REURTEGE-‘éSHEPS BETWEEN SELF-CONCEPT 0F ARlTHMi-ITE’J ABtii‘fl’ AND fiRl'i'i-IMETIG ACHIEVEMENT m A SE’LEC‘EES GROLEP 0F SEXYH GRADE STUDENTS Thesis for the Degree of Ph. D. MICHIGAN STATE UNWERSITY EDWARE} J. HAYES 1957 _.- F; ‘——55 L Us" RA R Y Michigan " rate Unzvcrmty rHtSIL‘I This is to certify that the I thesis entitled RELATIONSHIPS BETWEEN SELF-CONCEPT OF ARITHMETIC ABILITY AND ARITHMETIC A- CHIEVEMENT IN A SELECTED GROUP OF SIXTH GRADE STUDENTS presented by Edward J. Hayes has been accepted towards fulfillment of the requirements for Ph.D. degree in Educational Psychology lbw KW Major professor Date May 24; 1967 0—169 ABSTRACT RELATIONSHIPS BETWEEN SELF-CONCEPT OF ARITH- METIC ABILITY AND ARITHMETIC ACHIEVEMENT IN A SELECTED GROUP OF SIXTH GRADE STUDENTS by Edward J. Hayes 223.Problem The purpose of this study was to test the relationships between arithmetic self-concept and arithmetic achievement as indicated by the results of standardized achievement tests and by teacher evaluation of arithmetic competency in the on- going classroom milieu. Corollary to this purpose was the intent to study the independent variables included in this eXploration in relation to arithmetic self-concept. The Sample The research pOpulation consisted of a group of 161 Sixth Grade students from which a research sample of 144 children, for whom full data was available, was used. Pertinent literature was reviewed and cited within the study in terms of theoretical and practical aspects. Methodology Edward J. Hayes The eXperimental design was based primarily on a Pearson Product Moment Coefficient ef Correlation corre- lational analysis. The Eftest and the Mann-Whitney 9 test were used to analyze significance of differences between research groups for variables within the study. Results A summary of the results obtained from this study follows: 1. Students with high arithmetic self-concept achieved significantly higher in arithmetic, in terms of standardized arithmetic achievement test results, than did children with low arithmetic self-concept: g = 6.15, p_ =<.001. 2. There was no significant difference between the arithmetic self-concept of girls and arithmetic self-concept of boys, though the arithmetic self-concepts of girls was higher than the arithmetic self-concept of boys: 3.: .74, p =) .05. 3. Students with high arithmetic self-concept obtain- ed significantly higher teacher-assigned grades in arithmetic than students with low arithmetic self-concept: 2.: 5.87, p = < .001. 4. Girls received significantly higher grades in arithmetic than boys: E’= 1.90, E.=‘<'05‘ 5. There was a significant positive correlation be- tween scholastic aptitude, as measured by the California Edward J. Hayes Short-Form Test of Mental Maturity, and arithmetic self- concept: E -= .51, p_ =<.01. 6. There was a significant positive correlation be- tween arithmetic achievement test results and arithmetic self-concept: E. = .62, p =<.01. 7. There was a significant positive correlation be- tween teacher-assigned arithmetic grades and arithmetic self-concept: £’= .71, p_=‘<.01. 8. There was a significant positive correlation be- tween total academic grade point average and arithmetic self-concept: £’= .68, p = (.01. 9. There was a low, though significant, positive cor- relation between mothers' educational level for students in the sample and arithmetic self-concept: E = .23, p_=‘(.02. 10. There was no significant relationship between mothers' educational level for girls in the sample and arithmetic self-concept: _r_ = .13, p_ =>.05. 11. There was a low, though significant, negative cor- relation between a composite social-status ranking for children in the sample and arithmetic self-concept: £’= -.30, p II-<.01. 12. There was no significant relationship between social-status ranking and arithmetic self-concept for boys: _r_ 8 -.22, 3 => .05. 13. There was a significant positive correlation be- tween general self—concept of ability and arithmetic ._._-— _- — _— ‘— ——.— _— — _— _— _ _. achieveme scores: E 14. eral self cept of a 15. ' correlati' teacher-a. 16. '. Correlatic subjeCts a P = (.01. 17. I] latiORShi; for Child; ility, r = 18. 'I ily 5°Cia1 tY f0: bOy Genera ings ShOul t° the Stu ECOROmic a Edward J. Hayes achievement as indicated by standardized achievement test scores: g = .48, p ={.01. 14. There was no significant difference between gen- eral self-concept of ability of boys and general self-con- cept of ability of girls in the sample: £_= .06, E.=)"05° 15. There was a moderately high significant positive correlation between general self-concept of ability and teacher-assigned arithmetic grades: _r_ = .61, p =< .01. 16. There was a substantially high significant positive correlation between total grade point average for academic subjects and general self-concept of ability: £‘= .67, p =<.01. 17. There was a low, though significant, negative re- lationship between a composite family social-status ranking for children in the sample and general self-concept of ab- ility: E. = -.32, p =<.01. 18. There was no significant relationship between fam- ily social-status ranking and general self-concept of abili- ty for boys: 5 = -.20, 2 => .05. Generalizations which might be made concerning the find- ings should be viewed within specified limitations inherent to the study, which include grade level, geographic, socio- economic and ethnic-cultural restraints. This study should be considered primarily explorative, but with some replica- tive implications. The conclusions derived in the study have supported the i 14—“——‘—_—_-—_—‘_— Operatior metic sel to arithx achieveme grades. Thou using a p WOIK of s underlyim jUStify f1 Edward J. Hayes Operational hypotheses of the research, namely that arith- metic self-concept is significantly and positively related to arithmetic achievement as indicated by both arithmetic achievement test results and teacher-assigned arithmetic grades. Though conclusions are tentative, the rationale for using a perceptual frame of reference within the frame- work of self-theory seems feasible for study of factors underlying learning. The findings are sufficient to justify further study in this context. ACKNOWLEDGEMENTS To Dr. Don E. Hamachek, who provided direction and guidance as project director, I owe deep gratitude and appreciation. Dr. Hamachek was the source of support and encouragement throughout the course of the project. To the members of my doctoral committee, Dr. G. Marian Kinget, Dr. Lorraine Shepard and Dr. Robert L. Green grati- tude is due for guidance and assistance during the course of my doctoral program of study. Appreciation is also owed to Dr. John L. Johnson, presently of Syracuse University, who served as a member of my guidance committee during a crucial period in the develOpment of the dissertation project. To Mr. Jack Greenstein and Mr. Russell Christie, of the administrative staff of the Algonac, Michigan Community Schools, I wish to express my appreciation for making this study possible. To Mr. Bruce Rogers and Mr. Edward A. Holovka for con- sultation and technical suggestions in statistical methods and design, my thanks are extended. Their assistance is reflected in an important part of the study. Finally, to my wife Rose and my children Brian, Kathleen, Patrice, Karen, Kevin, Keith and Bruce I dedicate this work. RELATIONSHIPS BETWEEN SELF-CONCEPT OF ARITHMETIC ABILITY AND ARITHMETIC A- CHIEVEMENT IN A SELECTED GROUP OF SIXTH GRADE STUDENTS BY .'(‘\ V Edward J} Hayes A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Educational Psychology College of Education l967 Abstract Chapter I: ’ Nature 1 The Pro Researc Thecret 1 Need fc Definit 2 Overvie Suzzary Chapter II: Nature DeveIOp Relatio RQIatic Arithre Implica Classro Sunfiary Chapter III: Prepato The pod The San: SeleCti SElecti InstrUf PIECEdu PrcC9du I‘J . General PErati Arithre Hypothe G47 / l: 0/ q. 77."l g [P Jr— pl TABLE OF CONTENTS Abstract Chapter I: Introduction Nature of the Problem ------------------------- Research Objectives -------- - ------------------ Theoretical Considerations -—----—----——------- Need for the Study ---------- ————— - ........... - Definition Of Terms -------------------- ————— -- Overview of the Study ----- - ----- -_------- ..... Chapter II: Review of the Literature Nature of the Self-Concept -------------------- DeVGIOpment Of the Self-Concept - ----- - -------- Relation of the Self-Concept to Learning ------ Relation of Affective Characteristics to Arithmetic Achievement ----- - --------- --------- Implications of the Self-Concept for the Classroom ------------- ..................... --- smnmary --------------------—------------- ----- Chapter III: Design and Methodology The POpulation ---------— ------- --------------- The S amp le --------------------—-—------------- Selection of the Variables -—----——--- ......... Selection and Description of the Instruments Used ---------- ............... ----- Procedures for Collection of Data ------------- Procedures for Analyzing the Data ------------- summary ------------ ----- -------- -------- ------ Chapter IV: Hypotheses and Limitations of the Study General HypOtheseS ---------------------—------ Operational Hypotheses Related to the Arithn‘tetic Self-Concept ----------—-—---——-—--— Hypotheses Related to the General Self-Concept of Ability ----------------------- Limitations and Scope of the Study --------- -—- Summary --------- ----- - ----- - .................. Chapter V: Analysis of the Research Data Arithmetic Self-Concept Data --—-— ............. Hypotheses Related to the General Self-Concept of Ability -------- ............... ii 42 42 43 45 49 50 54 55 58 6O 6O 60 62 63 63 64 64 91 Other Surya: Chapter VI : Proble Desic: Surxaz Ccnclu Discus Reccr: iii Other Relevant Results ----- ---------- - -------- 99 Summary --- ------ - ------ ------ ----------------- 100 Chapter VI: Summary, Findings and Conclusions, Discussion and Recommendations 106 Problem -- --------- - --------------------------- 106 Design and Procedure ---------------- ----- ----- 107 Summary of the Findings ------------- -------- -- 109 Conclusions -- ---------------- - --------------- - 115 Discussion - ----------- --------—---- ------- ---- 119 Recommendations for Further Study - ------------ 124 BibliOgraphy: 126 Appendix: 138 3:1 h2 h3 $1 $2 $3 5:4 $6 $7 UV on a.) Number < for Tcta Number c for Sam; Analysis of Total Group fc hean Scc Self-Cor and Arit Upper ar Data CO: UPPer ar Arithret DAta Cor CPS?! a: Arithret Data CO? Concepts self'Cc; Data Cor TESt SCC AChiEVer Data CO? metic G1 ined in lned in Populaéj 5:1 5:2 5:3 5:4 5:5 5:6 5:8 LIST OF TABLES Number of Children in Teaching Groups for Total Population -------------- -------- ---- Number of Children in Teaching Groups for Sample POpulation ----------------—-- Analysis of Difference between Means of Total Group and Means of Sample Group for Research Variables - ------ --- -------- Mean Scores of 1.0., Arithmetic Self-Concept, Arithmetic Achievement and Arithmetic Grades According to Upper and Lower Quartile Levels ------ --------- Data Comparing Arithmetic Achievement for Upper and for Lower Quartile Divisions of Arithmetic Self-Concept Scores ----- ----------- Data Comparing Arithmetic Grades for Upper and Lower Quartile Divisions of Arithmetic Self-Concept Scores ---------- Data Comparing the Arithmetic Self- Concepts of Boys with the Arithmetic Self-Concepts of Girls --- -------- ------- Data Comparing the Arithmetic Achievement Test Scores of Boys and the Arithmetic Achievement Test Scores of Girls ------- ------- Data Comparing the Assigned Arithmetic Grades of Boys and the Assigned Arith- metic Grades of Girls -----_--_----- ........... Correlations between Arithmetic Self- Concept and Specific Variables Exam- ined in This Study for Total POpulation ------- Correlations Between Arithmetic Self- Concept and Specific Variables Exam- ined in This Study for Sample POPUlation ----------—--------- ---------------- iv ----- y 46 47 48 65 66 67 69 70 70 72 73 5-9 Different . Correlati l Concept_a Examines Grade PC; Group fr: 1 l 1 L hlo Correlati Self-Coho Variables Study - E Edi Correlati } Self-Conc 1 Variables Study - c 553 Differenc Of Correl ‘ Self-Cone ‘ Variables for BOYS 5‘13 Correlati Self-conC variableS Group I - Sell-‘Conc': Variab‘! i6 8 Stacy — G SSS Difffitenc Correlati COHCept: “Kamineda and II 5-16 COrre lat 0 variables Smule 5 .17 Di lfcerenc Variabl:s Differea no 0f CO orrel: 5:12 5:13 5:15 5:16 5:17 5:18 Differences between Coefficients of Correlation for Arithmetic Self- Concept and Specific Variables Examined in This Study for Sixth Grade POpulation and for Sample Group from Population -- ----------- Correlations between Arithmetic Self-Concept and Specific Variables Examined in This Study - Boys - -------------- -—----- Correlations between Arithmetic Self-Concept and Specific Variables Examined in This Study - Girls - -------------------------------- Differences between Coefficients of Correlation for Arithmetic Self-Concept and Specific Variables Examined in This Study for Boys and Girls in the Sample -- Correlations between Arithmetic Self-Concepts and Specific Variables Examined in This Study - Group I -- --------- - --------------- Correlations between Arithmetic Self-Concept and Specific Variables Examined in This Study - Group II ---------- ------- - Differences between Coefficients of Correlation for Arithmetic Self- Concept and Specific Variables Examined in This Study for Groups I and II --- ------------------ ----- Correlations between General Self- Concept of Ability and Specific Variables Examined in This Study - Sample -- ---------- - --------------- Differences between Coefficients of Correlation for General Self- Concept of Ability and Specific Variables Examined in the Study for Total and Sample Groups ------- Differences between Coefficients of Correlation for General Self- 74 75 76 77 78 79 80 81 82 Concept Variable for Eoys $19 Correlat Self-Cor as Measu Form Tea h20 Correlat ConCEpt Standard 5:21 Correlat Concept metic Gr 5:22 Correlat Concept Grade PO “23 Correlat Concept tional L 5224 Correlat Concept Economic 5:25 Correlat Oncept 5:26 COrrelat Concept Assigned Grade PO Correlat onceht ‘ ~umily S 5:19 5:20 5 21 5:22 5:23 5:24 5 25 5:26 5:27 5:28 5:29 vi Concept of Ability and Specific Variables Examined in This Study for Boys and Girls in Sample ------------------ Correlations between the Arithmetic Self-Concept and Scholastic Aptitude as Measured by the California Short- Form Test of Mental Maturity ------ ------------ Correlations between Arithmetic Self- Concept and Arithmetic Achievement on Standardized Arithmetic Achievement Tests ----- Correlations between Arithmetic Self- Concept and Teacher-Assigned Arith- me tic Grade S ------.---c-—----------.—- ————————— Correlations between Arithmetic Self- Concept and Sixth Grade Total Academic Grade Point Average - --------- ----- --------- --- Correlations between Arithmetic Self- Concept and Students' Mothers' Educa- tional Levels -- --------------------- -- -------- Correlations between Arithmetic Self- Concept and A Measure of Family Socio- Economic Status -- --------------- - ------------- Correlations between General Self- Concept of Ability and Arithmetic Sub-tests -- ------------- ------- ------------- -- Correlations between General Self- Concept of Ability and Teacher- Assigned Grades ------------------- ------------ Correlations between General Self- Concept of Ability and Total Academic Grade Point Average -- ------------- - ----------- Correlations between General Self- Concept of Ability and A Measure of Family Socio-Economic Status --- --------------- Differences between Coefficients of Correlation Among Arithmetic Self- Concept and General Self-Concept of Ability and Specific Variables Ex- amined in This Study for the Sample Group -- ----- - --------------------------------- 83 84 85 86 88 89 9O 92 93 94 96 98 vii Data of the Multiple Coefficients of Correlation between the Arithmetic Self-Concept Scale and Independent Variables in the Study ----- ------------------- 99 Partial Correlation Coefficients and Beta Weights for the Arithmetic Self-Concept Scale and Independent Variables in the Study ---------~--- ----------- 100 Data for the Multiple Coefficients of Correlation between the Arithmetic Applications Sub-test and Components of the Arithmetic Self-Concept Scale ---------- 102 Partial Coefficients of Correlation and Beta Weights for the Arithmetic Self-Concept Component Questions and Arithmetic Applications Sub-test -- -------- 103 44 A. The Pre-T Scale --- The Self- Student - The Self- Teacher - Stanford ‘ Californi O ”1 Id 0 LIST OF APPENDICES The Pre-Test of the Arithmetic Self-Concept The Self-Concept of Arithmetic Ability Scale - Student -- --------- ----- ------ ----------------- The Self-Concept of Arithmetic Ability Scale - Teacher -- ..................................... Stanford Achievement Test - --------- - ----- ----- California Short-Form Test of Mental Maturity - McGuire-White Measurement of Social-Status ---- Matrix of Intercorrelations ------------------- viii 138 142 151 154 156 158 164 CHAPTER I INTRODUCTION Nature of the Problem The technonomy of the twentieth century, character- ized by dramatic advancements in science and mechanics, places a high premium on education for its peOple. It has been said that the proliferation of knowledge during the present era is such that technical skills essential to many contemporary industrial, scientific and investi- gative Operations will be rendered obsolete in less than a decade. Demographers and planners estimate that approx- imately sixty per-cent of the peOple living at the end of this century will be employed in technically oriented po- sitions that do not even exist today. Automation, al- ready responsible for the elimination of more than 200,000 factory jobs annually, will leave the uneducated behind. Clearly there is in prospect a mandate for expan- sion of our educational objectives. There is a need for improved and increased education for greater numbers of our peOple to enable them to adapt to the demands of the technical epoch into which we are inexorably moving. This need arises when noted educators such as Conant (35) sug- gest that only about fifteen to twenty-five per-cent of our high school youth can profit by high level education in mathematics, science and foreign languages. Fundamental to technical adequacy is a competency in mathematics and basic to this is a good start in arithme- tic. However, Brown (17), in a recent publication of the National Council of teachers of Mathematics, reveals that though school enrollments have increased over the years the per-cent of student time spent on study of mathema- tics has steadily declined. Price (104), in The Revolu- tion in School Mathematics, writing about the impact of progress on mathematics in the school, indicates that too few students gain an adequate knowledge of mathematical concepts in school to prepare them for advanced technical training and education. As a consequence we are deprived of the technical personnel required to enable us to re- main competetive with other leaders in technology through- out the world. According to Woodby (145), in contempor- ary America many essential technical positions go unfilled because of the dearth of trained manpower. Recent enrollment figures in mathematics classes at the secondary levels do not reflect a cognizance of the need for more students trained in mathematical princi- ples. The need is amplified by the warning that the stu- dent who does not take at least three years of high school mathematics is so handicapped that many fields of study are permanently closed to him when he reaches college. That there is a shortage Of mathematicians has been emphasized by Price (104:10), writing Of the progress in mathematics and its implications for the schools. He writes: As a result of the revolution in mathematics there is an unprecedented demand for mathematic- ians and mathematics teachers -- it is impossi- ble to foresee a time when there will be an ade- quate supply. There is some reason that we are faced with this shortage Of competent mathematicians at all occupational levels ranging from education to scientific Operations and research. It is apparent that there is a need to investigate the reasons that students dO not elect math- ematical courses. TO maximize the educational potential- ities Of our students and to provide our economy with the trained personnel necessary for implementation Of our technical commitments the charge tO eXplore the possible causes Of student reluctance tO elect mathematics courses appears imperative. Investigation has been done into the cognitive areas Of arithmetic learning, but little research has been made of the affective correlates Of arithmetic competencies. Weaver and Gibb (141), writing in the Review Of Educa- tional Research, recommend examination Of personality variables as possible influences on the development Of difficulties in arithmetic (mathematical) growth. The Problem It is the purpose Of this study to investigate the relationship between a measure Of arithmetic self-con- cept and arithmetic achievement in a selected group Of sixth grade students. It is recognized by psychologists and educators that innate intellectual factors do delimit learning ability: however, during the course of individual psycholOgical examination Of children with learning problems it occurred to the writer that there is possibly an important rela- tionship between the concept a learner holds of self- ability in arithmetic and potentialities in relation tO arithmetic achievement. The strategy is to explore, isolate and describe relationships among measures Of arithmetic self-concept, general self-concept Of ability, arithmetic aptitude, arithmetic achievement, teacher evaluations and student variables related to 1.0.. sex, a designation Of family social-status, and the educational level Of the mother. Research Objectives Clinical, educational and social psychologists have, in recent years, attempted to explore the psychologically relevant configuration in which learning and behavior oc- curs through study Of the self-concept in a variety Of experimental designs. The specific Objectives Of the research are: 1. TO explore background information related to students' arithmetic self-concept which could be related tO arithmetic achievement. 2. TO investigate relationships among student res- ponses on an arithmetic self-concept scale; scores on a group achievement test; scores on a group ap- titude (I.Q.) test: teacher evaluations (current and cumulative) and rankings Of family social-status. 3. TO investigate differences in arithmetic ach- ievement that may exist between students with high arithmetic self-concepts and students with low arith- metic self-concepts. Theoretical Considerations Theoretical direction for the present investigation evolves from the social psychOlOgy Of Mead (92); the per- ceptual approach to individual behavior as delineated by Combs and Snygg (32), and the self-theory of Rogers (111). A basic postulate is that the capacity for intelli- gent behavior is dependent upon the state Of one's per- ceptual field. The self-concept Of learning abilities is seen as a functionally facilitating or limiting factor in academic achievement since a poor self-attitude im- plies a lack Of confidence that adversely affects general adjustment and proficiency in school. A corollary prOposition is that the self-concept functions as an integrating force in school achievement to the extent that a student will achieve at a level con- sistent with his perceptions Of his ability to achieve. Much motivation for sound learning comes from the child's drive to enhance his self-concept through positive self- achievement, and to protect it by avoiding failure. An adequate self-concept is related to high academic ach- ievement and an inadequate self-concept is related to low achievement. The self-concept is learned, it is a product of ex- perience and interaction with others in the social mil- lieu: it is dynamic and is in continuous involvement with living and with learning at school. Bases for these postulates will be established in Chapter II of this study. Need for the Study In education an understanding of the dynamics under- lying achievement gains and/or deficits is important to educators at the theoretical, practical and administra- tive levels. There is research evidence supportive Of the idea that the self-concept is an important factor in the determination Of how children act, learn and function in life. The thesis presented in this study is that de- finitive knowledge of the affective influences Operant in mathematical/arithmetic performance is essential to effective planning for individual differences that occur in children in the on-going teaching-learning process. There has been related research which examined relation- ships between self—concept and academic achievement, Brookover (16) and Weatherman (140). NO study, however, has attempted an in-depth probe Of the possible relation— ship between the arithmetic self-concept and measures Of arithmetic achievement. Definition Of Terms Self-concept is a personal perceptual framework that includes the individual's conscious attitudes and under- standings Of himself, his experiences and his capacities in relation to his surrounding world as motivation for behavior and learning. Arithmetic is defined as the understanding of rules for combining two or more numbers. It is the art of com- putation and the application of this art. Arithmetic self-concept refers to the individual's perception Of his understanding Of arithmetic functions: his expectations of future arithmetic capabilities: his appreciation of arithmetic as a socially relevant discip- line: his interest in arithmetic as a tool for present and future economic existence, and his aspiration for future use of arithmetic (mathematic) skills. Operationally the arithmetic self-concept refers to responses on items one through twenty-five on the arith- metic self-concept scale: an adequate arithmetic self- concept is equated with a score deviating above the med- ian on the arithmetic self-concept scale. General self-concept of ability refers to the stu- dent's perceptions Of his ability to do school work and his attitudes toward the evaluative-competetive aspects of school. Functionally the general self-concept of ability re- fers to the student's responses to the fifteen test items of the original Michigan S3332 University Self-Concept 25 Ability S5212 cited by Brookover and associates (16). Academic aptitude refers to those competencies spec- ifically related to academic success in terms Of measured potentialities. Functionally aptitude is defined as the student's score on a recently administered standardized test scale. The California Short-Form Tg§E_g£ Mental Maturity, 22221 _2_, £2.63 revision was the test used. Arithmetic achievement is the level at which compe- tency in arithmetic skills is empirically demonstrated by test scores. These scores were determined by perfor- mance on the arithmetic sub-tests of a group administered standardized achievement test taken by each child in the sample. The Stanford Achievement 2255! Intermediate I Batte '.l3§i revision was the achievement test used. The Arithmetic Applications sub-test score was designated as the score indicative of arithmetic achievement for general purposes. Teacher evaluation is regarded as the letter grade conferred upon students from teacher perceptions of arith- metic competency in the on-going classroom activity. Teacher evaluation is Operationally defined as the marks listed for arithmetic achievement in the student's permanent school record for the quarterly marking period during which the arithmetic self-concept scale was ad- ministered; the final semester mark and a cumulative av- erage of arithmetic marks received ingrades three, four and five. The total grade point average for academic subjects in the sixth grade was also computed. The Social-Status Index is a numerical designation representative of the socio-economic status of the fam- ilies of the children included in the study. Social-status is functionally defined as the com- posite of ratings made of socio-economic factors of fam- ilies of children in the study in terms of parent educa- tion, father occupation, type of dwelling and residential area. The McGuire-White Measurement 2; Social Status was the instrument used to yield this designation. Mothers' Education is the indication of the highest .grade in school attended by the mothers of children in the sample. I Operationally the mothers' educational level was re- presented by the_grade level of highest attendance listed 10 in the permanent school record of their children. Overview of the Study In Cha ter I, the introduction presents the rationale and the overall objectives of this study of the relation- ships between arithmetic self-concept and measures of arithmetic achievement and scholastic aptitude. Nature of the problem, statement of purpose, strategy, research objectives, theoretical considerations and need for the study are included in this section. In Chapter II literature and research relevant to the problem is reviewed. Theoretical concepts and defin- itions pertaining to the self-concept, develOpment of the self-concept, relation of the self-concept to learning, relation of affective characteristics to arithmetic ach- ievement and the classroom implications of the self-con- cept are included in this section. In Chapter III the research design and methodology are delineated. Prepatory activities, the population, and the sample are defined. Selection of the sample, sel- ection of the variables, selection and description of the instruments are explained. Finally, procedures for col- lecting and analyzing the data are outlined in this sec- tion. In Chapter IV contains the hypotheses and limita- tions of the study. The general hypotheses, Operational hypotheses and limitation and scope of the study are de- ll fined in this section. In Chapter V an analysis of the experimental results is made with appropriate tables to aid in the interpre- tation. Chapter VI is a summary of the findings and conclu- sions: a discussion of the findings and recommendations are included in this section. Summary The general purpose of this study was to investigate the relationships that may exist between a student's arithmetic self-concept and measures of arithmetic ach- ievement, scholastic aptitude and family social-status. The strategy was to explore, isolate and describe rela- tionships between the dependent variable, arithmetic self- concept, and the independent variables of I.Q., sex, arith- metic achievement, teacher grades and designations of family social-status. The principle hypothesis of this investigation was that there is a significant positive relationship be- tween arithmetic self-concept and arithmetic achievement. CHAPTER II REVIEW OF THE LITERATURE In recent years a number of research projects have been conducted to eXplore what aspects of the student's affective make-up operate in the learning process. Many of the current research projects reflect the growing in- terest of educational psychologists in the phenomenon of self-concept as an influence on behavior, and in partic- ular the behavior underlying academic achievement. The involvement of educational psychologists in the investi- gation of characteristics of the self-concept represents a move from the psychotherapeutic emphasis and direction that characterized much of the early research into the dynamics of self-concept. In regard to this transition, Combs and Saper (33), comparing teachers' perceptions of the helping relation with views held by therapists, con- cluded that there are common characteristics in_good help- ing relationships wherever they are found; and no appre- ciable differences exist between the percepts of teachers and the attitudes of therapists toward the helping rela- tion. For the purpose of the present investigation only 12' 13 those studies that are related to the development of self- concept and its role as a precursor of academic attitudes are reviewed. Nature of the Self-Concept After nearly half a century of relegation to virtual oblivion, self-theory has emerged with such vigor that one finds it threaded through the fabric of almost all current psychological thought. Patterson (100), tracing the development of Rogerian Self-Theory, comments that the self is today becoming of central importance in all theories of personality. In this regard, Arieti (5) sees the prominence accorded the self in modern psychological theory as testimony of the emergence of the psychology of personality as a field of scientific endeavor. Self-theory, during the era that James (663291) de- fined a man's self as the sum total of all that a man calls his own, was associated primarily with the labora- tory and the lecture hall. In a more modern era the im- petus was initially given to the deveIOpment and expan- sion of a theoretical frame-work by clinical psychologists and psychotherapists. In contemporary times, self-theory is being adopted by a growing number of educators and psychologists. For instance, Combs and Snygg (32), Jersild (69), Rogers (111) and Symonds (128) have been the guiding lights in the deve10pment of ideas enunciated by Cooley (36), Lecky (82) and Mead (92); they have adap- l4 ted constructs related to self-theory and, with numerous other colleagues, have added significantly to the practi- cal body of knowledge of that theory. Since a number of contemporary practioners and theo- rists in the fields of education and psychology have con- tributed materially to the accumulation of scientific knowledge relevant to self-theory there have been some semantic difficulties to overcome. There has been a ten- dency for writers to personalize concepts and refer to an idea they hold in common with others by an ideosyncratic term. In this respect, the terms ego and self are fre- quently used synonymously despite the distinction that Chien (28) and Smith (120) made between the two entities: they construe the ego as a motivational-cognitive struc- ture built up around the self and working interdepend- ently in reference to the self, but apart from awareness. Awareness is central to self-concept as defined by Combs and Snygg (32), Jersild (69) and Rogers (111). Some of the terms compatible with self and used in- terchangeably in self-theory are: the proprium used by Allport (2); the phenomenologiggl.§gl£'of Snygg and Combs (32): the gglffsystem delineated by Sullivan (126), and the gglffregard of Wylie (147). The important con- sideration of these terms, despite their verbarian dif- ferences, is the implication that all represent an ef- fort to understand the behavior of the individual from 15 his own perceptions of his position in the environment in which he moves. Self-concept is basic to self-theory and is that facet of personality that is, by definition,accessible to investigation. Self-concept has been defined in the following ways: Ausubel (6:273): The self-concept is an abstrac- tion of the essential and distinguishing character- istics of the self that differentiates an individ- ual's "self-hood” from the environment and from other selves. In the course of develOpment vari- ous evaluative attitudes, values, aspirations, motives and obligations become associated with the self-concept. Combs and Snygg (32:127): The self-concept is an individual's attempt to reduce his self organiza- tion to its essence so that he may be able to per- ceive and manipulate it effectively: it is the generalization of self which aids in perceiving and dealing with self. DeLisle (43:7): The self-concept is the under- standing which the individual has of himself in relation to his surrounding world as a motivating drive for his behavior. Dinkmeyer (41:183): The self-concept is really the individual's anticipation of hisgeneral ac- ceptance or rejection in a given situation. Fink (53:58): The self-concept is the attitudes and feelings a person has regarding himself. Jersild (69:116): The self-concept is a composite of a person's thoughts and feelings, strivings and hopes, fears and fantasies, his view of what he is, what he has been, what he has become, and his atti- tudes pertaining to his worth. Raimy (105:155): The self-concept is the map which each person consults in order to understand himself, expecially during moments of crisis or strife. It is what a person believes about him- self 0 16 Shostrom and Bremmer (116:138): The self-concept defines one's nature (I am), one's capacities (I can), one's values (I should or I shouldn't) and one's aspirations (I want to be). Thorne (134:110): Self-concept involves not only a continuing awareness of what one is in the pre- sent but also the record of what one has been in the past, and also what one would like to be in the future. Thus the self-concept consists of (a) memories and contexts stemming from past his- tory, (b) conscious awareness of current mental states and (c) expectancies or imaginings con- cerning what one might be in the future. The essence of these definitions is that self-con- cept is a constellation of one's conscious feelings and attitudes about himself in terms of four aspects ennum- erated by Symonds (128): 1) how a person perceives him- self: 2) what he thinks of himself: 3) how he values him- self: 4) how he attempts through various actions to en- hance or defend himself. Self-theorists, such as Combs (29), Jersild (68), Mead (92), Raimy (105) and Rogers (111), postulate that self-concept is the conscious core of an individual's be- havior that mediates behavior by serving as an intermed- iary between perceptions and behavior. There is research evidence to suggest that one behaves in ways which are consistent with his self-concept. For example, Lecky (82) found that what a person is able or unable to learn depends, to a large extent at least, upon how he has learned to define himself. In this respect, the rele- V‘ancy of self-concept to the understanding of inceptive, tYlpical and on-going behavior seems crucial -- particu- l7 larly in terms of student performance. Combs and Snygg (32) indicate that self-concept is a useful structure for studying the individual since it can be interpreted as a convenient approximation of the personality because it represents the most stable, impor- tant and characteristic self-perceptions of the individ- ual. The self-concept is a shorthand description of the innumerable discrete perceptions of self an individual possesses. Brownfain (19), in his doctoral study of the stabil- ity of self-concept, concluded that self-concept is an important point of reference for any evaluation a person makes of himself and thus is servicable to the work of understanding and predicting behavior. Benjamins (ll), in an investigation involving 48 high school students as subjects and designed to relate reactions in terms of self-conceptualization to measurable behavior, showed that behavioral reactions related to external influences on changing self-conceptualization can indeed be asso- ciated with measurable components of the self-picture. To Combs (29) the perceptual field is an organiza- tion continuously changed and modified by the perceptions occuring within it: Maslow (87) associates this phenomenon with the constant striving to self-actualization -- to express and to achieve potentialities. Carlson (25) and ‘Engel (49), in separate longitudinal investigations of 18 stability and change in self-concepts of adolescents, con- cluded that self-concept is a relatively stable dimension of the self that is related to the level of self-esteem. High self-esteem, according to their findings, is associ- ated with adequate school related social adjustment. Lazarus (80), in his book about psychological adjust- ment, writing of the conscious factors that influence per- sonality growth and adjustment cites the self-concept as an important factor in understanding individual behavior. He perceives self-concept, as a representation of an in- dividual's understanding of present behavior, as an im- portant construct through which those interested in under- standing human behavior can gain valuable insights. Such data, he suggests, can be gained through a self-report technique used to measure aspects of human personality related to interests and attitudes. Development of the Self-Concept Self-concept does not emerge, as Venus was supposed to have, full grown: it is a genetic structure that de- velops through a process of social interaction mediated by maturational factors. Mead (92), in this regard, theo- rized that though the self is not present at birth because of its social nature very early eXperiences play an im- portant role in its development. Anderson (4) relates, due to the relative helplessness of an infant, the first year of life is vital to the development of the self-con- 19 cept. Ausuble (6) also posits that social, indogenous and perceptual processes established in early life eXper- iences have an important influence in ultimate personal- ity development which includes the self-concept. Symonds (130), writing of the origins of personality, relates that even in the first year the mental processes of the infant are concerned with the reactions of others toward him and out of these reactions of others he builds his concept of himself. With regard to the impact of early experiences on the growth pattern, self-theory indicates that later stages of development are naturally dependent upon the beginning phases of self since early experiences do delimit the possibilities of later ones. The self-concept of each individual is unique and is a synthesis of the experiences and social interactions he incurs during the course of his life. Kelley (73), writing about the fully-functioning self, states that the self consists, at least in part, of the accumulated ex- periential background, or backlog, of the individual. Writers such as Combs and Snygg (32), Dinkmeyer (41) and Lecky (82) tell us that the way an individual is treated by parents and significant others in his life is an im- portant determinant in the development of the self-con- cept since the introjection of the values of others plays a significant role in the unfolding of the self. In res- pect to its development, however, Hilgard (64) suggests 20 that the self-concept is not an unfolding of an inevit- able pattern. Sarbin (114), analyzing the psychological genesis of the self, has schematized its deve10pment this way: Stage: 51. Somatic self (undifferentiated); age one month. 82. Receptor-effector self (differentiated- vocal): age three-to-four months. 83. Primitive construed self (differen- tiated to others): age six months. S4. Introjecting-extrojecting self (res- ponds to commands): age ten-to-fourteen months. SS. Social se1f(use of I, me, mine, etc.); age twenty-four months. Davids and Lawton (39), studying interrelationships among a child's self-concept and the child's concept of the mother in a group of normal pre-adolescents and a group of hospitalized emotionally disturbed children, de- duced that results lent empirical support to the theore- tical convictions that mother-child relations play a cru- cial role in theuformation of the child's self-concept. Medinnus (93), investigating self-acceptance in a group of college freshmen, found that self-acceptance is related to perception of maternal attitudes. His findings indicate that the mother exerts a greater influence than the father on the child's personality development because of the Child's identification with the mother: the findings obtained for both boys and girls. Pertinent to this influence of mothers, Drucker and Remmers (45), in an exploration of the environmental determinants or correlates of 21 adjustment through analysis of attitudinal responses of 1000 seventh and eighth graders to the SEA 22252 Inventory, concluded that the level of education of the mother was significant in relation to adjustment or lack of adjust- ment as indicated by the Basic Difficulties items on the Eggth Inventory: pf .01. Helper (63), studying indicants of parent reward as impinging upon children's self-descriptive behavior, also cites parental influences as an important factor in the shaping of the self-concept. Using a sample of 53 child- ren at the seventh and eighth grade levels he tested the degree of correlation between parental reward for emula- tion of like-sex parent and the child's self-concept. From the results Helper concluded that parental reward is positively correlated with self-concept modeling: 5f .40. Ausubel (6), Dinkmeyer (41) and Prescott (103) per- ceive the schools as significant in producing feelings about the self. Since self-concept is a product of ex- periences, what happens to the child during the time spent in school becomes an important factor, particularly since the school is a place where the child is constantly com- paring himself with others. Concerning this, Swift (127), evaluating the effects of early group experiences in nur- sery school, relates that peer influences operate in the develOpment of self-concept during very early school- oriented experiences. The way children react toward him 22 and the way other children perceive him influences a child's own perception of himself as a social person. Bandura and Huston (7), working with 48 nursery school children in an investigation to demonstrate that children learn a good deal of social behavior imitatively, inferr- ed from results of their study that children do diSplay a good deal of social learning of an imitative sort: nur- turance on the part of the model facilitated such imita- tive learning in the study. Pertinent to the school-re- lated influences affecting growth of the self-concept, Kipnis (75), studying the relationship between interper- sonal perception and the process of changing self-concepts concluded that the relationships between a subject's per- ception of his friends and changes in self-concept indi- cated strongly that interpersonal percepts and self-per- ceptions are closely related: pf .01. The self-concept is not static -- changes in the self-concept do occur and can be associated with matura- tion, social experience and learning. Beck (10) theorizes that though by the age of five or six attitudes toward self and others have taken shape they are amenable to change: and in this vein, Rogers (113) writes that under certain psychological conditions needed reorganization and restructuring of the self can be facilitated. Changes in self-concept can be effected through enabling one to alter his perception of self, his abilities and the values 23 which he holds by establishment of an atmosphere condu- cive to such change. The relative plasticity of self- concept is a construct of child development that should have pertinency, according to Symonds (128), for parents and teachers in relation to their roles in shaping child- ren's self-concepts. In this context Symonds (128:110) has written: Children respond not only to what is said to them and about them but also to the attitudes, gestures, and subtle shades of expression that indicate how parents and teachers feel. Indeed, these more subtle expressions may be taken by the child as representing the parent's true atti- tude toward him. Illuminating this point made by Symonds, Bower (15) has suggested that the psychological processes of children are subject to neither moral or logical restrictions. Most self-theorists, including Jersild (69), Mead (92) and Rogers (111) underscore the importance of social interaction on develOpment of self-concept. Indeed, Combs (30) has stated that no-one can evolve into anything re- motely resembling humanness without interaction and in- tercourse with others and the environment. Relation of the Self-Concept to Learning The self-concept, because it is fundamental to the understanding of behavior, is an important principle to the analysis of the affective antecedents of learning since, according to Combs and Snygg (32), the capacity for intelligent behavior and the development of abilities 24 are rooted in the self-concept. Lane (77), in a book on the education of human beings, perceived feelings and at- titudes as basic determiners of a learning experience: and Dinkmeyer and Dreikur (42), writing about the encourage- ment process in learning, suggested that what the child decides to do depends largely on his own concepts, his perceptions of himself and others and his method of find- ing a place for himself. Relative to this Lecky (82:178), in his compact but cogent book on self-consistency,wrote: Academic difficulties and social maladjustments are both conceived as due to resistance arising from the subject's conception of himself. If a student shows resistance toward a certain type of material, this means from his point of view it would be inconsistent for him to learn it. Amplifying this proposition, Dinkmeyer (41:184), in a book about the self written in the contextual framework of child development, suggests that self-concept serves to integrate and differentiate experiences: he has writ- ten: The self permits the child to act, to adjust, to do more than merely respond to specific stimuli. Murphy (97:75) further states in relation to self-con- cept and learning: The self is dependent upon attributes or traits, things that one can do, things that are acceptable or unacceptable to others, things that win status or prestige in a group. Jersild (70) has observed that the ability to learn is grounded in the individual's perception of his capac- ity to learn, and Rogers (111) adds that an individual's 25 attitude toward learning is related to his personal ad- justment. Allport (3) gives substance to this postulate in his thesis that learning is affected by self-awareness in terms of attention, judgement, memory, motivation, as- pirational level, productivity and personality traits: Ausubel (6) regards such temperamental and personality characteristics as facilitative to problem solving tech- niques. It has been suggested that poor self-attitudes imply a lack of confidence that adversely affects general adjustment to the environment and proficiency in school. Silverman (118), studying the relationship of self-es- teem to responsiveness to success and failure in which he was attempting to establish that high self-esteem sub- jects show little response to self-devaluating stimuli, concluded that there is a tendency for both high and low self-esteem subjects to limit their cognitive input to information which is congruent with their self-image. In a research project nuclear to the study of the academic relevancy of self-concept, Wattenberg (139), in research with kindergarten children devised to determine whether the association between self-concept and academic achievement had its origin in self-concept being causal to achievement, or if, by contrast, the experience of aca- demic success or failure played a part in the formation of self-concept, concluded that as early as kindergarten self-concept phenomena are antecedent to and predictive 26 of academic accomplishment -- at least in reading. Symonds (129) has stated that failure is intimately related to concept of self because it is a function of the level of aspiration which is an expression of self- appraisal. Bruck and Bodwin (20), examining this aspect of the association between self-concept and academic ach- ievement in 60 children referred to a child guidance clinic, found that a positive significant relationship exists between educational disability and immature self- concept as measured by the Self-Concept £2113 2f the Magh- gygg.gégflggg§: 5f .60. Borislow (13), in a study de- signed to investigate the importance of self-evaluation as a non-intellectual factor in scholastic achievement in a group of 197 college students, reached the conclusion that students who underachieve scholastically have a poor- er conception of themselves as students than do achievers: pf .01. In the same vein, Dabbs (38), working with 88 college psychology students and investigating the pro- cess underlying defensive resistance to attitude change, found that high and low self-esteem subjects differ in their characteristic mode of adjustment: either they ac- tively approach and attempt to cope with their environ- ment, or they react to it in a passive, noncOping manner. Rath (107), in a study of the relationships between self- concept and achievement in a group of college freshmen, decided that those who achieve, as well as those who do 27 not, do so as a result of the needs of their own self- system. Pink (53), studying the relationship between self- concept and achievement, concluded that a relationship does in fact exist between adequate self-concept and level of academic achievement. His conclusions appeared to be unquestionable for boys, but less definite for girls: 2 for boys= .01: p_for girls- .10. In relation to the impact of self-concept on academic achievement or underachievement, Jackson and Strattner (67) offer the tenet that a person with a positive atti- tude toward a subject learns more readily than a person with a negative attitude because the former possesses a more stable and elaborated perSpective into which to fit new material. In respect to this observation, Coopersmith (37) found that a personal conviction of adequacy provides the prerequisites for effective focussing of efforts even in stress-laden situations. Also in regard to this con- cept, Delisle (43) indicates that a significant aspect of the feeling about the self relates to the ability to ach- ieve in competition with others. Brookover and associates (16), in a longitudinal study, determined that self-concept of ability is signif- icantly related to school achievement of seventh grade boys and girls: £_was .57 for both boys and girls in the study. Haarer (58), in a similar research project, but 28 limited to delinquent and nondelinquent boys, obtained corresponding results in reference to the relationship be- tween general self-concept of ability and classroom ach- ievement. Washburn (138) submits that it becomes important to consider personality factors such as self-concept along with academic achievement as a predictor of long-range academic successes. Jersild (70:113) underlines this sen- timent about the learner with this counsel: If he is convinced he is stupid, this conviction is likely to close many avenues of life to him (the student) that might in reality be Open. Relation of Affective Characteristics to Arithmetic Ach- ievement Even though innate cognitive characteristics related primarily to intellectual abilities such as perception, language skills, intelligence and problem solving ability are acknowledged as a delimiting factors in learning po- tential, it is evident, from the study of learning prob- lems, that there are other influences of a non-cognitive sort that affect academic learning: these affective fac- tors appear to Operate in all areas of school-related learning including arithmetic. Some affective character- istics related to attitudes and values have been cited in relation to general learning in previous sections of this chapter. An attempt will be made to illuminate affective foundations of arithmetic learning in this section. 29 Horrocks (65:441), in an assessment of behavioral OOgnates Of arithmetic learning, indicates that readiness to learn and manipulate numbers is a function not only Of the individual's stage of intellectual develOpment, but also of his attitude toward numbers. Lerch (83), in this regard, in an article develOped to show that arithmetic instruction can change attitudes toward arithmetic, sub- mits that experiences in arithmetic classes play a signif- icant role in the develOpment of attitudes toward arith- metic: such attitudes are related to achievement in arith- metic, according to him. Concerning the inception of at- titudes toward arithmetic, Smith (119), tracing prospec- tive arithmetic teachers' attitudes toward arithmetic, learned that feelings toward arithmetic are develOped at all stages of our arithmetic instructional curriculum. Attitudes toward arithmetic were develOped in the major- ity of such students while they were in elementary school. That non-cognitive characteristics play an important role in the develOpment Of arithmetic seems indicated by Harap (61) who, in a controlled experiment stratifying children according to I.Q., found that 1.0. variance played a relatively insignificant role in the learning of specif- ic arithmetic tasks such as decimals: 2F .05. Bassham (9), studying relationships between pupil af- fective traits and achievement in arithmetic, found that attitudes and interests appear to act as catalysts in 30 arithmetic achievement. In his sample Of 159 pupils, over four times as many pupils with a poor attitude to- ward arithmetic were classified as .65 grades below ex- pected achievement: almost three times as many high-atti- tude pupils overachieved .65 grades. As a corollary to his study he suggests that attempts to favorably influence interest in arithmetic should be centered on the modifica- tion of pupils' perception of subject material in rela— tion to self by changing the perception or by changing the method of presentation. In relation to the develOpment of arithmetic attitudes, Aiken (l), exploring the effects of attitudes on performance in mathematics, concluded that attitudes toward arithmetic are the result Of ex- periences with parents and teachers. Combs and Snygg (32) posit that since behavior is a function of perception the factors which govern percep- tion will determine the nature and degree of ability which an individual possesses. And Cattell (26), writing about acquired motives and the learning process, tells us that attitudes tend to be organized in relation to the individual's self-concept. In this regard, Lindgren (85: 232) has written: If a child sees himself as a good reader, but an indifferent and hOpeless cipherer, the time and trouble spent on getting him to practice arithmetic skills will be to no avail. Combs and Snygg (32:57) suggest an explanation to his phen- omenon 8 31 To the child, the learning of arithmetic -- or whatever -- may seem to have nothing whatsoever to do with the quest for self-enhancement. Flora (55), writing about the things that cause child- ren to avoid mathematics, counsels that there is an ele- ment of fear in a child's rejection of arithmetic rather than difficulties intrinsic to the curriculum. Wilson (143). explaining why children avoid mathematics, suggests that the fear of arithmetic is conditioned. Concerning this idea, McDermott (89), in a doctoral study of some factors that cause fear and dislike of arithmetic and mathematics, concluded that attitudes toward arithmetic are attribut- able to teacher influences, parental indoctrination or sib- ling conditioning. Turlock (135), writing about emotional blocks in arith- metic, states that pupils who constantly fail mathematics have deflated egos and tend to develop attitudes of dis- like and hostility toward mathematics. Brookover and asso- ciates (16), in a longitudinal study of self-concept of ability and academic achievement, obtained a significant correlation coefficient between arithmetic self-concept and arithmetic grades in junior high school students: 5; .57 for both boys and girls. Pertinent to this, it would appear that the argument whether boys or girls achieve sig- nificantly higher in arithmetic has hardly been resolved: writers such as Ausubel (6) and Gibbs (56) indicate that boys do better, so far as arithmetic achievement is con- 32 cerned, than girls do at the elementary school level: Aiken (l), Weatherman (140) and Wozenkraft (146) counter that girls do better in arithmetic than boys do at the ele- mentary school level. Weaver and Gibb (141), in Review 2; Educational 527 search, recommend examination of personality variables as possible influences on the develOpment of mathematical learning. Witty (144) provides a rationale for investigat- ing the effects Of attitudes on arithmetic achievement in his charge that when meaningful methods of teaching arith- metic are used, changes in arithmetic attitudes take place: negative attitudes, according to him, become positive and positive attitudes become enhanced. He indicates that sig- nificant changes in arithmetical computation and reasoning are associated with changes in arithmetic attitudes. The importance of research in the field of mathematics is emphasised in the contention by Cairns (24) that capa- bility in mathematics is essential for simply relating one's self to the environment in which he must live, work and find enjoyment and satisfaction in a science oriented so- Ciety 0 Implications of the Self-Concept for the Classroom Despite the fact that such writers as Ausubel (6), Dinkmeyer (41) and Prescott (103) perceive the classroom atmosphere as significant in producing feelings about the self, few teachers are acquainted with self-theory, except 33 at the intuitive level, because it is a construct relative- ly new to education and for the most part has not been a part of teacher training. Since self-concept is a product of experiences, what happens to the child during the time spent in school becomes an important factor in its devel- Opment. Symonds (128) relates, in this context, that con- cepts of self are reflections of the attitudes expressed a person by others and sees this as an indication of the power that parents and teachers have in determining the kinds of selves that children will develOp. NO social institution outside the family exerts more important influences on the develOpment of self-concept than the school. The self-concept is an important construct in respect to effective classroom Operation, particularly when one takes cognizance of the point made by Ellis (48) that, due to the social aspects of self-evaluation, an individual is taught that because others dislike or disapprove of one when he fails to master something, he should accept this evaluation and make it his own. Self-concept is susceptible to the competitive and evaluative aspects of education. Lighthalls (84), writ- ing about the impact Of anxiety in relation to thinking and forgetting, indicates that threats to self-concept such as censure or invidious comparison causes children to be preoccupied with fears of failure. In respect to 34 the effect of anxiety on learning, Combs and Taylor (34), exploring the consequences Of a mild degree of threat on rote learning in a group of 50 college students, inferred that even mild self-perceived threat can seriously affect the adequacy of learning behavior. They concluded that, if even a mild perception Of threat can affect the ade- quacy of behavior, their findings represented a challenge that has far reaching implications for psychotherapy, edu- cation and social action. The group dynamics inherent to classroom activities provide potent Opportunities for influencing formation of self-concept. The importance of group forces as a social and educative determinant is suggested by findings in re- cent research projects by such investigators as Dittes (44), Kipnis (75), League and Jackson (81), Miyamoto (94) and Stotland and Cottrell (124). Davidson and Lang (40), studying children's percep- tions of teachers feelings in a group of 203 middle-school- aged children, found that teacher feelings, as the child internalizes them, have an effect on school adjustment. Staines (123), in his report of an eXperiment conducted abroad supportive Of this point, concluded that teaching methods can be adapted so that definite changes in self can be planned and effected in the classroom: 2f .01. According to Staines, the self can be deliberately pro- duced by teaching methods. The inference derived from 35 these findings is that it is essential that teachers com- municate positive feelings to their students and thus not only strengthen self-appraisal but also stimulate academic growth. Bostrom and associates (14) suggest one method Of reinforcing desired attitudes in the learning process: in studying grades as reinforcing contingencies in altering learning behavior in a sample of 127 university students, they concluded that good grades serve to reinforce atti- tudinal changes of a desired sort in emotionally charged learning circumstances. In this same tenor and particular- ly relevant to arithmetic, Lerch (83) suggest that it would seem that, if desirable attitudes toward arithmetic are to be develOped and undesirable attitudes are to be changed, arithmetic students should be assured of a certain mea- sure Of success. That significant others play an important role in the determination of self-concept and in the alteration of self-concept has been suggested in several recent studies: Helper (63), demonstrating this idea, in an experiment ap- plying reinforcement stimulus-response theoretical concepts with 50 boys, deduced that reward techniques can be an im- portant method for shaping self-concept and encouraging learning when a significant model is used. He used the like-sex parent in a situation in which emulative behavior of a desired sort was promoted: 5f .40. Bandura and Kupers 36 (8), in an unrelated research project involving 160 ele- mentary-schOOl-aged children and designed to investigate the transmission of patterns of self-reinforcement, showed that the behavior of models is influential in shaping pat- terns of self-control. They found that the introjection of attitudes was facilitated when a positive, prestigious model was used and reward for imitation of desired attitudes was an integral part Of the encouragement process. Videbeck (136) also concluded from an investigation of the effects Of reaction of others on the develOpment Of self-concept that self-conceptions are learned and that evaluative re- actions of others play a significant part in that learning process. LaPere (78), reporting on an experiment involving a treatment group and a placebo group Of ninth grade students in a counseling situation, indicates that self-concept can be improved with a collateral improvement in academic ach- ievement through working with parents and significant Others. Erickson (51) also, reporting on a study Of the normative influences of parents and friends upon academic success, reinforces the idea that parents and teachers do exert an important influence on self-concept and learning behavior of students. Jourard and Remy (72) obtained sub- stantiating results in a study of children's percepts of parent's attitudes in relation to self-concept. An important consideration in respect to the amena- 37 bility to change in self-concept is the tenet that there is some pliability to it, particularly during the school- age years. Piers and Harris (101), in an interim report on an on-going longitudinal study, indicate that there are fluctuations in self-concept between the third and tenth grades. Self-concepts of sixth grade children differ sig- nificantly from self-concepts Of third and tenth grade students according to their findings: 2? .01. Morse (96), also in a preliminary report of findings in a longitudinal study Of self-concept conducted by the University of Mich- igan, relates that as a child progresses through school his confidence diminishes and self-regard decreases. He implies that teachers need knowledge of the child's self- concept for almost every decision he makes in the on-go- ing classroom activities. Jordan (71:30) echoes this sen- timent in these words: It seems to me that without a concept which en- ables me to think at least of a self that can whistle, with all the problems of control, direction, and creativity that this involves, I must teach more or less blindly. Pertinent to the growth pattern inherent to self-concept, Erickson (50) prOposes that the period Of sexual latency is a period during which the child is most susceptible to school influences. Since self-concept and academic achievement-adjust- ment appear intimately related, it would seem that teachers should be sensitive to Combs' (30) indictment that we re- 38 ject too many peOple in our schools with a resultant in- adequacy Of self-concept and diminution Of COping mechan- isms. Prescott (103:379) has this to say about the re- sults of such rejection: An inadequate concept of self, so common in our culture, is crippling to the individual. When we see ourselves as inadequate we lose our 'can-ness.” There becomes less and less that we can do. The consequence Of an inadequate self-concept was illus- trated by Lafferty (76), who, in a study of values that defeat learning, found that significantly more children who underachieve seem convinced that they cannot, be their own effort, alter the course of a failing experience: 2? .05. Despite the fact that Taylor (132) indicates that findings in his doctoral dissertation about consistency of self-concept show that persons with positive self-concepts are more numerous than persons with negative self-concepts, we patently have a charge to work toward adequacy of self- concept in our educational processes. Teachers sometimes may interfere with this objective and in this regard Montagu (95) gave voice to the enjoinment that, in relation to the develOpment of self, teachers have a job to do on themselves before working on others. Kelley (73:13) reflects this feeling in these words: It is that the representatives of the school -- teachers and administrators -- often have their own ends to be served, not those of the learners. They act from their own fears, which cause them to dampen 39 and delimit the expanding personalities of the young thus defeating the very purpose for their being. The task suggested here, it should be pointed out, is not, according to self-theorists such as Combs (29) and Rogers (112), one of changing the student's behavior, but is one of creating a situation in which changes in perceptions will be encouraged and facilitated. Only as a child has the Opportunity to have experiences which will alter his concepts about his schoolwork, his teacher, his peers and other significant models can the concept of self be changed. With knowledge of self-concept educators could be more effective in the helping of children in school: with- out such knowledge, according to Allport (3), we would be unable to increase the breadth of learning and the trans- fer of effects. Perkins (102) states that the self can be used as a psychological construct that will enable teachers, counselors, parents and others to achieve with training deeper understanding and insights into the devel- Opment and behavior of children. In research of the literature related to self-concept and the learning process one sees important implications for the training of the teachers who will staff our class- rooms in future years. Preservice and in-service training projects should, for instance, take cognizance of Staines' (123) findings that teaching methodology can be adapted to effect definite changes in the self: he showed that teach- 40 ers can shape the self-concept through planned teaching experiences. Since changes in the self-picture are an in- evitable consequence of learning in every classroom, whether the teacher is aware of them or not, school-related influ- ences become an important consideration for educators -- particularly as Dinkmeyer (41), Rogers (111) and Symonds (130) apprise us that the more deeply attitudes are engrain- ed in self-concept through individual percepts, the more difficult they are to change. Changes do occur, however, at every stage of the child's develOpmental sequence. The conclusion would appear to be that the schools can best assure the develOpment of children into non-threat- ened, socially adjusted individuals by enabling them to gain effective and healthy group identification, self-know- ledge and self-acceptance. Summa The intent of this chapter has been to review the lit- erature Of self-theory with especial emphasis on self-con- cept as it is related to learning. A principle aim of this purpose is to present self-concept as a measurable complex of feelings that grows through social and school interactions and that is related to school achievement with particular reference to arithmetic achievement. An attempt has also been made to fit some affective and non-cognitive aspects of arithmetic learning into the framework related to elements of self-concept. 41 An evaluation of the literature indicates that the theoretical structure of self-theory is in a process of growth impelled by psychological and educational research. An attempt has been made to show the changing complexion ‘pf research into the construct Of self-concept by educa- tors, psychologists and sociologists. An inference made from the literature is that self-theorists, though moving toward agreement, are still confronted with semantic prob- lems. Recent empirical studies have demonstrated the per- tinency of self-concept to academic achievement. Accepting self-theory principles as discribed in this chapter, the hypothesis would appear to be that the learn- er's self-concept of his learning ability can be a facil- itating or interfering factor in academic achievement -- including arithmetic achievement. CHAPTER III DESIGN AND METHODOLOGY The design and methods used in this study are des- cribed under eight main headings: 1) Prepatory Activities: 2) POpulation: 3) Sample: 4) Selection of the Variables: 5) Selection and Description of the Instruments Used in this Study: 6) Procedures for Collecting Data: 7) Proced- ures for Analyzing the Data. Prepatory Activities In preparing for the present study a number of instru- ments theoretically designed to test the self-concept of children in relation to adjustment and/or academic achieve- ment was studied. Among the scales studied were instru- ments develOped and used by the following authors in school- related research with children: 1) Bowers (15): 2) Brookover (16): 3) Carlson (2S): 4) Fitts (54): 5) Lafferty (76): 6) Ringness (108) and 7) Rogers (109). Some, because of the findings of Laxer (79) and Lipsett (86) that discrepancy scores add little to the knowledge of an individual, were arbitrarily rejected because of their design. Study Of the design, format and intent Of the scales led to the dis- card of Others of the scales examined. The Michigan State 42 43 University_Self-Concept pg Ability S2212 used by Brookover and associates (16) in a series of longitudinal studies conducted through the Office of Research and Publications, Michigan State University under the auspices of the COOper- ative Research Program Of the Office of Education, United States Department of Health, Education and Welfare, was selected as the self-concept scale most amenable to modi- fication for the needs of the present study. Permission to modify the Michigan State University Self-Concept gfiAbility S2212 for needs inherent to this study was obtained from Dr. Wilbur Brookover, Principal Investigator, for COOperative Research Project NO. 845, “The Relationship of Self-Images to Achievement in Junior High School Subjects", conducted by the College of Educa- tion, Michigan State University. A self-concept scale, or self-report depending upon one's orientation, was selected because Ausuble (6) and Rogers (111) relate that a self-report is frequently the only way of directly investigating the subjective life of human beings, their attitudes, beliefs, motives and per- ceptions. Jersild (68) has concluded that, in terms of self-concept, the child has more capacity for understand- ing himself than educators, or others, have ever realized. Further, Dubin (46) asserts that the framework of the self- image cannot be inferred from an outsiders Observation. The Population 44 This study was conducted in the Algonac, Michigan Community Schools during the 1965-1966 school year and was terminated in June, 1966. The Algonac Community Schools serve the Village of Algonac and Environs. The Village of Algonac has a present pOpulation of approximately 3200 citizens, but the school system serves an additional pOpu- lation in excess of 2000 residents of neighboring residen- tial and rural communities. The population Of the Village prOper had remained relatively constant during the two dec- ades following World War II, but the surrounding area has been extensively develOped since that time, particularly during the 5 years immediately preceding the time of this study. The resort complexion of the area and its accessi- bility to the Detroit MetrOpOlitan area with recent high- way development and improvement programs contributed to this expansion. The Village of Algonac is primarily a resort community situated on the St. Clair River, a linking body of water on the St. Lawrence Seaway route. The major industry is boat building: however, many of the wage-earners of the area commute daily to jobs in the industrial centers of the Detroit, Michigan MetrOpOlitan complex. The school pOpulation consists of approximately 1800 children in grades kindergarten through twelve. There are three public and one parochial elementary school: one in- termediate school housing grades six through nine, and one 45 three year high school recently constructed. The district serves as a center for a cooperative vocational education- special education program. The curriculum is considered fairly representative of the type found in most urban school systems of comparable size in the United States. The ele- mentary curriculum includes: teaching of basic social skills: teaching of fundamental reading, writing, spelling and arith- metic skills: education for general science, social stud- ies (history, geography and civics), health, safety, phys- ical education, art education and music education. There is a planned and implemented articulation with and contin- uity between the curricula Of the elementary, intermediate and high schools through the Offices of a curriculum di- rector. Mentally retarded children are placed in special edu- cation programs Operated in the cOOperative program refer- red to above. The Sample The subjects for this study included the sixth grade pOpulation Of the Algonac, Michigan Community Schools. There was a total of 167 students in the sixth grade at the end of the school year in June, 1966. Of the 167, test data was obtainable for 161 in part, and for 144 gn_tg£g, Transfers and absences during periods Of testing accounted for the lack of data for some Observations in the sample. The children in the study were grouped into two sec- 46 tions through use Of random numbers. The students were placed in groups to be taught on a team - teaching basis. One section was designated an experimental group and the other section a control group. The differences in ap- proaches were largely semantic since the control group of approximately seventy students was taught in a block - time sequence with a team of two teachers sharing the instruc- tional responsibilities: the experimental group, composed of approximately ninety-five students, was taught by a team of three teachers sharing teaching responsibility. From each team one teacher - member was assigned the task Of teaching arithmetic. Thus, in this study, two teachers were involved in the teaching of arithmetic in the sixth grade. A preliminary report indicates no significant differ- ence between the groups so far as aptitude and achievement is concerned. Table 3.1 Number of Children in Teaching Groups for Whom Partial Data Was Obtainable N=161 W m (Time-Block) (Team Teaching) Total Boys 29 47 76 Girls 33 22 §_5_ 64 97 161 47 Table 3 02 Number of Children in Teaching Groups for Whom Total Data Was Obtainable N=l44 W m (Time-Block) _(Team-Teaching) Total Boys 24 42 66 Girls 2 £1 7_8_ 55 89 144 A statistical analysis was made of the differences between means for the dependent and independent variables for the group of 161 and the group of 144. NO significant differences were observed so the decision was made that the elemination of the 17 observations for which only par- tial data was Obtainable would not affect results of the study. Analysis was made through use of Michigan EEEEE University_Computer Laboratory MDSTAT Routine (Calcula- tion of basic statistics when missing data is involved). Results are shown in Table 3.3. 48 Table 3.3 Analysis of Difference between Means of Total Group and Means of Sample Group Total Group Sample Group Variable Mean Mean tftest ‘p N=161 =144 1. Family Social Status Rank 4.57 4.52 .33 NS 2. Mother's Educational Level 10.94 10.96 .07 NS 3. General Self- Concept 50.30 50.26 .05 NS 40 Arithmetic Computations 19.63 19.65 .02 NS 5. Arithmetic Concepts 17.08 17.25 .20 NS 6. Arithmetic Applications 20.69 20.74 005 NS 70 F111]. Scale 1000 104.79 104079 -0- 8. Language I.Q. 103.97 103.97 -0- 9. Non-Language 1.0. 104.97 104.97 ~0- 10. Number Values 7.51 7.51 -0- 11. Number PrOblemS 6022 6022 '0- 12. Current Arithmetic Grade Average 7.61 7.63 .06 NS 13. Semester Arith. Grade Average 7.53 7.53 -0- l4. Cumulative Arith. Grade Average 7.93 8.00 .23 NS 15. Total Grade Point Average 7.68 7.70 .06 NS 16. Arithmetic Se1f- ___QQEEERE: 88.94 89.18 .13 NS The sixth grade was chosen for this study for several reasons: 1) in the present situation the sixth grade rep- resented the only grade at the elementary level in which all children in one grade were concentrated in one area: 2) more important, the sixth grade is a period during which 49 skill repetition and reinforcement is diminished -- drill is, to a large measure, discontinued in basic tool areas. 3) The sixth grade represents a transitional level between elementary and secondary school programs. The group of sixth graders used in this study were selected to avoid curricular difficulties that might arise because of the arithmetic variable in terms of instruction- al methodology related to “modern math". “Modern math" had not yet been introduced into the curriculum of the Algonac Community Schools. Selection of the Variables The variables included in this study were chosen for practical as well as theoretical reasons: relationship of the dependent variable to the independent variables as in- dicated by related research, and the accessibility of data related to the specific variables. Factors that have been associated with academic achievement and child growth and develOpment were selected. Research cited in Chapter II suggests that there are several factors that appear to be related to academic ach- ievement of children. The variables used in this study were: 1) general self-concept of ability: 2) arithmetic self-concept: 3) sex of the child: 4) educational level of the mother: 5) family socio-economic ranking: 6) intelli- gence quotient: 7) academic achievement in terms of stan- dardized achievement test scores, and 8) teacher evaluation 50 of arithmetic competency. These variables were defined in Chapter I. Selection and Description of the Instruments Used 1. General Self-Concept of Ability was measured by the Michigan gtgtg_0niversity Self-Concept pf Ability S2213 in its original form. The scale consists of fifteen items measuring a student's perception Of his present and antic- ipated school ability, and of his feelings toward the eval- uative - competitive aspects of school. The scale was sel- ected for its established reliability and validity in terms of measuring attitude toward school desired for this study. 2. Arithmetic Self—Concept was measured by an exten- sion of the Michigan S2322 University Self-Concept‘gf Ability - Specific. The arithmetic self-concept scale in- cluded eight test items related to specific subject ability (arithmetic) from the original model augmented with seven- teen questions designed to measure additional arithmetic attitudes in accordance with theoretical formulations of Dutton (47). The arithmetic self-concept scale was devised to measure the students' perception of his understanding Of arithmetic functions: his relative competence in manip- ulation of arithmetic functions: his expectations of future arithmetic capabilities: his appreciation of arithmetic as a socially relevant discipline: his interest in arithmetic as a tool for modern existence: and his aspirations for future use of arithmetic skills. The scale consists of 51 twenty-five items. The scale was develOped to measure conceptual and attitudinal components of the self as delineated by Jersild (69:116). Validity and reliability tests were made on the instrument prior to use in the present research. 3. The family socio-economic ranking was determined through use of the McGuire-White Measurement 9; Social Status. The instrument provides an index of socialestatus through an averaging of rankings assessed for each family in the study on the basis of parent education, father oc- cupation, type of family dwelling and area in which the family lives. The social-status ranking was Obtained with the cOOperation of Mr. Russell Christie, the school prin- cipal, who has an intimate knowledge of the community and Of the residents Of the community. Areas of the community were visited to assess dwellings and residential areas. 4. Intelligence Quotient was Obtained for children in the sample during the regular school testing program. T22 California M £935.“). _'I_'g_s_t_ 9; Mental Maturity is de- signed tO yield an 1.0. that correlates highly with 1.0. scores on the full scale California 2225725 Mentalngthf ity. A language intelligence quotient, a non-language in- telligence quotient and a full-scale intelligence quotient are yielded by the test. The aptitude tests were adminis- tered by classroom teachers, under the direction of the Guidance Director, at the termination of the fifth grade 52 as the children were preparing to enter the sixth grade. 1132 California $1953 2223 _'I_'_e_§_t 9_f_ Mental Maturity, m g, 1.31:3}; E, l_9_6_3_ revision was used. 5. Academic achievement scores in arithmetic con- cepts, arithmetic computations and arithmetic applications sub-tests were Obtained from results of testing with the Stanford Achievement 2252, 2252 W, Intermediate II, 22E? plgtg_Battery:‘lg§4 revision. Raw scores were used for this study since converted scores would provide no addi- tional pertinent detail in terms of relative scores. The achievement battery was administered by the school princi- pal under the direction of the Guidance Director. Testing was done during three morning sessions and to the total sixth grade pOpulation in joint session to control for test variations as much as possible. 6. Teacher evaluations were Obtained for each child in the study for the marking period and semester during which the achievement tests and the self-concept scales were administered. Grades were obtained from the perman- ent school record. A cumulative grade point average for third, fourth and fifth grades were Obtained for all child- ren in the research sample through research of the perma- nent school records. To investigate possible differences that could arise in teacher appraisals, the two arithmetic teachers assign- ed to sixth grade arithmetic instruction were asked to re- 53 spond to a questionaire modeled on the arithmetic self- concept scale, and designed to yield a self-concept of arithmetic teaching ability. Because of the small number involved, statistical analyses of results were infeasible. Both teachers were male: both teachers had taught ele- mentary school classes four years: both teachers had taught arithmetic at the sixth grade level for four years: both teachers had participated in two methods courses in teach- ing of arithmetic since beginning to teach arithmetic. The teacher in Group I, the control group, differed from the teacher in Group II, the eXperimental group, in that he had a minor in mathematics at the undergraduate level, and the teacher in Group II had less than a minor in Mathema- tics. The teacher in Group I scored somewhat higher on the self-concept scale, but differed from the teacher in Group II only in that he ranked his satisfaction with his professional training higher in regards to mathematics pre- paration than did the teacher in Group II: and he express- ed a preference for teaching mathematics over other sixth grade subjects which the teacher in Group II did not share. The teacher in Group II reported he enjoyed teaching arith- metic only so much as he enjoyed teaching other sixth grade subjects. Oddly, the teacher in Group II ranked himself as one of the best in the teaching or arithmetic while the teacher in Group I ranked himself only above average in the 4! loitll El. Ibéfi 54 teaching of mathematics. Results indicated that teachergrades could be accep- ted at face value in terms of classroom arithmetic achieve- ment, and that teacher differences would not affect results significantly. 7. The educational level for mothers of each child in the study were Obtained from the permanent school re- cords. The permanent school records indicate the highest level of elementary school, secondary school or institu- tion of higher learning attended by the mother of each child in the study. 8. A total grade point average for all academic sub- ject areas at the sixth grade level were computed for each child in the study. The final sixth grade marks for each academic subject area were Obtained from research of the cumulative school record of each child in the study. Provisions were made in the data collection to analyze the above information on the basis of sex differences as well as on a total group basis. Girls were expected to have higher arithmetic self-concepts than boys have. Procedures for Collection of Data In the final month of the fifth grade the subjects were administered the California Short Form Test 2; Mental Maturit , Form g, Level II, 1963 revision as part of the school testing program. The tests were administered by the classroom teachers under the supervision Of the Direc- 55 tor of Guidance Services of the Coordinated Educational Program mentioned above. In the last month of the sixth grade, June 1966, the Stanford Achievement 2222, EEEE.EJ Intermediate I}, 1264 revision, Complete BatteEy was ad- ministered to the total sixth grade group in joint sessions in the assembly hall by the Intermediate School Principal under the supervision of the Director of Guidance Services. Testing was done during three morning sessions under close- ly controlled conditions. Four days after administrating the achievement battery, the Michigan S5232 University General Self-Concept pg Ability Sgglg and the arithmetic self-concept scale were administered to the children in the research samples by the author. Testing conditions were closely controlled, directions were explicit and the testing conditions involved were the same for each group. The language, non-language and full I.Q. scores were obtained from the California gaggg'gggm,Mental Maturity 2225' as were the number values and number problems sub- test results of the non-language portion Of the test. The arithmetic computations, arithmetic concepts and arithme- tic applications sub-test scores were used from the S232: ford Achievement Test battery. Procedures for Analyzing the Data The analysis Of data in this present study was done in these phases: The Pearson Product Moment correlational analysis of 56 the data to investigate relationships between the depend- ent and independent variables was used. The formula appears below. N .— t-1(xit'zi) (th'xj) rij- N ‘- 2 N " 2 (X ‘X ) (X 'X-) t=1 it i t=1 jt 3 McNemar (91) and Guilford (57) cite these assumptions an necessary to the use of coefficients of correlation in establishing relationships between two variables: 1. The measurements must be represented in at least an equal interval scale (it must be possible to use the Operations of arithmetic on the scores, i.e. add- ing, dividing, finding means, etc.). 2. The population must be bivariate and normal. 3. The Observations should be normally distributed. 4. The Observations must be homoscedastic (disper- sions must be approximately equal and symmetrical). S. The regression of scores must be linear. Normal distribution is not always necessary so long as the distribution is symmetrical and unimodal. Since the pOpulation and data in the present study satisfy these requirements a correlational analysis was utilized. The .05 level Of confidence was arbitrarily desig- nated the level of significance associated with rejection rIII ulv...lllu' | ILI! Ill, 57 of the null hypotheses. The significance of difference in arithmetic achieve- ment between students with high arithmetic self-concept scores and students with low arithmetic self-concept scores were tested by the Mann-Whitney U-Test as described by Siegel (117:121). The formula appears below: U-nln2 z= 2 V/(nl) (n2) (nl+n2+I_ 12 The upper and lower quartiles scores of the arithme- tic self-concept distribution were used to show differences in arithmetic achievement. Students with 1.0. scores i two standard deviations from the mean were excluded from the sample for this analysis. Since the distribution would not be considered normal a non-parametric test was selected. Siegel (117) considers the Mann-Whitney U-Test one of the most powerful of the non parametric tests, and sees it as a most useful alternative to the parametric tftest when the tetest assumptions do not hold. The .01 level of confidence was established as the level Of significance associated with rejection of the null hypothesis. The Estest as described by Walker and Lev (137:156) was used to analyze differences between girls' arithmetic self-concept and boys' arithmetic self-concept. The form- 58 ula is shown below: 56-3? t= 1 2 \/ Sle'HVz N1N2 The .05 level of confidence was chosen for accepting or rejecting the null hypothesis. An additional study Of the data was made through use of the Coefficient of Multiple Correlation, as described by Guilford (57:390), to analyze the amount of relation- ship between the dependent variable arithmetic self-concept and the independent variables in this study. It was sel- ected to test the relative contributions of the independent variables to arithmetic self-concept. A Coefficient of Multiple Correlation was also used to analyze the degree of relationship between test items in the arithmetic self-concept scale and arithmetic achieve- ment as indicated by achievement test scores of the arith- metic applications sub-test of the Stanford Achievement 2225. It was desired to test the relative contributions of the test items to arithmetic achievement. .05 was designated as the minimum level of signifi- cance 0 Summary The sample selected for this study included the sixth grade pOpulation of the Algonac, Michigan Community Schools. 59 An instrument was develOped to test the arithmetic self-concept of elementary aged children after prepatory investigation of self-concept scales used in similar stud- ies. The arithmetic self-concept scale was administered to all students in the pOpulation from which the sample was drawn. Related achievement, aptitude and socio-economic data were obtained for children in the sample. A design based on correlational analysis and a test of differences between means was used to study statistical significances among measures of arithmetic self-concept, academic achievement and scholastic aptitude. Relation- ships Of the arithmetic self-concept to variables of aca- demic achievement, scholastic aptitude and rankings of family socio-economic level were investigated. The .05 level of confidence was selected to indicate acceptance or rejection of the null hypotheses. CHAPTER IV HYPOTHESES AND LIMITATIONS OF THE STUDY On the basis of theoretical formulations and research findings discussed in Chapter II of this study the follow- ing hypotheses are made concerning the relationships be- tween arithmetic self-concept and measures of student apti- tude, achievement and socio-economic status: General Hypothesis Stated in general terms the fundamental hypothesis of the present study is: there is a significant positive rela- tionship between arithmetic self-concept and arithmetic achievement. Operational Hypotheses Related to the Arithmetic Self-Con- £222 H:l There is no significant difference in arithmetic achievement as measured by scores on a standardized achieve- ment test between those students with high arithmetic self- concept and students with low arithmetic self-concept as indicated by scores on the arithmetic self-concept scale. H:2 There is no significant difference in arithmetic achievement as indicated by teacher grades for arithmetic 60 61 performance in the on-going classroom instructional situ- ation between students with high arithmetic self-concept and students with low arithmetic self-concept as indicated by scores on the arithmetic self-concept scale. H:3 There is no significant difference between the arithmetic self-concept of boys and the arithmetic self- concept of girls when scores on the arithmetic self-concept scale are used as the criterion. H:4 There is no significant relationship between scores on the measure Of arithmetic self-concept and scores on a measure of scholastic aptitude (I.Q. scores). H:5 There is no significant relationship between scores on the measure of arithmetic self-concept and scores on the arithmetic sub-tests of the Stanford Achievement 2223' H:6 There is no significant relationship between scores on the measure of arithmetic self-concept and arith- metic grades. (Individual correlation coefficients will be computed for current, semester and cumulative arithmetic grades.) H:7 There is no significant relationship between scores on the measure of arithmetic self-concept and the Sixth Grade total grade point average for all academic areas. H:8 There is no significant relationship between student's scores on the measure Of arithmetic self-concept and their mother's educational level. 62 H:9 There is no significant relationship between student's scores on the measure of arithmetic self-concept and ranking of their family social-status as indicated by criteria delineated in the McGuire-White Measure 2; Social Status. Hypotheses Related tO the General Self-Concept of Ability H:10 There is no significant relationship between scores on the measure of general self-concept of ability and scores on the arithmetic sub-tests Of the Stanford Achievement 2335. H:11 There is no significant relationship between scores on the measure Of general self-concept of ability and arithmetic grades. (Individual correlation coeffic- ients will be computed for current, semester and cumula- tive arithmetic grades.) H:12 There is no significant relationship between scores on the measure Of_general self-concept of ability and the Sixth Grade total grade point average for all aca- demic areas. H:13 There is no significant relationship between student's scores on the measure of general self-concept of ability and ranking Of their family social-status as in- dicated by criteria delineated in the McGuire-White M23: 5252,23 Social-Status. The confidence level for these hypotheses was set at the .05 level of significance. 63 Limitations and SOOpe of the Study The scOpe of the present study is limited to consid- eratiOns pertaining to arithmetic self-concept and its re- lationships with arithmetic achievement, academic aptitude, teacher evaluations and personal-social variables. Other variables that were not included in the present study may contribute to the relationships studied, conjecturally speaking. The sample, considered fairly representative, consisted of a sixth grade pOpulation of an all-Caucasian elementary school situated in a semi-rural community in Lower Michigan. Since instruments measuring self-concept are, at best, approximations, an additional limitation should be consid- ered: namely, that it is advisable to consider the present exploration within the restriction implied by the construc- tion of research instruments. Summary In this chapter the general hypothesis, the Operational hypotheses and the limitations and scOpe Of the study have been detailed. The Operational hypotheses have been stated in the null form in each case. The scOpe of the study is circumscribed by the considerations pertaining to arithme- tic self-concept and its relationship with measures of arith- metic achievement, scholastic aptitude and personal-social variables included in the study. CHAPTER V ANALYSIS OF THE RESEARCH DATA Chapter V is a report of the analysis of the research data based upon the methodological and statistical proced- ures outlined in Chapter III. The analysis of the data is presented in two parts: first, relationships between arith- metic self-concept and arithmetic achievement: second, the coefficients of correlation between self-concept of general ability and arithmetic achievement. Analysis was predicated on the assumption that the independent variables of sex, socio-economic status, mother's education, self-concept of general ability, intelligence quotient, arithmetic achieve- ment and arithmetic grades are related to the arithmetic self-concept. The confidence level for this study was es- tablished at the .05 level of significance. Analyses were made for the total group and for boys and girls within the group. The statistical data for the correlational analysis were computed from programs develOped by the Michigan State University Computer Laboratory after data was coded on I.B.M. cards. Arithmetic Self-Concept Data 64 65 In an effort to accurately analyze differences be- tween the high arithmetic self-concept group and the low arithmetic self-concept group in terms Of arithmetic ach- ievement those students with extreme I.Q. scores in high and low ranges were eliminated from the sample for this analysis. It was theorized that the 1.0. factor could skew results, so those students whose I.Q. scores varied more than two standard deviations from the mean were not used in the test of the first two hypotheses Of this study. Table 5:1 below shows the means for the upper and low- er quartile divisions of the total sample, for males and for females. Table 5:1 Means of 1.0. scores, Arithmetic Self-Concept, Arithmetic Achievement and Arithmetic Grades Accord- ing to Upper and Lower Quartile Levels Up Quart ’Div LOw 'Quart Div Total Boys Girls Total Boys Girls N-35 N-l6 N219 N=35 N=15 N=20 Arith Sf Con 103.74 103.25 104.16 79.03 76.67 80.80 Arith Achieve 29.80 29.38 30.60 15.20 13.67 16.35 Arith Grade 10.06 9.13 10.79 5.74 5.20 6.15 fi=70*** The first major hypothesis tested was: There is no significant difference in arithmetic achievement, as mea- sured by scores on a standardized achievement test, be- tween those students with high arithmetic self-concept and 66 students with low arithmetic self-concept as indicated by responses on the arithmetic self-concept scale. Table 5:2 below indicates the Mann-Whitney U-Test re- sults of analysis of data related to H:l. Table 5:2 Data Comparing Arithmetic Achievement for Upper and for Lower Quartile Divisions of Arithmetic Self- Concept Scores Subjects Mean—Up- Mean fow- _z_ score p guart N=35 Quart N=35 Worm e e 6 e I?) p“ <0 GUI;— BOYS 29038 13.67 3.88 p3<0001* Girls 30.60 16.35 4.64 p-<.001* E'two-taIled test N-70 The data indicated that there was a significant dif- ference in level of arithmetic achievement between students with high arithmetic self-concept scores and students with low arithmetic self-concept scores, as indicated by scores on a standardized achievement test, when arithmetic self- concept scores from the upper and lower quartile divisions were used as the criterion. Differences for total group, for boys and for girls were found to be significant beyond the .001 confidence level when a two-tailed test was used. The results were anticipated in terms of theoretical principles evolved from research Of self-concept literature with particular reference to findings by Brookover and as- sociates (16). H:l was rejected on the basis of research data derived 67 in this investigation. The findings, however, do not pre- clude the possibility that individual students may achieve in arithmetic at a level at variance with their arithmetic self-concept score. The second major hypothesis suggested that there was no significant difference in arithmetic achievement, as indicated by teacher-assigned grades for arithmetic perfor- mance in the on-going classroom instructional situation, between students with high arithmetic self-concept scores and students with low arithmetic self-concept scores as in- dicated by responses on the arithmetic self-concept scale. Table 5:3 below indicates the results of analysis of data relevant to H:2 in terms of Mann-Whitney_U-Test re- sults. Table 5:3 Data Comparing Arithmetic Grades for Upper and Lower Quartile Divisions of Arithmetic Self-Concept Scores Subjects Mean Mean E'scores p U - uart Low- uart Mroup . 6 M . p-<.001*"' Boys 9.13 5.20 3.05 p=<.001* 61:13 10.79 6015 3.60 P3<0001* ftwo-tailed test N-70 Analysis of the data shows that there was a signifi- cant difference in arithmetic grades between students with high arithmetic self-concept scores and students with low arithmetic self-concept scores, as indicated by teacher- 68 assigned arithmetic grades, when arithmetic self-concept scores from the upper and lower quartile ranges are used as the criterion. Results of a two-tailed test of signif- icance of differences suggest that there is a high degree of difference for the total group, for boys within the group and for girls within the group used in the present study. The differences are significant beyond the .001 confidence level. H:2 was rejected on the basis Of data intrinsic to this study. Again, the data pertained to statistical re- lationships and group differences, not to individual cases. Hypothesis number three speculated that there was no significant difference between the arithmetic self-concept of boys and the arithmetic self-concept of girls when scores on the arithmetic self-concept scale are used as the crit- erion. Table 5:4 below indicates results of statistical anal- ysis of data pertinent to suppositions in H:3. (An Erratic- test was used to establish homogeneity of variance before the tftest analysis was undertaken: g? 1.11, pf>.05.) 69 Table 5:4 Data Comparing the arithmetic Self-Concepts of Boys with the Arithmetic Self-Concepts of Girls Scale Mean Mean t-test p Boys N=66 Girls N-78 " mm: Concept 88.14 90.06 .70 p=>.05* Gen Self-Conc Of Ability 50.30 50.23 .87 p=>.05* ‘Utwo-tailéd’test N-144 Statistical results indicated that there was no sig- nificant difference between the arithmetic self-concept of boys and the arithmetic self-concept of girls: there was no significant difference between the self-concept of gen- eral ability of girls and the self-concept of general abil- ity of boys. Differences were not significant at the .05 confidence level. Findings in this analysis, though not entirely consis- tent with exPectations predicated on findings by Brookover (16) and Weatherman (140), do suggest that girls have higher arithmetic self-concept scores than do boys. H:3 was supported: however, Eftest analysis of arith- metic achievement computed independently from objectives of the present study show that there was also no clear-cut difference in arithmetic achievement between boys and girls in this study. Independent tftest analyses do suggest that girls received higher teacher-assigned grades than did boys in this study. 70 Table 5:5 shows results of Eftest analyses of differ- ences between arithmetic achievement test scores for boys and for girls: Table 5:6 depicts results of Eftest analyses of differences between teacher-assigned grades for boys and forygirls. Table 5:5 Data Comparing the Arithmetic Achievement Test Scores of Boys and the Arithmetic Achievement Test Scores of Girls Scale Mean Mean fiftest p BO s N=66 Girls N=78 AEIEEIEBmp l7¥68 21:31 2.85 p-<.02?"' Arith Conc 17.62 16.94 .60 p->.05* Arith A l 21.00 20.51 .34 ‘p->.05* * two-tailed test N-l44 Table 5:6 Data Comparing the Assigned Arithmetic Grades of Boys and the Assigned Arithmetic Grades of Girls Grade Mean Mean Effest p BO s N-66 Girls N-78 Eur Arith M . 9 8.08 2.02 pa «6'5?— Sem Arith Gr 6.98 7.99 3.06 p-<.01* Cum Arith Gr 7.59 8.35 1.72 p->.05* rtwo-tailed test N-144 Results suggest that there was a significant differ- ence between arithmetic computations scores for boys and for girls: girls scored significantly higher on this sub- test than boys did in this study: the significance level was beyond .02. There was no significant difference be- tween scores on the arithmetic concepts and the arithmetic 71 applications sub-tests of the standardized achievement tests, however. Differences were not significant at the .05 level: boys achieved somewhat higher than girls on the arithmetic concepts and arithmetic applications sub-test. Findings show that there was a significant differences be- tween current teacher-assigned grades for boys and for girls in the study. The differences are significant be- yond the .05 level. There also was a significant differ- ence between semester teacher-assigned grades for boys and girls in the study. The differences were significant be- yond the .01 level. There was no significant difference between cumulative teacher-assigned grades for boys and for girls. Differences were not significant at the .05 level: girls had higher mean grades on a current, semester and cumulative basis than boys. Table 5:7 through table 5:18 contain coefficients of correlation and analyses Of differences between coeffic- ients of correlation for the various groups involved in the present study. The statistical analyses included in these tables are pertinent to questions essential to hy- potheses four through thirteen. 72 Table 5:7 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study Correlation Coefficients’fbr TOEaI POpuIation N=161 Variable rho p (two-tailed test) Student's Sex .06 p->.05* Family Social-Status -.30 p-<.01 Mother's Education .23 p=<.02 General Self-Concept of Ability .70 p-<.01 Arithmetic Computations Sub-test .62 p-<.01 Arithmetic Concepts Sub-test .61 p-<.01 Arithmetic Applications Sub-test .59 p-<.01 Full Scale Intelligence Quotient .51 p-<.01 Language Intelligence Quotient .43 p=<.01 Non-Language Intelligence Quotient .49 p-<.01 Number Values Sub-test .45 p-<.01 Number Problems Sub-test .47 p=<.01 Current Arithmetic Grade .71 p=<.01 Semester Arithmetic Grade .72 p-<.Ol Cumulative Arithmetic Grade .67 p-<.01 Total Academic Grade Point Average .68 p-<.01 * Not Significant at .05 level lull—i .: k .AHVJQ-Dw.a 73 Table 5:8 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study Correlation Coefficients for Sample Group N=l44 Variable rho p (two-tailed test) Student's Sex .06 p=>.05* Family Social-Status -.30 p=<.02 Mother's Education .23 p-<.05 General Self-Concept of Ability .71 p-<.01 Arithmetic Computations Sub-test .62 p-<.01 Arithmetic Concepts Sub-test .61 p=<.01 Arithmetic Applications Sub-test .59 p-<.01 Full Scale Intelligence Quotient .51 p=<.01 Language Intelligence Quotient .43 p-<.01 Non-Language Intelligence Quotient .49 p=<.01 Number Values Sub-test .45 p=<.01 Number Problems Sub-test .46 p=<.01 Current Arithmetic Grade .71 p-<.01 Semester Arithmetic Grade .72 p=<.01 Cumulative Arithmetic Grade .67 p=<.01 Total Academic Grade Point Average .68 p=<.01 * Not significant at .05 level 74 Table 5:9 Differences Between Coefficients Of Correlation for Arithmetic Self-Concept and Specific Variables Examined in This Study for Sixth Grade POpulation and for Sample Group from POpulation VariEEIes Correlated with Pop Samp gftesfp Arithmetic Self-Concept rho rho N=161 N=I44 Student's Sex .06 .06 -0- - Family Social-Status -.30 -.30 -0- - Mother's Education .23 .23 -0- - General Self-Concept of Ability .70 .71 .018 >.05* Arithmetic Computations Sub-test .62 .62 -0- - Arithmetic Concepts Sub-test .61 .61 -0- - Arithmetic Applications Sub-test .59 .59 -0- - Full Scale Intelligence Quotient .51 .51 -0- - Language Intelligence Quotient .43 .43 -0- - Non-language Intelligence Quotient .49 .49 -0- - Number Values Sub-test .45 .45 -0- - Number Problems Sub-test .47 .46 .009 >405* Current Arithmetic Grade .71 .71 -0- - Semester Arithmetic Grade .72 .72 -0- - Cumulative Arithmetic Grade .67 .67 -0- - Total Academic Grade Point Average .68 .68 -0- - * Not significant at .05 level 75 Table 5:10 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study Correlation Coefficienfs for Boys In Study Variable N=66 522‘ p (two-tailed test) Family Social-Status -.22 p=>.05* Mother's Education .32 p=<.01 General Self-Concept Of Ability .72 p-<.01 Arithmetic Computations Sub-test .56 p-<.01 Arithmetic Concepts Sub-test .56 p=<.01 Arithmetic Applications Sub-test .55 p=<.01 Full Scale Intelligence Quotient .47 p-<.01 Language Intelligence Quotient .37 p-<.01 Non-Language Intelligence Quotient .48 p-<.01 Number Values Sub-test .39 p-<.Ol Number Problems Sub-test .30 p-(.01 Current Arithmetic Grade .71 p-<.01 Semester Arithmetic Grade .70 p=<.01 Cumulative Arithmetic Grade .63 p-4.01 Total Academic Grade Point Average .66 p=<.01 * Not significant at .05 level 76 Table 5:11 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study CorreIaEIon Coefficienfs for Girls in Stfidy Variable N-78 £22] 2 (two-tailed test) Family Social-Status -.35 p=<201 Mother's Education .13 p->.05* General Self-Concept of Ability .70 p=<}01 Arithmetic Computations Sub-test .67 p-<.01 Arithmetic Concepts Sub-test .65 p-<.01 Arithmetic Applications Sub-test .63 p-<.01 Full Scale Intelligence Quotient .54 p-<.01 Language Intelligence Quotient .49 p-<.01 Non-Language Intelligence Quotient .49 p=<.01 Number Values Sub-test .49 p-<.01 Number Problems Sub-test .57 p-<.01 Current Arithmetic Grade .72 p-<.01 Semester Arithmetic Grade .73 p-<.01 Cumulative Arithmetic Grade .69 p-<.Ol Total Academic Grade Point Average .70 p-<.Ol * Not significant at .05 level 77 Table 5:12 Differences Between Coefficients of Correlation for Arithmetic Self-Concept and Specific Variables Examined in This Study for Boys and for Girls in Sample VarIEBIes Correlated with Boys Girls g-test p Arithmetic Self-Concept rho rho N266 fi=78 Family Socia1*Status -.22 -.35 .88 >.05* Mother's Education .32 .13 1.17 >.05* General Self-Concept Of Ability .72 .70 .24 >'.05* Arithmetic Computations Sub-test .56 .67 1.06 7%05* Arithmetic Concepts Sub-test .56 .65 .88 '205* Arithmetic Applications Sub-test .55 .63 .70 2505* Full Scale Intelligence Quotient .47 .54 .52 205* Language Intelligence Quotient .37 .49 .88 205* Non-Language Intelligence Quotient .48 .48 -0- - Number Values Sub-test .39 .49 .76 205* Number problems Sub-test .30 .57 1.61 '205* Current Arithmetic Grade .71 .72 .24 205* Semester Arithmetic Grade .70 .73 .35 E05* Cumulative Arithmetic Grade .63 .69 .64 205* Total Academic Grade Point Average .66 .66 -0- - * Not significant at .05 level 78 Table 5:13 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study CorrelatiOn COefficients for Group I Variables N=55 522 p (two-tailed test) Student's Sex .06 p=>.05* Family Social-Status -.37 p=<.01 Mother's Education .34 p=<.01 General Self-Concept of Ability .79 p=<.01 Arithmetic Computations Sub-test .57 p-<.01 Arithmetic Concepts Sub-test .66 p=<.01 Arithmetic Applications Sub-test .63 p-<.Ol Full Scale Intelligence Quotient .47 p-<.Ol Language Intelligence Quotient .41 p-<.Ol Non-Language Intelligence Quotient .38 p=<.01 Number Values Sub-test .31 p-<.02 Number Problems Sub-test .40 p=<.01 Current Arithmetic Grade .73 p=<.01 Semester Arithmetic Grade .75 p-<.01 Cumulative Arithmetic Grade .71 p-<.Ol Total Academic Grade Point Average .77 p=<.01 * Not significant at .05 level 79 Table 5:14 Correlations Between Arithmetic Self-Concept and Specific Variables Examined in This Study Correlation Coefficients for Group II Variable N=89 £22. 2 (two-tailed test) Student's Sex .04 p=>.05* Family Social—Status -.26 p-<.05 Mother's Education .19 p=>.05* General Self-Concept of Ability .66 p=<.01 Arithmetic Computations Sub-test .68 p=<.01 Arithmetic Concepts Sub-test .59 p=<.01 Arithmetic Applications Sub-test .59 p=<.01 Full Scale Intelligence Quotient .53 p-<.01 Language Intelligence Quotient .44 p=<.01 Non-Language Intelligence Quotient .55 p=<.01 Number Values Sub-test .51 p=<.01 Number Problems Sub-test .51 p=<.01 Current Arithmetic Grade .71 p=<.01 Semester Arithmetic Grade .72 p=<.01 Cumulative Arithmetic Grade .64 p-<.01 Total Academic Grade Point Average .62 p=<.01 * Not significant at .05 level 80 Table 5:15 Differences Between Coefficients Of Correlation for Arithmetic Self-Concept and Specific Variables Examined in This Study for Group I and for Group II in Sample VarIEEIe Correlated with Cr I Gr I! aftest p Arithmetic Self-Concept rho rho 14:55 N'=‘8’9 Student's Sex .06 .04 .12 >.05* Family Social-Status -.37 -.26 .35 >.05* Mother's Education .34 .19 .93 >.05* General Self-Concept of Ability .79 .66 .16 ‘K05* Arithmetic Computations Sub-test .57 .68 .10 >.05* Arithmetic Concepts Sub-test .66 .59 .64 :h05* Arithmetic Applications Sub—test .63 .59 .35 2305* Full Scale Intelligence Quotient .47 .53 .46 ‘z05* Language Intelligence Quotient .41 .44 .11 3.05* Non-Language Intelligence Quotient .38 .55 1.22 >.05* Number Values Sub-test .31 .51 1.40 >.05* Number Problems Sub-test .40 .51 .81 >.05* Current Arithmetic Grade .73 .71 .23 >.05* Semester Arithmetic Grade .75 .72 .35 >.05* Cumulative Arithmetic Grade .71 .64 .76 >.05* Total Academic Grade Point Average .77 .62 1.69 >.05* * Not significant at .05 level 81 Table 5:16 Correlations Between General Self-Concept of Ability and Specific Variables Examined in This Study LTOtal Samp Boys CirIs Variable POp POp rho rho rho rho NEI61 N=I44 N=66 N-78 Student's Sex .02* .01* -0- -0- Mother's Education .26*** .24** .30*** .18* Family Social-Status .30*** .32*** -.20***-.42*** Arithmetic Computations Sub-test Arithmetic Concepts Sub-test Arithmetic Applications Sub-test Full Scale Intelligence Quotient Language Intelligence Quotient Non-Language Intelligence Quotient Current Arithmetic Grade Semester Arithmetic Grade Cumulative Arithmetic Grade Total Academic Grade Point Average Arithmetic Self-Concept .49*** .46*** .42*** .40*** .38*** .35*** .61*** .63*** .55*** .66*** .72*** .43*** .46*** .42*** .40*** .38*** .35*** .61*** .62*** .55*** .67*** .71*** .49*** .43*** .39*** .40*** .36*** .37*** .62*** .64*** .59*** .72*** .72*** .50*** .48*** .44*** .4o*** .40*** .33*** .62*** .63*** .53*** .66*** .70*** *iNot signifiCant at .05 level ** Significant at or beyond the .02 level *** Significant at or beyond the .01 level 82 Table 5:17 Differences Between Coefficients of Correlation for General Self-Concept of Ability and Specific Variables Examined in This Study for Total Group and Sample Group Total Samp 2: (two- Variable Pop POp test tail rho rho test) N=I61 N=I44 Student's Sex 002* -001* 009 >.05* Family Social-Status -.30** -.32** .13 >.05* Mother's Education .26** .24** .27 i05* Arithmetic Computations > Sub—test .49** .48** .13 .05* Arithmetic Concepts Sub- test .46** .46** -0- - Arithmetic Applications sub-teSt .42** .42** -0- - Full Scale Intelligence Quotient .40** .40** -0- - Language Intelligence Quotient .38** .38** -0- - Non-Language Intelligence Quotient .35** .35** -0- - Current Arithmetic Grade .61** .61** -0- - Semester Arithmetic Grade .63** .62** .09 C05* Cumulative Arithmetic Grade .55** .55** -O- - Total Academic Grade Point Average .66** .67** .13 >.05* Arithmetic Self-Concept .72** .71** .13 >.05* * Not significant at .05 level ** Significant at or beyond .01 level 83 Table 5:18 Differences Between Coefficients of Correlation for General Self-Concept of Ability and Specific Variables Examined in This Study for Boys and Girls in Sample Eoys Girls z- prtwo- Variable Samp Samp {est tail rho rho test) N=66 N=78 Family Social-Status -.20* -.42** 1.50 >.05* Mother's Education .31* .18* .80 >’.05* Arithmetic Computations Sub-test .49** .50** .06 >.05* Arithmetic Concepts Sub- test .43** .48** .36 t05* Arithmetic Applications Sub-test .39** .44** .36 >.05* Full Scale Intelligence Quotient .40** .40** -0- - Language Intelligence Quotient .36** .40** .24 >.05* Non-Language Intelligence QuOtient 037** 033** 030 >005* Current Arithmetic Grade .62** .62** -0- — Semester Arithmetic Grade .63** .63** -0- - Cumulative Arithmetic Grade .59** .53** .54 :i05* Total Academic Grade Point Average .72** .66** .71 >.05* Arithmetic Self-Concept .72** .70** .24 >.05* * Not significant at .05 level ** Significant at or beyond .01 level 84 In hypothesis number four it was hypothesized that there was no significant relationship between scores on the measure of arithmetic self-concept and scores on a measure of scholastic aptitude (I.Q. scores). Data obtained in this study as indicated in Table 5:19 below dictate an inference that there was a relationship {- between arithmetic self-concept scores and scores on an intelligence quotient scale. Table 5:19 Correlations between the Arithmetic Self-Concept m- and Scholastic Aptitude as Measured by the California Short-Form Test of Mental Maturity CHM Samp Grp EOys Girls rho rho rho N=I44 NR6 £4778 PEII Scale I.Q. .57: .471» .54; Language I.Q. .43* .37* .48* Non-Language I.Q. .49* .48* .49* I *p-<.01 (two-tailed test) The relevancy of interaction of the I.Q.--arithmetic self-concept scale was indicated here. The results sug- gested a substantially significant positive relationship between I.Q. and arithmetic self-concept. The relation- ships were significant beyond the .01 level: Eggs were somewhat higher for girls than for boys. Results were an- ticipated in reference to findings by Ringness (108) that academic self-ratings are related to I.Q. levels. H:4 was rejected since it appeared that I.Q. and arith- metic self-concept were significantly related for the sample 85 in this study. Full scale I.Q. scores and Non-Language I.Q. scores were more highly correlated with arithmetic self-concept than Language I.Q. scores were. The fifth hypothesis stated: There is no significant relationship between scores on the measure of arithmetic self-concept and scores on the arithmetic sub-tests of the Stanford Achievement 2332. The correlations between arithmetic self-concept scores and scores on arithmetic sub-test of the standardized ach- ievement test shown in Table 5:20 suggested that there was a positive relationship between arithmetic self-concept scores and scores on arithmetic sub-tests of a standardized achievement test in this study. Table 5:20 Correlations Between Arithmetic Self-Concept and Arithmetic Achievement on Standardized Arithmetic Achievement Tests Arithmetic Samp Grp :Eoys Girls Sub-tests rho rho rho NEI44 N366 N378 AEIEHmeEic Comp .627: .56*“' .67; Arithmetic Conc .61* .56* .65* Arithmetic A291 .59* .55* .63* Nilil *pf<.01 (two-tailed test) Research findings in this study indicated that stu- dents' abilities to use arithmetic functions as demonstrated by standardized achievement test results were substantially and positively related to the arithmetic self-concept. H:5 was rejected since results showed statistically 86 significant positive relationships between arithmetic self- concept and arithmetic achievement tests. The relationships were significant beyond the .01 level. As in the previous analysis the correlations were higher for girls than for boys, but not a significant level. Though not predicted the differences were expected in terms Of a thesis by Ausubel (6) that boys, because they possess less derived status than do girls in our culture, tend to set higher levels of aspiration than girls do. Hypothesis number six predicted that there was no sig- nificant relationship between scores on the measure Of arithmetic self-concept and teacher-assigned arithmetic grades. Correlations between arithmetic self-concept and teacher-assigned arithmetic grades for this study are shown in Table 5:21. Table 5:21 Correlations Between Arithmetic Self-Concept and Teacher-Assigned Arithmetic Grades Mirking period Samp Crp Eoys “Cirls Arith Grade rho rho rho N=I44 N566 F78 Current Arithmetic Grade .71* .71* .72* Semester Arithme- tic Grade 072* 070* 073* Cumulative Arith- metic Grade .67* .63* .69* N-l44 *EF<’01 (two-tailed test) The findings derived from analyses of data related to 87 the assumptions of hypothesis number six demonstrate con- clusively that there was a positive significant relation- ship between arithmetic self—concept and teacher designated arithmetic grades for students in this study. The correl- ations were markedly high between arithmetic self-concept and arithmetic grades. The relationships were significant beyond the .01 level. The relationships were somewhat higher for recent and immediate grades than for cumulative grades, but the differences were not significant. The findings were anticipated in respect to theoret- ical precepts of Combs and Snygg (32) and Rogers (111) that one's experiences serve as a framework for the acqui- sition of self-concept and that self-concepts of children are responsive to new experiences: findings also are con- sistent with conclusions by Aiken and Dreger (1) that di- rect experiences in relation to mathematics contribute to math attitudes. H:6 was rejected since it was apparent that there was a marked significant positive relationship between the arithmetic self-concept and arithmetic grades for students in this study. In hypothesis number seven it was asserted that there was no significant relationship between scores on a measure of arithmetic self-concept and the Sixth Grade total grade point average for all academic areas. Table 5:22 delineates the coefficients of correlation 88 between arithmetic self-concept and total academic grade point average for students in this study. Table 5:22 Correlations Between Arithmetic Self-Concept and Sixth Grade Total Academic Grade Point Average Samp Grp Eoys Cirls rho rho rho N344 15:66 NETS TOtEl Academic Crade Point Average .68* .66* .70* Nil44 *pf<.01 (two-tailed test) The data indicated that there was a high positive re- lationship between arithmetic self-concept and total aca- demic grade point average for the Sixth Grade subjects in this study. The relationships were significant beyond the .01 level. The relationship was somewhat higher for girls than for boys, but not at a significant level. The findings were expected in terms Of direction pro- vided by Brookover (l6) and Werdelin (142) that mathemati- cal ability is one part of the total intellectual-academic field and therefore cannot be studied in isolation, but only in connection with other parts Of the field. H:7 was rejected since it was shown that there was a high positive significant relationship between arithmetic self-concept and total academic grade point average for subjects in this study. It was hypothesized next that there was no signifi- cant relationship between students' scores on the measure 89 Of arithmetic self-concept and their mothers' educational level. Coefficients of correlation for data derived from in- vestigation of hypothesis number eight are depicted in Table 5:23 below. Table 5:23 Correlations Between Arithmetic Self-Concept and Students' Mothers' Educational Levels Samp Grp Boys Cirls rho rho rho M44 N166 N278 Students' Mothers' Educational Level .23* .32* .13** N-144 :p§<.02 (two-tailed test) pf>.05 (two-tailed test) The findings suggested that there was a small positive relationship between arithmetic self-concept and the level of mothers' education for students in the present sample. The coefficients of correlation for the total sample and for boys in the sample were significant beyond the .02 level: the relationships for girls in the sample were not significant at the .05 level. The findings were not GXpected in regard to theoret- ical directions provided by findings of Drucker and Remmers (45): they are consistent with conclusions by McDermott (89) that fathers exert more influence on the develOpment of mathematical attitudes than do mothers, and inferences by Medinnus (93) that mother influences are higher for boys 90 than for girls: Boys appear more dependent on parental per- ceived attitudes due to the sociological and economic impact in terms Of achievement demands and general expectations. H:8 was tentatively rejected for the total sample group and for boys in the sample: but was sustained for girls in the study herein reported, since the relationships between arithmetic self-concept and level of mothers' education were not significant at the .05 level for girls. The assumption in hypothesis number nine was that there was no significant relationship between students' scores on the measure of arithmetic self-concept and ranking of their families' social-status as indicated by criteria in- cluded in the McGuire-White Measure pf Social Status. Results of investigation of data related to the asso- ciation between arithmetic self-concept and family social- status of children included in the study are illustrated in Table 5:24. Table 5:24 Correlations Between Arithmetic Self-Concept and A Measure Of Family Socio-Economic Status Samp Grp iEOys Girls rho rho rho N344 14:66 N378 Family Social-Status -.30; -.22** -.35’ N=144 *p?<.01 (two-tailed test) * pf>.05 (two-tailed test) The findings showed that there was a slight signifi- cant negative relationship between arithmetic self-concept 91 and a measure of family socio-economic status for subjects in the total sample group and for girls in the sample group: the relationships for boys were not significant. The rela- tionships for the total sample and for the girls were sig- nificant at the .05 level -- all coefficients of correla- tion were in a negative direction. The findings were not expected in reference to previous findings by Brookover (16) and Passy (99). The negative relationships may be explained by deficiencies in the social status scale used in the study in terms Of community makeup: however, Sherif and Sherif (115) advise that cases of dis- crepancy between actualgroup membership and psychological relatedness are not infrequent in the U.S. which is a good example of a highly differentiated society in which there are many individuals whose reference groups are not the groups with which they are actually associated in day-to- day living. Following this line of reasoning, Beck (10) theorizes that many children from homes with low economic- status have parents with middle-class identifications. H:9 was tentatively rejected for the total sample ’group and for girls within the sample group, but not for boys in the sample group, since the E for boys does not at- tain the .05 level. Hypotheses Related to the General Self-Concept of Ability Hypothesis number ten, the first major hypothesis per- taining to the general self-concept of ability, stated that 92 there was no significant relationship between scores on the measure of general self-concept Of ability and the arith- metic sub-test of the Stanford Achievement 2252! Table 5:25 shows the coefficients of correlation be- tween general self-concept of ability and arithmetic ach- ievement sub-tests Of the Stanford Achievement 2233 as com- puted from data obtained in the study. Table 5:25 Correlations Between General Self-Concept Of Ability and Arithmetic Achievement Sub-tests Arithmetié Samp Grp —Eoys 4—Cirls Sub-tests rho rho rho M44 m6 NETS Ktithmetic damp . 481 . 49“ JOE—‘— Arithmetic Conc .46* .43* .48* Arithmetic Appl .42* .39* .44* NEl44 fpf<.01 (two-tailed test) Information derived from examination of data pertinent to this hypothesis indicated that there was a substantial positive relationship between general self-concept of abil- ity and arithmetic achievement. As with other coefficients of correlation in the present project the relationships were higher for girls than for boys, but not at a signifi- cant level. The relationships were significant beyond the .01 level. The findings were expected in relation to research direction provided by Brookover and associates (16). H:10 was rejected since the evidence indicated that 93 there was a moderately high, significant, positive rela- tionship between the general self-concept of ability and arithmetic achievement as indicated by arithmetic sub-tests of an achievement scale for children in the study presently reported. Hypothesis number eleven stated that there was no sig- nificant relationship between scores on the measure of gen- eral self-concept of ability and teacher-assigned arithmetic grades. Correlational results Of analysis of data involved in the relationships between general self-concept Of ability and teacher-assigned arithmetic grades are shown in Table 5:26. Table 5:26 Correlations Between General Self-Concept Of Ability and Teacher-Assigned Grades Matking period Samp Grp -Tfioys Girls Arithmetic Grade rho rho rho Nil-T44 N366 N278 Current Arithmetic Grade .61* .62* .62* Semester Arithmetic Grade .62* .64* .63* Cumulative Arithmetic Grade 055* 059* 053* Nil44 *p§<.01 (two-tailed test) Data obtained from analysis of hypothesis number eleven revealed that there was a moderately high, significant, pos- itive relationship between the general self-concept of abil- ity and teacher-assigned arithmetic grades for achievement 94 in the on-going classroom situation for children in the study. The relationships were fairly constant for boys and for girls. Relationships were significant beyond the .01 level. The results were anticipated in relation to research direction provided from conclusions formed by Brookover and associates (16). H:11 was rejected because evidence suggested a sub- stantially high, significant, positive relationship between scores on the measure of general self-concept Of ability and teacher-assigned arithmetic grades representing aca- demic achievement for subjects in the present study. The twelfth hypothesis of the study averred that there was no significant relationship between scores on the mea- sure Of general self-concept of ability and the Sixth Grade total academic grade point average. Table 5:27 illustrates the relationships between the general self-concept of ability and the total academic grade point average for subjects in the study. Table 5:27 Correlations Between General Self-Concept of Ability and Total Academic Grade Point Average Samp Grp —T§Oys Girls rho rho rho N=I44 14:66 N=‘78 Total Academic Crdde Point Average .67* .72* .66* Ni144 02,4,01 (two-tailed test) 95 Results of the analysis Of these data prompted the conclusion that there was a substantial-to-marked signif- icant, positive relationship between the general self-con- cept of ability and total academic grade point average. As in the previous analyses the relationships were some- what higher for boys than for girls, but not at a signif- icant level. The relationships were significant beyond the .01 level. The findings were expected in terms of research direc- tions Obtained from findings by Brookover, gthgi (16). H:12 was rejected since the evidence related to hy- pothesis number twelve lead to the conclusion that there was a moderate-to-high correlation between general self- concept of ability and total academic grade point average for students in the study. The final major hypothesis of the present research study was: There is no significant relationship between students' scores on the measure of general self-concept Of ability and ranking of their families' social-status as in- dicated by criteria included in the McGuire-White Measure 2f Social Status. Results of investigation of data related to the asso- ciations between general self-concept of ability and family social-status of children included in the study are illus- trated in Table 5:28. 96 Table 5:28 Correlations Between General Self-Concept Of Ability and A Measure of Family Socio-Economic Status Samp Grp EOys Cirls rho rho rho N3l44 N366 N;78 EamilySociaI-Status ;:30* -.20" -.42’ N-144 :p- (.01 (two-tailed test) pf).05 (two-tailed test) The findings show that there was a definite, though slight, significant negative relationship between general self-concept of ability and a measure of family socio-econ- omic status for subjects in the total sample and for girls with the group. The relationships for boys were supportive Of the hypothesis. The relationships -- negative in direc- tion -- were significant beyond the .01 level for the total sample and for girls in the sample: for boys the relation- ships did not attain the .05 level. The findings were contrary to expectations in refer- ence to research directions suggested by findings of Brookover st 21 (16) and Passy (99). H:13 was tentatively rejected for the total sample group and for girls within the group, but not for boys within the group. The coefficients of correlation between _general concept of ability and family socio-economic status were in a negative direction. Table 5:29 shows an analysis of differences between ‘E's among arithmetic self-concept and general self-concept .II III" I Illilllu al- lllllll- l [y 97 of ability and specific variables examined in this study for the sample group. Analysis of the data that appears in Table 5:29 sug- gested that arithmetic self-concept correlated significantly higher with arithmetic sub-tests than the general self-con- cept Of ability does. The correlations between arithmetic self-concept and the specific variables included in this study weregenerally higher than the correlations between general self-concept of ability and the same variables in this study: the higher correlations for arithmetic self- concept and variables in the study obtain particularly in reference to the arithmetic criteria. The differences are significant beyond the .02 level when the S122.EE§E is ap- plied to the data. 98 Table 5:29 Differences Between Coefficients Of Correlation Among Arithmetic Self-Concept and General Self-Con- cept of Ability and Specific Variables Examined in This Study for the Sample Group variablesflEXamined i ASC GSCA 5? Sign This Study . rho rho test Test ** N=l44 N=l44 Mother's Education .23 .24 .08 ).05 Family Social-Status -.30 -.32 .17 >505 Arithmetic Computations Sub-test .62 .48 1.62 'ZOS Arithmetic Concepts Sub-test .61 .46 1.79 <.05* Arithmetic Applications Sub-test .59 .42 1.96 <.025* Full Scale Intelligence Quotient .51 .40 1.19 7.05 Language Intelligence Quotient .43 .38 .50 205 Non-Language Intelligence Quotient .49 .35 1.45 >.05 Current Arithmetic Grade .71 .61 1.53 205 Semester Arithmetic Grade .72 .62 1.53 ROS Cumulative Arithmetic Grade .67 .55 1.62 205 Total Academic Grade Point Average .68 .67 .17 >.05 N=l44 *ge<,os (two-tailed test) * pf<.02 (Sign Test analysis) 99 Other Relevant Results The multiple correlation analysis was used in the present investigation to study the degree of variation attributable to variables in the study. Results of the multiple correlational analysis in- volving arithmetic self-concept and the independent var- iables included in this study are expressed in Table 5:30 and Table 5:31. Partial Correlations and Beta Weights are listed for the variables. Table 5:30 Data Of the Multiple Coefficients of Correlation Between the Arithmetic Self-Concept Scale and Inde- pendent Variables in the Study R R2 R2 R .82 .67 .63 .79 NaIIT The results indicated that there was a definite marked positive significant relationship between the arithmetic self-concept and variables included in this study. The multiple correlation obtained between the dependent and independent variables suggested that arithmetic self-con- cept had a predictive relationship. More than two-thirds of the variance in the total arithmetic self-concept scores was accounted for in terms of the variance in the independent variables part scores: approximately thirty-three percent variance remained. 100 Table 5:31 Partial Correlation Coefficients and Beta Weights for the Arithmetic Self-Concept Scale and Independent Variables in the Study firtial Etta Variable rho Weights Student's Sex .018 .012 Family Social-Status .021 .015 Mother's Education .037 -.026 General Self-Concept of Ability .471 .444 Arithmetic Computations Sub-test .033 .036 Arithmetic Concepts Sub-test .116 .155 Arithmetic Applications Sub-test .017 .023 Full Scale Intelligence Quotient .032 .251 Language Intelligence Quotient .044 -.197 Non-Language Intelligence Quotient .023 -.094 Number Values Sub-test .062 .050 Number Problems Sub-test .073 .061 Current Arithmetic Grade .063 .108 Semester Arithmetic Grade .098 .185 Cumulative Arithmetic Grade .132 .145 Total Academic Grade Point Average -.ll7 -.l7l 143144 Partial Correlation: net correlation between two varia es w en e n uence Of one or more other vari- ables on their relation has been eliminated or allowed for. Beta Weight: the amount that each variable must be multiplied in order to make the multiple correlation with a criterion a minimum. 101 Examination of the regressions of the multiple cor- relation coefficient derived in the analysis indicated that the independent variables of General Self-Concept of Ability, the Semester Arithmetic Grade and the Arithmetic Concepts Sub-test of the Achievement test contributed sig- nificantly to the total arithmetic self-concept score: the partial correlations range from .44 for the General Self- Concept of Ability to .16 for the Arithmetic Concepts sub- test. The dimensions Of Mother's Education, Language and Non-Language Intelligence Quotients and Total Academic Grade Point Average with negative partial correlations added little to the total arithmetic self-concept score. For this study, the results indicated that Student Sex, Non-Language Intelligence Quotient and Family Social-Sta- tus contributed least to the total arithmetic self-concept score. The findings prompted the conclusion that the vari- ables examined in this study contributed significantly to the arithmetic self-concept with the exceptions of those variables cited above. Results of the multiple correlation examination in- volving the Arithmetic Applications Sub-test of the stan- dardized achievement test and the components Of the arith- metic Self-Concept Scale are depicted in Table 5:32 and Table 5:33. Partial Coefficients Of Correlation and Beta Weights are listed for the variables. 102 Table 5:32 Data for the Multiple Coefficients of Correla- tion Between the Arithmetic Applications Sub—test and Components of the Arithmetic Self-Concept Scale R R2 352 R .709 .503 .411 .641 N: The results Of the multiple correlational analysis suggested that there was a substantial relationship be- tween the Arithmetic Applications Sub-test scores and com- ponent questions Of the Arithmetic Self-Concept Scale. The multiple correlation Obtained between the dependent and independent variables suggested that the Arithmetic Self-Concept had a predictive relationship with this test of arithmetic achievement. Approximately fifty percent of the variance in the Arithmetic Applications Sub-test scores was accounted for in the variance in the component questions. Examinations of the regressions of the multiple coe- fficient of correlation derived in the analysis indicated that the questions pertaining to parent attitude toward student arithmetic grades: student understanding of arith- metic compared with other school subjects: ability to do multiplication problems and students' estimates of future ranking in college class in arithmetic contributed little to the total Arithmetic Applications Sub-test scores: par- 103 Table 5:33 Partial Coefficients of Correlation and Beta Weights for the Arithmetic Self-Concept Component Questions and Arithmetic Applications Sub-test Question Number Partial rho Beta Weights 1 .150 .227 2 .126 .214 3 .079 .094 4 .077 .080 5 .009 .011 6 -.123 -.114 7 .075 .096 8 .150 .184 9 -0173 -0215 10 -.005 -.006 11 -0202 “'0210 12 .028 .029 13 .091 .081 14 .074 .076 15 -.199 -.269 16 .055 .065 17 -0103 “0092 18 .076 .085 19 '0001 ‘0001 20 .160 .189 21 .007 .008 22 .087 .119 23 -.029 -.037 24 -.020 -.018 25 .082 .107 104 tial correlations for these questions were in a negative direction. Questions relating to students' evaluation of personal arithmetic ability compared with close friends: students' impressions of the relative difficulty involved in learning arithmetic in comparison with other school subjects and student comparison of arithmetic grades with grades received in other school subjects appeared to con- tribute most highly to the total Arithmetic Applications Sub-test score for subjects in this study. The partial correlations for these questions ranged from .150 to .160. The multiple coefficients of correlation yielded for this study indicated that, for the subjects in this study, the component questions of the Arithmetic Self-Concept have validity in relation to predicting arithmetic ach- ievement. Summary The non-parametric Mann-Whitney_U-Test was used to test the first two hypotheses of this study pertaining to the high versus low arithmetic self-concept in rela- tion to arithmetic achievement. Mann-Whitney_U-Test anal- yses indicated there were significant differences between arithmetic achievement levels for students with high arith- metic self-concept scores and students with low arithmetic self-concept scores. Hypothesis number three related to the differences between the arithmetic self-concept of boys and the arith- 105 metic self-concept of girls was tested by use of the t: test: tftest results showed no significant difference be- tween the arithmetic self—concept of boys and the arith- metic self-concept of girls for subjects in this study. Hypotheses related to the arithmetic self-concept and the general self-concept of ability and independent vari- ables intrinsic to the study were tested through correla- tional analysis using the Pearson Product Moment Coeffic- i223 pf Correlation formula. Results indicated signifi- cant relationships for all variables in the study except that related to student's sex. The correlations for fam- ily social-status were in a negative direction for both arithmetic self-concept and general self-concept of abil- ity. Multiple correlations indicated that the general self- concept of ability, semester arithmetic grades and arith- metic concepts sub-test scores contribute significantly to the total arithmetic self-concept score for subjects in this study. Multiple correlations also suggest that the arithmetic self-concept scale has validity in predicting arithmetic achievement for students in this study. CHAPTER VI SUMMARY, FINDINGS AND CONCLUSIONS: DISCUSSION AND RECOMMENDATIONS The final chapter of this study is organized into six main divisions: l) Restatement Of the Problem: 2) Re- cap of Design and Procedures: 3) Summary of Findings: 4) Conclusions: 5) Discussion and 6) Recommendations. Problem The primary purpose of this study was to investigate the relationships between arithmetic self-concept and var- iables related to arithmetic achievement in a selected group of sixth grade students. Subordinate purposes were: 1) to identify the variables that contribute most to the arithmetic self-concept score: 2) to isolate the compo- nents of the arithmetic self-concept scale that are most closely related to arithmetic achievement as indicated by scores on the arithmetic applications sub-test Of the Stage Eggg.Achievement 22253 An attempt was made to establish the self-concept as a measurable constellation of variables related, signifi- cantly, to the accumulated experiential background Of the individual. The school, it is theorized, contributes to 106 107 the develOpment of the self-concept in reference to the interpersonal relationships develOped with teachers and peers that impinge upon a child's perception of himself. The competitive aspects Of school related experiences have a particular impact on the develOpment of self-con- cept as does the evaluative aspects of educative procedure since it is built, in all its uniqueness for each individ- ual, in relation to others. The importance Of self-concept is suggested by the imputation by Hilgard (64) that all personality mechanisms imply a self-reference and such mechanisms are not understandable unless considered in the framework Of one's concept of himself. Eggign and Procedure Children of the sixth grade classes in the Algonac, Michigan Community Schools served as subjects and perti- nent data were collected during the last month of the fifth grade and the last month of the sixth grade. Data were collected on 161 sixth grade students for the study. The independent variables investigated in relation to their possible connection with the develOpment Of arith- metic self-concept were: student's sex, mother's educa- tional level, family socio-economic ranking, general self- concept Of ability, arithmetic achievement test scores, teacher-assigned arithmetic grades and total academic grade ‘point average for sixth grade subjects. For each student in the study a ranking of family 108 socio-economic status was obtained through use of the McGuire-White Measure pf Social Status. The California 22 t pf Mental Ability-Short Eggpland the Stanford Ash: ievement Eggt'were administered to students in the sample. Additional personal data for each child was Obtained from a research of the cumulative records of children in the investigation. Hypotheses numbers one, two and three were submitted to analysis using tests Of differences between means: The Mann-Whitney_U-Test was employed to measure the differences between means for the upper quartile and the lower quartile divisions of the arithmetic self-concept range for hypoth- eses one and two. The Mann-Whitney_U-Test was selected for these tests since the observations were considered to be not normally distributed. The tftest was used to analyze the differences between means pertinent to assumptions in hypothesis number three. The remaining hypotheses were examined through use of correlational analysis: the £225? 222 Product Moment Coefficient pf Correlation was the tech- nique used through the BASTAT program of the Michigan State University Computer Laboratory. In addition to the fundamental analyses of the major hypotheses, a multiple regression correlational analysis was made using the arithmetic self-concept as the depen- dent variable in the first 5, and the arithmetic applica- tions sub-test as the dependent variable in the second mul- 109 tiple correlation. The analyses were made through use of the £§2§£ program of the Michigan State University Computer Laboratory. The .05 level of confidence was established for re- jecting the null-hypotheses. Summary of_the Findings H:l There is no significant difference in arithmetic achievement, as measured by scores on a standardized arith- metic achievement test, between those students with high arithmetic self-concept as indicated by scores on the arith- metic self-concept scale and students with low arithmetic self-concept as indicated by scores on the same scale. Result: Hypothesis rejected. The findings indicated that there was a significant difference between achievement levels in arithmetic for children with high arithmetic self-concept and children with low arithmetic self-concept. The difference was significant beyond the .001 level. H:2 There is no significant difference in arithmetic achievement, as indicated by teacher-assigned arithmetic grades, between students with high arithmetic self-concept and students with low arithmetic self-concept as indicated by scores on the arithmetic self-concept scale. Result: Hypothesis rejected. The findings suggested that there was a significant difference between teacher- assigned grades in arithmetic for children with high arith- metic self-concept and children with low arithmetic self- 110 concept. The difference was significant beyond the .001 level. The findings from these two analyses give support to the thesis that arithmetic self-concept affects arithmetic achievement. H:3 There is no significant difference between the arithmetic self-concept of boys and the arithmetic self- concept of girls when scores on the arithmetic self-concept scale are used as the criterion. Result: Hypotheses accepted. Findings showed there was no significant difference between the arithmetic self- concept Of girls and the arithmetic self-concept of boys. Though girls did, as a group, have higher arithmetic self- concept scores than boys: the differences were not signif- icant at the .05 confidence level. The findings did not sustain the expectation that boys would have higher arithmetic self-concept scores than girls because of psycho-social implications related to dif- ferences in derived status between boys and girls in the American society. H:4 There is no significant relationship between scores on the measure of arithmetic self-concept and scores on a measure of scholastic aptitude (I.Q. score). Result: Hypothesis rejected. Analysis of data dem- onstrated that there was a significant positive relation- ship between I.Q. and arithmetic self-concept. The rela- 111 tionships were higher for girls than for boys for total I.Q. scores, language I.Q. scores and for non-language I.Q. scores. The correlations were significant beyond the .01 level. H:5 There is no significant relationship between scores on the measure of arithmetic self-concept and scores on the arithmetic sub-tests Of the Stanford Achievement TeSt0 Result: Hypothesis rejected. Data indicated that there was a significant positive relationship between arith- metic self-concept and standardized arithmetic achievement test results. Again, the results were higher for girls than for boys for arithmetic computations, arithmetic con- cepts and arithmetic applications sub-tests. The correla- tions were significant beyond the .01 confidence level. The findings suggest agreement with an idea by Brown (18) that the student's ability to learn arithmetic is as- sociated with his attitude toward arithmetic: and of an Observation by Brueckner (21) that arithmetic is not a gen- eral ability, that a high level of proficiency in one as- pect is not a guarantee of a correspondingly high level in other areas of arithmetic competency. H:6 There is no significant relationship between scores on the measure Of arithmetic self-concept and tea- cher-assigned arithmetic grades. Result: Hypothesis rejected. Findings indicated 112 that there was a significant positive relationship between the arithmetic self-concept and teacher-assigned grades. The correlations were significant beyond the .01 level. These findings would appear relevant to findings by Bostrom (14) that grades have a significant impact on learning behavior. H:7 There is no significant relationship between the arithmetic self-concept and the Sixth Grade total academic grade point average. Result: Hypothesis rejected. The data indicated that there was a significant positive relationship between the arithmetic self-concept and the Sixth Grade total academic grade point average. The correlations were significant be- yond the .01 confidence level. The results of this analysis would seem to be in ag- reement with thinking by Woodby (145) that success or mea- surable achievement in arithmetic has some correlation with achievement in other disciplines. H:8 There is no significant relationship between stu- dents' scores on the measure of arithmetic self-concept and their mothers' educational level. Result: Hypothesis rejected in part. Findings indi- cated that there was a significant positive relationship between the arithmetic self-concept and mother's level of education for the sample. The correlations for the total group and for boys within the group were significant be- junk IVE .IIIP, .l Mafia. — v I . r. ya 113 yond the .02 level: however, the correlation for girls with- in the group was not significant at the .05 level. The findings here appear related to a conclusion by Medinnus (93) that mothers influence the self-concept of boys more than of girls because of inferred cultural de- mands related to school achievement and general expecta- tions. H:9 There is no significant relationship between stu- dents' scores on the measure of arithmetic self-concept and ranking of their families' social-status as indicated by criteria delineated in the McGuire-White Measure g£_§27 gig; Status. Result: Hypothesis rejected in part. Data indicated that there was a significant negative relationship between the arithmetic self-concept and a ranking of family social- status. The correlation for the total group and for girls within the group were significant beyond the .01 confidence level: the correlation for boys within the group was not significant at the .05 level. The findings may be indicative of weaknesses in the social-status scale, but appear to be also reflective of a surmise by both Beck (10) and Sherif (115) that cultural permeability causes many children from low economic status home to have middle-class identifications. Dubin and Dubin (46) too, minimize the influence of the home eXperiences on attitudes, suggesting that the nature of the child and 114 his reaction to the psycholOgical field are critical de- terminants in creation of the self-image. H:10 There is no significant relationship between scores on the measure Of general self-concept of ability and arithmetic achievement as measured by sub-tests of the Stanford Achievement 2351;. Result: Hypothesis rejected. Findings indicated that there was a significant positive relationship between the general self-concept of ability and standardized arithmetic achievement test scores. The correlations were significant beyond the .01 confidence level and Obtain for arithmetic computations, arithmetic concepts and arithmetic applica- tions sub-tests. H:11 There is no significant relationship between scores on the measure of general self-concept of ability and teacher-assigned arithmetic grades. Result: Hypothesis rejected. The analysis of data indicated that there was a significant positive relation- ship between the general self-concept of ability and tea- cher-assigned grades. The correlations were significant beyond the .01 confidence level and held for the current, semester and cumulative arithmetic grades. H:12 There is no significant relationship between scores on the measure of general self-concept Of ability and the Sixth Grade total academic grade point average. Result: Hypothesis rejected. The findings indicated 115 that there was a significant positive relationship between the general self-concept Of ability and the Sixth Grade total academic grade point average. The correlations were significant beyond the .01 level. The findings appear consistent with deductions by Borislow (13) that students who underachieve scholastically have a poorer conception of themselves as students than do achievers. H:13 There is no significant relationship between students' scores on the measure of general self-concept of ability and rankings of their families' social-status as indicated by criteria delineated in the McGuire-White Measure pf Social Status. Result: Hypothesis rejected. The findings indicated that there was a significant, though negative, correlation between the general self-concept of ability and ranking of family social-status. The correlations were all signifi- cant beyond the .05 confidence level: the correlations for girls were slightly higher than the correlations for boys. Conclusions Literature related to the present study suggested that there would be a relationship between academic self-concept and academic achievement. The results of the study appear to lend empirical support to the theory that a relationship exists between the level of arithmetic self-concept and arithmetic achievement. 116 Of the thirteen null hypotheses in this study all but one were rejected: ten were unqualifiedly rejected: two were tentatively rejected, and one was sustained. In gen- eral, the null hypotheses set forth in this study were re- jected at a substantial level (beyond the .05 confidence level). The three hypotheses related to mother's educa- tional level and family social-status ranking, however, were only tentatively rejected: the findings related to family social-status were significant in a negative direc- tion. The findings for the total sample group and for boys and for girls within the sample group for this study were substantial and the correlations Obtained at a high level of significance for arithmetic achievement as indicated by standardized achievement test results and by teacher-assi- gned arithmetic grades. For this study, the conclusion that there was no significant difference between the arith- metic self-concept of boys and the arithmetic self-concept of girls was warranted: similarly, there was no signifi- cant difference between the general self—concept of ability of girls and the general self-concept of boys. The con- clusion that there was a difference between the arithmetic achievement of boys and the arithmetic achievement of girls was only tentatively supported. There was a difference, significant at the .02 level, for the arithmetic computa- tions sub-test of the standardized achievement test: there 117 was no significant difference for the arithmetic concept and arithmetic applications sub-tests. Sixth Grade girls appeared to receive somewhat higher grades in arithmetic than boys do: the difference was significant beyond the .05 level for current grades, but was not significant on a cumulative basis (marks over a three year period). Other conclusions that were deduced from the findings were: 1. It appears that there was a significantly high positive relationship between scholastic aptitude and arith- metic self-concept. Self-concept, specifically arithmetic self-concept, appeared to vary with intelligence: sex did not appear to be a determiner for differences among children in this study. The correlation between aptitude and arith- metic self-concept, ranging between .43 and .51 as they do for this study, were somewhat higher for the E's between I.Q. and general self-concept of ability which range be- tween .35 and .40. In reference to these correlations, it might be pointed out that Jersild (68) suggests that one views himself more on an emotional basis than on an intel- lectual basis. 2. Students who achieved scholastically in arithmetic had higher arithmetic self-concepts than children who did not achieve scholastically in arithmetic. Relationships were somewhat higher for teacher-assigned grades than for standardized arithmetic achievement test scores. The cor- 118 relations between arithmetic self-concept and arithmetic achievement ranged between .59 for arithmetic self-concept and the arithmetic applications sub-test to .72 for arith- metic self-concept and the semester arithmetic grade. Arithmetic self-concept appeared to be a good predic- tor Of achievement in arithmetic for children in the study. Recent arithmetic grades seemed to be somewhat more impor- tant in the formation of the arithmetic self-concept than past arithmetic grades. 3. Students' arithmetic self-concept was positively related to the Sixth Grade total academic grade point aver- age. The concept a student held of his arithmetic ability appeared at least partially contingent upon his concept Of his general school ability when teacher.assigned grades are considered. The correlation between arithmetic self- concept and total academic grade point average was .68. 4. For boys the mothers' educational level appeared to have some impact on the arithmetic self-concept, but for girls at the Sixth Grade level the mothers' educational level did not appear to be a determining factor in the de- velOpment of arithmetic self-concept. This conclusion was tentative since the correlations tended to be slight. 5. Family social-status ranking was negatively cor- related with the arithmetic self—concept and with the gen- eral self-concept of ability, and did not appear to work a significant influence on scholastic achievement (Efs were 119 between -.20 and -.33). This conclusion was not antici- pated since related literature and prior findings suggested that social-status would have a positive effect on self- concept. 6. The general self-concept of ability for children in this study was positively related to arithmetic self- concept and collaterally with arithmetic achievement in terms of both standardized achievement test scores and in terms of teacher assigned arithmetic grades. The correla- tions between general self-concept of ability and arithme- tic achievement was somewhat higher for teacher-assigned grades than for standardized arithmetic achievement test scores. Correlations for arithmetic achievement test scores ranged between .40 and .46 while the 5's for teach- er-assigned grades were between .55 and .62. 7. The general self-concept of ability was markedly correlated with Sixth Grade total academic grade point average. The general self—concept appeared to be a good predictor of general academic ability for Sixth Grade child- ren. The correlation was .71. 8. General self-concept of ability, arithmetic grades and arithmetic achievement scores contributed significant- 1y to the arithmetic self-concept. Discussion This study should be considered an exploratory inves- tigation with definite replicative possibilities. The study 120 was conducted using observations from a sample selected from one grade level in a single school community. The research was designed to test hypotheses pertinent to re- lationships between arithmetic self-concept and arithmetic achievement inherent in the study. Since the study was conducted in a single community with students from a designated grade level there are lim- itations intrinsic to findings of the study. NO effort was made to establish the sample as representative of the general pOpulation. In this investigation the focus has been on statisti- cal relationships and group implications, and not on indi- vidual cases. The findings, therefore, do not preclude the possibility that individual students might achieve at levels not consistent with their arithmetic self-concept. The conclusions derived from data in the study sup- ported the hypothesis that there is a significant positive relationship between arithmetic self-concept and arithmetic achievement for sixth grade students. In this investiga- tion, boys and girls with high arithmetic self-concept scored significantly higher in arithmetic achievement than boys and girls with low arithmetic self-concept. These relationships hold for the total group as well as for boys and girls within the group. The findings were consistent with the literature which indicated that various factors underlie academic self-con- 121 cept and student achievement is positively associated with\ perceptions of ability in specific academic areas, partic- ularly arithmetic achievement. The present findings, though part of an exploratory study, sustained some findings de- rived from a series of studies related to general self-con- cept Of ability by the College of Education, Michigan State University. The findings indicated that the Brookover Scale can be used at the upper elementary level as well as at the junior-senior high school levels. The study results tended to show that the factors of ability, attitude and experience entered significantly in- to the picture of arithmetic achievement. In respect to this, it would appear that the school plays an important role in the formation of self-concept and that the compet- itive aspects of the learning climate, the sensitivity Of children to reactions of peers and teachers in reference to successes and failures appear to be reflected in a child's self-regard. Correlations Obtained in this study were somewhat higher than those secured in previous studies in terms of relationships between arithmetic self-concept and arithmetic achievement in reference to arithmetic achieve- ment test scores and teacher-assigned arithmetic grades. Findings in this study suggested that family socio- economic ranking had a negligible effect on self-concept. There was evidence, also, that the level of mothers' edu- cation affected arithmetic self—concept Of boys, but not 122 of girls. These relationships, though statistically sig- nificant in a negative direction, were modest. It was apparent, from findings in the study, that gen- eral self-concept of ability had a high correlation with general academic achievement, including arithmetic achieve- ment. In this study the coefficients of correlation between arithmetic achievement test scores and teacher-assigned arithmetic grades ranged from .60 for the 3 between the arithmetic concepts sub-test and the cumulative arithmetic grade to .78 for the 5 between the arithmetic computations sub-test and the current arithmetic grade, and so tended to lend credence to the postulate the school marks have some ”face validity“ as a measure of ability of children in arithmetic. In respect tO these facts, the principal finding that arithmetic self-concept was positively and significantly related to teacher-assigned grades suggested that teacher-pupil relationships did affect pupil learning efficiency and so appeared to have implications for teacher education on a pre-service and in-service basis. There appeared to be suggestions for teachers in terms of the importance of grades on the formation the self-concept in the on-going classroom situation. The connotation derived would seem to be that grades have an important impact on the development of arithmetic self-concept. There seemed to be, furthermore, some inference to be drawn in terms of 123 devising methods Of studying particular elements of child- ren's self-concept of ability in specific academic areas. It is suggested, in results of this study, that it may be feasible to develop self-concept scales for other specific academic areas to increase the body Of knowledge of the motivating forces underlying achievement in those areas. A consequence of further study Of the implications of self- concept in relation to academic achievement might be the encouragement of positive self-concept formation and the diminution Of inadequate self-concept frequently generated in many school children because of the spectre of rejection implied in traditional reactions to failure in regard to school expectations. It should be pointed out that conclusions of the pre- sent study were limited to the variables measured pertain- ing to arithmetic self-concept and its relationships with arithmetic achievement, academic achievement, scholastic aptitude, teacher evaluation and family related variables. Other variables not included in the study may contribute to the relationships studied. Generalizations Of the find- ings should be made with caution since the sample, though considered fairly representative, did consist Of only the Sixth Grade pOpulation Of an all-Caucasian elementary school situated in a semi-rural north central community. Caution should be used, particularly, in the extrapolation of find- ings related to multiple regression coefficients of correl- 124 ation in the study since these statistical results are es- pecially sensitive to variations from group-to-group. Recommendations for Further Study On the basis of Observations and inferences made dur- ing the conduct and evaluation Of this study, the follow- ing recommendations are Offered with the hOpe that more knowledge pertaining to the dynamics involved in the aca- demic self-concept may be gathered: 1. In order to obtain more knowledge of the dynamics involved in the interplay between the arithmetic self-con- cept and the aspects of arithmetic achievement included in this study, the following questions are relevant for further study: In what group of pupils -- high, middle or low in arithmetic self-concept -- is the range of ach- ievement differences most pronounced? May we safely pre- dict that the range Of differences will be less among the high arithmetic self-concept group than they will be in the average and low arithmetic self-concept groups? Will differences be greater for boys or for girls? 2. To help determine the usefulness of arithmetic self-concept scale as a predictor Of arithmetic achieve- ment it would be well to replicate reliability and valid- ity studies following the design of this study with other pOpulations. Replications with other populations is also desirable to discover whether results derived from this study Obtain for other populations. 125 3. Replication of the study with a larger sample us- ing full intelligence quotient batteries rather than the short-form battery may enhance findings. 4. Replication of the study using the same variables, but involving samples from different grade levels from third through eighth grade might be helpful in determining whether F‘ the relationships between arithmetic self-concept and arith- metic achievement persists, or vacillates, throughout the middle school years. 5. A follow-up study of the children involved in the k- present study as they complete seventh and eighth grade would provide a longitudinal evaluation of the constancy of the arithmetic self-concept and may enable evaluations of areas in which changes occur. 6. It would be well to conduct a study, along the design of the present study, with underachievers and over- achievers in arithmetic to discover the relationships of arithmetic self-concept to the phenomena of underachieve- ment and overachievement in arithmetic. 7. It would be well to conduct studies involving treatment effects related to manipulation and modification of the arithmetic self-concept to determine how such treat- ments affect the level of arithmetic achievement. l. 2. 3. 4. 6. 7. 8. 9. 10. BIBLIOGRAPHY Aiken, L.R., Jr. and Dreger, R.M. “The Effect of At- titudes on Performance in Mathematics,” Jour. g: Educ. Psych. Vol. 52, No. 1, (1966), pp. l9-24. Allport, G.W. Becoming. New Haven: Yale University Press, 1955. Allport, G.W. Pattern and Growth in Personality. N.Y.: Holt, Rinehart and Winston, I961} Anderson, C.M. ”The Self-Image: A Theory of the Dy- namics of Behavior," Reprinted from Mental Hy: iene, Vol. 36, (1952), pp. 227-244 in Tfie Self In Growth, Teaching_and Learnin , Edited—5y HEmacHeE, D.E. EnglefiOOd Cliffs, N.J.: Prentice- Hall, Inc., 1965. Arieti, 8. American Handbook sf Psychiatry. N.Y.: Basic Books Inc., l§5§. Ausubel, D.P. Theo and Problems of Child Develop- ment. N.l.: Grune and StrattOfi, I958. Bandura, A. and Huston, Aletha C. "Identification as A Process Coincidental Learning,“ Jour. g: Abn. and SOCe PSXChe V010 63' NO. 2' (lggIS' pp. 3I1- Bandura, A. and Kupers, C.J. ”Transmission of Pat- terns Of Self-Reinforcement Through Modeling," Jour. of Abn. and Soc. Psych.,VOl. 66, No. 1, 71964) 7957-1-97 "— Bassham, H., Murphy, M. and Murphy K., "Attitudes and Achievement in Arithmetic," The Arithmetic Teacher, Vol. 11, No. 2, (1964). Beck, R.H., Cook, W.W., Kearney, N.C. Curriculum in the Modern Elementary School, 2nd Ed. Englewood Cliffs, N.J.: Prentice-Hall Inc., 1960. 126 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 127 Benjamins, J. "Changes in Performance in Relation to Influences Upon Self-Conceptualization," Jour. of Abn. and Soc. Psych. Vol. 45, (1950), pp. 47322130 ."'"' "‘" """'"'" Bloom, B.S. Stabilitv and Change in Human Cha racter- istics. N.Y.: John Wiley and Sons, 1964. Borislow, B. ”Self-Evaluation and Academic Achieve- ment," Reprinted from Jour. Of Couns. Psvch., Vol. 9, (1962): in The Self“ 1n Growth, Z‘eaching and Learnin , Edited—Ey_ Hama.EEek, D. E. Engla- wood Cliffs, N. J.: Prentice-Hall, Inc., 1965. Bostrom, R.M., Vlandis, J.W. and Rosenbaum, M.C. "Grades As Reinforcing Contingencies in Attitude Change," Jour. of Educ. Psych., Vol.52, No. 2, (1961) . pp 112315?" Bower, E. M. Earlv Identification of Emotionally Hanoi- canped CHiIdren 1n SchooI.Spr1ngfie1d, IlI.: C as. C. Thomas, PuEI., I960. Brookover, W. B., Paterson, A., Thomas, S. Self-Concept of Abilit3 and School Achievement: The Rele tion- Efiips of Self:lmace to AEhieverent 1n _Junior Hi Squectg. E. Lan51ng, Micnigan: Ofiice of R :— search and Publication, Michigan State University, 1962. Brown, K.E. "The Drive to Improve School Mathematics," The Revolution in School Mathematics. Wash., D.C.: Nat'l Council of Teacne rs of Mathlenatics, 1961. Brown, L.E. and Abell, T. L. "Research in the Teaching of Elementary School Mathematics," The Arithmetic Teacher, Vol. 12, No. 7, (Nov., 19657_'pp.547; 57r— Brownfain, J. J. Stabilit of the Self-Concept as A Dimension of Persona11ty. “PH. 5. DissertatiOn, Univ. of Michigan, I950: Publication # 2385. Bruck, M. and Bodwin, R.F. "The Relationships between Self-Concept and the Presence and Absense Of Scholastic Underachievement," Jour. of Clin. Psych., Vol.58, (1962), pp. 18I- I82. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 128 Brueckner, L.J. and Bond, G.L. The Diagnosis and Treatment of Learning_Diff1culties. N.Y.: Ap- pleton-Century-Crofts, Inc.,—1955. Bryan, M.B. "The Stanford Achievement Test, 1964 Re- vision." Sixth Mental Measurements Yearbook, Buros, O.K. (Ea.5 Highland Park, N.J.: The Gryphon Press, 1965. Buros, O.K. (Ed.) Sixth Mental Measurement Yearbook. Highland Park, N.J.: The GrypHon Press, 1965. Cairns, S.S. “Mathematics,” The Case for Basic Educa- tion, Koerner, J.D. (Ed.$ Boston: Little, Brown ana C00, 1959. Carlson, R. "Stability and Change in the Adolescents' Self-Image,“ Child Development, Vol. 36, No. 3, (1965)! PP. 659-6670 Cattel, R.B. Personality, A Systematic Theoretical and Practica Stu y. N.Y.: McGraw-HiII, 1955. Chance, E. ”Measuring the Potential Interplay of Forces Within the Family During Treatment,” Child Development, Vol. 26, (1955), pp. 261-256. Chein, I. "The Awareness of Self and the Structure of the Ego," Psychological Review, Vol. 51, (1944) , pp. 364-314. _— Combs, A.W. ”Clinical Practice and Personality Theory: A Phenomenological Approach to Adjustment Theory," Jour. of Abn. and Soc. Psych., Vol. 44, No. 1, W9 ) 713572 9:33.— Combs, A.W. ”The Individual." Speech delivered to the National Convention 2: the American Society for Curriculum Development, 1962. Combs, A.W. “A Perceptual View of the Adequate Person- ality," Perceiving, Behaving, Becomin . Wash., D.C.: Yearbook 1962: Association for gupervision and Curriculum Development, 1962. Combs, A.W. and Snygg, D. Individual Behavior, Rev. Ed. New York: Harper and‘Brothers, 1959. Combs, A.W., SOper, D.W., Courson, C.C. ”The Measure- ment of Self-Concept and Self Reports," Educ. and Psych. Meas., Vol. 23, No. 3, (Aug., 1963). 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 129 Combs, A.W. and Taylor, C. "The Effect of the Percep- tion of Mild Degrees of Threat on Performance," Jour. 2£.Abn. and Soc. Psych., Vol. 47, No. 2, (1956). pp- 425-325- Conant, J.B. The American High School Today. New York: McGraw-Hill Book 50., Inc., 1959. Cooley, C.H. Social Organization. New York: Scribner, 1909. Coopersmith, S. ”Relationship Between Self-Esteem and Sensory Perceptual Constancy," Jour. g: Abn. and Soc. Psych., Vol. 68, No. 2, (1934!, pp. 217-271. "”5 Dabbs, J.M., Jr. "Self-Esteem, Communicator Charac- teristics, and Attitude Change," Jour. of Abn. and Soc. Psych., Vol. 69, No. 2, (Aug.,-T9645, pp. I73-19 a 1,1. Davids, A. and Lawton, Marcia J. "Self-Concept, Mother Concept, and Food Aversion in Emotionally Dis- turbed and Normal Children,” Jour. of Abn. and ggg, Psych., Vol. 62, No. 2, YI§31)7‘p§T'3o§=§14. Davidson, H.K. and Lang, G. "Children's Perception of Their Teacher's Feelings,” Jour. of Exoer. EdUCo' V01. 29' (DeCo' 1960), pp. 103:1190 Delisle, F.H. 5 Stud of the Relationshi s of the Self-Concepgpfo Adjhgfment in A SeIectEd'Grouo 2f College Wofiah. Unpfibl. 9525} 515. MSU, I933. Dinkmeyer, D.C. Child Development: The Emercin Self. g/ Englewood Cliffs, N.J.: Prentice-HaII, Inc., J 1965. Dinkmeyer, D.C. and Dreikurs, R. Encouraging Children to Learn: The Encouragement firocess. Englewood_ EIiffs, N.J.: Prentice-Hall, Inc., 1963. Dittes, J.E. “Attractiveness of Group As A Function of Self-Concept and Acceptance by Group," Jour. g£.Abn. and Soc. Psych., Vol. 59, No. 1, (I999), FPO—77.87:— — Drucker, A.J. and Remmers, H.H. "Environmental De- terminants of Basic Difficulty Problems," Jour. of Abn. and Soc. Ps ch., Vol. 47, No. 2, (Apr., I952}, pp. 379-382. 130 46. Dubin, R. and Dubin, E.R. ”Children's Social Percep- tion: A Review of Research,” Child Development, V01. 36' N0. 3, (Sept., 1965), pp. “ o 47. Dutton, W.H. ”Measuring Attitudes Toward Arithmetic,” Elementary School gggrnal, Vol. LV, No. 1,(Sept., 1954)! PP. 24-310 48. Ellis, A. Reason and Emotion i2_Psychotherapy. New York: Eyle Sfuarf, I962. 49. Engel, Mary. "The Stability of Self-Concept in Adol- escence,“ Jour. of Abn. and Soc. Psych., Vol. 58' NO. 2' Pp. ZII;ITEO 50. Erickson E.R. ”Youth and the Life Cycle,“ Reprinted from Children, Vol. 7, No. 2 (1960), in The Self ig,Growa, Teaching_and Learning, Edited-By’5.E. HamacHeE. -— 51. Erickson, E.L. A Stud ‘2; the Normative Influence pg Parents and‘Friends u on Academic Achievement. Unpublished Ed. D. Dis. MSU., I935. 52. Erickson, L.H. ”Certain Ability Factors and Their Ef- fect on Arithmetic Achievement,“ The Arith. Tea- cher, Vol. 5, No. 6, (1958), pp. 297L293. 53. Fink, M.B. “Self-Concept As It Relates to Academic Achievement,” Calif. Journ. of Educ. Res., Vol. 13, No. 2, (Mar., I962), pp.-97-63. 54. Pitts, W.H. Tennessee Self-Concept Scale. Nashville, Tenn.: Counselor Recording and Tests, 1965. 55. Flora, B. ”Why Won't They Take Mathematics?” Peabod Journ. of Educ., Vol. 37, No. 6, (May, 1960), '—'3'pp. 39354777 56. Gibbs, E.G., Mayor, J.R., Truenfelds, E. “Mathematics," Encyclopedia of Educational Research, Washington, 0' E a, 1W1. 57. Guilford, J.P. Fundamental Statistics in Ps cholo and Education. New YorE: McGraw-filll Book Co., lnC., o 58. Haarer, D.L. A Comparative Study of Self-Concept of Ability 9e ween Ins 1tut1onalized Delinquent—- Bo s and Non-Delin uent Boys Enrolled in Public Sc ool, UnpuBl. PH.5. Bis. MSU.,l964. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 131 Hall, C.S. and Lindzey, G. Theories of Personality. New York: John Wiley and Sons, Inc., 196I. Hamachek, D.E. The Self in Growth, Teaching and Learning, Selected Readings. Englewood Cliffs, .J.: Prentice-Hall, Inc., 1965. Harap, H. and Mapes, C.E. ”The Learning of Decimals in an Arithmetic Activity Program," Jour. g: Educ. Res. Vol. 38, (1936), pp. 686-693. l!”- Harsh, C.M. and Schrickel, H.G. Personalit Develoo- ment and Assessment. New York: THe Ronald Press Eompafi§7 1950. Helper, M.M. ”Learning Theory and the Self-Concept," Jour. gf Abn. and Soc. Psych., Vol. 51, (Sept., : I333), pp7_l84:l§47—-’ E .! I... Hilgard, E.R. ”Human Motives and the Concept of Self," The American Psychologist, Vol. 4, (Sept., 1949), 5'57 3771-333. Horrocks, J.E. Assessment of Behavior. Columbus, Ohio: Chas. E. Merrill-Books, Inc., 1964. James, W. The Principles of Ps choloo , Vol. 1. New York: Henry Holt and-5o., I396. Jackson, P.W. and Stratner, N. ”Meaningful Learning and Retention: Noncognitive Variables," Rev. Of Educ. Res., Growth, Dev. and Learning. Vol. 3?, No. 5, (I965), pp. 5l3-535. Jersild, A.T. £2 Search of Self. New York: Bur. of Publ., Teacher's College, Columbia University, 1952. Jersild, A.T. Child PsychOIOgy, 5th. Ed, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1960. Jersild, A.T. The Ps cholo of Adolescence, 2nd. Edi- 1 tion. New York: The Macfilllan Co., 1963. Jordan, J.A., Jr. ”A Self That Can Whistle," Harvard Educational Review, Vol. 34, No. l, (1964), pp. 22.3 3 o Jourard, S.M. and Remy, R.M. "Perceived Parental At- titudes, The Self and Security," Jour. g: Cons. PSYCho' V01. 15' (1955), pp. 364-3660 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 132 Kelley, E.C. "The Fully Functioning Self," Perceiv- ing, Behaving, Becoming: 1962 Yearbook. Wash., 5.5.: AssTn. for Supv. and Curr. Dev., 1962. Kelley, T.L., 23 a1. Manual Accompanying Stanford Achievement fast, Intermediate Battery,‘I964. Kipnis, Dorothy M. "Changes in Self-Concepts in Relation to Perception of Others,” Jour. gfi Personality, Vol. 29, (1961), pp. 444-465. Lafferty, J.C. "Values That Defeat Learning,” The Inter-Institutional Seminar in Child Develoo- menE. Dearborn: The Edison leE., l9 . Lane, H.A. 9§_Educating Human Beincs. Chicago: Follett Publ. Co., 1963. LaPere, J.M. "A Study of the Impact of the Parent on the Child's Concept of Self as It Relates to Academic Achievement,” A paper presented as part of: A S moosium on Im roving Academic Achieve- ment-Through SEudEnE's Self-Concept Enhancement. Ehicago: NaE'l. Educ. Res. Ass n., 1964. Laxer, R.M. ”Self-Concept Changes of Depressive Patients in General Hospital Treatment," Jour. of Cons. Psych., Vol. 28, No. 3, (1964), pp. Lazarus, R.S. Adjustment and Personality. N.Y.: McGraw-Hill Book Co., 1961. League, Betty Jo and Jackson, D.N. "Conformity, Veri- dicality, and Self-Esteem,” Jour. of Abn. and SOC. PSXCho' V01. 68' NO. 1' (19643-3- ppo 113-115. Lecky, P. Self-Consistenc : A Theor of Personality. New York: e S oe ring Press,-l961. Lerch, H.H. ”Arithmetic Instruction Changes Pupils' Attitudes Toward Arithmetic,“ The Arith. Teacher, V01. 1' NO. 3' (1960), pp. 117-II90 Lighthalls, F.F. “Anxiety as Related to Thinking and Forgetting,” Bulletin 50: What Research Sa 5 to the Teacher. Wash., D{5T: The American E uca--—' EEEhEl‘kEEEarch Association. Lindgreen, H.C. Educational Psycholocv in the Class- room. New York: doHn Wiley andfgodg, lnc., I958. 133 86. Lipsett, L.P. "A Self-Concept Scale for Children and Its Relationship to the Children's Form of the Manifest Anxiety Scale," Child Development, Vol. 29' NO. 4' (1958), pp. 463-Z:I¢ 87. Maslow, A.H. Motivation and Personalit . New York: Harper and Brothers, Publishers, 1954. 88. Matthews, R., Hardyck, C. and Sarbin, T.R. ”Self- Organization As A Factor in the Performance of Selected Cognitive Tests,” Jour. of Abn. and §_9_<_:_. Psych., Vol. 48, (1953 , ppf'B'oUT-Tofi“ 89. McDermott, L.A. A_Studv pf Some Factors That Cause Fear and Dislike 2£_Mathema5ics. UnpuEl. Ed. . fils.,'h§U,'I§56. 90. McGuire, C., White, G.D. ”The Measurement of Social Status,” Research Paper in Human Deve10pment #3 (Rev.) Dept. of EducT—Psych.,—The University 3? Texas: March, 1955. 91. McNemar, Q. Ps chological Statistics. New York: John Wiley and Sons, Inc., 1962. 92. Mead, G.H. Mind, Self and Societv. Chicago: Univ. of Chicago Press, I934. 93. Medinnus, G.R. ”Adolescents' Self-Acceptance and Per- ception of Their Parents,” Jour. of Cons. Psvch., Vol. 29, No. 2, (1965), pp.'T§U-1‘S'Z. 94. Miyamoto, S.F. and Dornbusch, S.M. ”A Test of Inter- actionist Hypotheses of Self-Conception," The Amer. Jour. of S22., Vol. 61, No. 5, (196577_pp. 399346§T__' '— 95. Montagu, A. "Quest for Self,“ Childhood Education, V01. 41' N00 1’ (1964), pp. 3-9. 96. Morse, W.C. "Self-Concept in the School Setting," Childhood Education, Vol. 41, No. 4, (1964), pp. 97. fiurphy, G. Human Potentialities. New York: Basic Books, 1958. 98. Mussen, P.H. and Porter, L.W. ”Personal Motivation and Self-Conceptions Associated with Effective- ness and Ineffectiveness in Emergent Groups,” Jour. of Abn. and Soc. Psvch., Vol. 59, No. 1, pp ."‘23=77'." " " ' 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 134 Passy, H.A. 'Socio-Economic Status and Mathematical Achievement,” The Arith. Teacher, Vol. 11, No. 7, (1964), pp. 469-475. Patterson, C.H. "The Self in Recent Rogerian Theory,” Jour. g; Indiv. Psych., Vol. 17, No. 1, pp. 5-12. Peirs, E.V. and Harris, D.B. "Age and Other Correl- ates of Self-Concept in Children," Jour. of Educ. Psych., Vol. 55, No. 2, (1964), p .‘91-95. Perkins, H.V. ”Factors Influencing Change in Child- ren's Self-Concepts,“ Child Development, Vol. 29, No. 4, pp. 221-230. Prescott, D.A. The Child in the Education Process. New York: McGraw-Hill_Book 50., 1957. Price, G.B. "Progress in Mathematics and Its impli- cations for the Schools,” The Revolution in School Mathematics. Washington, D.C.: Natidhal Council 0 eac ers of Mathematics, 1961. Raimy, V.C. “Self-Reference in Counseling Interven- tion,” Jour. of Cons. Psych., Vol. 12, No. 3, (1948)! pp. 133;163. Rapaport, D. "Mathematics -- Logical, Psychological and Pedagogical,“ The Arith. Teacher, Vol. 9, Rath, R.M. ”The Role of Self-Concept in Achievement," Jour. g: Eerr. Educ., Vol. 27, (1959), pp. 265- Ringness, T.A. “Self-Concept of Children of Low, Aver- age and High Intelligence," Amer. Jour. of Ment. Def., Vol. 65, No. 4, (1961)_-, pp. 453-5461? Rogers, C.R. Measuring Personality_Adjustment in Children: ContributiEns’to Educatibn, No._158. New York: Columbia University Press, 1931. Rogers, C.R. “Some Observations on the Organization of Personality,” The American Psychologist, Vol. 2’ (1947), pp. 35§‘§68. Rogers, C.R. Client-Centered Theran : Its Current Practices, Implications and Theory. New York: Houghton, M1 in o., 1 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 135 Rogers, C.R. “The Place of the Person in the New World of the Behavioral Sciences,“ The Pers. and Guid. Journ., Vol. 39, No. l, (19605,pp. ZZZ-WET. Rogers, C.R. "The Organization of Personality,” Per- sonalit : Readings and Theor and Research, a. 5 . BeImonE, Ca1if.: out we 1, E.A. and Mir aum, M Wadworth Publ. Co., 1964. Sarbin, T.R. ”A Preface to A Psychological Analysis of the Self,“ Psychological Review, Vol. 59, (1952), pp. 11-22. .V‘hlfl.‘ '1 Sherif, M. and Sherif, C.W. An Outline 23 Social Psy- chology, Rev. Ed. New York: Harper and Brothers, Shostrom and Bremmer. D namics 23 the Counseling Pro- cess. New York: McGraw-HiIl Book Co., Inc., 1937. a: :_.,-.‘ " 1; Siegel, S. Nonparametric Statistics for the Behavioral Sciencesinew York? McCraw-fiiIl Book Co., Inc., Silverman, I. "Self-Esteem and Differential Respon- siveness to Success and Failure," Journ. of Abn. 2332 E22. PSXCho' V01. 69' NO. 1' pp. IIS-II9.—_ Smith, F. "Prospective Teachers' Attitudes Toward Arithmetic,‘ The Arith. Teacher, Vol. 11, No. 7, (1964), pp. 47171777— ’— Smith, M.B. “The Phenomenological Approach in Person- ality Theory: Some Critical Remarks,“ Journ. g; Abn. and Soc. Psych., Vol. 45, (1950),pp. 516- Snygg, D. and Combs, A.W. Individual Behavior. New York: Harper, 1949. Snygg, D. and Combs, A.W. "The Phenomenological Ap- proach and the Problem of 'Unconscious' Behavior: A Reply to Dr. Smith,” Journ. of Abn. §g§.§gg. Psych” Vol. 45, (1950),——pp. 52335767 Staines, J.W. ”The Self-Picture As A Factor in the Classroom,“ Reprinted from: Br. Journ. of Educ. Ps ch. Vol. 28, 1958, in The-EelfT'n GrBVtIT Teacfiin and Learnin , EdT-fiamacHeET'fi.E. Engla- wooa CIiffET'N. .: Prentice-Hall, Inc., 1965. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 136 Stotland, E. and Cottrell, N.B. "Self-Esteem, Group Interaction and Group Influences in Performance," Jour. g£ Pers., Vol. 29, (1961), pp. 273-284. Stotland, E. and Zander, A. "Effects of Public and Private Failure in Self-Evaluation," Jour. pg Abn. and Soc. Psvch., Vol. 56, No. 2, 11958 , pp. 233-237- Sullivan, H.S. "Beginnings of the Self-System," Per- sonalit : Readings i2 Theory and Research, E37~ By SoutEwelI, E.A. and Mir aum, M. Bélmont, Calif.: Wadsworth Publ. Co., Inc., 1964. Swift, J.W. "Effect of Early Group EXperience: The Nursery School and Day Nursery," The Review of Child Development, Vol. 1, pp. 277. Symonds, P.M. The Ego and the Self. New York: Apple- ton-Century anH‘CrofEs, I951. Symonds, P.M. What Education Has to Learn from Psy- choloo . New York} Bur. of PuBT., TeachersT Col- Iege, Columbia University, 1960. Symonds, P.M. "Origins of Personality,” Teachers' Col. Record, March, 1960, pp. 301-3177 Taylor, C. and SOper, A.W. "Self-Acceptance and Ad- justment," Jour. pf Cons. Psych., Vol. 16, No. 2, (1952). Taylor, D.M. Consistency of the Self-Conce t. Unpubl. Ph.D. Dis., Vanderbilt—UniverSity, 195%. Thorndike, R.L. and Hagen, E. Measurement and Evalu- ation in Ps choloo and Education. New York: John Wiley and Sons, Inc., 1955. Thorne, F.C. Personality, A_Clinical Eclectic View- point. Brandon, Vt.: Jour of ClinicaI PsycHoI- ogy Press, 1961. Turlock, M.K. ”Emotional Blocks in Mathematics," The hiath. TeaCher' V01. 50'.NO. 8' pp. 572-576. Videbeck, R. "Self-Concept and the Reaction of Others," Sociometry, Vol. 23, (1960), pp. 351-359. Walker, H.M. and Lev, J. Statistical Inference. New York: Henry Holt and Co., 1953. 137 138. Washburn, W.C. "Patterns of Self-Conceptualization in High School and College Students," Jour. of Educ. Psych., Vol. 52, No. 3, (1961), pp. 1233131. 139. Wattenberg, W.W. and Clifford, C. "Relation of Self- Concept to Beginning Achievement in Reading," Child Development, Vol. 35, No. 2, (1964), pp. 140. Weatherman, R.F. A Stud 2£_the Relationships of Mobility to Academic Self-ConceptandAcaHEmic Achievemefit in Sixth Grade Children. Unpfibl. £25.15. Dis., WSW 4. 141. Weaver, J.P. and Gibb, E.G. ”Research in Elementary School Mathematics,” Review 2f Educ. Res., Vol. 34, (1964), pp. 273-285. 142. Werdelin, I. The Mathematical Ability: Experimental and Factorial Studies.tCOpen agen, Den.: Hakan- OHIssons, Publishers, 1958. 143. Wilson, G.M. ”Why Do Pupils Avoid Mathematics in High School?” The Arith. Teacher, Vol. 8, No. 4, (1961), pp. 4543458. 144. Witty, P.S. ”Role of Attitudes in Children's Failures and Successes," Nat'l. Educ. Ass'n. Jour., Vol. 37, (1948), pp. 422-423. 145. Woodby, Lauren. The Low Achiever in Mathematics. Wash., D.C. 5.5. Dept. oftHealth, Education and Welfare, U.S. Office of Education, 1965. 146. Wozenkraft, M. ”Are Boys Better than Girls in Arith- metic?” The Arith. Teacher, Vol. 10, No. 8, (1963). pp. W902" " 147. Wylie, R.C. The Self-Conce t: A Critical Survev 2: Pertinent Research LiEerature. Lincoln, Nebraska: Univ. oftNébraska Press, 1961. APPENDIX A The Pre-Test of the Arithmetic Self-Concept Scale Introduction Responses to preliminary scale items by a sample of sixth grade students randomly selected from the elementary school pOpulations of three separate school communities, in the winter of 1965, provided fundamental data for re- fining the arithmetic self-concept scale. The Pre-test Sample Responses to the arithmetic self-concept scale and the self-concept of general ability scale were obtained for 116 sixth graders from the schools involved in the pre- test sample. Current marking period grades in arithmetic were also obtained for the students in the pre-test sample. The 116 students represented approximately one-half of the total sixth grade pOpulations of the three school districts involved. Statistical Analysis The sample was divided into quartiles using arithmetic grades as the criterion: the upper quartile children whose grades were B or above: the lower quartile included obser- 138 "7'7 1‘. 139 vations whose grades were D or below. The significance of the difference between means of scores on the arithmetic self-concept scale for the upper and lower quartiles was analyzed through use of the non-parametric MaggéWhitney U-Test as outlined in Siegel (117). A Coefficient of Reliability was obtained for the arithmetic self-concept scale of the Spearmanfigggyg Prthecy Formula detailed in Thorndike and Hagen (133). Administration and Scoring The combined scales totalling 41 items was adminis- tered to three sixth grade classes. Classes were selected from three different semi-rural school communities repre- senting a cross-section of socio-economic levels. The self-concept of general ability scale consisted of fifteen multiple choice questions devised to explore one's conception of one's ability to do well in school. The arithmetic self-concept scale included twenty-six multiple choice questions related to one's attitudes to- ward arithmetic. Student responses to the questions of the two scales were scored on a five point scale. Respon- ses representing the low self-concept answer were weighted one and responses representing the high self-concept an- swers were weighted five. Validity Although validation of self-concept scales is diffi- 140 cult, scores obtained for the pre-test sample confirmed expectations that there would be a significant difference in achievement, represented by arithmetic grades, between children with high arithmetic self-concept scores and children with low arithmetic self-concept scores. A mean of 105 was obtained on the arithmetic self-concept scale for students with grades in the upper quartile of the sample compared with a mean of 77 for students whose arithmetic grades fell in the lower quartile as determined by teacher evaluations. The Mann-Whitney U-Test was used to analyze differences because the scores could not be considered to be normally distributed since extremes in quartile ranges were used. The results are shown in the table below: Table I Differences Between the Arithmetic Self-Concept Mean Scores for Upper and Lower Quartile Groups (one- Groups N AS-C Mean ‘5 p tailed test) Upper Quartile 36 105 Lower Quartile 36 77 5.67 (.0005 N372; The arithmetic self-concept scale was considered to have empirically demonstrated practical validity, at least in terms of arithmetic grades that reflect arithmetic sta- tus in the classroom for the pre-test sample. 141 Reliability Reliability data for the arithmetic self-concept scale is given in Table II. The results represent the odd-even split-half reliability coefficient as computed through use of the Spearman-Brown Prophecy Formula. 2:. = 2.1:.“ 1+5}: is Table II Reliability Coefficient for the Arithmetic Self- Concept Scale 5. £11 9 Arith. Self-ConC. .60 .75 (.001 Scale-Split-half In relation to the findings, the arithmetic self-con- cept scale was considered to have reliability for the pre- test sample since 25(2001. Pre-Test Results The pre-test data were used to refine the pre-test instrument into an improved questionaire for the study of relationships between arithmetic self-concept and arithme- tic achievement. Phrasing was brought more into line with sixth grade comprehension on the final questionaire than they were on the original instrument, and one question that showed little discrimination was eleminated on the revised scale. APPENDIX B Please print your name, birthdate, sex, school name and grade in the prOper blanks below: Name I I Last First Middle Birthdate Boy Girl Age Month Date Year Name of School Grade In the spaces below check the highest grade your mother completed in school: 16156156136 [3636 ”66: grade chool Hi Sgnool goglege This questionaire is part of a study project designed to help us to understand what school students in your grade think of their school ability, especially in arithmetic, and of their future plans in life. Please answer the questions as though you were describing yourself tglygurself. Answer each question just as honestly as you can. You should make the answers show how on really think about how well ygg are able to do in scfiaal and especially in arithmetic. Remember, there are no right or wrong answers as such -- only how on feel about the answers to the ques- tions. The answers s ould not show how you would like them to be, but should show how Easy are to you now. Read each question and the answers that follow it carefully. Select the answer that best tells how on feel about the question and place an 'x' as the directions tell you. Please read the directions for each part of the question- aire carefully before you answer. Each boy and girl is different and ou should say what is true for on. So be sure to pick t e answer that best shows how you eel or think about each question. ‘ 142 143 If you have any trouble, or if you have any questions about any word or statement raise your hand and someone will help you. No one will be shown your answers. They will not be shown to your teacher and your answers will not affect your school marks in any way. When you are told to start, follow the directions careful- ly and then begin. Take your time -- there is no hurry. Place an 'X' in the box in front of the statement that best answers each question for ou. Answer all the ques- tions. (You will have one “X” ior each question). 1. How do you rate yourself in school ability compared with your close friends? am the best am above average am average am below average am the poorest 3. b. C. d. 6. Htfidetd 2. How do you rate yourself in school ability compared with those in your class at school? a. b. C. d. 8. am among the best am above average am average am below average am the poorest HHHHH 3. Where do you think you will rank in your high school \graduating class? a. Among the highest b. Above average c. Average d. Below average e. Among the lowest 4. Do you think you have the ability to complete college? a. Yes, certainly b. Yes, probably c. Not sure either way d. Probably not e. No 5. 6. 7. 8. 9. 144 Where do you think you would rank in your class in college? a. Among the highest b. Above average c. Average d. Below average e. Among the lowest In order to become a doctor, lawyer or college teach- er work beyond four years of college is necessary. How likely do you think it is that you could complete work beyond four years of college? a. Very likely b. Somewhat likely c. Not sure either way d. Unlikely e. Most unlikely Forget for a moment how others mark your work. In your own opinion, how good do you think your school work is? a. My work is excellent b. My work is good c. My work is average d. My work is below average e. My work is much below average What kind of marks do you think you are able to get, if you work? a. Mostly A's b. Mostly B's c. Mostly C's 6. Mostly D's e. Mostly E's How important to you are the marks you get in school? d. a. Very important b. Important c. Not really important Doesn't matter to me at all 10. 11. 12. 13. 14. 15. 145 ‘. How important is it to you to be high in your class in marks? a. b. C. d. Very important Important Not really important Doesn't matter to me at all How do you feel if you don't do as well in school as you know you can? a. b. C. d. I feel very badly I feel badly I don't feel especially badly It doesn't bother me at all How important is it to you to do better than others in school? a. b. C. d- Very important Important Not really important Doesn't matter to me at all Which statement best tells how you feel? Ela- :jb- 1:34 1:14 I like to get better marks than everyone else I like to get better marks than almost everyone else I like to get about the same marks as every- one else I don't care about any special marks In your schoolwork do you try to do better than others? a. b. C. d. All of the time Most of the time Once in awhile Never How important to you are good marks compared with other things you get out of school? Eja- 53b- cad. Good marks are most important things in school Good marks are among the important things in school Some other things in school are more impor- tant Good marks don't matter to me at all 146 Put an ”x" in the box under the statement which best an- swers the question for you. (You will have an “X” for each line.) 1. How do you rate your ability in arithmetic compared with the ability of your close friends? I am among I am below I am I am above I am among the poorest average average average the best CI] [:3 C3 1:] [:1 2. How do you rate your ability in arithmetic compared with those in your class at school? I am among I am below I am I am above I am among the poorest average average average the best C3 C3 C3 [:1 12:] 3. Where do you think you would rank in your high school graduating class in arithmetic? Among the Below Average Above Among the lowest average average highest [:3 [:1 [II] II] [:3 4. Do you think you have the ability to do college work in arithmetic? No Probably Not sure Yes, Yes, not either way probably certainly [:3 [:3 C3 C3 [:3 5. Where do you think you would rank in a college class in arithmetic? Among the Below Average Above Among the lowest average average highest [:3 C] C] 1:] E3 6. How likely do you think it is that you could complete advanced work beyond college in mathematics? Most Unlikely Not sure Somewhat Very unlikely either way likely likely [:1 C] [:3 [II [:3 147 7. Forget for the moment, how others mark your work. In your own opinion how good do you think your work is in arithmetic? My work is My work is My work is My work is My work is much below below average good excellent average average [:1 E: [I] [:3 E3 8. What kind of marks do you think you are capable of getting in arithmetic? Mostly Mostly Mostly Mostly Mostly E's D's C's B's A's C: [:1 [:1 [:3 CI] 9. How do you rate your ability to do word problems in arithmetic compared with others in your class in school? I am among I am below I am I am above I am among the poorest average average average the best [:1 [:1 :3 CI! C: 10. How well do you feel you are able to understand arith- metic compared with other school subjects you study? Not well . Poorly Well Fairly Very well at all enough well indeed CI] [:3 C3 [:1 [:3 11. Do you think arithmetic is as difficult to learn as other subjects you study in school? It is the It is It is no It is not It is one most quite more diffi- difficult of the difficult difficult cult than easiest others [:1 1:! {:1 1:] CI! 12. How do you rank arithmetic in practical everyday use- fulness compared with other school subjects you study? It is the It is not About the It is very It is the least very use- same as useful most useful ful others useful CI] [:1 E: E3 [:3 148 13. How often do you think of arithmetic problems out- side of school? Never [:3 [:3 Very seldom Occasion- CZ] Quite often Very ally often [I [:3 14. How do you rate the study of arithmetic compared with other school subjects, such as language or social studies, that you study in school? It is the It is not least important important [:1 [:1 About the It is It is the same as important most other important subjects [:1 [:1 [:1 15. How do you rank your ability to do long division problems compared with others in your class at school? I am among the poorest average [:1 [:3 I am below I am I am above I am among average average the best [:3 1:3 [:3 16. How sure of yourself do you feel in the solving of arithmetic problems compared with work in other subjects you study in school? Very Not sure unsure of myself of myself [:2] :3 About the I am sure I am very same as of myself sure of other myself subjects [:1 CI] [:1 17. If you had a choice, how much time would you want to spend studying arithmetic in school? Much less Less time time than than at at present present [:3 [:I :3 About the More time Much more same as at than at time than present present at present [:1 1:3 149 18. How well do you rank your ability to do long addition problems in arithmetic compared with others in your class at school? I am among I am below I am I am above I am among the poorest average average average the best [:3 1:3 [:3 [:1 [:3 19. What marks do you think your parents would say you are capable of getting in arithmetic? Mostly Mostly Mostly Mostly Mostly E's D's C's B's A's [:3 [:3 C: 1:] CI! 20. How do your marks in arithmetic compare with marks you receive in other school subjects you study? They are They are They are They are They are among the below average above among the poorest average average best [:1 [:1 CI! [:3 12:1 21. How do you rank your ability to do long multiplication problems in arithmetic compared with others in your class? I am among I am below I am I am above I am among the poorest average average average the best 1:] CI] [:1 1:3 1:3 22. What marks do you think your teacher would say you are capable of getting in arithmetic? Mostly Mostly Mostly Mostly Mostly E's D's C's B's A's CI} [:1 [:3 [:1 [:1 23. How do you rank your ability to do difficult subtraction problems in arithmetic compared with others in your class in school? I am among I am below I am I am above I am among the poorest average average average the best [:1 [I] [:3 [:3 E3 150 24. Do you think that the study of arithmetic is as important for future living for you as much as other subjects you study in school are important? It is the It is not It is about It is some- It is the least as important as important what more most important as other as other important important subject subjects subjects than other subject subjects [:3 [:1 [:1 E: C] 25. How do you rank your ability to do work involving fractions in arithmetic compared with others in your class in school? I am among I am below I am I am above I am among the poorest average average average the best [:2] [:3 [:3 [:3 [:1 Thank you for your help APPENDIX C Name How many years have you taught school? years. Have you had occasion to teach arithmetic before? yes no How long have you taught arithmetic? years. What is your professional training in mathematics? Have a major 3 have a minor 3 have less than a minor in mathematics . Approximately how many college courses have you had in mathematics? How many courses in the teaching of mathematics? . Have you taken any coursework in mathematics since beginning to teach mathematics , ? How many courses? . yes no Please place an 'X' in the box under the statement which best describes how you feel about the statement. 1. How do you rank the importance of arithmetic to future living in comparison with other courses taught at this level of school? It is the It is not It is about It is among It is the least very the same as the most most important important others important important I: I: 2. How would you rank your understanding of arithmetic concepts and processes in comparison with other subject material you teach or have taught? Among the Below Average Above Among the poorest average average best I] C.J— I] 151 152 3. If you had your choice, how much time do you think should be devoted to the teaching of arithmetic at this grade level? Much less Less time About the More time Much more time than at same amount than at time than present of time present at present 4. How would you rank the rapport between you and your students in the teaching of arithmetic? It is among It is It is It is It is among the poorest below average above the best average average 5. How would you rank your professional training for teaching arithmetic? Very poor Poor Adequate Good Very good I] 6. How would you rank your personal sense of satisfaction in teaching arithmetic compared with other academic sub- jects taught at this grade level? It is the It is not About as Somewhat It is the least as satisfy- satisfy more satis- most satis- satisfying ing as ing as fying than fying sub- others others others ject of all 3 :l 7. How do you think your students would rank you in terms of competency in teaching of arithmetic in comparison with the teaching of other subjects? Among the Below Average Above Among the poorest average average best C] I: :1 153 8. How would you rank your ability to teach arithmetic at this grade level? I am among I am I am I am above I am among the below average average the best poorest average I] C] D 9. How well do you like teaching arithmetic in compari- son with other school subjects you teach or have taught? It is It is About the It is above It is among the below same as average among the poorest average others best I] 10. How would you rank your ability to teach arithmetic in comparison with other teachers of arithmetic at this grade level? ‘ I am among I am I am I am above I am among the below average average the best poorest average [:i :1 li'illlllll Illll.lliilliillllll APPENDIX D Stanford Achievement Test, Intermediate Battery, Form W Norms and Population The norms are derived from a nation-wide testing sampling, during February and March, 1963. A total of 264 school systems drawn from 50 states participated; 850,000 pupils were tested as part of this program. Public schools (integrated, segregated white, and segregated negro), pri- vate non-sectarian and private sectarian were included in the sample. The schools were selected from areas repre- senting nine geographic locations prOportioned to 1960 census reports. Validity Content or curricular validity was sought by examining apprOpriate courses of study and textbooks as a basis for determining the skills, knowledges, understandings, etc. to be measured, according to publishers. The purpose of Stan- ford Achievement Test is to provide dependable data con- cerning pupil achievement in important skills. The content of the final forms of the test was selec- "154 155 ted from the total body of material tried out experimental- ly in such a way that the final tests conform to the origi- nal specifications with respect to content. Reliability The manual presents odd-even split half reliability coefficients using the Spearman-Brown and the Kuder- Richardson reliability coefficients. According to Bryan (22), reliability coefficients derived from the Spearman- Brown and the Kuder-Richardson formulas have a median of .88. Reliability E's ranging from .88 to .90 indicate the test may apprOpriately be used to evaluate group dif- ferences in ability in various subject areas, for purposes of planning individualized instruction, grouping pupils for instructional purposes, determining and evaluating rate of prOgress, and evaluating achievement. The reliability coefficients are based on a sample of 1000 cases drawn randomly from 76 school systems testing in all grades 1-9 in national standardization. Bryan (22) says this about the arithmetic sub tests: In providing a measure of that phase of the traditional mathematics curriculum known by the gen- eral term "arithmetic", the Stanford Achievement Test continues to be outstanding among’tests of its kind. APPENDIX E California Short-Form Test of Mental Maturity, 1963, S-Form, Level 2 Norms and Population The 1963 revision of the §£2££7§2£fl embodies a com- pletely new scaling and norming from the 1957 E22537§25§3 A basic change in this connection is the use of chronolOgical age groups for normative data and deviation units for all de- rived scores, except mental age. The 1963 revision of the Short-Form was scaled to the 1960 revision of the Stanford- Binet, Form L-M, to obtain the total I.Q. and corresponding mental age. A total of 38,793 cases from schools represent- ing seven geographic regions of the United states was used for scaling and norming of the 1963 revision. Validity Norming and scaling involved testing procedures which included comparison with other intelligence or mental abili- ty tests, comparison to the 1957 edition, articulation, and try-outs of norms in complete school systems, in addition to scaling of the M'm series to the Stanford-m £2- telligence £2212“ For the most part scaling and other data processing were concentrated on cases in the national age- 156 157 group pOpulation from just below the first percentile rank to just above the 99th percentile rank, i.e., I.Q.‘s from 63 to 137 inclusive. The California Short-Form Test 2£.Menta1 Maturity is a one-period adaptation of the Cali- fornia Tg§t_g£ Mental Maturity. The Shggtfgggm, 1963 Re- vision consists of seven test units, each representing a different mental exercise: The Non-Language Section con- sists of four units measuring mental capacities related to the recognition or logical analysis of abstract relation- ships. The Language Section consists of three units samp- ling the ability to comprehend verbal and numerical con- cepts and the extent and accuracy of recall. It provides information about the functional capacities that are basic to learning, problem-solving and responding to new situa- tions. The §22£ETEEE§ serves both survey and analytical purposes for educators, counselors, psychologists and em- ployers in a variety of testing situations. Reliability Stanley, quoting statistics included in the "Reliabili- ty Report" issued by the California Testing Bureau, writes in Euros, Sixth Mental Measurements Yearbook (23) that reli- ability coefficients for the Short-Form using the Kuder- Richardson formula range from .94 for the language portion, .89 for the non-language portion and .95 for the total I.Q. scores. APPENDIX F The Measurement of Social-Status: McGuire-White Indices of social-status and family life style are de- scribed in this present paper and directions are given for their calculation. The status index approximates the posi- tion of a person with some frame of reference peOple employ to place one another. Human behavior tends to vary some what according to status. The relationship between "what one feels, thinks and does“ and ”where one fits in,” however, is not a direct one. Social roles are a functional aspect of status. Role behaviors appropriate to sex, age-grade, and social-status are learned according to place and through time. As a consequence of role experiences according to status system- atic variations in cognitive discriminations, in cathetic attachments, and in value-apprehensions appear and persist unless changed to accompany a shift in status. An index is useful in placing subjects in subclasses of sample pOpulations for various kinds of behavior research. Comparisons can be made among the several sub-samples in an investigation to determine just what are the probable sources of variation in behavior. In broad terms, the sources of 158 159 variation can be looked upon as biolOgical differences (age-sex), cultural patterns (life-styles...), social characteristics (status, roles), and psychological attributes (e.g. motives and attitudes). The status indices described below are based upon questions commonly asked by peOple who are seeking to ”q "place” one another. Each index depends upon a combina- tion of ratings from three or more scales. Index of Social-Status -- Short Form 0 Occupation Rate 1 to 7 on 00 Scale E Education Rate 1 to 7 on ED Scale A Dwelling Rate 1 to 7 on DA Scale Area The Rating Scales Dwelling Areas 1. Select residential area (or acres) of highest repute in the community. Such an area usually is set apart and does not exist in every community. 2. Status areas of high repute: homes vary in size but they are set upon well-kept grounds which afford some privacy; only a few highly-valued apartments. 3. Preferred residential areas where there are few if any pretentious homes but dwellings and grounds have a great deal of care: good apartment buildings. llill'll‘l'll'lr'll'lll’?‘ I 1 1| 4. 7. 160 Average residential neighborhoods with no deterior- tion; dwellings are relatively small and unpretentious but neat in appearance; "respectable" homes. Dwelling areas which are beginning to deteriorate; some families ”don't know how to take care of their place”; business or industry entering in outside of the neighborhood shOpping center which characterized '3" or "4". An area which has deteriorated considerably but is not p- a slum; "run-down" and the reputation is "low"; small ' businesses and industries are interspersed. Slum area (or areas) of the community; neighborhood is 5 in bad repute, although an occasional dwelling may be well-kept; other homes are "shacks". Educational Attainment l. 3. 5. 7. Completed appr0priate graduate work for a recognized profession at highest level; graduate of a generally recognized, high status four-year college. Graduate from a four-year college, university or pro- fessional school with a recognized bachelor's degree, including four-year teacher colleges. Attended college or university for two or more years; junior college graduate; teacher education from a nor- mal school; R.N. from a nursing school. Graduate from high school or completed equivalent sec- ondary education; includes various kinds of "post-high" business education or trade school study. Attended high school, completed grade nine, but did not graduate from high school; for persons born prior to 1900, grade eight completed. Completed grade eight but did not attend beyond grade nine; for persons born prior to 1900, grades four to seven would be equivalent. Left elementary or junior high school before completing grade eight; for persons born prior to 1900, no educa- tion or attendance to grade three. Hma muosuo on .955 II II 80.3 913 madam a... 3.513 JR .665 I. .I 3 8on mo mumsao .sxun .mmuss .smeuflmmwu Hm um .muoumummo wusmmn um pmsam> HmEHom .meumw .um ..Hnums .pup pmaaaxm psm msosmmamu «mxao avasvo no @000 so musmsoa swamowaom ou .ouums .ooum .mxao .uum mafia mmmswmsm .v .>ummsm osB usmmm .ompE s3osx ooo.omm suflz umumfi m.umonm pommwa mQOn was .mamu Ho mm mo smEmmHMm ou ooo.oaw swE .mmnss mo msoumsmmo mmmfl>uwmsm no .xao Hmumom no mummsn um msam> pmuosmwmmu dams pmufls suw3 um mxuo3 0:3 ..sfiam ousd no sosmns muflswm Ho .umsommu mumszo Eumm .Husoo aHmEm sumac xsmm Em «C .Hmz mumsamsm com wpmno .m umuwu3 mmnmmp mafia swans .uouflpm «spa musmmm H» w saws m>wuom sm m>ms Immamm Mooum one meow ooo.ooaw muwsuo psm psm mmwuummoum ..umm ammu mo .>m9m no on ooo.omm smanmuswa mmw>umm5m oss ..umm .Hsmsfl ..mE vamp w as pmsasb .umsommu muousummo psmq .usmussooos .wmo .umms mmmswmsm Hoosom am .N spaaau: %.umonm Hamsu .smmno msumum .nsm no xsmn muflssafioo .umsm sum mauomufip mmfl>nm mo nommlomxm Ihnmmoo no so mswpsmm .uOmmomoum Imsm uos 0p os3 «msanmmme no .Hm um mmum ImpIImHOE Ho .uwmsfl mumsBOUsma Ho Hmmmmmzms mo II mm>wu ooo.ooaw um msm .mmpsn Emmy smEmHusmw wouflpw ammo Isomxw mos Hm>.man mmq uwhzmq .H manomm Ehmm w0fl>uwm Hmaaoo msHm Hmaaoo muwsz soammmswmsm muoumaumoum HmsOAmmmmoum mDZHM 02¢ mdm>mq umZOHfideUUO NoH Homams u.usmn .smEOB smumumms nsuom .mos mumxno3 Imumuumsgmslmss .mamspmaaflxmss .mpsms Hawa no msHE ..mux3 usmumwz caummaoa asmE QOnIppo «Honda >>mmm smumxwmun3mq pmusmmm: an .pum spm mmm mumsumw mommmuuflms swasoum3 psm mama mosmumflmbsm w mumuwm3 smEmmsosmnm3 .smEmmpmup Ho ooo.mm .mumuonma Esau .mnm>flup pmaaflxm op musmumwmmm up pmsam> pmsmflHQMDmm xosuu «mumxuo3 sowuospoum psw hufisgm Ho .mummmouomumsm psm mea muouomm pmaaflxmleOm mmmswmsm .w .uso muss. .muxs .xm .cms ooo.mm os3 meumm mo smmemun .smEHHmmmH on ooo.nw mumszo HmEHOM mus .uomum .smEmmpmuu .mummo husmmn psm .msosm um pmsHm> .mEwa poom .Hmnumn pwaaflxm on Imam» .mxuoao mumoouw wuflsvm no so musmsmB.smEm0flHom moansmsmms .wxnmao muoum meflov mmmsflmsm .m .meama. snags: .uxsaoumz .11 .mm II. II ooo.osw munmdoumu.uwcoo mm .um ..m:m mm .Hm .um 0» ooo.mw amusmu mo .Hoawsu .sMAowHuome .mwuoum .ummp sw mamomm up pmsam> .meo .HosBo .swmummo ..mumu .Humz mwamm .usmmm uwxowu muwsvm no Ipsma HHmEm mowaom .swEmHom anmmmmxooa ..Hmosmum mmmsamsm .c 163 According to the authors, it should be remembered that the indices of class-status or of life styles made by using the criteria intrinsic to the McGuire-White Measurement QE’Social-Status are only approximations, probably correct 80 or 90 percent of the time. I I l t I I I... ‘t In... I 'l..| l {I . l ha ma ma «H ma NH oo.H mm. mm. Nb. an. mv. oo.H 55. mm. mm. vm. oo.a om. mm. mm. oo.H mm. mm. oo.H mm. oo.H usmocooumamm chumsanasa wmmnm>¢ #sflom wpmuw Hmuoa wpmno UHDoEspflud w>HumHsEsU mpmnw oauossuwnm umumoamm mpmnm cavmesufius uswuusu mEmHsoum nonfisz mmsHm> umnasz .O.H mmmsmsMQIsoz .O.H mmmsmsmq .0.H mamom Hash msOADMOHHmms oaumfisufiud mumwosou oapwEsuwua msoflumusmEoo oaumgsuflus ummosOUIwme Hmumsww sowumospm w.umsuoz xmpsH Hmwoom xmm Ha mv. vv. mv. ow. Nv. mm. oo.H .hH .mH .mH .ca .MH .NH .HH .oa .m .h .w .v .m .N voa OH m m b m m av. mv. Hm. mm. ow. No. Hm. vm. mm. 5w. mm. mp. mm. mm. mm. mm. Hm. mm. mm. 5v. mm. mm. mm. mm. mm. mv. pm. an. an. mm. mm. mm. vm. mm. Hm. mv. ow. ow. um. vv. ov. wv. oo.H me. am. am. mm. mm. oo.H mm. mm. mm. mv. oo.H mm. mm. mm. oo.H mm. mm. oo.H hm. oo.H msowumamuuoouwusH mo xfluumz O xHflzmmmd an. up. mm. me. am. mm. mm. mm. mm. ov. mv. mv. mv. oo.H mm. Hm. mm. mm. mm. vm. mm. mm. Hm. om. mm. mm. am. am. oo.H om.l mm.l Hm.l mm.l oq.l mm.l mm.l mm.l mm.l mm.l mm.l hN.I mm.l mm.l v¢.l oo.H mo. om. pa. ma. ma. mo. ma. no. mo. 00. mo. mo. mm. Ho.l ma. mo.l oo.H NH ma ma Va ma NH Ha OH lull"! I I I: . . I l'iIlI‘ll111.llll!i‘ii ' 'l-w'v v‘v—v F'wv F‘- v " 1‘ Irv—VW‘ Y' r- "I7111111117111“111713