s ) . I t represented the a r i t h m e t i c sum of the second and t h i r d scores. The second score represented the t o t a l number of conceptual e r r o r s made by students. 1 7 8 There were 2 7 d i f f e r e n t items i n the concepts and vocabulary portion of the diagnostic instrument (See Tables'D.l and D . 2 , Errors i n Concepts). A check under the mastery column counted as no error ( 0 ) , under the p a r t i a l column as one erro r ( 1 ) , and under the remedial column as two errors ( 2 ) . This meant that the highest possible score was 5 4 and represented a measure of conceptual errors, rather than mastery of concepts. The t h i r d score 1 represented 7 5 3 d i f f e r e n t opportunities f o r a subject to make procedural or resultant errors i n solv i n g the 4 8 equations (See Tables D.l and D . 2 , Errors i n Equations.) . This was not the expected 1 0 0 8 ( 2 0 e r r o r categories x 4 8 equations + 4 8 i n c o r r e c t f i n a l answers) due to the nature of the construction of the equations. For example, many of the equations d i d not include brackets, hence the opportunity to make a d i s t r i b u t i v e error was not present. Combining the number of errors made on the second score ( 5 4 ) and the t h i r d score ( 7 5 3 ) gave a t o t a l f o r the f i r s t score of 8 0 7 . This represented the t o t a l number of errors that any one student could possibly make on the diagnostic instrument. The reason f o r creating these error scores was due to the many comparisons among errors i n concepts, procedures, and r e s u l t s which could occur on the diagnostic instrument. An i n f l a t e d Type I (alpha) error could occur when making-multiple comparisons, causing i d e n t i f i c a t i o n of diagnostic symptoms which were not s i g n i f i c a n t . To l i m i t Type I error, only those errors which s i g n i f i c a n t l y c o r r e l a t e d ( p< . 0 5 ) with the t o t a l number of errors, and then with e i t h e r the t o t a l conceptual errors or the t o t a l e rrors i n equations were discussed (See Tables D.l and D . 2 ) . when Table D.l Correlations between Concept Scores and Error Scores C o n c e p t s T o t a l Number E r r o r s i n E r r o r s i n o f E r r o r s Concepts E q u a t i o n s V a r i a b l e i d e n t i f y 38* 54*** 28 meaning 37* 22 36* E x p r e s s i o n s i d e n t i f y + 34* 31 31 i d e n t i f y - 25 29 20 i d e n t i f y x 50** 61*** 40* i d e n t i f y / 16 47** 04 meaning b lank + -14 13 -21 meaning b lank - 20 46** 09 meaning b lank x 15 46** 04 meaning b lank / 49** 68*** 37* meaning v a r i a b l e + 04 47** -08 meaning v a r i a b l e - 06 45** -06 meaning v a r i a b l e x 36* 7 4 * * * 20 meaning v a r i a b l e / 49** 5 9 * * * 40* E q u a l i t y i d e n t i f y 19 41* 10 use 41* 51** 33* meaning 12 47** 00 Eguat ions meaning g i v e n c o n t e x t 37* 63*** 25 meaning w i t h o u t c o n t e x t 50** 66*** 40* g i v e n meaning c r e a t e 7 4 * f t « 63*** 68*** C o e f f i c i e n t s ^ i d e n t i f y 18 20 16 L i k e - T e r m s i d e n t i f y 12 01 14 g r o u p 35* 27 33* I n v e r s e s w r i t e o p p o s i t e s 52** 47** 47** w r i t e r e c i p r o c a l s 60*** 65*** 51** I d e n t i t y E lements z e r o 46** 46** 46** one 46** 59*** 36* A l l c o r r e l a t i o n s l i s t e d have been m u l t i p l i e d and rounded t o two s i g n i f i c a n t d i g i t s . ft P < .05 ** P £ .01 « « * P < .001 by 100 1 79 Table D.l Correlations between Concept Scores and Error Scores C o n c e p t s T o t a l Number E r r o r s In E r r o r s i n ' o f E r r o r s Concepts E q u a t i o n s V a r i a b l e i d e n t i f y 38* 54*** 28 meaning 37* 22 36* E x p r e s s i o n s i d e n t i f y + 34* 31 31 i d e n t i f y - 25 29 20 i d e n t i f y x 50** 61*** 40* i d e n t i f y / 16 47** 04 meaning b lank + -14 13 -21 meaning blank - 20 46** 09 meaning b lank x 15 46** 04 meaning blank / 49** 68*** 37* meaning v a r i a b l e + 04 47** -08 meaning v a r i a b l e - 06 45** -06 meaning v a r i a b l e x 36* 74*** 20 meaning v a r i a b l e / 49** 59*** 40* E q u a l i t y i d e n t i f y 19 41* 10 use 41* 51** ' 3 3 * meaning 12 4 7 * * 0 0 E q u a t i o n s meaning g i v e n contex t 37* 63*** 25 meaning wi thout contex t 50** 66*** 40* g i v e n meaning c r e a t e 7 4 * * * 63*** 68*** C o e f f i c i e n t s i d e n t i f y 18 20 16 L i k e - T e r m s i d e n t i f y 12 01 14 g r o u p 35* 27 33* I n v e r s e s w r i t e o p p o s l t e s 52** 4 7 * « 47** w r i t e r e c i p r o c a l s 60*** 65*** 51** I d e n t i t y Elements z e r o 46** 46** 46** one 46** 59*** 36* A l l c o r r e l a t i o n s l i s t e d have been m u l t i p l i e d and rounded to two s i g n i f i c a n t d i g i t s . * P < .05 ** P £ .01 *** P < .001 by 100 1 80 Table D.2 Correlations between Error Types and Error Scores E r r o r Types T o t a l Number E r r o r s In E r r o r s i n o f E r r o r s Concepts E q u a t i o n s P r o c e d u r a l z e r o a n n e x a t i o n 00 -16 06 i d e n t i t y c o n f u s i o n 22 15 22 l i k e - t e r m s ( c o n j o i n i n g ) 38* 24 37* p a r t i a l d i s t r i b u t i v e -22 -25 -19 o r d e r of o p e r a t i o n s 07 -27 17 a d d i t i o n p r o p e r t y = 44** 08 49** m u l t i p l i c a t i o n p r o p e r t y = -07 -26 -01 c o e f f i c i e n t -07 -33* 01 R e s u l t a n t C o m p u t a t i o n a l a d d i t i o n / s u b t r a c t i o n 69*** 29 73*** m u l t i p l i c a t i o n / d i v i s i o n 43** 05 49** b a s i c f a c t s 23 16 23 f a u l t y a l g o r i t h m 21 10 22 wrong o p e r a t i o n 56*** 19 61*** S i g n a d d i t i o n / s u b t r a c t i o n 63*** 20 68*** m u l t i p l i c a t i o n / d i v i s i o n 39* 20 40* r u l e - b a s e d 58*** 28 60*** d i s t r i b u t i v e 38* 21 38* O t h e r m e c h a n i c a l / p e r c e p t u a l 13 -11 19 random • -17 -27 -12 i n c o m p l e t e s o l u t i o n s 58*** 40* 56*** f i n a l answers 96*** 52** 9 4 * * * A l l c o r r e l a t i o n s l i s t e d have been m u l t i p l i e d and rounded to two s i g n i f i c a n t d i g i t s . * P < .05 * * P < .01 *** P < .001 181 the i n t e r a c t i o n s between the conceptual and equation errors were analyzed, only those errors which s i g n i f i c a n t l y c o r r e l a t e d (p<.05) to a l l three error scores were discussed. These procedures helped to ensure that the c o r r e l a t i o n s discussed were meaningful i n terms of measurement p r i n c i p l e s . A summary of the frequencies, 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 conceptual errors or procedural and resultant errors are presented i n Tables D.3 and D.4. 1 82 Table D.3 Conceptual Error Frequencies, Means, Standard Deviations, and S i g n i f i c a n t Differences among Achievement Levels High Achiever Average Achiever Low Achiever Combined Groups Total X Total X Total X Moan S.D. F test CONCEPTUAL Var iables Identify 5 0.42 11 0 .92 16 1.33 0 .89 0 .35 meaning 1 0.08 2 0 .17 4 0.33 0 .19 0 .05 * Expressions Symb+ 0 0.00 2 0 .17 2 0.17 0 .11 0 .02 Symb- 1 0.08 0 0 .00 2 0.17 0 .17 0 .04 * Symbx 3 0.25 5 0 .42 4 0.33 0 .33 0 .11 Symb/ 5 0.42 4 0 .33 8 0.67 0 . 47 0 .21 Blnk + 4 0.33 7 0 .58 8 0.67 0 . 59 0 .24 Blnk- e 0.50 5 0 .42 11 0.92 0 .61 0 .31 Blnkx 2 0.16 7 0 .58 10 0.83 0 .53 0 .30 Blnk/ 15 1.25 14 1 .16 22 1.83 1 .42 0 .78 Varb+ 6 0.50 6 0 .50 10 0.83 0 .61 0 .32 Varb- 4 0.33 6 0 .50 10 0.83 0 .56 0 .28 Varbx 5 0.42 13 1 .08 10 0.83 0 .81 0 .58 • Varb/ 13 1.08 13 1 .08 19 1.58 1 .25 0 .86 * E q u a l i t y Identity 4 0.25 6 0 .50 8 0.67 0 .50 0 .17 use 1 0.08 3 0 .25 3 0.25 0 .19 0 .05 meaning 3 0.25 4 0 .33 8 0.67 0 .42 0 .19 « Equations with 14 1.16 14 1 .16 17 1. 4 2 1 .25 0 .88 * without 12 1.00 15 1 .25 19 1.58 1 . 28 0 :97 * * « create 9 0.75 13 1 .42 20 1.67 1 .17 0 .99 ftftft C o e f f i c i e n t s coef f 18 1.50 15 1 .25 15 1.25 1 .33 0 .94 Like-Terms compare 19 1.58 19 1 .58 20 1.67 1 .61 1 .23 group 14 1.16 18 1 .50 16 1.33 1 .33 1 .27 Inverses Opposites 6 0. 50 7 0 .58 7 0.58 0 .56 0 .27 ft Reciprocals 5 0.42 9 0 .75 12 1.00 0 .72 0 .38 ft* Identity Elements zero 9 0.75 10 0 .83 14 1.16 0 .92 0 .45 ft* one 7 0.58 9 0 .75 12 1.00 0 .78 0 .37 ft « Abbreviations of concepts l i s t e d in t h i s table are in the same order as those l i s t e d i n Table D.l. » p i .05 »* P i .01 «** p < .001 183 Table D.4 Error Type Frequencies, Means, Standard Deviations, and S i g n i f i c a n t Differences among Achievement Levels H i g h A c h i e v e r A v e r a g e A c h i e v e r Low A c h i e v e r Combined G r o u p s T o t a l X T o t a l X T o t a l X Mean S.D. K t e s t PROCEDURAL z e r o a n n e x 7 0. 42 12 0, .67 5 0 .42 0. ,50 0 .66 i d c o n f u s e 3 0. 33 13 0. .92 9 0 .75 0. .67 0 .79 l i k e t e r m s 33 3. 33 21 2. .33 35 3 .17 2. .94 2 .93 p a r t l d l s t 2 0. 17 1 0. .08 1 0 .08 0. .11 0 .32 o r d e z o p e r 5 0. 50 1 0. .08 1 0 .17 0. .25 0 .50 +prop= xprop= 24 2. 00 26 2. , 17 35 2 .67 2 . 28 2 .99 6 0. 50 11 0, .83 4 0 .42 0. .58 0 .97 c o e f £ 3 0. 50 8 0. ,67 8 0 .58 0 . .58 0 .91 tESULTANT C o m p u t a t i o n +- 46 2. 92 61 4 , .58 74 4 .25 3. .81 2 .55 x / 9 1.92 30 2. .50 43 3 .33 2. .58 1 .63 b s c f c t 14 1. 17 19 2. .17 25 2 .33 1. .89 1 .51 f l t y a l g 20 0. 83 31 1. .83 21 1 .67 1. .44 1 . 16 wrngoper 26 2. 50 52 4 , 33 58 4 .08 3. .64 2 .53 S i g n +- 55 4. 00 77 4, .58 79 5 .33 5, .03 2 .69 x / 10 1. 33 14 2. .92 28 3 .08 2 . 44 1 .87 r u l e b a s e 30 1. 75 40 2, .42 37 1 .92 2. .03 1 .67 d l e t r l b 2 0. 50 0 0. ,00 1 0 .67 0 . .53 0 .81 O t h e r mechper 39 2. 42 39 2 , . 00 46 1 .83 2, .06 2 .02 random 18 1. 75 20 1. .83 20 2 .33 1, .97 1 .96 l n c o m p l t 112 4. 08 148 3. .00 160 5 .83 4 , . 36 4 .40 tlnlans 181 10. 25 262 14 , .00 290 17 .25 13. .83 8 .69 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 .05 « * p i .01 * » • p i .001 1 84 P a r t 3: C o r r e l a t i o n s o f E r r o r s w i t h O t h e r M e a s u r e s Because of the possible number of i n t e r c o r r e l a t i o n s among the demographic data, IQ, and mathematics achievement variables and the ideations held and errors made on the diagnostic 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 with the t o t a l number of erro r s and with e i t h e r t o t a l number of errors i n concepts or t o t a l number of errors i n equations were discussed. Correlations between these three scores and the demographic, IQ, and achievement variables were presented i n Table D . 6 . A l l three of the mathematical achievement variables (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 to a l l instrument scores (refer to 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 occurred which r e l a t e d errors to these mathematics achievement v a r i a b l e s . Some of these errors were r e l a t e d to a l l three, while others r e l a t e d to only one or two of the mathematics achievement v a r i a b l e s . The crosstabulations of demographic data and concepts of algebra and the errors made i n solving equations r e s u l t e d i n a number of s i g n i f i c a n t c o r r e l a t i o n s . These were presented i n Tables D .6 and D . 7 . Table D.5 Correlations between Error Scores and Demographic IQ and Mathematical Achievement Variables Variables Error Scores Total Conceptual Equation Grade -48** -28 -48** Age -26 -08 -28 Gender 08 -18 14 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 at 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 Mathematical Achievement V a r i a b l e s (p<.05) Conceptual Grade Age Gender IQ Marks Lankton CSMS V a r i a b l e i d e n t i f y meaning Expressions Symb+ Symb-Symbx 32 Symb/ Blnk+ -49 Blnk-Blnkx -37 Blnk/ Varb+ Varb-Va rbx Varb/ E q u a l i t y i d e n t i f y use meaning Equations wit h without create C o e f f i c i e n t s c o e f f L i k e Terms compare group 33 Inverses oppos -60 r e c i p -43 I d e n t i t i e s zero one •46 29 45 29 36 38 52 35 46 42 33 32 35 31 35 44 50 53 37 40 46 35 49 42 34 38 -34 48 36 33 51 67 29 39 50 60 44 44 51 30 36 37 35 28 47 29 34 46 39 29 44 40 39 53 50 41 31 .35 31 * 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 d i g i t s and represent hundredths. Table 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 for E r r o r Types and Demographic, IQ, and Mathematical 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 r o c e d u r a l zeroannex 43* IDconfuse -33 l iketerms p a r t l d i s t -32 orderoper +prop= -35 xprop= 35 32 coeff 32 R e s u l t a n t C o m p u t a t i o n a l +- -36 x/ bsc fc t - 31 -35 32 f l t y a l g 34 29 wrngoper S i g n +- -33 -31 x/ 38 rulebased 39 -52 d i s t r i b O t h e r mechper random 33 4 4 incomplt 62 -37 -49 -47 F i n a l A n s w e r - 54 - 37 -67 - 33 -46 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 d i g i t s and represent hundredths. 188 P a r t 4: I n t e r c o r r e l a t i o n s among O t h e r M e a s u r e s w i t h C o n c e p t s a n d E r r o r s Table D.8 presents the i n t e r c o r r e l a t i o n s among the demographic data, IQ, and mathematics' achievement v a r i a b l e s . Within the demographic data, only the expected 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 occurred (r=0.43, p<0.01). IQ showed highly 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 measures of mathematical achievement (Marks: r=0.56, p<0.001; Lankton: r=0.55, p<0.001, CSMS: r=0.54, p<0.001). The measures of mathematical 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 themselves (Marks/Lankton: r=0.60, p<0.001; Marks/CSMS: r=0.45, p<0.01; Lankton/CSMS: r=0.65, p<0.001). The CSMS was also 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 (r=0.64, p<0.001). Of the demographic data, only grade l e v e l was s i g n i f i c a n t l y c o r r e l a t e d with the diagnostic c h e c k l i s t . This seemed due not to students' conceptions, but rather to the errors made while so l v i n g equations (See Table D.5). Those errors 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: rule-based sign errors (r=0.39, p<0.05) and incomplete solutions (r=0.63, p<0.001). The 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 and equation errors are presented Table D.9. Table D.8 Int e r c o r r e l a t i o n s of Demographic Data, IQ, and Mathematical Achievement Grade Age Gender IQ Marks Lankton CSMS Grade 1.00 0. .43 * 0. .00 0, .34 0 . 41 0, .35 0, .64 ** Age 1. .00 0 . 10 0 , .26 0 , .25 0 , .26 0 . 17 Gender 1. .00 0 . 16 0, .08 0 , .03 - 0. .02 IQ 1. .00 0, .56 ** '0: .55 * * 0 .54 * * Marks 1. .00 0 , . 60 ** 0, .45 * Lankton 1. .00 0. .65 ** CSMS 1, .00 * S i g n i f i c a n t at p<.01 ** S i g n i f i c a n t at p<.001 1 9 0 Table D . 9 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 and Error Types 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ?4J» 1 1.00 35 35 -39 2 1.00 36 34 36 39 -37 -40 3 1 .00 39 46 43 4B 39 33 -40 -38 4 1.00 45 66 46 46 59 40 53 ' 35 -23 -40 5 1.00 42 39 35 46 -47 -46 6 1.00 56 56 43 45 -47 -46 7 1.00 66 41 38 41 a 1.00 67 50 59 -43 9 1.00 54 44 48 41 -39 -36 -40 -59 -66 10 1.00 37 35 -35 -39 -46 -55 11 1.00 42 -45 -37 -39 -47 12 1.00 75 -39 -34 -41 13 1.00 -36 -46 14 1.00 36 36 15 1.00 53 41 88 16 1.00 58 81 61 40 37 63 17 1.00 63 57 37 46 18 1.00 45 35 37 53 19 1.00 51 38 57 20 1.00 21 1.00 50 52 22 1.00 34 23 1.00 65 24 1.00 * Correlations l i s t ed have been rounded to two signif icant d ig i t s and represent hundredths. Significant correlation levels 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 variables. Their def ini t ions are: l=Variable Identif icat ion 2=Variabl'e Meaning 3=Identify Mul t ip l ica t ion Synbol 4=Blank Division Expressions 5=Variable Mul t ip l ica t ion Expressions 6=Variable Division Expressions 7=Meaning of Equations Given Context 8=Meaning of Equations Without Context S=6iven Meaning-Create Equation 10=Additive Inverse l l=Mul t ip l ica t ive Inverse 12=Additive Identity 13=Hultiplicative Identity 14=CoBibining Unlike Teras 15=Addition Property of Equality 16=Coaputational Add/Subtract l7=Coaputational Mult iply/Divide 18=Coaputational Wrong Operation 19=Sign Add/Subtract 20=Sign Mult iply/Divide 21=Sign Rule-Based 22=Sign Dist r ibut ive 23=Inconplete Solutions 24=Final Answers P a r t 5: 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 , S t r u c t u r e , a n d M a g n i t u d e Figures D.l, D.2, and D.3 present the i n t e r a c t i o n e f f e c t s among complexity and structure, complexity and magnitude, and structure and magnitude, r e s p e c t i v e l y . Tables D.10, D . l l , and D.12 give the error means for complexity, structure, and magnitude, re s p e c t i v e l y . 192 Figure D . l Phase 5: Interaction E f f e c t s of Equation Complexity and Structural Format O) Z3 #~—* > l CL CU CT) i_ > 1.0 .9 .8 .7 . 6 .5 .4 .3 .2 .. 1 0 i r~—J Complexity •S1 = Variable l e f t of e q u a l i t y , l e f t of constants •52 = Variable l e f t of e q u a l i t y , r ight of constants 53 = Var iable r ight of e q u a l i t y , l e f t of constants 54 = Variable r igh t of e q u a l i t y , r igh t of constants 1 93 Figure D.2 Phase 5: Interaction E f f e c t s of Equation Complexity and Numerical Magnitude GJ > I <v CT «o > 1.0 .9 .8. .5 .3 .2 ,1 0 1 —7. J Compiexliy N1=Numerical Magnitude 1 N2=Numerical Magnitude 2 N3=Numerical Magnitude 3 •N4=Numericai Magnitude 4 ( |n| < 20) ( 20 < |n| < 50) ( 50. < |n| < 100) ( 100 < |n| < 1000) . 1 9 4 Figure D .3 Phase 5: Interaction E f f e c t s of St r u c t u r a l Format and Numerical Magnitude 00 CO Z3 •—t > I 0) cn a; > < 1.0 .9 .8 .7 . 6 .5 .4 . o .2 .1 0 2 3 Structure — N1=Numerical Magnitude 1 — N2=Numerical Magnitude 2 — N3=Numericai Magnitude 3 N4=Numerical Magnitude 4 |n| < 20 < |n| < 50 < |n| < 20) 50) 100) ( 100 < |n| < 1000) Table D . 10 Equation Complexity Error Means Error Types Levels of Complexity* C l C2 C 3 P r o c e d u r a l ZeroAnnexation IDconfusion Liketerms OrderOperations P a r t i a l D i s t r i b u t i v e +prop= xprop= C o e f f i c i e n t R e s u l t a n t C o m p u t a t i o n a l Add/Sub Mult/Div BasicFacts FaultyAlgorithm WrongOperation S i g n Add/Sub Mult/Div Rulebased D i s t r i b u t i v e O t h e r Mech/Perceptual Random Incomplete F i n a l A n s w e r 0 . 0 0 0 . 0 3 0 . 0 3 0 . 0 0 0 . 0 3 0 . 2 5 0 . 0 3 0 . 00 0 . 2 2 0 . 3 6 0 . 0 8 0 . 0 6 0 . 5 3 0 . 7 2 0 . 1 7 0 . 47 0 . 0 0 0 . 0 8 0 . 0 6 0 . 3 6 1 . 3 6 0 . 0 3 0 . 0 3 0 . 6 1 0 . 0 3 0 . 0 0 0 . 3 9 0 . 0 0 0 . 0 3 0 . 2 5 0 . 1 7 0 . 1 4 0 . 0 8 0 . 1 1 0 . 2 5 0 . 3 6 0 . 0 6 0 . 2 2 0 . 0 6 0 . 1 9 0 . 3 9 1 . 4 7 0 . 0 0 0 . 0 9 0 . 61 0 . 0 0 0 . 0 6 0 . 3 6 0 . 1 8 0 . 1 5 0 . 2 1 0 . 1 5 0 . 0 6 0 . 0 3 0 . 2 1 0 . 3 3 0 . 2 4 0 . 1 2 0 . 1 8 0 . 1 8 0 . 09 0 . 47 1 . 44 Cl= Complexity Level 1 one-step equations C2= Complexity Level 2 two-step equations C3= Complexity Level 3 multi-step 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 Levels of S t r u c t u r a l Format* SI S2 S3 S4 P r o c e d u r a l ZeroAnnexation 0. .11 0 .03 0 . ,03 0. .33 IDconfusion 0. .08 0. .00 0. .39 0. .19 Liketerms 0. .97 0. .50 0. .53 0. .94 P a r t i a l D i s t r i b u t i v e 0. .03 0. .03 0 . 06 0. .00 OrderOperations 0. .08 0. .06 0 . 08 0. .08 +prop= 0. .56 0. .47 0 . 50 0. .75 xprop= 0. .25 0. . 14 0 . 08 0. .11 C o e f f i c i e n t 0. . 17 0. .06 0 . 06 0. .31 l e s u l t a n t l o m p u t a t i o n a l Add/Sub 1. .25 1. .33 0 . 78 0 . . 44 M u l t / D i v 0 . 50 0 . 58 0 . 89 0 . 61 Bas icFacts 0 . 39 0. .53 0 . 47 0 . 50 Fau l tyAlgor i thm 0 , .39 0. .50 0. .36 0. .19 WrongOperation 0. .89 0 . 89 1. .17 0 , . 69 ! i g n Add/Sub 1. .00 1. .81 1. .28 0 . . 94 M u l t / D i v 0, .31 0. .86 0. .72 0 , .56 Rulebased 0 .39 0. .56 0. .64 0 . 44 D i s t r i b r i b u t i v e 0 .22 0, .25 0. .03 0 .03 I t h e r Mech/Perceptual 0 .72 0. . 64 0. .42 0. .31 Random 0 .31 0 , .58 0. .47 0 . 61 Incomplete 1 .48 0 .78 0, .92 1 .19 " i n a l A n s w e r 3 .58 3 .33 3. .36 3. .56 * Sl=Structure 1 (var iable l e f t of e q u a l i t y , l e f t of number) S2=Structure 2 (var iable l e f t of e q u a l i t y , r i g h t of number) S3=Structure 3 (var iable r i g h t of e q u a l i t y , l e f t of number) S4=Structure 4 (var iable r i g h t of e q u a l i t y , r i g h t of number Table D.12 Numerical Magnitude Error Means Error Types Levels of Numerical Magnitude* Nl N2 N3 N4 P r o c e d u r a l ZeroAnnexation 0.03 IDconfusion 0.14 Liketerms 1.1.9 P a r t i a l D i s t r i b u t i v e 0.03 OrderOperations 0.08 +prop= 0.97 xprop= 0.19 Co e f f i c i e n t . 0.83 R e s u l t a n t C o m p u t a t i o n a l Add/Sub 0.67 Mult/Div 0.67 BasicFacts 0.28 FaultyAlgorithms 0.17 WrongOperation 0.83 S i g n Add/Sub 1.2 8 Mult/Div 0.75 Rulebased 0.64 D i s t r i b u t i v e 0.39 O t h e r Mech/Pereceptual 0.31 Random 0.33 Incomplete 1.22 F i n a l A n s w e r 4.28 0.03 0.03 0.42 0.28 0.11 0.14 0.67 0.56 0.53 0.00 0.06 0.03 0.03 0.11 0.03 0.44 0.53 0.33 0.06 0.17 0.17 0.22 0.14 0.06 0.92 0.97 1.25 0.36 0.78 0.78 0.31 0.61 0.69 0.22 0.08 0.97 0.92 0.97 0.92 1.25 1.56 0.94 0.39 0.56 0.75 0.61 0.44 0.33 0.00 0.06 0.08 0.79 0.64 0.44 0.28 0.58 0.78 1.36 1.14 0.64 3.44 3.33 2.78 Nl=Numerical Magnitude 1 ( |n| < 20) N2=Numerical Magnitude 2 ( 20 < |n| < 50) N3=Numerical Magnitude 3 ( 50 < |n| < 100) N4=Numerical Magnitude 4 ( 100 < |n| < 1000) APPENDIX E DIAGNOSTIC CHECKLIST FOR ALGEBRA ( L i n e a r One V a r i a b l e E q u a t i o n s ) DIAGNOSTIC CHECKLIST FOR ALGEBRA LINEAR ONE VARIABLE EQUATIONS Objectives Items C o n c e p t s A.1 Variable What do you call the x in 5 + x? A.1.1 Identity Variable W n a t d o e s j t s ( a n d f o f ? A.1.2 Meaning of Variable Tell me how the variables shown are related. 7w + 22 - 109 H o w d 0 y o u Know? 7n + 22 = 109 7 A.2 Expressions A.2.1 Identify Symbols Group these cards by operations so ones representing a + 4 4 + a , n e same expression 4a 4(a) 4xa 4«a are together. a_4 (Cards should be arranged a/4 a 4 4Ja" 4- l n *• in<"ca,9<1 orda') 4 A.2.2 Meaning of Expressions Read each expression aloud and show its meaning Containing • 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 X + 4 4-X J L X-=-4 4 A.3 EquaMty A.3.1 identity Symbol 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 Objectives Items S a n Comments A.3 Equality (continued) A.3.3 Meaning of Equality explain the meaning of each. 7 = 4 + 3 1 2 x 2 = 19 + 5 13 = 13 (listen lor reversal'. 4 + 3-7) How do you know? Have you seen this before? A.4 Equations A.4.1 Given equation with context create meaning 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. A.4.2 Given equation without context create meaning x + 19 = 47 3x + 4 = 10 Read each equation aloud. 2(x + 1) -6 Make up a word problem that someone might have been thinking of when they wrote 'he equation. x + 21 =39 4x + 7 = 23 3{x + 2) = 15 A.4.3 Given meaning create equation 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? DIAGNOSTIC CHECKLIST FOR ALGEBRA LINEAR ONE VARIABLE EQUATIONS OJ ro CJ ; E E Objectives Items s g Comments B. Vocabulary B.1 Coefficient For each expression, tell me the coefficient of n: (Hint: Remember coefficient means the number in front of the letter.) 3n 4 + Sn 7 - 2 n - n (watch for 9n) (watch for 5n) (watch tor negative) I B.2 Like Terms B.2.1 Given the term From the set cards, choose the like terms for 3x: for comparison -3x 3 x 3x 17x 3a B.2.2 Choose from a group Organize the cards into groups of like terms: of expressions - 5 a 5x 5a x 5 - 5 5n B.3 Inverses j B.3.1 Additive (Opposites) Write the opposite (additive inverse) of each j expression on your paper. 7 x 19a -25n 4 x - 9 B.3.2 Multiplicative (Reciprocals) Write the reciprocal (multiplicative inverse) of each expression on your paper. 1. ^ 2 -6x 3 5x B.4 Identities Solve each equation. Place the answer on your paper. B.4.1 Additive (Zero) 8 + n = 8 n + 3 = 3 - 3 + n = - 3 What do you notice about the answers? (point to the numbers and say) Does it matter what numbers we use? (point to the letters and say) Could we use variables (letters)? Do you know any other special properties of zero? B.4.2 Multiplicative (One) 4 x n = 4 n x - 5 = - 5 7n = 7 What do you notice about the answers? (point to the numbers and say) Does it matter what numbers we use? (point to the tetters and say) Could we use variables (letters)? Do you know any other special properties of one? I — |o I I C i p g < 8 3 i 5 I I 2 ro s c s •a «• 0 6" 3 03 ~ ro x -* (/> c 2 S a o > 2 a o a-o If; O CO 8-1' =• £-1 <° •< B 0 i O C -& s < > DO oo s r™ *£• m E" -n O 33 > O z o H m O c > H o z (/) > O z m r O m QJ 33 > O X rn O H Correct Final Solut ion zero annexation identity contusion like terms (conioming) partial distributive order of operations +propeny ol equality X properly ol equality coefficient errors TJ 30 O O m o c 33 > r -v -basic facts faulty Jgori thm .IS 4 wrong operation rule based it/) -ic distributive 33 m v> c r--H z mechanical /perceptual^ random incomplete 202 DIAGNOSTIC CHECKLIST P R O C E D U R A L R E S U L T A N T B~ M> m • R M an* M C o m p u t a i i p n a l S i q n O l h e r FOR ALGEBRA c ,9 Di > > LINEAR ONE S s 111111 s VARIABLE EQUATIONS l\\llUl\ u £ >• * o 8 8 -S » * : ! « ^ g • . t o e ~ ai h - 7. 1 ... 5 ; ; ; ; ; " ° S . 1 Objectives Items O ~ - . - - X O * * » T X it) ~i H E s S i S 9 ^ 5 o v aj ra -c = Q « > y E " 5 ° " 5 $ E S S -= 9 ' •!• ? 5 £ ? o ••' * + K -J 5 C 5 • Comments 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 T • - — I of equality 0 = - 7 - - 1 2x - - 1 7 - 2x J . _ . . .. 711-165x+ "613+ ~175x= - 582 1 ! E i.2 On both sides - 2x - ~8 = 5x-6 of equality 49- -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 13(x-3)= 2x-9 i j of equalitv 52x + - 16 = 15(x - 6) \ -9 + 24x= -9(-10 + x) 66x + 44 = 26(x - 26) E.2 2 On both sides 10(x+ -5l = 20(x-2) of eouality "9(x + 8) = 8(x + -9) . 1 10(10-xl = -9(-9 + x) i 144(6 -x) = 160(5 - x) 

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