A STUQ‘Y OF THE CORRELATION BETWEEN THE ACADEMIC PREPARATION OF TEACHERS OF MATHEMATICS AND THE MATHEMATICS ACHIEVEMENT OF THEIR STUDENTS IN KINOERGARTEN THROUGH GRADE EIGHT TImsIs Im» II" Daqm oI DII. D. MECIIIGAII STATE UNIVERSITY William Morrison Rouse, Jr. 1967 emu-“Ann. \ZIA In; «A. .n 7' ‘ XI A’Iic‘lzigan State Univcz'sgty (as a ‘ J o THESIS, MIMI 1293 10221 55 I III III I! III! II III I I! I! II III III! This is to certify that the thesis entitled A STUDY OF THE CORRELATION BETWEEN THE ACADEMIC PREPARATION OF TEACHERS OF MATHEMATICS AND THE MATHEMATICS ACHIEVEMENT OF THEIR STUDENTS I'N KINDERGARTEN THROUGH GRADE EIGHT presented by William M. Rouse, Jr. has been accepted towards fulfillment of the requirements for Ph.D. degree inMathematics Education < / Date 1 August 19' 0-169 . u— ”Hy—u N--. « .-.¢- ABSTRACT A STUDY OF THE CORRELATION BETWEEN THE ACADEMIC PREPARATION OF TEACHERS OF MATHEMATICS AND THE MATHEMATICS ACHIEVEMENT OF THEIR STUDENTS IN KINDERGARTEN THROUGH GRADE EIGHT by William M. Rouse, Jr. The Purpose A relationship may exist between the amount of mathe- matics preparation of the set of teachers who are respons- ible for the mathematics instruction of a student over a period of years and the subsequent mathematics achievement of that student at the end of that period. The purpose of the study was to determine the extent of such a relation- ship over periods of time encompassing the first five years, the first seven years, and the first nine years of formal elementary school education. The Procedure These three periods of time were called grade level periods, and they included: 1. kindergarten through the middle of grade four 2. kindergarten through the middle of grade six 3. kindergarten through the middle of grade eight. Three categories of teacher mathematics preparation were considered: William M. Rouse, Jr. 1. high school mathematics preparation 2. college mathematics preparation 5. total mathematics preparation (high school, college, and in-service mathematics combined). Three aspects of student mathematics achievement were considered: 1. arithmetic reasoning 2. arithmetic fundamentals 3. total arithmetic (reasoning and fundamentals combined). The three grade level periods, the three categories of teacher mathematics preparation, and the three aspects of student mathematics achievement resulted in 27 combinations for comparison. Each combination was con- cerned with a category of teacher mathematics preparation, an aspect of student mathematics achievement, and a parti- cular grade level period. Teaching experience was in- cluded in each of these 27 combinations to provide a bench mark against which teacher mathematics preparation could be compared. Adjustments were made for variations in student intelligence. Teacher data were collected by means of a question- naire. Student data were collected from examination of permanent school records. The averages of the values of William M. Rouse, Jr. the teacher characteristics for the teachers of each stu- dent were compared to the student's mathematics achieve- ment for each grade level period. An electronic digital computer was used to accomplish the matching of each stu- dent with his particular set of teachers. The computer was also used for the calculation of multiple regression statistics for each of the 27 combinations of teacher and student characteristics. Conclusions Teaching experience. A low positive correlation existed between the achievement in arithmetic fundamentals of eighth grade students and the amount of teaching experi- ence of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade eight. No correlation existed for arithmetic reasoning at that grade level period. No correlation existed for arith- metic achievement at the other two grade level periods. Teacher high school mathematicsgpreparation. No cor- relations existed for grade level periods kindergarten through the middle of grade four and kindergarten through the middle of grade six. A low positive correlation existed between the arithmetic achievement of eighth grade students and the amount of high school mathematics prepara- tion of the teachers responsible for their arithmetic William M. Rouse, Jr. instruction from kindergarten through the middle of grade eight. Teacher college mathematicsgpreparation. A low nega- tive correlation existed between student arithmetic achieve- ment and teacher college mathematics preparation for grade level periods kindergarten through the middle of grade six and kindergarten through the middle of grade eight. No correlation existed for grade level period kindergarten through the middle of grade four. Teacher total mathematicsgpreparation. A low negative correlation existed between student arithmetic achievement and teacher total mathematics preparation for grade level periods kindergarten through the middle of grade four and kindergarten through the middle of grade eight. No corre- lation existed for grade level period kindergarten through the middle of grade six. A STUDY OF THE CORRELATION BETWEEN THE ACADEMIC PREPARATION OF TEACHERS OF MATHEMATICS AND THE MATHEMATICS ACHIEVEMENT OF THEIR STUDENTS IN KINDERGARTEN THROUGH GRADE EIGHT BY William Morrison Rouse, Jr. A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1967 ACKNOWLEDGEMENTS The investigator is grateful to several persons with- out whose help this study could not have been conducted. Mr. Gary Fisher, Principal, Henry R. Pattengill Junior High School, Lansing, Michigan, cooperated in the procurement of student data. Mr. Eugene Richardson, Director, Department of Teacher Education and Certification, State of Michigan, provided outstanding assistance in the procurement of teacher data. Dr. Edward Remick, Consultant in Research, Lansing School District, provided continuing assistance and cooperation throughout the course of the study. Dr. Julian Brandou, Acting Director, Science and Mathematics Teaching Center, Michigan State University, provided serv- ices and facilities which greatly expedited the preparation of the thesis. Finally, the investigator wishes to acknowledge the service and assistance rendered by his doctoral guidance committee members, Dr. Charles Blackman, Professor of Edu- cation, Dr. Edward Nordhaus, Professor of Mathematics, Dr. Max Smith, Professor of Education, and Dr. William Walsh, Professor of Education and chairman of the committee. William M. Rouse, Jr. ii CHAPTER TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . I. II. III. IV. V. VI. THE PURPOSE OF THE STUDY. . . . . THE SCOPE OF THE STUDY. . . . . . Teacher Mathematics Preparation Student Mathematics Achievement Periods of Grade Levels Covered Secondary Factors Included. . . THE DELIMITATION OF THE STUDY . . Limitation of Factors . . . . . Limitations of Student Outcomes Limitations of Determination of Relationship. . . . . . . . . SOME ASSUMPTIONS OF THE STUDY . Cumulative Effect of Teacher Char- acteristics . . . . . . . . . Delayed and Immediate Effects . Effect of Compensating Teacher Char- acteristics . . . . . . . . . Effect of Variability in Student Growth. . . . . . . . . . . . THE IMPORTANCE OF THE STUDY . . . The Issue of Mathematics for Elemen- tary School Teachers. . . . . The Lack of Evidence Concerning Mathematics Preparation for Elemen- tary School Teachers. . . . . THE ORGANIZATION OF THE STUDY . . II. REVIEW OF THE LITERATURE. . . . . . . . iii '0 9: LG (D U1 IF “>04 (N NNNH I-‘ I—“ P (OODCDO) 10 10 14 16 17 TABLE OF CONTENTS - Continued CHAPTER I. RESEARCH RELATED TO TEACHER PREPARATION. Studies at the Secondary School Level and Above. . . . . . . . . . . . . . Studies at the Elementary School Level General Observations Regarding Teacher Preparation Studies. . . . . . . . . Specific Observations Regarding Teacher Preparation Studies. . . . . Summary. . . . . . . . . . . . . . . . II. RESEARCH RELATED TO TEACHER EXPERIENCE . The Studies. . . . . . . . . . . . . . Summary. . . . . . . . . . . . . . . . III. PROCEDURE . . . . . . . . . . . . . . . . . . I. THE DESIGN OF THE STUDY. . . . . . . . . II. THE SETTING OF THE STUDY . . . . . . . . III. THE PROCUREMENT OF THE RAW DATA. . . . . The Student Data . . . . . . . . . . . The Teacher Data . . . . . . . . . . . IV. THE TRANSFORMATION OF THE RAW DATA . . . The Student Data . . . . . . . . . . . The Teacher Data . . . . . . . . . . . The Collation of the Student and Teacher Data . . . . . . . . . . . . Summary. . . . . . . . . . . . . . . . IV. STATISTICAL RESULTS. . . . . . . . . . . . . . I. THE DISTRIBUTIONS. . . . . . . . . . . . II. THE RELATIONSHIPS. . . . . . . . . . . . Simple Correlations. . . . . . . . . . Multiple Correlations. . . . . . . . . V. SUMMARY. . . . . . . . . . . . . . . . . . . . iv Page 17 17 25 35 38 42 42 43 47 48 48 55 57 57 60 64 64 65 68 75 76 76 81 81 85 116 TABLE OF CONTENTS - Continued CHAPTER II. III. IV. THE PURPOSE OF THE STUDY. . RELATED STUDIES . . . . . . THE PROCEDURES OF THE STUDY The Design. . . . . . . . The Data. . . . . . . . . THE RESULTS OF THE STUDY. . Teacher Mathematics Preparation . . Teacher Experience. . . . Student Intelligence. . . VI. CONCLUSIONS AND IMPLICATIONS. . . I. II. III. IV. VI. STUDENT INTELLIGENCE. . . . TEACHER EXPERIENCE. . . . . Conclusions . . . . . . . Implications. . . . . . . TEACHER HIGH SCHOOL MATHEMATICS ARATION . . . . . . . . . Conclusions . . . . . . . Implications. . . . . . . TEACHER COLLEGE MATHEMATICS Conclusions . . . . . . . Implications. . . . . . . TEACHER TOTAL MATHEMATICS PREPARATION Conclusions . . . . . . . Implications. . . . . . . PREPARATION SUGGESTIONS FOR FUTURE RESEARCH . . . Replication in Other Subject Matter Areas 0 O O O O O O O O Page 116 116 118 118 120 121 121 121 122 123 123 123 123 124 126 126 126 132 132 134 136 136 137 138 138 TABLE OF CONTENTS - Continued CHAPTER BIBLIOGRAPHY APPENDICES Page Replication in a Non-traditional Mathematics Setting. . . . . . . . . 138 Replication Involving Special College Mathematics Courses. . . . . . . . . 139 Use of Maximum Rather than Average Values . . . . . . . . . . . . . . . 139 Effects of Teacher Interest. . . . . . 140 Immediate vs. Delayed Effects. . . . . 141 Concluding Statement . . . . . . . . . 141 . . . . . . . . . . . . . . . . . . . 143 . . . . . . . . . . . . . . . . . . . 154 vi TABLE II. III. IV. VI. VII. VIII. LIST OF TABLES Page Arithmetic means, standard deviations, and measures of skewness and kurtosis of eight characteristics associated with 129 students from kindergarten through grade four. . . . . 78 Arithmetic means, standard deviations, and measures of skewness and kurtosis of eight characteristics associated with 128 students from kindergarten through grade six . . . . . 79 Arithmetic means, standard deviations, and measures of skewness and kurtosis of eight characteristics associated with 128 students from kindergarten through grade eight . . . . 80 Coefficients of simple correlation of the measures of eight characteristics associated with 129 students from kindergarten through grade four. . . . . . . . . . . . . . . . . . 82 Coefficients of simple correlation of the measures of eight characteristics associated with 128 students from kindergarten through grade six . . . . . . . . . . . . . . . . . . 83 Coefficients of simple correlation of the measures of eight characteristics associated with 128 students from kindergarten through grade eight . . . . . . . . . . . . . . . . . 84 Multiple regression statistics, teacher high school mathematics and student arithmetic reasoning, kindergarten through grade four. . 86 Multiple regression statistics, teacher high school mathematics and student arithmetic fundamentals, kindergarten through grade four 87 Multiple regression statistics, teacher high school mathematics and student total arithme- tic, kindergarten through grade four. . . . . 88 vii LIST OF TABLES - Continued TABLE X. XI. XII. XIII. XIV. XVII. XVIII. XIX. Multiple regression statistics, teacher col- lege mathematics and student arithmetic reasoning, kindergarten through grade four. . Multiple regression statistics, teacher college mathematics and student arithmetic fundamentals, kindergarten through grade four Multiple regression statistics, teacher col- lege mathematics and student total arithme- tic, kindergarten through grade four. . . . . Multiple regression statistics, teacher total mathematics and student arithmetic reasoning, kindergarten through grade four . . . . . . . Multiple regression statistics, teacher total mathematics and student arithmetic funda— mentals, kindergarten through grade four. . . Multiple regression statistics, teacher total mathematics and student total arithmetic, kindergarten through grade four . . . . . . . Multiple regression statistics, teacher high school mathematics and student arithmetic reasoning, kindergarten through grade six . . Multiple regression statistics, teacher high school mathematics and student arithmetic fundamentals, kindergarten through grade six. Multiple regression statistics, teacher high school mathematics and student total arithme- tic, kindergarten through grade six . . . . . Multiple regression statistics, teacher col- lege mathematics and student arithmetic reasoning, kindergarten through grade six . . Multiple regression statistics, teacher col- lege mathematics and student arithmetic fundamentals, kindergarten through grade six. viii Page 89 9O 91 92 93 94 95 96 97 98 99 LIST OF TABLES - Continued TABLE XXI. XXII. XXIII. XXIV. XXVI. XXVII. XXVIII. XXIX. XXXI. Multiple regression statistics, teacher col- lege mathematics and student total arithme- tic, kindergarten through grade six. . . . . Multiple regression statistics, teacher total mathematics and student arithmetic reasoning, kindergarten through grade six. . Multiple regression statistics, teacher total mathematics and student arithmetic fundamentals, kindergarten through grade six Multiple regression statistics, teacher total mathematics and student total arithme- tic, kindergarten through grade six. . . . . Multiple regression statistics, teacher high school mathematics and student arithmetic reasoning, kindergarten through grade eight. Multiple regression statistics, teacher high school mathematics and student arithmetic fundamentals, kindergarten through grade eight. . . . . . . . . . . . . . . . . . . . Multiple regression statistics, teacher high school mathematics and student total arith- metic, kindergarten through grade eight. . . Multiple regression statistics, teacher col- lege mathematics and student arithmetic reasoning, kindergarten through grade eight. Multiple regression statistics, teacher col- lege mathematics and student arithmetic fundamentals, kindergarten through grade eight. . . . . . . . . . . . . . . . . . . . Multiple regression statistics, teacher col- lege mathematics and student total arithme- tic, kindergarten through grade eight. . . . Multiple regression statistics, teacher total mathematics and student arithmetic reasoning, kindergarten through grade eight. ix Page 100 101 102 103 104 105 106 107 108 109 110 LIST OF TABLES - Continued TABLE Page XXXII. Multiple regression statistics, teacher total mathematics and student arithmetic fundamentals, kindergarten through grade eight. . . . . . . . . . . . . . . . . . . . 111 XXXIII. Multiple regression statistics, teacher total mathematics and student total arith— metic, kindergarten through grade eight. . . 112 LIST OF FIGURES FIGURE 1. The 27 Multiple Regression Combinations. . 2. Definition of Variables. . . . . . . . . . 3. Information Contained in the Three Sets of Cards. . . . . . . . . . . . . . . . . . . xi Page 52 53 74 LIST OF APPENDICES APPENDIX Page A. DESCRIPTION OF STATISTICS. . . . . . . . . . 155 B. QUESTIONNAIRE AND COVERING LETTER. . . . . . 159 C. HIGH SCHOOL MATHEMATICS CATEGORY VALUE SCALE 162 D. COLLEGE MATHEMATICS CATEGORY VALUE SCALE . . 163 E. TOTAL MATHEMATICS CATEGORY VALUE SCALE . . . 164 F. A FLOW CHART OF THE TRANSFORMATION PROGRAM . 166 xii CHAPTER I INTRODUCTION I. THE PURPOSE OF THE STUDY Is there a relationship between a student's mathema- tics achievement level and the amount of mathematics studied by the set of teachers responsible for his instruc- tion in mathematics from kindergarten through grade eight? If such a relationship exists, what is its magnitude? 1 Does a high level of student mathematics achievement accom- pany a high level of teacher mathematics preparation? It was the purpose of this study to answer such questions. Essentially the study was an attempt to determine the correlation that might exist between student mathematics achievement at the end of a period of several years of instruction and the mathematics preparation of the teachers who were responsible for that instruction. II. THE SCOPE OF THE STUDY Teacher Mathematics Preparation Three categories of teacher mathematics preparation were considered in the study: 1. teacher high school mathematics preparation 2. teacher college mathematics preparation 3. teacher total mathematics preparation. The third category was a combination of teacher mathema- tics preparation in high school, in college, and in in- service mathematics education programs. In each category only mathematics subject matter courses were considered. Student Mathematics Achievement Three aspects of student mathematics achievement were considered: 1. student achievement in arithmetic reasoning 2. student achievement in arithmetic fundamentals 3. student total arithmetic achievement. The third aspect was a combination of student achievement in arithmetic reasoning and arithmetic fundamentals. Periods of Grade Levels Covered Three grade level periods were considered in the study: 1. kindergarten through the middle of grade four 2. kindergarten through the middle of grade six 3. kindergarten through the middle of grade eight. Secondary Factors Included Teaching experience. Although the primary concern of the study was with teacher mathematics preparation and student mathematics achievement, teaching experience was included for the purpose of providing a bench mark against which teacher mathematics preparation could be compared. This provided a means of determining the relative size of the relationship between teacher mathematics prepara- tion and student mathematics achievement. Student intelligence. It is commonly accepted that the intelligence of a student plays a significant role in the determination of his success in school. It was felt that no meaningful measure of the correlation of teacher mathematics preparation and student mathematics achieve- ment could be obtained if it were ignored. III. THE DELIMITATION OF THE STUDY Limitation of Factors Many factors probably exist which either make direct contributions to the mathematics achievement of students or which are at least concomitant with such achievement. It is commonly accepted that certain intrinsic factors such as intelligence, motivational level, and emotional adjustment are Closely related to student achievement in any subject. The courses of study of teacher education institutions offer evidence of the importance placed on extrinsic factors such as instructional materials, the school curriculum, and the behavior of teachers. Anderson stated, "...pupil accomplishment is affected by many factors which we are unable to measure satisfactorily at the present time and it is questionable if we ever will."1 It was not the purpose of this study to identify or to measure these other factors. Limitation of Student Outcomes There are many desirable objectives toward which elementary schools strive. The mathematical competence of students is but one of them. Consideration of factors related to this one student outcome would not constitute a sufficient basis for the structuring of teacher educa- tion programs nor for the establishment of educational policies within a local school district. Nevertheless, knowledge of the causes, effects, and relationships regarding mathematics instruction is desirable and has utility in conjunction with other considerations in the overall design of educational enterprises. Limitation of Determination of Relationship With regard to the study of relationships, Lavin stated: When a significant association is found between some predictive variable and academic performance, the .question arises as to whether the predictor is a determinant of performance in the causal sense... the observation of an association between two vari- ables does not, in itself, establish the presence of a causal relationship. ...certain steps can be taken that at least help to support the validity of Causal interpretations. One procedure involves the establishment of time sequences among variables. It follows from the 1H. M. Anderson, "A Study of Certain Criteria of Teaching Effectiveness," The Journal of Experimental Education, 23:44, September, 1954. assumption that in a causal relationship, the inde- pendent or causal factor will precede the dependent factor in time.... While determination of proper time sequence helps to support causal interpreta- tions, it does not establish them with certainty... even if the predictor variable is shown to precede the criterion, the correlation may still be de- termined by another unknown factor.2 Specifically, in terms of the present study, the mere fact that the mathematics preparation of the teachers preceded the mathematics instruction and testing of the students, is insufficient to establish that a cause and effect relationship existed between the two factors. It is pos- sible that some undetermined third factor was the cause, and that they were merely concomitant results. It appeared that the determination of cause and effect was beyond the scope of this study, and therefore the study was limited to the determination of the correlation of the factors. IV. SOME ASSUMPTIONS OF THE STUDY It should be noted that although it was not an objec- tive to attempt to determine the existence of a cause and effect relationship between teacher mathematics preparation and student mathematics achievement, there exists the pos- sibility that such a relationship actually exists. This was taken into consideration in the design of the study and influenced certain basic assumptions. 2D. E. Lavin, The Prediction of Academic Performance (New York: Russel Sage Foundation, 1965), pp. 40.41. Cumulative Effect of Teacher Characteristics It was assumed that if it were important to consider how the mathematics preparation of a student's most recent teacher related to his mathematics achievement, then it should be important to consider how the mathematics prepa- ration of his complete set of teachers related to his mathe- matics achievement. If a student's achievement level during a certain year depends upon the achievement level which he attained the previous year, and if his achievement level that year resulted from the mathematics preparation of the teacher who taught him that year, then the mathematics preparation of the teacher of the earlier year had an ef- fect upon his mathematics achievement during the later year. This argument can be extended to include all of the teachers who taught the student from his entry into kinder- garten. Howard Taylor, writing in the Twentyrseventh Yearbook of the National Society for the Study of Education, Part II, commented: In general, pupils and classes are not trans- formed in one semester or even in several. But there does seem to be a sort of differential pressure varying with the estimated ability of the teacher which, quite independently of other factors, paral- lels the variations in final achievement. It af- fects only very slightly the outcome of each semester of instruction, but its unique Character suggests that the sum total of teacher influence on a given child or class would constitute very import- ant data for the prediction of total elementary- school achievement.3 3H. R. Taylor, "The Influence of the Teacher on Rela- tive Class Standing in Arithmetic Fundamentals, and Reading Ryans has also indicated that the influences of previous teachers upon present pupil behavior should be considered.4 The difficulty of investigating this cumulative effect was suggested by Taylor when he wrote: In general, each child or class in the course of an eight-year period is exposed to a sort of aver- age teaching ability, in that there are about as many poor as good teachers, and the effects of inferior instruction tend to cancel the effects of superior teaching in each individual case. Thus "quality of teaching received" is a very unstable "trait“ of a child or class, and it is almost im- possible to identify, whereas intelligence and attainment-to-date, being present each year to about the same degree, are increasingly stable character- istics which can be readily recognized and evaluated.5 It should be noted that these comments by Ryan and Taylor appear to be based largely on conjecture, since they were made in conjunction with reports of studies which were not concerned with the cumulative effects to which they referred. Nevertheless, this assumption regarding the importance of the cumulative measure of teacher characteristics re- flected the primary interest of the investigator and was basic to the design of the present study. Comprehension," The Twenty:seventh Yearbook of the National §ociety_for the Study of Education, Part II (Bloomington: Public School Publishing Co., 1928), p. 109. 4D. G. Ryans, "Teacher Personnel Research," The Cali- fornia Journal of Educational Research, 4:24, January, 1953. 5H. R. Taylor, "Teacher Influence on Class Achieve- ment," Genetic Psychology Monographs, 7:159, February, 1950. Delayed and Immediate Effects Ryans pointed out another difficulty in determining relationships between teacher characteristics and student outcomes. When does a given teacher's influence really take effect? Is it at the time a pupil is in the teacher's class, or may it be at some time after a pupil has left the particular teacher and has gone on to another teacher, or perhaps has left the school behind? To the extent that the effect of a teacher may be delayed, or latent, the measurement of such an effect at any given time is (a) contaminated by carry-over effects of previous teachers and (b) in- complete, because some of the present teacher's influence is still to be felt. Such delayed effects give rise to questions concerning the proper weighting of the characteristics of the various teachers of a particular student. Should the earliest or the latest teachers in the set receive the greatest weight, or are there other considerations which should be used in determining the appropriate weights? Because this infor- mation could not be determined, it was assumed that the characteristics of all the teachers of a particular student were equally important. Therefore the characteristics of all the teachers were weighted equally. Effect of Compensating Teacher Characteristics It was assumed that many factors, including teacher Characteristics, probably influence student achievement. With regard to this study, differing teacher characteristics BRyans, oc. cit. _— may have enabled one of two teachers with the same amount of mathematics preparation to more greatly influence the mathematics achievement of a student than the other. It was assumed, however, that although a teacher with little mathematics preparation may have compensated for it with other characteristics, this condition did not occur in general, and over the total group of students the effect was negligible. Effect of Variability in Student Growth Gleason reported a "marked tendency for high physical variability [in growth] to be accompanied with lower achieve- ment."7 It may have been that differences in growth rates, individually or collectively, produced differences among students in their susceptibility to teacher influence. A student experiencing a rapid growth rate while studying with a specific teacher may have been influenced differently than he would have been, had he been undergoing a slower rate of growth. Such an effect may have distorted the results of this study. However, it was assumed that this effect was negligible and could be disregarded in this initial investigation. 7G. T. Gleason, "A Study of the Relationship Between Variability in Physical Growth and Academic Achievement Among Third and Fifth Grade Children," (Doctor's thesis, University of Wisconsin, 1956, 167 pp.), Dissertation Abstracts, 17:563, 564, No. 3, 1957. 10 V. THE IMPORTANCE OF TEE STUDY The Issue of Mathematics for Elementary School Teachers In higher education it has been traditional that the teacher of mathematics have a thorough academic preparation in his subject. In fact the Ph.D. in mathematics has long been regarded as the minimum teaching certificate for pro- fessors of mathematics. At the high school level there has developed a slightly modified emphasis on the importance of thorough mathematics preparation for those who would be teachers of mathematics. Buswell stated in 1948 that "Many high schools now require a level of specialization that corresponds to a Master's degree in the subject being taught."8 However, at the elementary school level the situation is completely different. Dyer, Kalin, and Lord commented: "...little knowledge of mathematics is ex- pected,even'officially.of prospective [elementary] school "9 teachers. Buswell also declared, "At present high-school teachers have a reserve of scholarship in the subject they teach which elementary school teachers are not able to I10 match.‘ It is not uncommon to hear voiced the opinion 8G. T. Buswell, "Scholarship in Elementary-School Teaching," The_§lementary School Journal, 48:242—244, January, 1948. 9H. S. Dyer, R. Kalin, and F. M. Lord, grpblems in Mathematical Education (Princeton: Educational Testing Service, 1956), p. 13. 1°Buswell, loc. cit. 11 that college preparation beyond mathematics methods courses and student teaching is of little consequence for the success of the elementary school teacher. For example, Kranes reported the following. At a recent meeting on teacher training, an ele- mentary school principal stated that the primary function of his teachers was "to understand the Child's needs." To a question, "Will children learn the three R's from a teacher whose education has been primarily in the field of child development?" he re- plied, "Yes, since children learn anyway."ll Supposedly, in pursuing his own education, the elementary school teacher acquires a sufficient understanding of the mathematics concepts which he will teach. On the other hand, many groups and individuals have advocated the study of mathematics by prospective elemen- tary school teachers. In 1930 Buckingham declared, "If teachers cannot escape teaching language, arithmetic, and geography, they should not as students be permitted to escape the professional study of these subjects."12 In 1939 Morton advocated the development of the mathematics backgrounds of elementary school teachers. He proposed a study of algebra and geometry in high school and six to 3 ten semester hours of mathematics in college.1 During the 11J. E. Kranes, "The Child's Needs and Teacher Train- ing," School and Society, 88:155, March, 1960. 12B. R. Buckingham, "Training of Teachers of Arithme- tic," Report of the Committee on Arithmetic (Chicago: The National Society for the Study of Education, 1930), p. 324. 13R. L. Morton, "Mathematics in the Training of Arith- metic Teachers," The Mathematics Teacher, 32:106-110, March, 1939. 12 1940's the Commission on Post-War Plans of the National .14 Council of Teachers of Mathematics and the National Commission on Teacher Education and Professional Standards 15 recommended that of the National Education Association at least one content course in mathematics be required of all prospective elementary school teachers. In 1956 Stipanowich obtained Opinions from a jury of 65 specialists in mathematics education from educational institutions in 32 states. This jury was unanimous in favoring the require- ment of some mathematics courses in the programs of ele- mentary education majors, and the majority favored the prerequisite of at least two years of high school mathe- matics for entrance into an elementary education program.16 Typical of the faith placed in the contribution of teacher education and understanding to student mathematics achieve- ment are the following statements. 14The National Council of Teachers of Mathematics, "Guidance Report of the Commission on Post-War Plans," The Mathematics Teacher, 40:315-339, November, 1947. 15K. G. Young, "Science and Mathematics in the General Education of Teachers," The Education of Teachers as Viewed by the Profession (Washington: National Commission on Teacher Education and Professional Standards, National Edu- cation Association, 1948), pp. 146-150. 16J. Stipanowich, "Mathematical Training of Pros ec- tive Elementary-School Teachers," The Arithmetic Teac er, 4:240-248, December, 1957. 13 A firm grasp of basic arithmetical concepts and processes is essential to teach arithmetic meaning- fully.17 The careful preparation of prospective teachers in mathematics subject matter is a prerequisite to an improved program in elementary schools. This point of view has been presented consistently in the writings of research workers in the field of arithme— tic for the past two decades.18 Poorly prepared teachers are not likely to provide the stimulus which will inspire their pupils to acquire this knowledge [of arithmetic] and arouse in them the desires to pursue other branches of mathematics.19 In 1960 the Panel on Teacher Training of the Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America (the PTT of the CUPM of the MAA) recommended that 12 semester hours of mathematics should be the minimum preparation in mathematics for elementary school teachers and should be followed with a mathematics methods course and student teaching experience. The Panel on Teacher Training specified that this mathematics should be presented in courses specially designed for future elementary school teachers and should not be the courses normally 17J. C. Bean, "Arithmetical Understandings of Elemen- tary School Teachers," The Elementary School Journal, 59: 447, May, 1959. 18A. K. Ruddell, W. Dutton, and J. Reckzeh, "Background Mathematics for Elementary Teachers," Twenty-fifth Yearbook of the National Council ofggeachers of Mathematics (Washing- ton: The Council, 1960), p. 297. 19E. Fulkerson, "How Well Do 158 Prospective Elemen- tary Teachers Know Arithmetic?" The Arithmetic Teacher, 7:146, March, 1960. 14 0 Some institutions have intended for mathematics majors.2 implemented or are considering the implementation of these recommendations.21 Ostensibly those individuals who advocate the study of mathematics by prospective teachers do so because of their faith in its ability to increase the classroom effective- ness of the teachers, while those who do not advocate such preparation infer that such training is of little conse- quence. Each of these two positions is based upon intuitive judgment rather than upon empirical evidence. With some persons seemingly discounting and others advocating mathe— matics preparation, it would seem appropriate to determine evidence which might help resolve this controversy. The Lack of Evidence Concerning Mathematics Preparation for Elementary School Teachers Although many studies have been made of the relation- ship between the academic preparation of prospective elemen- tary school teachers and their subsequent effectiveness in the classroom, relatively few have been directed at prepara— tion in specific subject matter areas, and fewer still in the particular area of mathematics. This situation has been 20Panel on Teacher Training of the Committee on the Undergraduate Program in Mathematics, "Recommendations of the Mathematical Association of America for the Training of Mathematics Teachers," The Amegican Mathematical Monthly, 67:982-991. December, 1960. 21C. E. Hardgrove and B. Jacobson, "CUPM Report on the Training of Teachers of Elementary School Mathematics," The American Mathematical Monthly, 70:870-877, October, 1963. 15 described in Chapter II. Many of the researchers simply compared the general academic grade point averages of students in college, or their grade point averages in professional course work, with some criterion of teaching effectiveness. Very often this criterion of effectiveness consisted of a rating by a supervisor or a principal, or performance on some written examination instrument designed to measure overall teacher competence. Such criteria are subjective in nature and are not very direct indicators of the teacher's effect on students. In the study being re- ported the criterion of teacher effectiveness, the mathe- matics achievement of the teacher's students, is very ob- jective and much more ultimate in nature. Of course, the selection of an objective achievement test requires a sub- jective judgment, but it was assumed that this selection was accomplished in an appropriate manner, particularly since the objectives of the test Chosen were consistent with the objectives of the instructional program. Furthermore, among the existing reports of research only a very few deal with the cumulative effect of more than one teacher on the achievement of the student. Of those studies which have considered this cumulative effect, none has concentrated specifically on the area of mathematics. The present study attempted, although perhaps in a rather gross manner, to consider the cumulative contribution of al_ of the student's mathematics teachers to his achievement over a period of nine years of study. 16 The importance of this study may be summed up, then, as the provision of some information about the relation- ship of teacher preparation in mathematics to student achievement in mathematics, which has not before been determined by the use of an objective, relatively ultimate criterion of teacher effectiveness over a period of time which encompasses the students' first nine years of formal education. VI. THE ORGANIZATION OF THE STUDY In Chapter I an attempt has been made to define and delimit the purpose of the study, to identify some assump- tions relative to the study, and to offer a validation of the study's importance. Chapter II consists of a review of the related literature and a summarization of the re- sults and conclusions. Chapter III is a description of the procedures used in this investigation. This descrip- tion includes the design and the setting of the study, the procurement of the raw data, and the transformation of the raw data into forms consistent with the research design. The specific statistical values and the results of hypothe- sis testing are reported in Chapter IV. Chapter V is a summarization of the study to that point. Chapter VI is the concluding chapter and contains the conclusions and the implications of the study. Certain selected and rele- vant items will be found in the Appendices. CHAPTER II REVIEW OF THE LITERATURE This Chapter consists of two parts: research related to teacher preparation and research related to teacher experience. The teacher preparation studies are subdi- vided into studies at the secondary school level and above and studies at the elementary school level. I. RESEARCH RELATED TO TEACHER PREPARATION Studies at the Secondary School Level and Above In 1931 Ullman1 compared the principals' ratings of 116 first-year secondary school teachers with their general academic marks and with their major subject marks, and reported correlation coefficients of .30 and .20, respec- tively. He did not report separate correlation coefficients by subject area. In 1935 Stein2 made a study employing a criterion of student achievement. He compared the subject matter 1R. R. Ullman, The Prognostic Value of Certain Factors Related to Teaching Success (Ashland, Ohio: A. L. Garber Co., 1931), 133 pp. 2H. L. Stein, "Teacher Qualifications and Experience and Pupil Achievement," (Master's thesis, University of Manitoba, 1935), 144 pp. 17 18 preparation of 272 teachers in one, two, and three room high schools in rural Manitoba with the achievement of their students in the respective subject areas. A corre- lation coefficient of .025 was reported for teacher training and student achievement in algebra, and a co- efficient of -.007 for teacher training and student achievement in geometry. Both of these coefficients were interpreted as being indicative of no correlation. Although Stein was careful to make adjustments for vari- ation in student intelligence, he did not use a true pupil growth criterion. His measurements of pupil achieve- ment consisted of a post-test, with no pre-test to provide a measure of the gains made in achievement by the students during their association with the teachers. In the same year Lancelot3 concluded that differences exist in the effectiveness of instructors, which is measur- able in the subsequent achievement of their students. He studied students who were enrolled in a sequence of mathe- matics courses for college engineering majors. He main- tained that certain instructors were more effective with better students, and others were more effective with poorer students. He did not attempt to identify the instructor differences which accounted for or which were associated with these variations in instructor effectiveness. 3W. H. Lancelot, "A Study of Teaching Efficiency as Indicated by Certain Permanent Outcomes," The Measurement of Teaching Efficiency (New York: Macmillan Co., 1935), pp. 1-69. 19 Rostker4 tested the students of 28 seventh and eighth grade teachers in non-departmentalized schools at the beginning and the end of a school year. He compared the resulting gain in achievement with the teachers' knowledge of subject matter and concluded that the two were signifi- cantly related to each other. In 1950 Lins5 administered pre-tests and post-tests to the students in 27 classes taught by 17 high school teachers, and obtained a measure of the pupil gain in achievement. He compared this gain with the grade point averages earned by the corresponding teachers in their major and minor areas of specialization. He reported a correla— tion coefficient of .552 between grade point average in the major and pupil achievement gain, and a correlation coefficient of .444 between grade point average in the minor and pupil gain in achievement. Also in 1950 Schunert6 reported a study which involved 102 elementary algebra classes and 94 plane geometry classes enrolling a total of 3,919 pupils in 73 schools. However, his analyses were performed on subsamples of this 4L. E. Rostker, "The Measurement of Teaching Ability, Study Number One," The Journal of Experimental Education, 14:6—51, September, 1945. 6L. J. Lins, "The Prediction of Teaching Efficiency," The Journal of Experimental Education, 15:2-60, September, 1946. 6J. R. Schunert, "The Association of Mathematical Achievement with Certain Factors Resident in the Teacher, in the Teaching, in the Pupil, and in the School," (Doctor's thesis, University of Minnesota, 1950), 269 pp. 20 population. He compared the final algebra achievement of ten Classes taught by teachers having less than two years of college mathematics with the final algebra achievement of ten classes taught by teachers having more than two years of college mathematics. Adjustments were made for variations in mental ability and initial achievement in mathematics. Although the results of comparison slightly favored the teachers with the lesser amount of mathematics, he concluded that there was no significant difference in the achievement of the students in the two groups. He also compared the final geometry achievement of 12 Classes taught by teachers having less than two years of college mathematics with the final geometry achievement of 12 classes taught by teachers having more than two years of college mathematics. Again he concluded that no signifi- cant difference existed for the two groups. For his study Nelson7 asked a group of secondary school principals to identify their superior teachers of mathematics. In comparison with other mathematics teachers, the most capable teachers had undergraduate majors in mathematics. The teacher differences between a group of high schools whose students made the greatest gains in mathematics achievement over a three-year period as measured by the 7T. S. Nelson, "Factors Present in Effective Teaching of Secondary School Mathematics," (Doctor's thesis, Univer- sity of Nebraska Teachers College, 1959, 393 pp.), Disser- tation Abstracts, 20:3207, 3208, No. 8, 1959. 21 Ipwa Tests of EducationalpDevelopment, were investigated by Sparks.8 He Chose 20 schools from the upper 15 per— cent and 20 schools from the lower 15 percent in perform- ance on the state-wide test, and paired them according to the mean ninth grade composite scores on the test at the beginning of the three-year period. He reported that the teachers who taught in the high achievement schools had credit for more semester hours of mathematics as college undergraduates than had the teachers in the low achieve- ment schools. Lindstedt9 compared the number of university mathe- matics courses taken by high school mathematics teachers to the scores of their students on a final examination in ninth grade mathematics. He reported that there existed no sig- nificant differences in the examination scores of students taught by teachers classified on the basis of amount of mathematics preparation. However, the criterion used was final achievement and not gain in achievement. Both graduate and undergraduate mathematics prepara- tion by high school teachers was related to student mathematics achievement according to a study conducted by 8J. N. Sparks, "A Comparison of Iowa High Schools Ranking High and Low in Mathematical Achievement," (Doctor's thesis, State University of Iowa, 1960, 255 pp), Disserta- tion Abstracts, 21:1481, 1482, No. 6, 1960. 9S. A. Lindstedt, "Teacher Qualification and Grade IX Mathematics Achievement," The Alberta Journal of Education, 6:76-85, June, 1960. 22 Leonhardt.1° He identified six schools whose tenth grade geometry students ranked high on the Cooperative General Mathematics Test for High School Classes, and six schools whose geometry students ranked low. Schools were paired from each group on the basis of comparable mean IQ of the student bodies, but no pre-tests were administered to pro- vide a measure of achievement gain. 11 studied 45 first year algebra teachers and Garner their 1163 students. Each pupil was given a comprehensive algebra examination at the beginning and end of a school year.- The number of hours of college mathematics for which the teachers had credit, bore a significant relation- ship to the gains in algebra which were made by the stu- dents. A study of the relationship between the understandings of arithmetic and geometry possessed by the seventh grade teachers in nine New York City junior high schools and the mathematics achievement of their students was made by 11* 10E. A. Leonhardt, "An Analysis of Selected Factors in Relation to High and Low Achievement in Mathematics," (Doctor's thesis, University of Nebraska, 1962, 307 pp.), Dissertation Abstracts, 23:3689,3690, No. 10, 1963. 11M. V. Garner, "A Study of the Educational Back- grounds and Attitudes of Teachers Toward Algebra as Related to the Attitudes and Achievements of Their Anglo-American and Latin-American Pupils in First-Year Algebra Classes of Texas," (Doctor's thesis, North Texas State University, 1963, 158 pp.), Dissertation Abstracts, 24:189, No. 1, 1963. 23 Peskinla in 1964. She reported that such understandings and achievement were significantly related to each other. Neillla studied 43 junior high school teachers and their 1,477 academically talented students in New York City and Philadelphia. He concluded that the length of the teacher's academic preparation was related to the student's achievement level. To summarize, it appears that most of the studies which have been made of the relationship between the mathematics preparation of secondary school teachers and their subsequent effectiveness in the classroom, point to a significant positive correlation between the two. This has been true whether the criterion of effectiveness was a rating by administrators or a gain in student mathematics achievement, and whether the preparation was considered in terms of grade point averages, number of courses, or levels of understanding. Studies at the Elementary School Level Most of the earlier studies were comparisons of the general academic preparation of teachers and their general 12A. S. Peskin, “Teacher Understanding and Attitude and Student Achievement and Attitude in Seventh Grade Mathematics," (Doctor's thesis, New York University, 1964, 179 pp.),-Dissertatign Abstract§, 26:3983, 3984, No. 7, 1966. 13R. D. Neill, "The Effects of Selected Teacher Vari— ables on the Mathematics Achievement of Academically Tal- ented Junior High School Pupils," (Doctor's thesis, Columbia University, 1966, 316 pp.), Dissertation Abstracts, 27:997—A, No. 4, 1966. 24 classroom effectiveness. More recently certain researchers have directed their efforts toward teacher preparation in the specific area of mathematics and the effectiveness of teachers in the teaching of this subject. Dggdies of qenegal preparation and effectiveness. The earliest study found was conducted by Meriam14 in 1906. He compared the general effectiveness of 1,185 teachers who had attended some type of teacher training institution between 1898 and 1902, to their academic course scholar- ship. Effectiveness was rated by principals, superintendents, or practice teaching supervisors. He reported a correlation of .251. In 1924 Whitney15 studied 1,156 graduates of 12 normal schools, who comprised stx percent of all 1920 normal school graduates in the United States. His criterion of teaching effectiveness was a mutual rating made by faculty peers in each school. He reported a correlation coefficient of .073 for teaching effectiveness and general academic marks in normal school. l‘J. L. Meriam, Normal School Education and Efficiency in Teaching (Teachers College Contributions to Education, No. 1. New York: Teachers College, Columbia University, 1906), 152 pp. 15F. L. Whitney, "The Prediction of Teaching Success," The Journal of Educational Research Monographs, No. 6 (Bloomington: Public School Publishing Co., 1924). 85 pp. 25 Jacobs16 in 1928 asked a group of elementary school principals to rate their teachers as to their effectiveness. He then compared the educational backgrounds of 50 of the teachers rated good and of 50 of the teachers rated poor. He was unable to find a correlation between the educational backgrounds of the teachers and their effectiveness in the Classrooms. 7 gave pre—tests and post-tests of arithmetic Taylorl achievement to the students in the ten different half-grades from the first semester of fourth grade to the second semester of eighth grade in nine elementary schools during the first semester of the academic year 1923-24. He also obtained the age and a measure of the intelligence of each of these students, and secured ratings of the students' teachers- The ratings were made by the school principals and the head of the school research department. He then compared the means of the pre-test scores, 105, and the respective teacher ratings for each class with the post-test scores. He reported that all four factors contributed to the 16C. L. Jacobs, The Relation of the Teacheg's Education to Her Dfifectiveness (Teachers College Contributions to Edu- cation, No. 277. New York: Teachers College, Columbia University, 1928), 97 pp. 17H. R. Taylor, "The Influence of the Teacher on Relative Class Standing in Arithmetic Fundamentals and Reading Compre— hension," The Twenty-Seventh Yearbook gTithe NationaT_§oCiety of the Study of Educationi Part II (Bloomington: Public School Publishing Co., 1928), pp. 97-110. 26 final achievement during the semester, but that a high rated teacher was the least important. Taylor concluded: ...it is conceivable that the cumulative influ- ence of all the different elementary-school teachers with which each Child or class comes into contact would have greater weight in a regression equation for the prediction of total final achievement than any other of the four factors studied.18 In 1936 Odenweller19 reported a study of the prediction of teaching effectiveness. He compared certain character- istics of 560 elementary school teachers with their rated effectiveness as determined by their principals and super- visors. He reported correlation coefficients of .293 and .281 for correlations of effectiveness with the teachers' general college marks and college subject matter marks, respectively. Gathercole20 compared general normal school scholarship and ratings of effectiveness made by school superintendents. He determined a correlation coefficient of .238 for the two factors. 18Ibid., p. 106. 19A. L. Odenweller, ggedicting the Quality of Teaching (Teachers College Contributions to Education, No. 676. New York: Teachers College, Columbia University, 1936), 158 pp. 20F. J. Gathercole, "Predicting the Quality of Teaching: A Study of the Relation of High School Marks, Intelligence, Standardized Test Scores, and Normal School Standing to Teach— ing Success," (Master's thesis, University of Manitoba, 1946), 129 pp. 27 The Teacher Characteristic Study21 which was jointly conducted by the American Council on Education and the Grant Foundation was reported by Ryans. All third and fourth grade teachers in four communities (275 women) were studied. No significant relationship was found between the amount of college training and a composite evaluation of effectiveness made independently by three trained ob- servers. SOper22 compared pupil gains in general school achieve- ment with the training of teachers. The subjects were 128 teachers and their 2,656 students in grades four, five, and six. The teachers were dichotomized according to the amount of general academic and professional training they had had. To provide a measure of achievement gain for each pupil, the students were given a pre-test and a post— test of achievement. The means of pupil gain scores for each class were compared with the amount of the teacher's training with adjustments made for variations in intelligence. 21D. G. Ryans, "A Study of the Extent of Association of Certain Professional and Personal Data with Judged Effec- tiveness of Teacher Behavior," The Journal of Experimental ,Dducatigg, 20: 67-77, September, 1951; also "Teacher Personnel Research," The California Journal of Educational Research, 4:19-27, 73-83, January, 1953. 22E. F. Soper, "A Study of the Relationship Between Certain Teacher-School Characteristics and Academic Progress, As Measured by Selected Standardized Tests, Of Elementary Pupils in Grades Four, Five and Six of New York State Public Schools in Cities Under 10,000 Population," (Doctor's thesis, Syracuse University, 1956, 135 pp.), Dissertation Abstracts, 17:570,571, No. 3, 1957. 28 Soper reported that pupils attained higher mean gains in Classes taught by teachers with less training, and that this was significant at the five percent level of confidence. Standlee and Popham23 found no significant relationship between teaching effectiveness as indicated by principal ratings and the general academic grade point averages of teachers. In 1959 a study was made by McCall and Krause‘g4 of 73 teachers and their students in sixth grade classes in rural and urban North Carolina. A conglomerate pupil growth criterion was used, which included gains made in reading, writing, and arithmetic, as well as in work skills, personal relations, reasoning, and recreation. Adjustments were made for variations in student intelligence, drive, home environ- ment, class size, and attendance. Several teacher character- istics were compared with this criterion of effectiveness. The teachers' knowledge of subject matter produced a zero correlation. good growth was produced by teachers whose college grade point averages were below 90 percent; very small gains were produced by teachers whose averages were above 90 percent. McCall and Krause summarized their find- ings regarding teacher preparation in a positive manner: 23L. S. Standlee and W. J. Popham, "Preparation and Performance of Teachers," The Indiana University School of Education Dulletin, 34:1-48, November, 1958. 24W. A. McCall and G. R. Krause, "Measurement of Teacher Merit for Salary Purposes," The Journal of Education- al Research, 53:73-75, October, 1959. 29 "'Training' was somewhat better as a criterion than drawing shuffled names out of a hat."25 In 1960 Standlee and Popham reported another study involving some Characteristics of 880 public school teachers, both elementary and secondary. In that study two criteria of teaching effectiveness were used, performance on the Minnesota Teaching Attitude Inventory and relative ranking by building principals. "Neither the professional nor the academic preparation of teachers was significantly related to either of the two criteria of teaching performance."26 127 studied 55 teachers Over a period of seven months Hei of grades four, five, and six and the pupils in their classes. The general pupil achievement gain over this period was com— pared to the liberal arts knowledge of the teachers. Heil reported that a definite negative relationship seemed to exist between the two, although it did not quite attain the five percent confidence level. In 1956 Chung-Phing Shima8 conducted a study similar in 25Ibid., p. 75. 26L. S. Standlee and W. J. Popham, "Teacher Variables Related to Job Performance," Psychological Reports, 6:458, June, 1960. 27L. M. Heil, Characteristics of Teacher Behavior and Competency Related to the Achievement of Different Kinds of Children in Several Elementary Grades (New York: Brooklyn College, 1960), 119 pp. 28Chung-Phing Shim, "A Study of the Cumulative Effect of Four Teacher Characteristics on the Achievement of Elemen- tary School Pupils," The Journal of Educational Research, 59:33,34, September, 1965. 30 design to the one presently reported. He attempted to determine the cumulative effect of college grade point average, possession of a degree, possession of certification, and teaching experience on general student achievement over a period of five years. The 89 teachers who taught 214 students while they were in attendance in grades one through five of a semi-rural school district, were dichotomized on each of four characteristics: having a college grade point average above or below 2.5, having a B.A. degree or not, being certified to teach or not, and having more or less than ten years of teaching experience. IQ scores were used in making adjustments for variations in intelligence, and the criterion of teaching effectiveness was total achieve- ment in arithmetic, language, and reading as measured by the California Achievement Tests. Each teacher character- istic was compared with the achievement criterion. Chung- Phing Shim concluded: The general implication is that according to the findings of this study, there is no significant dif- ference in pupil achievement to support the idea that an elementary school teacher has to be a superior stu- dent in college, to have a degree, to be fully certi- fied, or to have many years of experience in order to be successful as far as measureable pupil achievement is concerned.29 All of the studies reported in this section, were con- cerned with the general preparation of teachers or their general classroom effectiveness. None was directed at the 29Ibid., p. 54. 31 relationship between the preparation or understanding of teachers in the specific discipline of mathematics and the effectiveness of the teachers in the task of teaching that specific subject. The investigators who conducted the following studies were interested in this topic. Studies of specificipreparation and effectiveness. Smail30 in 1959 compared certain teacher characteristics with mean arithmetic achievement gain over a period of one academic year. His subjects were 2,438 students enrolled in grades four, five, and six, and their 97 teachers. He reported that no significant correlations were indicated between the teacher's understanding of basic mathematical concepts and arithmetic achievement gain, nor between the number of mathematics courses completed by the teacher and arithmetic achievement gain. 1 asked a In 1960 Barnes, Cruickshank, and Foster3 group of 66 elementary school principals to rate their fourth grade teachers on their mathematics instructional ability and to classify each of them as superior, good, or fair. For the 102 teachers involved, information was ob- tained concerning the number of years of elementary school 30R. W. Smail, "The Relationship Between Mean Gain in Arithmetic and Certain Attributes of Teachers," (Doctor's thesis, State University of South Dakota, 1959, 151 pp.), Dissertation Abstracts, 20:3654, No. 9, 1960. 31K. Barnes, C. Cruickshank, and J. Foster, "Selected Educational and Experience Factors and Arithmetic Teaching," The Agithmetic Teacher, 7:418-420, December, 1960. 32 teaching experience, the number and type of high school mathematics courses taken, the number of college mathema— tics courses taken, and attitude toward high school mathe— matics. No significant correlation was found between the number and type of high school mathematics courses taken and the principal's rating of mathematics instructional ability. The same was true concerning the number of college mathematics courses taken. However, the teachers who were rated superior in their ability, indicated a higher degree of interest in their high school mathematics courses than did either those rated good or fair. Houston32 in 1961 compared the relative effectiveness of two methods of providing in-service mathematics education to elementary school teachers. One group of teachers was instructed by means of television programs and another group by means of face-to-face lecture-discussion. Half of each group received supplementary consultant services in addition to the instruction received by their respective complete groups. The criterion of effectiveness by which the two methods were compared was the amount of growth in arithme— tic and the amount of change in mathematics interest attained by the students of the participating teachers during the 32W. R. Houston, "Selected Methods of In-Service Educa- tion and the Mathematics Achievement and Interest of Elemen— tary School Pupils," (Doctor's thesis, University of Texas, 1961, 215 pp.), Dissertation Abstracts, 23:157, No. 1, 1961. 33 course of the 24 week period of teacher education. Houston reported that both methods were equally effective means of providing in-service mathematics education as measured by the criterion. In 1962 Basshamss reported a study of 28 sixth grade teachers and their 620 students over a period of seven months. Multiple correlation techniques were used with teacher understanding of mathematics, student intelligence, student pre-test of reading achievement, and student prefer- ence for arithmetic activities used as independent variables, and post-test arithmetic achievement used as the dependent variable. Teacher understanding was measured by the use 3‘ Bassham of Glennon's Test ongathematical Understanding. indicated interest in the relationship of teacher under- standing and student achievement in general, and also in regard to this relationship as it concerned the high and low intelligence levels of students. He reported that teacher understanding of mathematics "explained approximately one- fourth of the variation among pupils in their efficiency of 33H. Bassham, "Teacher Understanding and Pupil Effi- ciency in Mathematics--A Study of Relationship," The Arithme— tic Teacher, 9:383-387, November, 1962. 34Glennon's test, an 80 item multiple choice instru- ment, was designed to measure those mathematical understand- ings which he considered fundamental to the algorithms com- monly taught in the elementary school. The test can be found in: V. J. Glennon, "A Study of the Growth and Mastery of Certain Basic Mathematical Understandings on Seven Educa- tional Levels," (Doctor's thesis, Harvard University Gradu- ate School of Education, 1948), 190 pp. 34 utilizing pre-experimental period abilities."35 Bassham made separate comparisons of teacher mathematics under- standing to student mathematics achievement for all students, for all students with 105 below the mean of the entire group, and for all students with IQS above the mean of the entire group. He reported correlation coefficients of .274,.097, and .417 for these comparisons, respectively. In 1965 Moore36 reported a study similar to that of Bassham, but which resulted in a contradictory conclusion. Moore compared the mathematics understanding of 11 sixth grade teachers and 10 fourth grade teachers with the gain made in mathematics achievement by their 508 students over a period of one semester. Teacher understanding was also measured by Glennon's test. Adjustments were made for variation in student intelligence. He reported that no significant relation existed between teacher mathematics understanding and student mathematics achievement gain. To summarize, it appears that the studies which have been conducted for the comparison of teacher preparation and teacher effectiveness at the elementary school level, have resulted in a lack of concensus. Of the very few studies which specifically related teacher mathematics I”, 3992, cit., p. 387. 36R. E. Moore, "The Mathematical Understanding of the Elementary School Teacher as Related to Pupil Achievement in Intermediate-Grade Arithmetic," (Doctor's thesis, Stanford University, 1965, 90 pp.), Dissertation Abstracts, 26:213, 214, No. 1, 1965. 35 preparation to student mathematics achievement, only one was found which demonstrated a positive correlation between the two factors. General Observations Regarding_Teacher Preparation Studies Differences in objectives. It appears that few studies have been conducted which deal with the relationship of the mathematics preparation of elementary school teachers and the mathematics achievement of their students. Since Meriam conducted his pioneering study37 of teaching effectiveness in 1906, many other investigators have directed their efforts to this general subject. However, the great majority of these investigators have dealt with the general academic preparation of teachers at some school level, or with the specific academic preparation of teachers in the secondary schools. It should be mentioned that the objectives and purposes of these studies differed. Some researchers were attempting to rate teaching merit. Others were trying to predict teach- ing success. Still others were endeavoring to predict sgggent achievement. Yet all showed a common concern for the relationship between teacher background and teacher effective- ness. Differences in criteria of teachihg effectiveness. The criteria of teaching effectiveness have varied considerably 37Meriam, loc. cit. 36 in keeping with the general evolution of teaching effective— ness research. Earlier researchers usually used a measure of teaching effectiveness based upon a subjective rating by a supervisor, a principal, or a superintendent. Subse— quently, as rating techniques were improved by the develop— ment of rating checklists and teaching inventories, these were used to obtain the desired measure. More recently the measure of effectiveness has been made in terms of pupil outcomes or behaviors. Although there does seem to be a trend toward the use of student growth criteria for determining teaching effectiveness,38 there is by no means universal acceptance of any one type of criterion as optimum. Howsam has pointed out39 that of 138 studies of teacher effectiveness summar— ized in 1948 by Barr,4o only 19 used a measure of student gain as a criterion. Howsam also noted that Mitzel and Gross41 found only 20 such studies in 1956. Writing in the 38P. J. Eccles, "The Relationship Between Subject Matter Competence of Teachers and the Quality of Science Instruction in the Elementary School," The Alberta Journal of_§ducational Research, 8:238-245, December, 1962. 39R. B. Howsam, Who's A Goodeeacheg? Problems and Progress in Teacher Dyaluation (Burlingame, California: Joint Committee on Personnel Procedures of the California School Board Association and the California Teachers Associ- ation, 1960), p. 19. 40A. S. Barr, "The Measurement and Prediction of Teacher Efficiency: A Summary of Investigations," The Jour- nal of Experimental Education, 16:203-283, June, 1948. 41H. E. Mitzel and C. F. Gross, A CriticanReview of theiDeveTopment of Pupil Growth Criteria in Studies of Teacher Effectiveness (Research Series, No. 31. New York: Board of HIgher Education, CCNY, 1956), 28 pp. 37 Encyclopedia of Educational Research in 1957 Mitzel said: More than a half-century of research effort has not yielded meaningful, measurable criteria around which the majority of the nation's educators can rally. No standards exist which are commonly agreed upon as phg criteria of teacher effectiveness.4 That the type of criterion chosen is an important factor in the outcome of research studies has been attested to by numerous comparisons of rating, inventory, and growth 5 criteria, such as those by Taylor,43 Barr,44 Rostker,4 Lins,46 Von Haden,47 Anderson,48 and McCall and Krause.49 42H. E. Mitzel, "Teacher Effectiveness," The Encyclo- pedia of Educational Research, Third Edition (New York: Macmillan Co., 1960), p. 1481. 43H. R. Taylor, "Teacher Influence on Class Achievement," Genetic Psychology Monographs, 7:81-175, February, 1930. 44A. S. Barr, e§_ai:, "The Validity of Certain Instru- ments Employed in the Measurement of Teaching Ability," The Measurement of Teaching Efficiency (New York: Macmillan Co., 1955). Pp. 107, 108, 115. 45L. E. Rostker, "The Measurement and Prediction of Teaching Ability, School and Sociehy, 51:30-32, 1940. 46Lins, loc. cit. 47H. I. Von Haden, "An Evaluation of Certain Types of Personnel Data Employed in the Prediction of Teaching Efficiency," The Journal of Experimental Education, 15:61-84, September, 1946. 48H. M. Anderson, "A Study of Certain Criteria of Teaching Effectiveness," The Journal of Experimental Educa- tion“ 23:41-71, September, 1954. 49MCCall and Krause, loc. Cit. 38 These investigators reported very low correlations (some even negative) between pairs of the various types of criteria. Evidently the variation in the criteria used in the past studies accounts in part for the variation re- ported in the correlation of teacher mathematics prepara- tion and student methematics achievement. Specific Observations Regarding Teacher ggeparation Studies As one peruses the literature related to the comparison of teacher mathematics preparation and student mathematics achievement, it appears that no general conclusions can be drawn. The reports of studies do not seem to indicate the existence of a pattern or common characteristic. However, if one considers the studies conducted at the secondary school 'level separately from those conducted at the elementary school level, a pattern emerges. The majority of the studies at the secondary school level indicate a significant positive correlation between the mathematics preparation of the mathe- matics teacher and the mathematics achievement of the stu- dent. Of the very few similar studies made at the elementary school level, that is, those which specifically related teacher mathematics preparation to student mathematics achievement, only one was found which demonstrated a positive correlation between the two factors. That study was the one conducted by Bassham. 39 Basshamso and Moore51 reported divergent results, even though their research designs were very similar. They studied approximately the same number of teachers of approx- imately the same number of students at approximately the same grade level over periods of time which were not too dissimilar. Both utilized controls for variation in student intelligence. Both gave pre-tests and post-tests of student mathematics achievement. Both used the same test to measure teacher mathematics understanding. The only major aspect on which the studies differed, and perhaps the one which accounts for the conflicting results, was the criterion of teacher effectiveness. Moore's criterion was a strict pupil growth measure, obtained by subtracting the pre-test score from the post-test score of each student. Bassham's criterion was the level of post—test achievement with the level of pre-test achievement being used only as another variable in the multiple regression equation. Although both research designs made allowances for the student's pre-experiment ability, they did so by different means, which may account for the discrepancy. Strictly speaking, Houston's study52 did not provide evidence that in-service teacher mathematics preparation 5°Bassham, loc. cit. 51Moore, loc. Cit. s2Houston, loc. cit. 40 results in increased mathematics achievement on the part of students. The objective of that study was to determine the relative effectiveness of two methods of providing in— service mathematics education to teachers. Appropriately, the effectiveness of each method was measured by the mathe- matics achievement gain of the teachers' students during the course of the experiment, and subsequently these gains were compared. That the two methods of education were equally effective was attested to by similar achievement gains in the students of the teachers taught by the two methods. However, one would expect a certain amount of gain in achievement over a period of time whether the teachers had received in-service instruction or not. Since no student control group was established in which the teachers received no instruction, there is no indication that either group of students exceeded the amount of achieve— ment gain which could normally be expected without in-service education. Perhaps the study by Barnes, Cruickshank, and Foster53 would have resulted in similar conclusions anyway, but there remains the question concerning the reliability of ratings as a criterion of teacher effectiveness. Raters tend to have an overall opinion of the value of a person whom they are rating and to rate separate characteristics 53Barnes, Cruickshank, and Foster, loc. Cit. 41 accordingly. As early as 1920 Thorndike commented on this "halo" effect: The writer has become convinced that even a very capable foreman, employer, teacher, or department head is unable to treat an individual as a compound of separate qualities and to assign a magnitude to each of these in independence of the others.54 Knight and Franzen55 referred to this condition as a spread of "aura." Taylor also commented on the situation: ...it may be that the ratings used to measure the capacity of each teacher to influence the achievement of her pupils are not so much ratings of teaching ability as they are indications of the general reputa— tion a teacher bears for cooperativeness, educational up-to-datedness, and disciplinary success-~all of which may or may not be closely related to the measured achievement of the pupils.56 There are many worthwhile objeCtives of education. It may be that in some cases the best method of assessing the teacher's attainment of objectives is by means of ratings. However, when the objective is the development of the stu- dent's mathematical competence, it would seem that a measure of that competence on the part of the student would provide the best measure of the teacher's success. 54E. L. Thorndike, "A Constant Error in Psychological Ratings," The Journal of Appiied Psychology, 4:28, 29, _March, 1920. 55F. B. Knight and R. Franzen, "Pitfalls in Rating Schemes," The Journal OT Educational Psychology, 13:204- 215, April, 1922. 56Taylor, pp, cit., p. 97. 42 Summary In the way of summary, the related studies seem to indicate that with regard to secondary school mathematics instruction, there exists a significant positive correla- tion between teacher preparation and student achievement, but at the elementary school level no such correlation exists. Furthermore, no studies were found which compared the mathematics preparation of the teachers with whom they studied during the first nine years of their school experi— ence . II. RESEARCH RELATED TO TEACHER EXPERIENCE Like the situation reported above regarding teacher preparation and teacher effectiveness, the reports of stud- ies regarding the relationship of teacher experience and teacher effectiveness also appeared to be contrary to each other. Unlike teacher preparation and teacher effective- ness, however, it seemed impossible to classify them in such a way that these discrepancies were resolved. Attempts at such classification were made according to grade level studied, the type of criterion used to judge teaching effectiveness, and whether general teaching effectiveness or mathematics teaching effectiveness were the main consider- ation. None of these classification schemes seemed to recon- cile the differences which were reported. Therefore, the studies related to teacher experience have been organized 43 below according to the general conclusions reported by the investigators. The only studies reported are those which deal in a general way or in a specific way with mathe- matics instruction. ThejStudies Both Stein57 in 1955 and Leonhardt58 in 1962 reported negative Correlations between teacher experience and stu— dent achievement. Both of these studies were concerned with instruction at the secondary level, but Stein's study59 was concerned with general instructional effectiveness, while Leonhardt's60 concentrated specifically on mathema- tics instruction. In 1922 Knight61 compared ratings of teaching effective- ness of 38 high school teachers and 118 elementary school teachers with amount of teaching experience. The ratings were made by supervisors, peers, pupils, and the teachers themselves. The correlation coefficients for high school and elementary school teachers were .172 and .010» 57Stein, loc. cit. 58Leonhardt, loc. cit. 59Supra, p. 17. (Since many of the studies which were concerned with teacher experience, have been described above in the section,'RESEARCH RELATED TO TEACHER PREPARATION. cross-references have been made for the reader's convenience.) 6°Supra, p. 22. 61F. B. Knight, Qpalities Related to Success in Teaching (Teachers College Contributions to Education, No. 120. New York: Teachers College, Columbia University, 1922). 67 pp. 44 respectively. Knight concluded that these correlations were too low for prognostic purposes. Concerning their study62 of elementary school instruc- tion, McCall and Krause reported, "Classes taught by teach- ers whose experience in teaching ranged from twenty to thirty-one years showed relatively little growth as com-y pared with classes with less experienced teachers."63 However, they concluded that generally years of service showed a zero correlation with the pupil growth criterion that they used.64 Several investigators have reported positive correla- tions between teaching effectiveness and teaching experi- ence. Schunert65 found that algebra classes taught by teachers with more than eight years ofpexperience exceeded the achievement of classes taught by teachers with less experience. Using pupil growth as a criterion of effective— ness, Soper66 concluded that teachers with more experience produced higher mean academic scores. Barnes, Cruickshank, and Foster67 using principal ratings to measure the 62%, p. .28. 63McCall and Krause, pp, pip,, pp. 74, 75. 64M” p. 75. 65Schunert, pp, pip,, p.233; ppppp, p. 19. 68Soper, Tpp, pip,; ppppp, p. 29. 67Barnes, Cruickshank, and Foster, pp, cit., p. 430; supra, p. 31. 45 effectiveness of teachers, concluded that years of teach- ing experience improve the quality of instruction. Although, as mentioned above,68 Lindstedt did not utilize a pupil growth criterion but merely final pupil achieve- ment, he reported that teacher competence increased in proportion to years of teaching experience. "Teachers with 5 to 9 years of experience are more competent than teachers with only 3 or 4 years of experience.... Teachers with 10 or more years of experience are more competent than teachers with less experience...."69 There also exists some evidence that supports the notion that the correlation between teaching experience and teaching effectiveness is non-linear, that is, that five years of experience toward the end of a teacher's career does not produce the same amount of Change in his effectiveness as does five years of experience at the beginning. In 1906 Meriam wrote, "After a year or so, experience seems to contribute little, if any, to effi- Ciency. That is, teachers with two years' experience have as high a rank on ratings as those with five, ten, or u 70 7]. concurred with this fifteen years' experience. Davis 68Supra, p. 21. 69Lindstedt, pp, cit., p. 83. 7oMeriam,]pp. cit., p. 11; supra, p. 24. 71H. M. Davis, The Use of State High Schoolpramina- tions as ainnstpument fop_Judging Work ngTeachers (Teachers College Contributions to Education, No. 611. New York: Teachers College, Columbia University, 1934), cited by W. I. Ackermann, "Teacher Competence and Pupil Change," Harvapijdppational Review, 24:273-289, Fall, 1954. .46 in 1934. He compared the teaching experience of 796 teachers to the scores of their 13,460 students on state- wide subject matter examinations in Minnesota. He re- ported that pupils of teachers with two or more years of teaching experience were more successful than were pupils whose teachers had only one year of experience. However, the pupils with teachers having more than two years of experience performed approximately the same as those whose teachers had only two years of experience. Ruediger and Strayer72 compared certain teacher characteristics of elementary school teachers to their general merit as measured by ratings made by their building principals. They found a correlation coefficient of .36, but also con- cluded that "...a teacher in the grades reaches first class efficiency in about 5 years...maintains this effi- Ciency for about 20 years, and...after about 25 years of H73 74 service he begins to decline. Boyce replicated the study of Ruediger and Strayer at the secondary school level and reported similar results. 7"“W. C. Ruediger and C. D. Strayer, "The Qualities of Merit in Teachers," The Journal of Educational Psychology, 1:272-278, 1910. 731515., p. 277. 74A. C. Boyce, "Qualities of Merit in Secondary School Teachers," The Journal of Educatioin Psychology, 3:144-157, March, 1912. 47 Summary To summarize the research concerning the relationship of teacher experience and teacher effectiveness, it seems that although several investigators have been able to de- termine positive correlations between the two, the relation- ship is not as pronounced as one might suppose. In fact there is considerable evidence that the relationship is non-linear and that after a time an increase in experience may result in a decrease in effectiveness. CHAPTER III PROCEDURE This chapter consists of descriptions of the design of the study, the type of raw data obtained, the sources and procedures for obtaining the raw data, and the con- version of the raw data into transformed data suitable for statistical descriptions and tests. I. THE DESIGN OF THE STUDY The basic design of the study consisted of the compari- son of the mathematics achievement of a group of students at a particular grade level with the amount of mathematics preparation of all of the teachers who taught the students up to that grade level. In order to make allowances for variation in student ability levels, student intelligence was introduced into the design. To provide a second teacher characteristic for the relative comparison of teacher mathe- matics preparation, the amount of teacher experience was introduced. Since the design called for the relative com- parison of student mathematics achievement to three differ— ent factors, the techniques of multiple correlation and regression were used. In terms Of a multiple regression equation, the independent variables were teacher mathematics 48 49 preparation, teacher experience, and student intelligence, while the dependent variable was student mathematics achievement. The objective of the study was the comparison of each student's mathematics achievement with certain character- istics of the pp; Of teachers who taught him. Therefore, it was necessary to define the teacher variables in such a way that their values were cumulative measures of each of the characteristics of each set of teachers rather than measures of each of the characteristics of each of the individual teachers. It was necessary to transform the raw data regard- ing individual teachers into cumulative measures for sets of teachers. It was decided that arithmetic means would be used as the cumulative measures. This transformation of the raw data has been described in Section IV of this chapter. When initiated, the study was designed to identify a group of students who had been examined in mathematics achievement when in the eighth grade, and to compare their mathematics achievement scores with certain characteris- tics of the teachers who had taught them mathematics from grade one through grade eight. However, in the process of data collection, it was discovered that this same group of students had been examined in mathematics achievement *when in the fourth grade and again in the sixth grade, and that these mathematics achievement scores were also 50 available. Therefore it was possible to duplicate the procedure of the study at these two additional levels, and thus obtain a developmental picture of the relationship under investigation. Because the data were obtainable, the study was also extended to include the kindergarten for those students who had attended it. Thus the design describ— ed in the paragraphs above, was applied to three periods in the elementary school careers of the students: 1. kindergarten through grade four 2. kindergarten through grade six 3. kindergarten through grade eight. There were also three different measures of student mathematics achievement employed in the study: 1. an arithmetic reasoning score which indicated achievement in mathematical reasoning 2. an arithmetic fundamentals score which indicated achievement in computational skills 3. a total arithmetic score which indicated general achievement in mathematics. Also when initiated, the study was designed to con- sider only that teacher mathematics preparation which had been accomplished when the teachers were enrolled in college. Because of the subsequent difficulties in the procurement of data regarding the college mathematics preparation of the teachers, as described below in Section II, a questionnaire was used to secure these data. 51 It therefore became practical to also obtain data regard- ing the high school mathematics preparation of the teachers and their in-service mathematics preparation. So, data were collected for three categories of teacher mathematics preparation: 1. high school mathematics preparation 2. college mathematics preparation 3. total mathematics preparation including in-service mathematics preparation. Since there existed student achievement scores of three grade level periods, three measures of student mathematics achievement for each grade level period, and three cate- gories of teacher mathematics preparation, it was possible to apply the techniques of multiple correlation and regres- sion to 27 combinations of student and teacher character- istics. These 27 combinations have been represented in Figure 1 using functional notation. In each combination the variables within the parentheses are the independent variables. Figure 2 indicates the definition of the vari— ables. The following statistics were computed.‘ For each variable: 1. the arithmetic mean 2. the standard deviation 3. a measure of skewness 1Appendix A contains descriptions of these statistics. 52 REAS4 = F(TEX4, ST IQ, THSM4) FUND4 = F(TEX4, ST IQ, THSM4) TOTAL4 = F(TEX4, ST IQ, THSM4) REAS4 = F(TEX4, ST IQ, TCM4) FUND4 = F(TEX4, ST IQ, TCM4) TOTAL4 = F(TEX4, ST 10. TCM4) REAS4 = F(TEX4, ST IQ, TTM4) FUND4 = F(TEX4. ST 10. TTM4) TOTAL4 = F(TEX4, ST IQ, TTM4) REAS6 = F(TEX6, St IQ, THSM6) FUNDS = F(TEX6, ST IQ, THSM6) TOTAL6 = F(TEXS, ST 10. THSM6) REAS6 = F(TEX6, ST IQ, TCM6) FUND6 = F(TEX6, ST IQ, TCM6) TOTALS 2 F(TEXS, ST 10. TCM6) REAS6 = F(TEXS, ST IQ. TTM6) FUNDS = F(TEXS. ST 10. TTM6) TOTALS = F(TEX6, ST 10. TTM6) REASS = F(TEX8, ST IQ, THSM8) FUNDS = F(TEX8, ST 10. THSM8) TOTALS = F(TEX8, ST IQ, THSM8) REAS8 = F(TEX8, St IQ, TCM8) FUND8 = F(TEX8. ST IQ, TCM8) TOTALS = F(TEXS, ST IQ, TCM8) REASS = F(TEX8, ST IQ, TTM8) FUNDS = F(TEX8, ST IQ, TTM8) TOTALS = F(TEXS, ST IQ, TTM8) FIGURE 1 THE 27 MULTIPLE REGRESSION COMBINATIONS 53 mafimfiHm¢> ho ZOHBHZHMWD N MMDOHh ucmfiuosv mucmmflaamusfl ucmpsum A.>Hmsomoamsm pmsflmmp mum mxms .mxmav “sow mpmum nmsounu :muummumpsflx .mocmflummxm umnummu COOS A.>Hmsomoncm pmeemmp mum page .mzoe .mzmme .mzee .pzoe .mzmmev usom mpmum cmsounu couummHOUCHx .COHumummmHm mUHumEmnumE Hmuou Hwnummu CDT: Doom mpmum smsousu smuummnmpcax .COHumummoum moaumEmnumE mmmaaoo Hmnommu cam: Hsom mpmum nmsounu :munmmnmpcflx .soflumnmmmum mUHumEmnumE Hoonom 50H: umnommu cmmz A.>Hmsomoncm emcemmp mum mqeeoe .mOZDm .mmemm .mqeeoe .mOZDm .mmemmv nsom mpmnm .ucmEm>mH£om UHDOESuHHm Hauou .mnoom pawEmomHm mpmum ucmpsum Hoom mpmum .ucmEm>mfl£Um mamucmEmDGSM uaumenuwum .muoom ucmEmUmam mpmum ucmpsum Hdom mpmum .DGOEO>0H£UD mGHGOmmmn Uflumenuaum .wuoom ucwEmUmHm mpmnm DGOU5um OH 8m «Xmfi #288 «EUR meZB dfidEOB fiQZDh fimdmm 4. 54 a measure of kurtosis For each pair of variables: 1. the simple correlation coefficient For each of the 27 combinations of variables: 1. 2. 3. 4. 5. the number of observations the coefficient of multiple correlation the coefficient of multiple determination the standard error of estimate an F-test value for testing the hypothesis that none of the sum of the squared deviations from the mean of the dependent variable is accounted for by the independent variables. an approximate confidence probability for the F-test value. For each of the variables in each of the 27 combinations: 1. 2. the multiple regression coefficient the standard partial regression coefficients (beta weights) an F-test value for testing the hypothesis that the variable does not account for any of the vari- ation in the dependent variable above that accounted for by the remainder of the independent variables and the mean of the dependent variable a t-test value for testing various hypotheses concerning the multiple regression coefficient an approximate confidence probability for the F- test value and the t-test value 55 6. the partial correlation coefficient 7. the coefficient of multiple determination which applies to the rest of the set of variables if the variable is deleted from consideration (the delete). All statistics were computed by means of the BASTAT Routine and the LS Routine2 of the Control Data Corporation 3600 Computer of Michigan State University. II. THE SETTING OF THE STUDY The population of the study was comprised of a portion of the students and teachers of the Lansing School District, Lansing, Michigan. This district is located in an urban, industrial area which grew in population from approximately 113,000 in 1957 to approximately 131,000 in 1966. This period of time was when the students of the study were ad- vanced from kindergarten to grade eight. The district en— rolls children of various ethnic, racial, and socio-economic backgrounds, whose distribution among the attendance dis- tricts varies according to residential patterns within the city. The students of the study received their instruction 2The BASTAT Routine and the LS Routine are part of a series of statistical computer programs prepared by W. L. Ruble and M. E. Rafter of the Agricultural Experiment Sta- tion, Michigan State University, and stored in the memory . section of the university's Control Data Corporation 3600 (Computer. The routines were designed to calculate statistics commonly used for DAsic §TATistics and Least §guares statis- tics. 56 in self-contained classrooms from kindergarten through grade six, and in departmentalized, junior high school classrooms in grades seven and eight. The choice of a junior high school from among the five junior high schools of the district was based pri- marily upon two considerations, (1) the stability of the population within the attendance district, and (2) the heterogeneous nature of the population with regard to ethnic, racial, and socio-economic backgrounds. Since information was needed concerning the students from the time they first entered school until the time they com— pleted the eighth grade, only students could be used who had attended the schools of the Lansing School District exclusively. Thus it was desirable to choose a junior high school whose attendance district included areas in which population mobility was minimal. It also seemed desirable to choose a junior high school whose attendance area included students with different backgrounds. These two considerations were in conflict to some extent, since some of the children who added to the heterogeneous nature of the student pOpulation, came from families that were relatively mobile. However, the school which was chosen, Henry R. Pattengill Junior High School, seemed to provide a reasonable compromise between the two. The attendance district included Caucasians, Negroes, and persons of Mexican ancestry, and included a sufficient number of 57 families whose children had attended schools of the Lansing School District since kindergarten. The class used in the study was the one which had most recently completed the eighth grade. The total membership of this class was 357. III. THE PROCUREMENT OF THE RAW DATA The Student Data By far the easiest data to obtain were the data which described the Students. These data were collected during a period of approximately three weeks during the summer of 1966. A request for permission to use the official records of the Lansing School District was submitted in care of Professor George Myers of the Student Teaching Office, College of Education, Michigan State University. The ap- proval of this request was recommended by Dr. Edward Remick, Consultant in Research, Lansing School District. Permission was subsequently granted by Mr. Robert Lott, Director, Division of Secondary Education, Lansing School District. In the State of Michigan each local school district is required to maintain a cumulative, permanent record for each student whom it enrolls. With the COOperation of the principal of Henry R. Pattengill Junior High School, Mr. Gary Fisher, and his office staff, the permanent record of each of the students was examined. Of the 357 students in the 58 class only 206, or approximately 58 per cent, had enrolled exclusively in the schools of the Lansing School District from kindergarten (or from first grade in those cases in which the students did not attend kindergarten) through grade eight. Except in a few cases in which complete test results were not available, the examination of the permanent records and the junior high school mathematics teachers' class books yielded the following information for each of the 206 students: 1. an arithmetiCLreasoning score, an arithmetic funda- mentals score, and a total arithmetic score result- ing from the administration of the California .Achievement Tpstss when the student was in grade four_ 2. an arithmetic reasoning score, an arithmetic funda- mentals score, and a total arithmetic score result- ing from the administration of the California Achievement Teppp4 when the student was in grade six 3. an arithmetic reasoning score, an arithmetic funda— mentals score, and a total arithmetic score resulting 3E. W. Tiegs and W. W. Clark, California Achievement Tests, Elementary Level, 1957 Edition (Los Angeles: Califor- nia Test Bureau, 1957). 4Loc. cit. 59 from the administration of the California Achieve- ment Tests5 when the student was in grade eight 4. an intelligence quotient resulting from the admin- istration of the California Test QT Mental Matprity6 when the student was in grade two 5. an intelligence quotient resulting from the admin- istration of the California Test of Mental Maturity7 when the student was in grade four 6. an intelligence quotient resulting from the admin- istration of the Califophia Test of Mental Maturity8 when the student was in grade six 7. the name of each teacher by semester with whom the student studied mathematics through grade eight. This comprised the raw student data. An examination of these data revealed that the 206 students had been taught by a total of 273 teachers. The procurement of information concerning these teachers was the next step in the study. 5‘8. W. Tiegs, and; W. W. Clark, California Achievement Tests, Junior HithLevel, 1963 Edition (Los Angeles: California Test Bureau, 1963). 6E. T. Sullivan, W. W. Clark, and E. W. Tiegs, California Tept of Mental Matupity, Primary Level, 1957 Edition (Los Angeles: California Test Bureau, 1957). 7E. T. Sullivan, W. W. Clark, and E. W. Tiegs, li ornia Tept of Mental Maturityy,ElementarygLevel, 1957 Edition (Los Angeles: California Test Bureau, 1957). 8Loc. Cit. 60 The Tepcher Data The procurement of the teacher data consumed the major portion of the time and effort expended on the study. When the study was initiated, it appeared that the collection of teacher data would be a straightforward process similar to the collection of student data. The collection of teacher data was to have been accomplished by the examination of the teacher personnel folders on file with the Lansing School District. The investigator assumed that each teacher's folder contained a transcript of the mathematics courses for which the teacher had credit. This assumption was confirmed by personnel of the Lansing School District. When permission to use the official records of the district had been granted, it appeared that it would be a simple matter to examine the transcripts and to determine the amount of mathematics preparation of each teacher. However, when the personnel folders were carefully examined, it was found that only a very few of them actually contained the desired transcripts. Evidently, a short time before the study was initiated, the personnel officers had inaugurated a new policy under which the transcripts of teachers were examined for purposes of salary schedule ad- vancement and then returned to the teachers involved. Nevertheless, some useful information was obtained from the personnel folders. This information included the number of years of teaching experience which each teacher 61 had at the time she was associated with the students of the study. Since the required information could not be obtained from the personnel folders, another source of teacher data was explored. This source was the records of the Depart- ment of Teacher Education and Certification of the Michigan Department of Education. Exceptional cooperation and assistance was provided by Mr. Eugene Richardson, the di- rector of the department. Approximately three weeks were spent by the investigator working with department personnel in examining the teacher certification records of the State of Michigan. The locating of each teacher's folder was facilitated by the use of information obtained from the personnel records of the Lansing School District, since the certification records were filed according to type of certificate and year of certification. However, in terms of total information this source of data did not prove very fruitful. Those teachers who had graduated from approved Michigan teacher education institu- tions had been granted teaching certificates upon the recommendations of their respective institutions. For these teachers it was not necessary for their transcripts to be examined by personnel of the certification department, and therefore their transcripts were not on file. Furthermore, not all teachers who entered Michigan from other states had had to present transcripts for purposes of certification; 62 Michigan had entered into reciprocity agreements with selected states and had agreed to grant teaching certifi- cates on the face value of teaching certificates granted earlier by those states. Despite these limitations, con- siderable teacher information was obtained about the teachers in the study. The only remaining recourse for securing the required teacher data was to employ a questionnaire. The use of a questionnaire demanded knowledge of the current addresses of the teachers. This presented a sizable problem because many of the teachers had retired or resigned since teach— ing the students involved in the study. Only 126 of the original 273 teachers were still teaching in the Lansing School District. Of the remaining 147 former teachers many had moved away. Some had been gone from the Lansing area for eight years. Thus the immediate objective temporarily shifted from the procurement of teacher data to procurement of teacher addresses. Several sources were used, with possible leads from one source checked out with other sources. These sources of addresses included the following: 1. an extensive collection of telephone directories in the Lansing Public Library 2. an extensive collection of city directories in the office of the Chamber of Commerce of the Greater Lansing Area 63 3. a collection of old personnel directories of the Lansing School District for the years from 1956 through 1966 4. conversations with school secretaries, school principals, and former colleagues of the missing teachers 5. the alumni office of the college from which the teacher graduated 6. the Alumni Office of Michigan State University 7. the Married Housing Office, Michigan State Uni- versity.9 The teachers who had moved away were scattered as far as Germany, Wales, Turkey, Nigeria, and Indonesia. A questionnaire was carefully developed by designing, discarding, and revising repeatedly over a period of several weeks, during which time advice was secured from persons experienced in the use of questionnaires. A copy of the final result has been included as Appendix B. It may be noted that this questionnaire included items extraneous to this study; they were included for the purpose of providing information which was desired for a future study. The initial return for the questionnaire from all teachers being studied, both those still teaching and those no longer teaching, was approximately 50 percent. However, 9The last two sources were particularly fruitful because the proximity of Michigan State University and the Lansing School District resulted in the enrollment of the teacher or the teacher's spouse in the university. 64 through dilligent follow-up efforts including written ap— peals, telephone conversations, and personal interviews, the final return was raised to approximately 88 percent. These percents are reported here for their possible aca- demic interest; neither the percent nor the number of re- turns was meaningful in terms of their statistical value. This fact resulted from the distribution of the teachers and the students. Some teachers had taught several of the students, while othershad taught only one. Also, some of the teachers had taught the students longer than one year. The meaningful number was the number of students for whom complete information was available. That is, the size of the statistical population was the number of stu— dents for whom information was attainable concerning them- selves and concerning the teachers who taught them. The process of matching the teacher information with the ap- propriate student information has been described in the next section. IV. THE TRANSFORMATION OF THE RAW DATA The_§hudent Dpta Two transformations were performed on the student data. The first was necessitated by the fact that some of the mathematics achievement scores had been entered in the stu- dent permanent records in the form of percentiles and some in the form of grade placement scores. Because computations 65 were to be performed with the scores, it seemed more appro- priate for them to be in grade placement form than in per— centile form. Therefore, the appropriate conversion tables in the test manual10 were used to convert the percentile scores into the corresponding grade placement scores. The second transformation of the raw student data was performed on the intelligence quotients. Intelligence quo- tients had been recorded for most students when they were enrolled in grades two, four, and six. Because of probable errors in measurement, it was assumed that a measure of central tendency of each student's intelligence quotients would provide a more accurate measure of his intelligence than would any single one of them. Therefore, the arithme- tic mean was determined for each student's set of recorded intelligence quotients and used as the measure of his intel- ligence. The Teacher Dpta There were essentially two teacher characteristics for which data were needed, the amount of teaching experi- ence and the amount of mathematics preparation. The amount of teaching experience was measured in years. A scale was developed for the measurement of mathematics preparation. The amount of mathematics preparation could have been measured by clock hours of instruction, by term or semester 10E. W. Tiegs and.W. W. Clark, Manual, California Achievement TestsL Elementary Leveli 1957 Edition (Los Angeles: California Test Bureau, 1957), 62 pp. '66 hours of credit, or by the number of courses taken. However, it seemed appropriate to develop a different type of scale which would reflect the amount of commitment to mathematics which each teacher had made, as well as the number of courses taken. Because of the sequential nature of mathe- matics courses and the increasing sophistication of the con- tent within the sequence, it was decided that a scale should be established which would indicate the progress of each teacher within the sequence. Since teachers in preparation sometimes enroll for courses similar in content to those for which they have previously earned credit, the mere ppppT of courses would not indicate the extent of the subject matter they had studied. Therefore, three scales called mathematics category value scales were developed: 1. for assigning measures of the amounts of high school mathematics studied 2. for assigning measures of the amounts of college mathematics studies 3. for assigning measures of the amounts of mathematics studied from the high school through the college to the in-service education of the teacher. These scales have been included as APPENDIX C, APPENDIX D, and APPENDIX E, respectively. For each of the categories (high school, college, and total) the raw data were examined and measures assigned according to the respective category value scales. 67 For example, if a teacher had studied general mathematics, business mathematics, algebra, and geometry in high school, had studied algebra in college, and had participated in an in-service workshop in mathematics, then his high school mathematics category value would have been Tppp, his college mathematics category value would have been three, and his total mathematics category value would have been Tiyp, It should be noted that on each scale, 0 was used to indicate that no information was available, rather than to indicate that the teacher had studied no mathematics in that category. This tactic proved to be useful later in elimi- nating from the study those students for whom complete teacher information was not available. Although the actual number of teachers was 273, the practical number was 315. This resulted from the fact that many of the teachers had had contact with certain students over a period of more than one year. For example, some teachers taught students for two consecutive years when the students repeated a course. Other teachers taught in differ- ent grades on different years due to the teaching assignments for which they were scheduled. During the time between these student contacts, some of these teachers were exposed to additional mathematics instruction, thus changing their amounts of in-service mathematics preparation. Also this affected their respective amounts of teaching experience. A convenient means of accounting for these changes in 68 teacher characteristics was to treat each subsequent contact as a new and distinct teacher. In this way the statistical teacher population was extended to 315. The Collation of the Student and Teacher Data There were two tasks which had to be accomplished in order to provide a cumulative measure of each teacher char- acteristic for the complete set of teachers that taught each student over each of the grade level periods. First, it was necessary to match each student with the specific set of teachers that taught him over each grade level period. Second, it was necessary to determine the arithmetic mean of the values of each teacher Characteristic for each set of teachers. These two tasks were further complicated by the fact that some students had changed teachers in the middle of an academic year or had repeated a course. With 206 students and 315 teachers involved, the two tasks would have been practically impossible without the use of an electronic digital computer. The preparation of computer data cards. The first step was the assigning of an identification number to each stu- dent and each teacher in the study. Then three sets of computer coding sheets11 were prepared. The first set con- tained for each student: 11A computer coding sheet is a form upon which data are recorded and from.which a keypunch operator prepares computer data cards. 1. his 2. his 3. his 4. his 5. his 6. his 7. his 8. his 9. his 10. his 11. his 69 identification number mean intelligence quotient arithmetic reasoning score for grade four arithmetic fundamentals score for grade four total arithmetic score for grade four arithmetic reasoning score for grade six arithmetic fundamentals score for grade six total arithmetic score for grade six arithmetic reasoning score for grade eight arithmetic fundamentals score for grade eight total arithmetic score for grade eight. The second set contained for each teacher: 1. his 2. his 3. his 4. his 5. his identification number teaching experience value high school mathematics category value college mathematics category value total mathematics category value. The third set contained for each student: 1. his 2. the identification number identification number of each teacher who taught him each semester from kindergarten through the middle of grade eight, including any semesters which he repeated. These coding sheets were submitted to the User Service Office, the Computer Center, Michigan State University. Trained and experienced keypunch operators prepared three 70 sets of computer data cards, corresponding to the three sets of coding sheets. These operators then verified the computer data cards to ensure that the data had been cor- rectly transferred from the coding sheets to the computer data cards. The preparation of the computer program. A computer program was designed and written in FORTRAN IV language for the purpose of instructing the Control Data Corporation 3600 Computer of Michigan State University to perform the desired collating and averaging of the raw data. This pro- gram was written by the investigator with the technical assistance of the programming consultants of the Computer Center, Michigan State University. To provide additional safeguards to the validity of the program, the program was submitted to trial runs after which the results obtained from the computer were checked by the investigator. These trial runs.confirmed that the computer was properly pro- grammed to perform the desired transformations of the data. A flow chart of this transformation program has been in- cluded as APPENDIX F. The operation ogithe computerpprogram. The program directed the computer to read the data from the three sets of computer data cards and store this information in its memory section. The following information was stored in memory for each of the 206 students: 11. 12. his his his his his his his his his his his the him 71 identification number arithmetic reasoning score for grade four arithmetic fundamentals score for grade four total arithmetic score for grade four arithmetic reasoning score for grade six arithmetic fundamentals score for grade six total arithmetic score for grade six arithmetic reasoning score for grade eight arithmetic fundamentals score for grade eight total arithmetic score for grade eight mean intelligence quotient identification number of each teacher who taught each semester from kindergarten (or grade one if he did not attend kindergarten) through the middle of grade eight, including any semester which he repeated. The following information was stored in memory for each of the 315 teachers: 1. 2. 3. 4. 5. his his his his his identification number teaching experience value high school mathematics category value college mathematics category value total mathematics category value. Then the computer was directed by the program to con- sider the teacher identification numbers for the first student, one at a time in sequence. The computer used these teacher identification numbers to locate the teaching 72 experience values for each of the teachers. As each teach- ing experience value was located by the computer in its memory section, the computer entered it as an addend into a cumulative sum of teaching experience values. As the computer progressed one semester at a time from the first semester of kindergarten, it divided the cumulative sum by appropriate divisors to compute the arithmetic mean of teaching experience values for certain grade level periods. Thus, the arithmetic means for teaching experience values were determined for the teachers in each of the following grade level periods: 1. kindergarten through the middle of grade four 2. kindergarten through the middle of grade six 3. kindergarten through the middle of grade eight. Note that each set of semesters ended in the middle of a grade. This was necessary since the achievement tests used for comparison were administered in the middle of grades four, six, and eight. Next the program directed the computer to repeat these operations using the next teacher characteristic, high school mathematics preparation, in the place of teaching experience. Subsequently the computer was directed to per- form these operations for all four of the teacher charac- teristics. At this point the collating and averaging was accom- plished for only the first of the 206 students. Therefore, 73 the computer repeated all of these cycles for the second student, for the third student, and for each student in sequence until the collating and averaging was accomplished for all of the students. Finally the computer controlled the operation of card preparation equipment in the preparation of three sets of computer data cards. Each card in the first set contained information on one of the students and the cumulative measure of each of the characteristics of the teachers who had taught him from kindergarten through the middle of grade four. Each card in the second set contained information on one of the students and the cumulative measure of each of the characteristics of the teachers who had taught him from kinder— garten through the middle of grade six. Each card in the third set contained information on one of the students and the cumulative measure of each of the Characteristics of the teachers who had taught him from kindergarten through the middle of grade eight. Figure 3 indicates the information contained in each of the three sets of computer data cards in addition to the student identification numbers. Defini— tions of the variables used in Figure 3 have been listed in Figure 2, page 53. Some additional provisions of the progppm, Although the basic operation of the computer has been described, the program contained some additional provisions. The program provided for the inclusion of repeated semesters whenever 74 Information contained in Set Number One $288 #208 ¢mea wxmfi OH Em mA¢HOB mQZDm madmm mA.0 0H 8m 00000.6 Nfimwd.0 0mmfid.0 mam00.0 mmNMH.0 dxma 00000.fl thmd.01 >0mma.0 m0¢m0.0 mm0m0.0 Hhfiho.o dimmfi 00000.H mmm>m.0 mmmma.0I mmdfio.ou Nmnmo.0I 05¢OH.0I mmmm0.0I «SUB 00000.H mmdmh.0 hmmhm.0 ma¢0a.0I N0¢HH.0 00m¢0.0I mm¢50.0I ¢¢0N0.0I $298 #298 #206 SmeB fixma 0H:Bm ¢A¢BOB SQZDM #mfimm .Hsom mpmum nmsounu cmuummumpcwx Eouw mucmpsum mNH nua3 pmumaoommm woeumflumuumumno usmflw mo mwusmmoe may «0 sowumamunoo OHQEHm mo mucmememoo .>H magma 83 00000.6 mwémm 00000.fi 0Hmm>.0 mnzam 00000.a mmeom.o mnwmm.0 madfioa 00000.H omdmw.0 mmmom.0 thdb.o 0H Em 00000.d mea>a.o N00>H.0 mam>a.o #mw¢d.0 mxma 00000.H memma.0u NOH0d.0 Hmmmo.o maomo.0 >0m¢0.0 mmeB 00000.H emomm.0 emmmo.ou mm¢00.01 emeoa.0I 0mooa.on oomm0.0| @208 00000.H 0>ad>.o memmm.o a0a¢o.ou aemm0.0 H0m>0.ou ommm0.0I ma0>0.01 @285 mzaa @209 mzmma mxma 0H Em 0A4BOB mnzam mmmmm .xflm mpmum amdouau cmuummumpcax Eonm mucmpsum mma nua3 UTDDAUOmmm mUHumHuqumumno unmflm mo mmusmmwa m3» m0 cowumamuuou DHQEHD mo magmaoflmmmoo .> magma 84 00000.6 madam 00000.6 mmmmm.0 mozam 00000.6 mmonm.0 m6mfim.0 mndaoa 00000.6 owmm>.o wOM6>.o 605mm.0 OH 8m 00000.6 hemmm.0 60mm”.o N6¢mm.0 mnemm.0 mxua 00000.6 mmmw6.o memmm.0 Nm>6¢.o ememn.o m00¢¢.0 02mme 00000.6 mNNNO.o mm¢m6.0I mm606.0I mmmm6.01 >m¢m6901 0m6>N.01 0209 00000.6 whoom.0 666mmwo Ndmmo.01 mmom0.0n 6mmm6.0I m0>>6.OI mwmom.0I @298 @299 @209 mzmme mxfla 0H Em mA manna 85 coefficients of these three categories of teacher prepara- tion were more pronounced. A low positive correlation coefficient resulted for teacher high school mathematics preparation and student mathematics achievement, and low negative correlation coefficients resulted for both teacher college mathematics and teacher total mathematics prepara- tion and student mathematics achievement. The greatest relationship appeared to be between teacher high school mathematics preparation and student mathematics achievement for grade level period eight. The correlation coefficients for teacher experience and student mathematics achievement were all positive. By grade level periods the smallest correlation coefficients resulted for kindergarten through the middle of grade four, and the largest resulted for kindergarten through the middle of grade eight. As one might expect, much larger correlation coeffi- cients resulted for student intelligence and student mathe- matics achievement. This was true with respect to the arithmetic reasoning scores, the arithmetic fundamentals scores, and the total arithmetic scores for all three grade level periods. Multiple Correlations Tables VII through XXXIII display the multiple regres- sion and correlation statistics for the 27 combinations 86 .DODOHOU OHOB OHQMHHD> MH :06ums65umump 06m6u6sfi mo DCOHUHMMOOU * moo. n Sueeenonoue oooooemeoo Aeom.oe n ooeoo poooum moeemeoe. u Somehomo mo uouuo ouooooum 65mm. u c06umc6EHmump m6m6u6ss mo usm6uwmmmoo mmmn. n GOHDMHUHHOU 06m6uasfi mo ucm606mmmoo mm6 u m:06um>ummno mo HUQEDZ «6.0 mnnw.m 66mm.6u 6mm¢m06m.6r BZdeZOO mmwm0.0 www6>.0 moo.o nmmm.NM6 mm6m.66 mmmm>.o 6mme6¢m0.o OH Em m6>mm.o mmmmo.o m>.o ommo.o momm.o 6mm60.0 60m6o¢00.o «Nae mon~m.o 6mmm0.on mm.o mom6.6 m>>0.6l ommmo.01 mmmwmmm6.01 dSmEB .mmoo .HHOU .noum m56m> $56m> #30603 ucm606wmmou .msoo ummulm ummunu mumm c06mmmummm OHQD6HD> *mum6ma HMHuumm .Hsom mpmum smsounu cmuummumpGHx .mc6aommmu 06umE£u6um ucwpsum pom mUHDMEmnumE 6oonom £m6£ Honommu .m06um6umum COHmmonmmu O6m6u652 .HH> O6QMB 87 .UODOHOD OHOB manm6um> MH o06ums6Eumuop @606u6se mo usm606mmwoo* moo. n MU666QMQOHQ musmp6mnoo mmmm.m6 n m56m> ummuum mommm6wn. u OumEHumm mo Hound pumpcmum mmwm. n :06ums6aumump OHQHDHDE mo unm6o6mmmoo Name. u COHumHmuHoo $606u69e mo ucm6o6mwmoo mm6 u m:06um>ummno mo HOQEDZ 00.0 6066.m 06mm.6 6mmhbumm.6 BZéBmZOO mmm00.0 mmmm¢.0 m00.0 6560.06 mmhm.m 0066m.0 6m6>0mm0.0 OH Em mbmdm.0 bmmN0.0I m>.0 N6~0.0 mmmm.0u 0m6N0.0| 0mmm0600.01 «Mme mmmem.0 «m6m0.01 6m.0 bmm0.6 mmm0.6l wmmmo.01 6000N006.0I 62mm8 .mmoo .HHOU .QOHQ m56m> Ho56m> usmHOB ucm606mmmoo *mumHmQ HM6uHmm .mcoo ummulm ummulu mumm c06mmmummm m6nm6um> ucmpsum pom mUHUDEUSDME Hoosom £06: Hmsomou .usom mpmum nmsounu :muummuopc6x .mHOHQOEMOssm U6umfinu6um .mo6um6umum s06mmmumou O606u6sz .HHH> O6QMB 88 .pmum6mp mum3 m69m6um> NH :06umc68Hmump m6m6u6sfi mo ucm606mmm00* moo. u >UH66QDQOHQ OUGOUHMGOO mmom.wm u 036m> umwuwm 6m6>6m>m. u mumEHDmm mo Hound pumwcmum omme. u COHDMCHEMODDD O606u658 mo uCOHOHMMOOO Summ. n COHDMHOMHOU m606u658 mo ua0606mmwoo 0N6 u mC06um>umeo mo Honesz mm.0 m>m0.0 mmm6.0 monm6666.0 Ezmamzoo 0mm60.0 momwm.0 m00.0 mmdm.¢06 mmmm.06 mmmmm.0 mmmm6m60.0 0H em momme.0 mmm00.0 mm.0 n000.0 mmm0.0 ~5600.0 mm6¢m000.0 exms mmmme.0 0mm06.01 mm.0 m>6¢.6 >0m6.6l nmomo.0| moonwmm6.0| 62mma .mmoo .HHOU .Qoum O56m> OsHm> unm6m3 ucm6o6mmmoo *mumama 6M6uumm .mooo ummulm ummulu mumm a06mmmummm OHQD6um> .Hsom mpmum nmsousu cmuummumpc6x .06umenu6um 6muou ucmpsum Dam mo6um505uma Hoosom £063 Hmsummu .m06um6umum COHDDOHmOH m606u652 .xH magma 89 .pmum6mp muo3 O6QMHHM> M6 GOHDDCHEHODOU OHQ6UHDE mo usm606mmmoo* moo. n mueeenononm moomoemooo ammo.oe u ooeo> uoooum mommmmmn. u wumEHDmm mo HOHHO unannoum memm. u soaumc6finmump m6m6u6se mo DCO6UHMMOOU meme. u COHDDHOuHOU w606u655 mo ucm6o6mmmoo 0N6 n ma06um>uomno mo Hmnfiflz 60.0 mm60.> emwm.ml mmm6mmm>.6l BZfiBmzoo N0m60.0 N6m6>.0 m00.0 0m06.mM6 65mm.66 6>m6>.0 mmmmmm00.0 0H 9m mnemm.0 mmmm0.0 6>.0 0N¢6.0 005m.0 mommo.0 mm>m>m00.0 dxma m0mmm.0 memwo.0I mw.0 0006.0 000>.0I mmm60.0I mamm6m66.ou «SUB .mmoo .HHOU .Qoum ms6m> osHm> u£06m3 ucw606wmmoo *Oum6ma HD6uHmm .MCOO umOunh ummuuu mumm s06mmmummm OHQD6HC> ucmpsum Dam moflumsosume omm6600 Honommu .mowum6umum GOHDmOHmOH mam6u6sz il .usom mpmum nmsouzu cwuummHODCHx .mGHGOmmmH 06umE£u6Hm .x m6nme 90 .0000600 0003 06Q06um> 06 CO6ums6EH0u0U 06060658 00 ua0606mm0ou* 000. u >6666nmnoum 00c006mcoo 0006.06 u 0560> um0u|m 0mmmooww. n 000E6uw0 mo 60660 Oumwcmum mmmm. u ooeoooHSHouoo mameoeos 0o Dooeoemmooo 0000. u so6um60nuoo 0606u658 mo 0:06U6mm0oo 006 u mGO6DM>u0on mo H0QEDZ 00.0 6000.0 6>>0.6 6000000N.6 92090200 00060.0 60N0¢.0 000.0 0>00.06 0>Nm.0 0N¢0¢.0 06000000.0 OH 90 >0NON.0 006N0.0I 0>.0 >000.0 006N.0I 0N060.0I 60>00000.0| ¢XMB 00N6N.0 00066.0I 06.0 0600.6 0000.6: «0606.0: N060060N.0I 0209 .m0oo .HHOU .Qoum 0560> 0560> 0:0603 6:06U6mm0oo *060600 6060600 .6:00 um0ulm um0ulu 000m c06mm0n00m 06906Hm> .6500 00060 £05065» c0unm0H0GC6x .0606s0Empssm U6u0enu6uw 0:00:60 ps0 mo6umE0£umE 0006600 H0som0u .0060066000 c06mm0n00n 0606u652 .Hx 06908 91 .0000600 0603 0600660> 06 0060006060000 06060608 00 000606000OO* moo. ":00666nmnoue ooomoemooo 0000.00 u 0060> 0000I0 60006060. u muos60mm mo uouum oumoomum 0006. u 0060006860000 06060600 00 00060600000 6000. n 00600606600 06060600 00 00060600000 006 u 000600>60mno 00 606002. 60.0 0006.0 6000.0I 06066>06.0I 92080200 50060.0 60050.0 000.0 6060.606 0000.06 00000.0 00660060.0 06 90 00006.0 65000.0 00.0 0000.0 6500.0 06000.0 00000600.0 6x08 00006.0 00006.0: 60.0 0666.6 0006.6! 00000.0: 00005606.0| 6209 .0000 .6600 .0060 0060> 0060> 000603 00060600000 *000600 6060600 .0000 0000I0 000010 0000 0060006000 0600660> Ii Ii .6000 00060 0000600 000600600060 .0600000660 60000 0000000 000 00600500008 0006600 6000000 .0060060000 0060006006 06060602 .66% 06609 92 .0000600 0603 0600660> 06 0060006060000 06060605 00 00060600000* moo. u mueaflnonoum mooooflmooo omoo.e6 u mo0m> uomuum mmomeoee. n mumefluom 0o uouuw ouooomum 0000. n 0060006560000 06060600 00 00060600000 0005. u 00600606600 06060600 00 00060600000 006 n 000600>60000 00 600002 06.0 0005.6 0560.6: 00606500.6: 82090200 00560.0 00605.0 000.0 0006.006 0605.66 00605.0 00600000.0 OH 90 00000.0 00600.0 05.0 0650.0 0050.0 00060.0 60066600.0 0x09 00000.0 66066.0: 06.0 0005.0 0000.6: 00006.0: 66000060.0: 6298 .0000 .6600 .0060 0060> 0060> 000603 00060600000 *000600 6060600 .0000 0m00:0 0m00:0 0000 0060006000 0600660> .6000 00060 0000600 000600600060 .006000006 0600000660 0000000 000 00600000008 60000 6000000 .m060m6000m oo6mmmumwu 06060652 .HHHx 06006 95 .0000600 0603 0600660> 06 0060006060000 06060600 00 00060600000* 000. u 00666000060 0000060000 0000.66 n 0060> 0m00:0 00000005. u 00006000 00 60660 06000000 5000. u 0060006060000 06060600 00 00060600000 0060. M 00600606600 06060600 00 00060600000 006 k 000600>60m00 00 600002 00.0 0000.0 6560.0 00506500.6 82080200 00000.0 00600.0 000.0 6000.06 0600.0 56660.0 00000000.0 06 90 06000.0 66000.0: 55.0 0600.0 0600.0: 00000.0: 00060600.0: 6N08 00060.0 00006.0: 06.0 0560.0 6505.6: 00006.0: 00600000.0: 6288 .0000 .6600 .0060 0060> 0060>. 000603 00060600000 *000600 6060600 .0000 0m00:0 000010r 0000 0060006000 0600660> u, Ill .6000 00060 0000600 000600600060 .060000000000 0600000660 0000000 000 00600000000 60000 6000000 .0060060000 0060006006 06060602 .>Hx 06008 94 .Umumamv muo3 mannaum> ma coaumcflfiumumfl mamauass mo ucmfioflmmmoo* 3.0 moo. u MuHHflQMQOHm mocmwwmqou m>m>.>m u msHm> ummumh 0Nmmmfinw. u mumfifiumm mo nounm Unmccmum mm>¢. n coaumcflfiumumw mHmHUHSE mo ucmfioflmmmou 0000. u coaumamuuou mamfluase mo ucmfioammmoo mma u mcowum>ummno mo umnfisz mmmn.0 mamm.0 #mm¢fimm¢.0 Bzm¢a0.0 00600.0 m00.0 mmmm.moa amm¢.0H mmmmm.0 00¢mmm¢0.0 0H Em mmmhwqo 05600.0 mm.m $000.0 00H0.0 mma00.0 mom¢N000.0 ¢XHB mmmm¢.0 nOmwa.0I 00.0 nmdm.m mmmm.fil 0mmma.0u amomhomm.01 $299 .mmou .Hnoo .Qoum 05Hm> msam> unmflm3 ucmfioflmmmoo *mumamn Hafiuumm .mcou ummuum ammulu mumm scammmummm mHQMHHm> .Haom mwmum nmsounu cmunmmumccflx .Uflumfinuwum Hmuou ucmwsum 0cm moflumfimsume Hmuou Honomwu .moflumflumum coammmummu mamfiuasz \|l JII 5x 383 95 .Umumamc mum3 mHQmHHm> MH coaumaHEumumU mHmHUHsE mo ucmflofimmmoo* mam moo. u >ufiaflnmnoum mucmcflwcoo wo~m.¢¢ n mSHm> ummuum «mmmn. u mumaaumm mo uouum cumucmum whfim. u COHuMCHEHmumU meHUHDE mo ucmflowmwmoo mmfiu. u COHumHmuHOU mamfluase mo ucmfioammmoo mma u mCOHum>umeo mo umflfisz no.0 homo.o wfimd.oa owofimmmd.0I BZdBmZOU #momo.o 0¢0H>.o m00.0 mmmm.omd H0¢N.Hd mm>a>.0 moo¢wmoo.o OH Em wwwam.o m¢0m0.0 m>.o flmda.o hmmm.o hmamo.o m>>mmm00.o oxma mfihfim.0 hammo.0| «v.0 ommfi.o ommm.0I m¢mm0.on mmmmamoo.01 mzwms .mmoo .Huoo .Qoum mSHm> wSHm> uanmB ucmwoflmmmoo *mumHmQ Hafiuumm a .mcoo ummurm ammulu muwm scammmummm mHQMflHm> xfim mwmum nmsounu cmuummumvcax .mCHcommmu UHumesuflum ucmcsum 0cm mUHuaEmnumE Hoonom 50H: Hmnummu .mowumflumum coammmummu mamfluasz . H>X ma QMB 96 .Umumamc mumz magmaum> ma coaumcflfiumumv mamfluasfi mo unmaoflmmm00* moo. u muaaflnmnoum mocmuflmaoo mmao.mm u msam> ammuum m¢mmmamm. u mumefiumm mo uouuo nuancmum 055m. u GOHuMCHEHGumU mHmHuHDE mo ucmwoflmmmoo 0dfio. u coaumHmHHOU mamfluasfi m0 ucmfluwmmmou mNH u macaum>ummno mo HmQEsz Ho.o ¢mam.m Hmmm.m m>m¢¢>mm.a BZdBmzoo oa>mo.o «mem.o moo.o H¢m0.wm mmmm.m mommm.o m>mmmmm0.0 OH am homnm.o. mmmmo.o mm.o momm.o momm.o mamwo.o Nmmuomao.o mxma amonm.o momao.o 00.0 omao.o N¢md.o 00000.0 odmmnmao.o mZmflE .mmoo .uuoo .Qoum msHm> msHm> unmflw3 unmaoflmmmoo *mumamo HMHuHmm .mcoo ammulm ummutp muom coammmummm mHQMHHm> ll '1 .me mvmnm nmsonnu cmuummumvcflx .mamucmEMUGSM Uaumenuflum ucmvsum cam moHumEmnumfi Hoonom 30H: umnommu .mofiumflumum GOHmmmHmmu mamfluasz .HH>N magma 97 .omumamo mnm3 mHQMHHm> ma coaumcflfiumumo mamfluHSE mo u:mauammm00* moo. u muflaflnmnoum mocmofimcoo oom¢.m¢ u maam> ammuum ppmhooao. u mumsflumm we gonna oumocmum ommm. u cowumcflfiumumn mamfiuasfi mo quHUHmmmOU ommw. H coaumeuuoo mamfluase mo ucmwoflmmooo mma n m:0aum>umm£0.mo Hmnfidz NN.0 000m.a NFNN.H mammwdmm.o Ez¢smzov mo¢no.o nmddo.o moo.0 mmmd.>ma m>>m.aa dmmah.o mommmm¢0.o 0H um omamm.o mammo.o m¢.o ommm.o ohm>.o amm¢0.0 bmommmoo.0 oxma Nwmmm.o owmdo.ou Hm.o ahwo.o ohHN.OI moma0.0u bmmmaom0.0I 02039 .mmoo .HHOU .Qoum wsHm> mSHm> pnmflm3 ucmfioflmmmou *mumHmQ Hmwuumm .mcoo umwulm ummuuu mumm Goflmmmummm mHQMHHm> .xwm momum nmsounu :muummumvcflx .owumazuflum Hmuou ucmonum cam mUHumEmnumE Hoonom now: umnommu .mofluwflumum scammmummu mamfiuazz .HHH>N magma 98 .vmumamo mumz mHQMHnm> ma GOHumcflEHmuwo mHmHuHSE mo ucmHUHmmwoo* moo. u mummmnmnoum mocmommaoo mmoo.m¢ u msHm> ummulm mohmmmhm. u mumEHumm mo uouum oumocmum ommm. u coaumcflfiumumo mHmHuHsE mo unmaoammmoo mfimo. u coaumHmHHou mamwuase mo ucmfloflmmmoo mma n mCOHum>umeO mo Hmnfidz no.0 0000.0 mdmo.01 N¢>ammao.0| Badamzoo mommo.o m¢¢a>.o moo.o whom.mmd mobm.dd omflan.o moshwmmo.o OH am ¢¢¢Nm.o Nomm0.o 65.0 mmma.o mwom.0 mammo.o mmmfioooo.o oxma mandm.o mmmNH.0| mH.0 ¢>N0.N mmm¢.HI mamm0.0u mommam>m.01 0208 .mmoo .HHOU .Qoum 05Hm> m5am> usmfiw3 ucmfloflmumoo *mumamn amauumm .mcoo ummulm umwulu muwm coammmummm 0HQMHHm> .xflm mcmum :msounu :muummumocflx .mCHcommmu Uflumfinuflum unmosum Una mUHumEmnumE mmwaaou Hmnommu .moflumflumum scammmummm mamfluasz .xHx magma 99 .omumamo mum3 mHQMHum> ma coaumcflEHmumo mHQHuHSE mo ucmfloflmmmoo* moo. u muflamnmnoum moamommcoo 0000.00 u msHm> ummulm Ha00m5a0. n mumeflumm mo uouum oumocmum a000. u cowumcflsuwumu mamfluasa mo ucmfloammmou «H00. u aoHumHmHHoo mamfluasfi mo ucmfioflmmmoo 00H u macaum>ummn0 «o qu592 000.0 $¢00.0a 00H0.d 0NNON¢00.N 92090200 0H000.o 00000.0 000.0 0050.05 NH50.0 50500.0 00000000.0 OH um 00600.0 60000.0 00.0 0500.0 0000.0 00000.0 000000H0.0 0x09 H0050.0 H0HNH.0I 5H.o 0H00.H 0000.al 05000.0: 00¢0N5om.ou 0209 .%000 .Huoo .Qoum wSHm> wsHm> unmflm3 uanUHMMmOU *mumamo amauumm .mcou ummulm ummuuu mumm scammmummm mHQMHHm> .xflm womum smdousu cmuummuwocfix .mamucmfimocsw Uflumenuflum unmosum 0cm moflumEmSumE mmwaaoo umnommu .moflumflumum scammmummu mamwuasz .xx manna 100 .Umuwamc mum? wHQMHHm> 09 :09umc9fiumum0 mHQHuHSE mo ucm9o9mmmOU* moo. n muflaflnmnouo mucmofimcoo 5000.59 u m59m> ummuum 00090090. u mumfi9umm mo Houum Unmvcmum 0000. n a09umc95umumv mHQHuHSE mo ucm909mmm00 9005. u GOHHMHmHHoo mHQHuasE mo ucm909mmw00 009 u m:09um>ummno mo quEsz 00.0 0000.9 0090.0 00000009.9 92090200 00000.0 90595.0 000.0 0950.909 0909.99 00095.0 00009090.o OH 90 09900.0 00000.0 59.0 5090.0 0005.0 00090.0 00099000.0 0x09 09000.0 00099.0: 09.0 0550.0 0000.9: 09009.0: 99009590.0| 0209 .0000 .Huoo .Qoum m59m> msam> u£09m3 quHUHMMmoo *mumawa HM9unmm .0200 ummulm ummuuu mumm c09mmmummm mHQMHHm> .X9m mnmum nmsounu :0uum0um0c9x .09umezu9um Hmuou ucmwsum cam mo9umaw£uma mmmaaou umnommu .mUHumHumum :09mmmu0mu mHQHuHDE .Hxx m9nm9 101 .0000900 0003 09909H0> 09 0090009500000 09090908 00 00090900000* 000. n >0999Q0QOHQ 0000090000 pom0.m9 u msam> um¢0:0 00090005. u 00069000 00 00090 00000000 090m. u :090mc9200000 00090955 00 0c090900000 9900. u :090mamunoo 09090955 no 0:090900000 009 u 000900>H00£0 00 000052 00.0 0900.0 9090.0 00900000.o 92090200 00000.0 90095.0 000.0 0009.009 9009.99 00595.0 00005000.o 0H 90 50900.0 00000.0 05.0 9099.0 0000.0 50900.0 05900000.0 0x09 09590.0 95009.0: 09.0 0900.9 0999.9: 59500.0: 00090990.0: 0299 .0000 .0000 .9000 0090> 0090> 000903 00090900000 , *000900 9090000 .0000 0000:m 0000I0 0000 0090000000 0990900> :IHI |lr .090 00000 0000000 00000000009x .009000000 0900000900 0000000 000 00900800000 00000 0000000 .0090090000 0090000000 09090902 .HHxN 09909 102 ”0000000 0003 0900000>.00gcompma0000000 000000segmo 0:0000000o0* 000. u 20999000000 0000090000 5005.00 u 0090> 0000:m 09900090. N 00009000 00 00000 00000000 9000. u 0090009000000 09090900 00 00090900000 0090. u 00900900000 09090900 00 00090900000 009 n 000900>00000 00 000002 000.0 0999.09 0000.0 09000590.0 92090200 05000.0 00900.0 000.0 0000.09 0000.0 50000.0 05000000.0 09 90 09050.0 09000.0 00.0 0000.0 5000.0 00000.0 99999090.0 0x09 90050.0 00009.0: 00.0 0090.9 0999.9: 00000.0: 90000009.0: 0299 .0000 .0000 .0000 0090> 0090> 000903 00090900000 *000900 9090000 .0000 0000:m 0000:0 0000 0090000000 0900900> .090 00000 0000000 000000000090 .090000000000 0900000900 UGflUSpm Cam WUHUM—uwSUME HMHOU HGSUMQH .0090090000 0090000000 09090902 .HHHNN 09009 105 .cmumamv mHmB wHQMHHm> ma cowumcflfiumumo mamwuasfi mo ucwfloflmmmoo* moo. u auflaflnmnoum mocmvflmaoo momo.>w n msam> ammuum mmummmdm. u mumeflumm mo uouum vumnamum mmmm. u cowumaflfiumumn mamauasfi mo usmauammmoo pmmp. u coflumamuuoo mamfluass mo ucofloflmmvoo mNH u mcowum>ummno mo Hmnfidz, no.0 ooma.m ommm.m mmmmmomm.a BZdezco >mwmo.o mmma>.o moo.o mdma.mma m¢m¢.aa mbma>.o mmnmmmfio.o OH am mmomm.o «Nmmo.o m¢.o ¢>m¢.o mmow.o oo¢wo.o mmmwomoo.o mxma Ndnmm.o momma.ou Nd.o nmow.m mamm.al Ndmmo.OI HmNNmNmH.OI mSBB .mmoo .uuoo .Qoum 05Hm> msam> unmflm3 ucmfloflwmmou *mumamn amauumm .mcoo ummuIm ammulu mumm scammwummm mHQMHHm> .xflm mcmum nmsous» cmuummnmncwx .UHumfinufium Hmuou pawnsum can moHuMEmsumE Hmuou umnommu .mUHumHumum scammmummn mamauasz .>Hxx magma 104 .Umumamv mum3 magma Hm> ma coaumcHEHmuww mamfluasfi mo ucmfloflmwmoo* moo. n mafiafinmnoum mocmoflmcoo moo¢.o« u msHm> ammunm momoommo.a u mumsflumm mo uouum oumocmum wmmm. u cowumcafiumumv mHmHuHDE mo ucmfloflmmmoo >¢nb. u GOHUMHGHHOU mamfluase mo ucmfloflmwmoo mma n macaum>ummno mo M09852 moo.o Nwmm.OH mdfim.ml mmmmmmm>.mu BZdBmzoo mmmwm.o- mmmow.o moo.o ommo.ab mmm¢.m mommm.o mm0>0d50.0 OH Em mmmmm.o moamd.o wd.o om¢m.d mmmm.a ammmo.o mfimoommo.o mxma H0¢m¢.o Nmoom.o moo.o mmam.md mmom.m o¢mmm.o onmmmmmm.o mzmms .mmou .Huoo ..Qoum wsHm> msam> usmfim3 ucmfloflmmmoo *mumama Hmauumm .mcoo ummulm ummulu mumm coammmummm mHQMHHm> .usmflm wwmnm nmsounm :muummuwvcflx .mcacommmu Uflumesuwum unmodum 6cm moflumfimnume Hoosom swan umsommu .mUflumflumum scammmnmmu mamfluasz .>xx manna 105 .omumamu mumB manmwum> NH GOHuMGHEHmumU mamfluasfi mo ucmflofimmmoo* moo. u huHHHQwQOHQ mucmoflmcoo flam¢.om u msHm> ammulh fiwmmdwdo.d n mumEHumm mo Houum oumvcmum nmwm. u coaumcHEHmumc mamfluase mo uamfloflmmmoo dado. n coaumHmuHou mamfluase mo ucmfluammmoo mNH u mGOHum>HmeO mo Hmnfidz Ho.o mmom.m m>mm.ml 00¢mmm¢m.NI BZdamzou «humm.o amomm.o. moo.o >mmw.>n maam.m mm¢om.o mmmomano.o 0H em Namnm.o mmmwd.o mo.o mmoo.¢ maoo.N moana.o moammdmo.o mxma m¢¢mm.o mmomm.o do.o m¢¢m.m Nmmm.m mmmmd.o dddfianmm.o mzmma .mmoo .uuoo .Qoum 09Hm> msHm> usmflm3 ucmfloflmmmoo *wumamn amaunmm .mcou ummuum ummunu mumm scammwummm mHQMflum> .unmflm momma nmsounu cwuummumccwx .mamucwfimvasm UHumESUHHm unmUSHm @cm moaumEmsumE Hoosum swan umnommu .moflumflumum coammmummu mamwuasz .H>NN magma 106 .vmumamo mnm3 manmwum> ma coaumcwfiumumc mamfluase mo ucmHUHmmmOU* moo. u wuflaflnmnoum mococflmaoo fimm¢.>m u 05Hm> ummulm HNNdOomm. u mumefiumm mo uouum Unmocmum mamm. u coaumcHEumumo mamfluase mo ucmfloflmmmoo mmow. u coaumeuHou mamwuase mo uan0flmmmoU oma u mGOfium>ume0 mo Hwnfisz moo.o memo.m mma0.m| «mommndm.ml Bzmo.m mmhmfi.o Honsowmo.o mZmSB .mmoo .Huou .Qoum msam> 05Hm> unmfim3 ucmfloflmmmou *mumHmQ amauumm .mcoo ummuuh ummulu mumm scammoummm wHQmHHm> .usmflw momma nmsounu couummumocwx .oaumenuflum Hmuou unmosum Cam moflumEmnumE Hoonom nod: Hmsummu .muaumfiumum coammmummu mamwuasz .HH>xx manna .Umumamv mumB mHQMHHm> “H GOHuMGHEHmumo mamfluasfi mo ucmfloflmmmoo* 107 moo. u huHHHQMQOHm mocmvfimcou odhd.m¢ u msam> umoulm mmmmmmmo.fi u mumEHumm mo Houum Unmocmum omam. u GOHuMGHEHmumo mamfiuasfi mo ucmflowmmwou Oman. n coaumawuuoo mamfluHSE mo uanUHmmwoo mmfi u mcoflum>ummno mo Hmnfisz Hm.o mmmd.o 0550.0 domddhhh.o BZ.¢ fimod.ml bound.on ammooohm.ou @208 .mmoo .Huou .Qoum 05Hm> m5am> usmdmz ucmfloflmmmou *mumama Hafiuumm .maoo ummulm ummulu mumm scammmmmmm wHQMHHm> .usmam womum smacks» cmuummumvcflx .mCHcommmu UHuGESuHHm unmosum cam woau~fimnume mmwaaoo Hmnommu .mowumflumum coflmmmummu mamwuasz .HHH>NN magma 108 .Umumamv mum3 mannaum> «H coaum:HEHmum© mamfluasfi mo unmmUflmmmoo* moo. u muflafinmnoum mocmoflmcoo mmm¢.m¢ u mSHm> ummunm hmomm0¢o.a u mumfiaumw mo uouum Unmocmum omNm. u coaumcHEHmumo mamfluase mo ucmfloflmwmou moan. u coaumamuuoo mamwuade mo unmfiodmmwoo mud n mGOHum>umeO mo umnasz Hm.o omoo.o momo.OI mmhmmmmo.on Bz¢fimzoo Nommd.o mommo.o moo.o mmmm.mo mm>~.m ddmmo.o wmmomwbo.o OH Em mmfidm.o Nm0>d.o mo.o >ma>.m Homm.d mommd.o mmmmnomo.o mxua m¢¢mm.o mammo.0| om.o dmom.d mwmo.d| o¢moo.0| mammoo>m.01 @299 .mmoo .unoo .Qoum msHm> msam> usmfim3 ucmfioflmmmoo *mumamn HMHuHmm .mcoo ummulm ummulu mumm scammmnmmm mHQMHHm> .unmflm momnm nmsounu :muummumocflx .mamucmamocsm owumanuwum unmosum cam moHumEmnumE mmmaaou Honommu .muaumflumum scammmnmwu mamfluasz. .NHxx magma 109 .Umumamv meB manmwum> ma coaumcflanwumn wamfiuazfi mo ucwfloflmm000* moo. u muflaflnmnoum wuamvwmcou mmmw.mm u msam> ummulm mdmomamm. n muwfiflumm Mo Houum Unmonmum doom. H COHumcflfiumumv wamfluadfi mo UGGHUHMHGOU fimfio. u cowumHmHHoo mamfluasfi mo unmfiowmmwoo mma n mGOHum>umeO mo Hmnfisz «0.0 Homo.o mnmm.o mmhammmm.o Badamzoo momma.o mmdmm.o moo.o momm.mOH >H@¢.OH nmomm.o mmmmmmho.o OH Em mmmdm.o Hamma.o mo.o mm>a.m Hmm>.a Nmmda.o Nmmmh¢wo.o mxma mmomm.o mm>wa.on oa.o HNm>.N mmmo.al mommo.OI mmammmmm.ou @209 .mmoo .Huoo .Qonm msam> mSHm> unmfim3 ucmHUHmmmOU *mumamn Hafiuumm .mcoo umwulm ummuup mumm coflmmmummm mHQmwum> I‘ I I 4" I‘ .unmflm mcmum smsousu qmuummumocfix .UHumfinuHum Hmuou unmosum cam mUHumEmsumE mmmaaoo Hmsummu .mUHumHumum coammmummu mamfluanz .xxx magma 110 .Umumamo mu03 manmwum> ma GOHuMCHEkumo wamfiuase mo quHUflmmmoo* moo. u muflaflnmnoum moamvflmcoo mmmm.n¢ n msHm> ummuIm mommmmmo.a u mumEHpmm mo uouum Unmocmum Noam. u SOHUMCHEHmumU mHmHquE mo quHUflmmmoo ooan. w coaumamuuoo mamfluasa mo ucmwoflummoo mma n mGOHum>uwmno mo umnfisz om.o momm.a momm.a Noaomamo.m Bz¢9mzou mm>ma.o mmmfio.o moo.o mmmm.om dmmm.m #am¢m.0 ##mmmmoo.o 0H Em Nmmom.o _ fifimda.o ma.o N¢m>.a Nomm.a >¢omo.o ommmmmmo.o oxma d0¢mdwo NommH.OI no.0 fimom.w a¢am.ml mwmma.01 nm>¢m¢m¢.0I 0289 .mmou .HHOU .Qoum msam> mSHm> usmflmB ucmfioflmmmou *mumamn Hmwuumm .mcoo ammulm ummuuu mumm scammmummm mHQMHHm> .unmfim mvmum nmsounu cmuummumccflx .mcflcommmu UHumESuwnm unmoaum cam moflumfimnume Hmuou Hmnumwu .woflumfiumum coammmummn mamfluasz .Hxxx magma 111 .omumamv mum3 manmwum> ma cowumaflfiumumn mamfluasfi mo quHUflmmmoo* moo. u mafiawnmnoum mocmcflmaoo Noa>.>¢ n msam> ammulm nmmmommo.a u mumEHumm mo Houum Unmocmum omnm. n :OHumaHEHmumo mamfluase mo pamflofiummoo owns. u GOHuMHwHHOU mHmHuHSE mo ucmwofimmmou mma u mcoaum>nmmno mo Hmnfidz mm.o ohm>.o Nomm.o momwmmmmn.a BZdezoo onmha.o Hmoom.o moo.o deo.mm mmm>.m mmmmm.o Nomm¢>>o.o 0H Hm nmamm.o owwhd.o mo.o mmmm.m mmnm.a mmama.o Nommmamo.o mxms m¢¢mm.0 mm¢ma.0u 0H.0 mnm0.m hdwb.dl ma>0a.0n m>>0mahm.0I 0288 .mwou .uuou .Qoum msHm> msam> unmam3 uanUHmmmoo *mumamo Hafiuumm .mcoo ummuah ammulu mumm coammmummm magmaum> .unmflm mvmum nmsounu cmuummumocax .mamucmEMGGSM Uflumfinufinm unmosum cam moflumsmnume Hmuou Hmnomwu .moflumflumum scammwumwu mamwuadz .HHNxN magma 112 .Umumamw mum? mHQMHum> ma coaumcHEHoumo mamfluasfi mo ucmflofimmmoo* moo. u mafiaflnmnoum mucmvflmcou amm0.¢m u wsam> ummulm mhmaadmm. u mumfiflumm mo Houum unmocmum ooom. u coflumaHEHmumo mamfiuasa mo unmfloammmoo bump. n cowumamuuoo mamfluadfi mo uamfloflmmmoo mma u chflum>Hmmfio m0 umnfisz. mN.o floam.d owma.d nooommmo.a Bz¢9mzoo o¢mma.o m¢mmo.o moo.o maam.moa mmm¢.oa mmwwm.o mmmammoo.o 0H Em Ndfimm.o mamma.o no.0 om»w.m m¢mm.a nomad-o mmommw¢o.o mxme mmomm.o ammma.0| «0.0 momm.¢ mmma.ml >m>ma.0u dwaamwm¢.0| 0299 .mmoo .unou .Qoum mSHm> msam> unmam3 quHUHmmmou *mumamn Hafiuumm .mcou ammulm ummulu muwm scammmummm manmflum> 1 .uzmflw momum nmsounu cwuummumvcwx .UflumeSUHum anuou ucmusum Ucm mUHumEMSHME Hmuou ngomwu .moflumfiumum coammmummu mamfluanz .HHHxNX magma 113 of characteristics described in Chapter III. Statistics for grade level period kindergarten through grade four are displayed in Tables VII through XV. Statistics for kindergarten through grade six are displayed in Tables XVI through XXIV. Statistics for kindergarten through grade eight are displayed in Tables XXV through XXXIII. In each of these 27 tables an aspect of student mathematics achieve— ment is compared with a category of teacher mathematics preparation, teacher experience, and student intelligence. Four statistics which are especially descriptive of the relationships are the beta weight, the partial corre- lation coefficient, the coefficient of multiple correlation, and the delete. The comparison of the beta weights in each combination provides an indication of the relative importance of each of the characteristics of the combination. The partial correlation coefficient provides an indication of the extent of the relationship between the corresponding independent variable and the dependent variable, taking into account any existing effects of the other independent vari- ables. The coefficient of multiple determination provides an indication of the percent of variation in the dependent variable which can be attributed to those independent varia- bles under consideration. The delete is the coefficient of multiple determination which results for the dependent varia- ble and the remainder of the independent variables after the corresponding independent variable is deleted from considera- tion; it indicates the remaining percent of the variation 114 in the dependent variable which can be accounted for with- out the corresponding independent variable. Thus, a com- parison of the delete for each independent variable with the coefficient of multiple determination for a particular combination provides an indication of the relative im- portance of the corresponding independent variables. The confidence probabilities for grade level periods kindergarten through the middle of grade four and kinder- garten through the middle of grade six indicate that the only correlational statistics which can be relied upon are those for student intelligence. A comparison of the coef- ficients of multiple determination and the deletes reveals that student intelligence accounted for nearly all of the variation in the dependent variables. As was indicated by the coefficients of simple corre- lation, the grade level period kindergarten through the middle of grade eight presented a statistical description which was different than those of the other two grade level periods. In most of the nine combinations the confidence probabilities were much better. In all nine combinations a low positive correlation coefficient resulted for teacher experience and student mathematics achievement. Low posi- tive correlation coefficients resulted for teacher high school mathematics preparation and student mathematics achievement. However, teacher college mathematics prepara- tion and teacher total mathematics preparation each had low 115 negative correlation coefficients with respect to student mathematics achievement. CHAPTER V SUMMARY I. THE PURPOSE OF THE STUDY A relationship may exist between the amount of mathe- matics preparation of the set of teachers who are response ible for the mathematics instruction of a student over a period of years and the subsequent mathematics achievement of that student at the end of that period. The purpose of the study was to determine the extent of such a relation- ship over periods of time encompassing the first five years, the first seven years, and the first nine years of formal elementary school education. II. RELATED STUDIES Although many studies have been made of the relation- ship between the academic preparation of teachers and their subsequent effectiveness in the classroom, none was found which dealt with the specific objectives of the present study. Each of the earlier studies differed from the present investigation in that it concerned one or more of the following aspects: 116 117 1. the general academic preparation or background of teachers rather than their preparation or back- ground in the specific academic area of mathematics 2. the general academic achievement of students rather than their achievement in the specific academic area of mathematics 5. the use of a relatively subjective criterion of teaching effectiveness rather than one of a more objective nature 4. the use of a relatively proximate criterion of teaching effectiveness rather than one of a more ultimate nature 5. the consideration of the preparation or background of only one teacher per student rather than the cumulative preparation or background of a set of several teachers per student. In regard to this last aspect, only one study was found which considered this cumulative characteristic of a set of teachers, and that study was concentrated on the relation— ship of general academic preparation of teachers to the general academic achievement of their students. Some of the studies were conducted at the secondary school level and some at the level of the elementary school. Of those secondary school studies which were focused specifically on the mathematics preparation of teachers and the mathematics achievement of students, the majority 118 indicated a positive correlation between the two. Of the few similar studies at the elementary school level, only one was found which indicated such a correlation. III. THE PROCEDURES OF THE STUDY Theersign The three periods of time considered in the study were referred to as grade level periods. These grade level periods were: 1. kindergarten through the middle of grade four 2. kindergarten through the middle of grade six 5. kindergarten through the middle of grade eight. Three categories of teacher mathematics preparation were considered: 1. high school mathematics preparation 2. college mathematics preparation 5. total mathematics preparation thigh school, college, and in-service combined). Three category value scales were developed and used to assign to each teacher a measure of mathematics preparation in each of these three categories. Then the set of teachers who had taught a particular student over a particular grade level period, was identified. For each of the three categories of teacher mathematics preparation, the arithmetic mean of each set of teachers was computed and used as a cumulative measure of teacher mathematics preparation. 119 Three aspects of student mathematics achievement were considered: 1. arithmetic reasoning achievement 2. arithmetic fundamentals achievement 5. total arithmetic achievement. These three aspects were measured in terms of the three corresponding scores which resulted from the administration of the leijornia Achievement Tests. The three grade level periods, the three categories of mean teacher mathematics preparation, and the three aspects of student mathematics achievement, resulted in 27 combinations. Each combination was concerned with a cate- gory of teacher mathematics preparation and an aspect of student mathematics achievement for a particular grade level period. To each of these 27 combinations was added acumulative measure of the teachers' teaching experience. This factor was included in order that its relationship to student mathematics achievement could be used as a bench mark against which to compare the relationship of teacher mathe- matics preparation and student mathematics achievement. Also included in each of the 27 combinations was a measure of student intelligence. This measure was the IQ resulting from the administration of the California Test of Mental Maturity, This factor was included to provide a means of adjusting for the effect of variations in indi- vidual student abilities. 120 Therefore, each of the 27 combinations was concerned with four factors: 1. teacher mathematics preparation 2. teacher experience 3. student intelligence 4. student mathematics achievement. The techniques of multiple regression and correlation were applied to each of the 27 combinations of factors. The first three factors were used as the independent variables and the last factor was used as the dependent variable. The Data The student population consisted of 206 students who completed the eighth grade of a junior high school in June, 1966. From kindergarten through grade eight these students had attended only the schools of the local school district in which the junior high school was located. Raw data were secured for each of these students by an examination of each student's permanent record folder. The teacher pOpulation consisted of the 275 teachers who taught these students from kindergarten through grade eight. The main source of raw data for these teachers was a questionnaire. A return of 88 percent was obtained for this questionnaire. These data were transformed by the use of an elec- tronic digital computer. The transformation consisted of the matching of the raw teacher data with the apprOpriate 121 raw student data and of the computing of the desired arith- metic means for the resulting sets of teachers. The trans- formed data consisted of 129 observations for kindergarten through the middle of grade four, 128 for kindergarten through the middle of grade six, and 128 for kindergarten through the middle of grade eight. IV. THE RESULTS OF THE STUDY The statistical results of the study were computed from the transformed data by means of an electronic digital computer. These results have been tabulated in Chapter IV. Teacher Mathematics greparation Twelve of the 27 comparisons of teacher methematics preparation and student mathematics achievement resulted in statistics which were at least significant at the ten percent level. Low positive partial correlation coefficients re- sulted for teacher high school mathematics preparation and the three aSpects of student mathematics achievement at grade level period eight. Low negative partial correlation co- efficients resulted for three of the comparisons of teacher college mathematics preparation and student mathematics achievement, and for six of the comparisons of teacher total mathematics preparation and student mathematics achievement. Teacher Experience Six of the 27 comparisons of teacher experience and student mathematics achievement resulted in low positive 122 partial correlation coefficients. All were at least sig- nificant at the ten percent level. Student Intelligence All 27 comparisons of student intelligence and student mathematics achievement resulted in high positive partial correlation coefficients which were at least significant at the one-half percent level. CHAPTER VI CONCLUSIONS AND IMPLICATIONS “I. STUDENTTINTELLIGENCE It was not the purpose of this study to investigate the relationship between student intelligence and achieve- ment. Student intelligence was included for the purpose of adjusting for variations in student ability. However, the results were consistent with the general belief that student intelligence is directly associated with student achievement. II. TEACHER EXPERIENCE Conclusions The factor of teacher experience was of secondary con- cern in this study. It was included as a teacher character- istic which could be used for the comparison of the factor of teacher mathematics preparation. 1. Kindergarten through the middle of gradequur, No evidence was determined which would indicate the existence of a relationship between the arithmetic achievement of fourth grade students and the amount of teaching experience of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade four. 123 124 2. Kindergarten through the middle or grade six. No evidence was determined which would indicate the existence of a relationship between the arithmetic achievement of sixth grade students and the amount of teaching experience of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade six. 3. Kindergarten through the middle gr_grade eight. Statistical evidence was determined which indicates the existence of a low positive correlation between the achieve- ment in arithmetic fundamentals of eighth grade students and the amount of teaching experience of the teachers responsible for their arithmetic instruction from kinder- garten through the middle of grade eight. The statistics suggested the existence of a similar relationship between achievement in arithmetic reasoning and the amount of teach- ing experience, but these statistics were not significant enough to warrant reliability. Thus it appears that over a nine year period, teaching experience is more closely related to student competence in arithmetic fundamentals than to student competence in arithmetic reasoning. Implications If the relationship between teacher experience and stu— dent mathematics achievement is actually one of cause and effect, then the conclusion that teaching experience is more closely related to student competence in arithmetic funda- mentals than to student competence in arithmetic reasoning, 125 suggests the following. As a teacher gains experience he becomes more proficient in the teaching of the manipulative procedures involved in arithmetic algorithms, but he does not make similar gains in his proficiency in teaching stu- dents to reason in arithmetic problem situations. This condition may result from instruction which emphasizes computational facility at the expense of concept develop- ment. As the teacher concentrates on computational facil- ity, he benefits from his experience and learns the teaching behaviors which result in greater student success in compu- tation. However, without a similar concentration on con- cepts the teacher is not confronted with the theoretical basis of mathematics and does not develop additional insights which result in increased student reasoning ability. This concentration on the teaching of computational facility and the lack of emphasis on the development of conceptual under- standing characterize the traditional school mathematics programs. Because the students of this investigation began school in 1957, they had progressed to the upper elementary grades under a traditional program before the influence of the national revolution in the teaching of mathematics introduced even a semblance of new mathematics education with its emphasis on concepts into their experience. 126 III. TEACHER HIGH SCHOOL MATHEMATICS PREPARATION Conclusions 1. Kindergarten thrgggh the middle of grade four. No evidence was determined which would indicate the existence of a relationship between the arithmetic achievement of fourth grade students and the amount of high school mathe- matics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade four. 2. Kindergarten through the middle or_grade six. No evidence was determined which would indicate the existence of a relationship between the arithmetic achievement of sixth grade students and the amount of high school mathematics preparation of the teachers responsible for their arithmetic instruction from.kindergarten through the middle of grade six. 3. Kindergarten through the middle of grade eight. Statistical evidence was determined which indicates the ex— istence of a low positive correlation between the arithmetic achievement of eighth grade students and the amount of high school mathematics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade eight. Implications Importance of_low correlations. In comparison with stu- dent intelligence, neither teacher high school mathematics 127 preparation nor teaching experience result in high corre- lations with student mathematics achievement. Statistically speaking, neither the means of the teacher mathematics preparation values nor the means of the teacher experience values account for very much of the variation in the stu- dent mathematics achievement values. This does not mean, however, that these teacher factors are unimportant and can be ignored by those who are interested in the improvement of teaching effectiveness. For example, consider the standard partial correlation coefficient which resulted for REASS Larithmetic reasoning achievement at grade eight) and THSM8 (teacher high school mathematics preparation for the set of teachers from kindergarten through the middle of grade eight). This coefficient was only .50062. Yet it can be demonstrated1 by the use of the appropriate multiple J'Let Group A be a set of students taught from kinder- garten through the middle of grade eight by teachers whose average high school mathematics preparation consisted of one year of algebra and one year of geometry. Then for these teachers THSM8 = 4. Let Group B be a set of students taught from kinder- garten through the middle of grade eight by teachers whose average high school mathematics preparation consisted of three semesters of algebra and three semesters of geometry. Then for these teachers THSM8 = 6. Let the students of both Group A and Group B be of average intelligence so that ST IQ = 110.758. Let the teachers of both Group A and Group B have an average amount of teaching experience so that TEX8 = 11.903. The appropriate multiple regression coefficients for THSM8, TEX8, ST IQ, and the constant term are .823, .056, .071, and -5.766, respectively. The resulting multiple regression equation is: 128 regression equations, that this low correlation affects sizeable differences in eighth grade arithmetic reasoning. If intrinsic factors such as student intelligence and motivation account for the larger part of the variation in the achievement level of students, then one should not expect extrinsic factors such as teacher characteristics to play such a spectacular role. However, if these rela- tionships actually involve cause and effect, then it is through these extrinsic factors, which are more controllable, that the effectiveness of instruction can perhaps be im- proved. Immediate vs. dalayed effects. It seemed odd that no correlation appeared forteacher high school mathematics preparation and student arithmetic achievement until the ,REASB==.823(THSM8) + .036(TEX8) + .071(ST IQ) + -3.766. The equations obtained for Groups A and B by substi- tuting the values above are: REASSA = .823(4) + .036(11.903) + .081(110.758) + -3.766 REASBB .823(6) + .036(11.903) + .O71(110.758) + -3.766. After simplification the resulting values are: REASBA = 7.819 REASBB = 9.465. The standard error of estimate is 1.053, so the pre- dicted grade placement scores of two-thirds of Group A and two-thirds of Group B would be between 7.819 1.1.053 and between 9.465.: 1.053, respectively. That is, two-thirds of the grade placement scores for Group A would fall between 6.8 and 8.9, while two-thirds of the scores for Group B would fall between 8.4 and 10.5. 129 Students reached eighth grade. Perhaps this eighth grade correlation resulted from the immediate effects of the seventh and eighth grade teachers, and perhaps it resulted from the delayed effects produced by teachers in the early elementary grades. If the behavior of a teacher produces an effect on a student, some of this effect may be apparent immediately and some of this effect may be apparent only after the passing of many years. Perhaps these delayed effects are controlled by the maturation process so that the effects of a particular teacher lie dormant within a student until a time when the student's growth has created a condition suitable for their manifestation. The design of this study was not suitable for investigating this problem. The screening of teacher candidates. It has been demonstrated by the use of the results of this study, that high school mathematics preparation bears an important rela- tionship to the classroom effectiveness of teachers of elementary school mathematics. However, in the opinion of the investigator, this does not provide justification for the establishment of entrance requirements by teacher edu- cation institutions, which would bar teacher candidates who have not completed a relatively large amount of high school mathematics courses. This position is supported by several reasons . 130 First, the present study was based upon the conditions which existed in the educational system of the past. Within that system it was customary to require one year of algebra and one year of geometry of those high school students who were enrolled in a college preparatory program. On the other hand, more advanced high school mathematics courses were considered to be elective. The student was allowed to study these elective courses if he were interested in fur- ther study of mathematics. In some small high schools, of course, the limited enrollment and the limited staff re- sulted in a more restrictive curriculum in which no elective mathematics courses existed. However, in general, the amount of high school mathematics studied is at least in part a measure of the teachers' interest in mathematics. Perhaps this interest on the part of some teachers and lack of interest on the part of others was subsequently transmitted to the students of the teachers and affected their arithme- tic achievement. That is, perhaps a cause and effect relationship existed in which teacher interest in high:school mathematics was the antecedent and teacher high school mathe- matics preparation and student mathematics achievement were concomitant consequents. To set high mathematics admission standards for entrance into teacher education programs would be to make the advanced high school mathematics courses required rather than elective and thus change the conditions upon which the conclusions of this study were based. 131 Second, the fact that the development of student compe- tence in mathematics is but one of many public school ob- jectives, means that the institutions responsible for the education of teachers cannot afford to exclude otherwise acceptable candidates on the grounds that they will not be highly effective in attaining this one objective. However, this does not mean that these institutions should not try to counsel accepted candidates into»specializations in which they will reach their highest potentials. With some school systems practicing departmentalization of subject matter in the elementary schools, there is a need for teachers with extra competence to teach in Specific subject matter areas such as mathematics. Those teacher candidates with relative- ly large amounts of high school mathematics preparation could be encouraged to develop academic majors in mathematics. Third, before such drastic action were undertaken as barring teacher candidates because of low amounts of high school mathematics preparation, additional evidence would be needed. The design of this study would need to be refined, and the study replicated in order to validate the conclusions. The recruitment and aasignment or_teacher§, What impli— cations for local school districts arise from this study? In the recruitment of new teachers for a district, the personnel officers should give consideration to the extent of the high school mathematics courses completed by teacher applicants. Certainly this does not mean that these officers 132 should consider only this one factor nor that it must be given greater weight than other factors. A particular candi- date may have qualities which would compensate for a lack of high school mathematics preparation. However, the results of this study strongly imply that students can be expected to reach a higher level of arithmetic achievement when taught by teachers having a relatively large amount of high school mathematics preparation, than they can be expected to attain when taught by teachers with less preparation. In the placement or assignment of teachers the admin- istrative officers of a district should attempt to place teachers with large amounts of high school mathematics prep- aration, in positions which will capitalize upon this asset. That is, if the elementary schools are departmentalized, these teachers should be assigned in such a way that at least the major part of their instructional efforts will be in the teaching of mathematics. This of course presupposes that these teachers will be content with the assignments and that they possess no other characteristics which would suggest that they could make greater contributions outside the area of mathematics instruction. IV. TEACHER COLLEGE MATHEMATICS PREPARATION Conclusions 1. Kindergarten through the middle of gradegrour. No evidence was determined which would indicate the existence 133 of a relationship between the arithmetic achievement of fourth grade students and the amount of college mathematics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade four. 2. Kindergarten through the middle orggrade six. Statistical evidence was determined which indicates the ex- istence of a low negative correlation between the total arithmetic achievement of sixth grade students and the amount of college mathematics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade six. Similar relationships were indi- cated for achievement in arithmetic reasoning and in arith- metic fundamentals, but these relationships were not as pronounced. 3. Kindergarten through the middle of grade eight. Statistical evidence was determined.which indicates the ex- istence of a low negative correlation between the achieve- ment in arithmetic reasoning of eighth grade students and the amount of college mathematics preparation of the teachers responsible for their arithmetic instruction from kinder- garten through the middle of grade eight. A similar rela- tionship was indicated for achievement in arithmetic .fundamentals, but this relationship was not as pronounced. 134 lmplications gailpre of teacher:education programs. These conclue sions should not be interpreted as evidence that college mathematics preparation of elementary school teachers should be abandoned. However, they do indicate the failure of teacher education programs to provide mathematics prep- aration capable of modifying teacher behavior to an extent which was measurable in terms of student growth criteria. In the traditional mathematics setting which characterized most of the mathematics instruction of the students of this study, the instructional emphasis was on computational facility rather than on conceptual understanding. As indi- cated in Section II, TEACHER EXPERIENCE, the confrontation of the tasks of teaching computational facility over a period of years may have developed the teachers' effectiveness in attaining that objective. Any mathematics courses which the teachers might have studied in college contributed little to the development of this skill in teaching the manipula- tions of arithmetic algorithms. Perhaps in the next few years, if the principles of new mathematics education are truly implemented in the schools, a replication of this study will yield completely different results. If emphasis is placed on teaching for conceptual understanding, then the success of students will probably measure the college mathe- matics preparation of their teachers. 135 Special mathematics courses for elementary school teachers. As indicated in Chapter I, the Panel on‘Teacher Training of the Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America has advocated college mathematics courses specifically designed for future elementary school teachers. At the present time teacher education institutions are offering such courses to prospective teachers. However, when the teachers of the study were in college, only one course was typically offered which was specifically intended for prospective elementary school teachers. That course was usually called "arithmetic for teachers" or-"general mathematics." Those teachers in the investigation, who had studied mathematics beyond that minimum course, had available to them only courses designed for students majoring in mathematics. Such courses as the traditional first course in calculus with its emphasis on algorithms were evidently of little value in developing effectiveness in the teaching of arith— metic to children. In fact, the negative correlations obtained in this study suggest that such courses may have actually been detrimental. Thus the conclusions regarding teacher college methematics preparation tend to support the recommendations of the Panel on Teacher Training that prospective elementary school teachers should be given specialized college mathematics courses in order to improve their subsequent effectiveness in the classroom. 136 V. TEACHER TOTAL MATHEMATICS PREPARATION Conclusions 1. Kindergarten through the middle ofggrade four. Statistical evidence was determined which indicates the existence of a low negative correlation between the achieve- ment in both arithmetic reasoning and arithmetic funda- mentals of fourth grade students and the amount of total mathematics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade four. 2. Kindergarten through the middle of grade six. Similar relationships were indicated for kindergarten through the middle of grade six, but these relationships were not as pronounced. 3. Kindergarten thrgggh thermiddle orpgrade eight. Statistical evidence was determined which indicates the existence of a low negative correlation between the achieve— ment in arithmetic reasoning of eighth grade students and the total mathematics preparation of the teachers responsible for their arithmetic instruction from kindergarten through the middle of grade eight. A similar relationship was indi— cated for achievement in arithmetic fundamentals, but this relationship was not as pronounced. 137 Implications Teacher high school mathematics preparation provided a measure of the amount of mathematics studied by the teachers when they were in high school. Teacher college mathematics preparation provided a measure of the amount of mathematics studied by the teachers when they were in college. Teacher total mathematics preparation was in- tended as a measure of the overall mathematics background of the teachers, including mathematics studied in high school, in college, and in in-service programs while em- ployed by the school district. As a group the correlations between teacher total mathematics preparation and student mathematics achievement were the most negative of all the correlations obtained in the study. Since teacher total mathematics preparation included the in-service mathematics education of the teachers in addition to their high school and college mathematics education, the more negative correlation suggests that the in-serVice programs may have actually been detrimental to the effectiveness of the teachers. A possible explanation of this negative correlation lies within the area of teacher attitudes. Most of the in—service mathematics education which the teachers re- ceived consisted of concentrated, after-school workshops designed to familiarize traditionally oriented teachers with the principles of new mathematics education. 138 Attendance was semi-compulsory. Some of the workshops were conducted by representatives of textbook publishers and some by supervisory personnel of the school district. It may have been that negative attitudes toward mathematics instruction were developed or reinforced by this experience. At any rate, this study suggests that in-service mathematics education of teachers can have harmful effects on the mathe- matics achievement of their students. VI. SUGGESTIONS FOR FUTURE RESEARCH There exist some additional implications of this study in the form of suggestions for future investigation of the relationships between teacher characteristics and student achievement. These suggestions are offered below. Raplication in Othgr Subject Matter Areas A fairly obvious suggestion for future research is the replication of this type of study in subject matter areas other than mathematics. It would seem that a study of the characteristics of teachers might be especially appropriate in other subject matter areas which possess cumulative or sequential qualities, such as reading or science. Replication in a NonjrraditionalrMathematics Setting It has been suggested in Sections II and IV of this chapter, that the aspects of mathematics which are emphasized 139 in a particular school program, may affect the interrelated- ness of teacher mathematics preparation, teaching experience, and student mathematics achievement. The mathematics in- struction of the students of this study was generally tradi- tional in nature and emphasized computational facility. Different results might be obtained if this study were repli- cated in a few years with a group of students whose mathe- matics instruction emphasized conceptual understanding as advocated in the new mathematics programs. Replication lnvplving Special College Mathematics Courses It has been suggested in Section IV of this chapter that most of the college mathematics courses available in the past were unsuitable for the preparation of elementary school teachers. Upon the recommendation of the Mathematical Association of America many teacher education institutions are developing sequences of mathematics courses which are specifically designed for elementary education majors. A replication of this study in a few years, which would in- volve teachers who studied within these new courses, would probably yield important information regarding the effec- tiveness of the courses. Use orrMaximum Rather Than Average Values An assumption underlying the present study was that teacher mathematics preparation was additive over the set of teachers that taught a particular student. The amounts of 140 mathematics preparation of individual teachers were added and this sum (actually the mean, for it exhibits the addi— tive quality) for a set of teachers was compared with the mathematics achievement of their student. Perhaps teacher mathematics preparation should not be treated as an addi- tive factor. Perhaps it is not the cumulative sum of prep- aration that is important, but the maximum preparation that is important. It may be that it is more important for a student to be exposed to one or two teachers with extensive mathematics preparation even if his other teachers have very little mathematics preparation, than for him to be exposed to a whole set of teachers with only an average amount of mathematics preparation. A study suggested by this is one in which the individual mathematics preparation value of each teacher would be considered, the greatest one chosen from the set of the student's teachers, and this maximum value compared with the student's mathematics achievement. Effects oraTeacherplnterest The effect of the teacher's interest in mathematics has been alluded to several times in this chapter. Although several studies have been made in which teacher interest in mathematics has been compared with student mathematics achievement, none has been made which treats interest as a cumulative characteristic of the complete set of a student's elementary school teachers. If such a study were attempted, it might be difficult to obtain an accurate measure of 141 teacher interest, but the results of such a study would probably be of importance in the recnuitment and place- ment of teachers. Immediate vs. Delayed Efrects It has been stated in Section III of this chapter that the design of this study was not suitable for investigating the relative importance of immediate and delayed effects of teacher characteristics. In considering the set of nine teachers that taught a student from kindergarten through the middle of eighth grade, how much weight should be given to the characteristics of the primary grade teachers? Should they be weighted less because they are further re— moved from the measurement of the student's achievement in the eighth grade and may thus have less of an immediate effect? Or, should they be weighted more because of a de- layed effect which they might have? Information regarding this problem should be very important in the placement of teachers within a school system. The determination of an appropriate weighting scheme might considerably alter the apparent relationships between teacher mathematics prepara- tion and student achievement in mathematics. Concluding Statement In view of the dearth of knowledge concerning the re- lationship between student mathematics achievement and teacher mathematics preparation, particularly at the 142 elementary school level, it would seem that research in this area is urgently needed. Very little research has been con- ducted on the long range effects of mathematics instruction. In the coming years such research will be complicated by the current ferment in mathematics education. However, if mathematics instruction is to have a more substantial base than supposition and conjecture, then it would seem appro- priate to observe the long range, longitudinal effects on student outcomes. BIBLIOGRAPHY 143 BIBLIOGRAPHY A. BOOKS Barr, A. S., er_al,, "The Validity of Certain Instruments Employed in the Measurement of Teaching Ability," The Measurement of Teaching Efficiency. New York: Macmillan Co., 1935. Pp. 71-141. Buckingham, B. R., "Training of Teachers of Arithmetic," Twenty-Ninth Yearbook orpthe National Society for the Study of Education. Bloomington: Public School Pub- lishing Co., 1930. Pp. 319-408. Davis, H. M., The Use of State High School Examinations as an Instrument fgerudging Work of Teachers. Teachers College Contributions to Education, No. 611. New York: Teachers College, Columbia University, 1934. Dyer, H. S., Kalin, R., and Lord, F. M., Problems in Mathe- matical Education. Princeton: Educational Testing Service, 1956. 50 pp. Frazier, B. W., er al,, National Survey of the Education of Teachers. Bulletin 1933, No. 10, Vol. V. Washington: United States Government Printing Office, 1935. 484 pp. Gibb, E. G., Mayor, J. R., and Truenfels, E., "Mathematics," Encyclppedia of Educational Research, Third Edition. New York: Macmillan Co., 1960. Pp. 796-804. Grossnickle, F. E., "The Training of Teachers in Arithmetic," Elrtieth Yearbook of the National Society for the Study of Education, Part II. Chicago: University of Chicago Press, 1951. Pp. 203-231. Heil, L. M., Characteristics prpTeacher Behavior and Compe- tency Related to the Achievement of Different Kinds of Children in Several Elementary Grades. New York: Brooklyn College, 1960. 119 pp. Howsam, R. B., Who's a Good Teacher? greblems and Progress in Teecher Evaluation. Burlingame, California: Joint Committee on Personnel Proceedings of the California School Board Association and the California Teachers Association, 1960. 48 pp. 144 145 Jacobs, C. L., The Relation of the Teacher's Education to Her Effectiveness. Teachers College Contributions to Education, No. 277. New York: Teachers College, Columbia University, 1928. 97 pp. Knight, F. B., Qualities Related to Success in Teachipg. Teachers College Contributions to Education, No. 120. New York: Teachers College, Columbia University, 1922. 67 pp. Lancelot, W. H., "A Study of Teaching Efficiency as Indi- cated by Certain Permanent Outcomes,“ The Measurement of Teaching Efficiency. New York: Macmillan Co., 1935. Pp. 1-69. Lavin, D. E., The Prediction of Academic Performance. New York: Russel Sage Foundation, 1965. 182 pp. Meriam, J. L., Normal School Education and Efficiencyiin Teaching. Teachers College Contributions to Education, No. 1. New York: Teachers College, Columbia Univer- sity, 1906. 152 pp. Mitzel, H. E., "Teacher Effectiveness," Encyclopedia of Educational Research, Third Edition. New York: Macmillan Co., 1960. Pp. 1481-1485. Mitzel, H. E. and Gross, C. F., A Critical Review of the Development of Pupil_§rowrh Criteria in Studies of Teacher Effectiveness. Research Series, No. 31. New York: Board of Higher Education, City College of New York, 1956. 28 pp. Morsh, J. E. and Wilder, E. W., ldentifying the Effective lnstructera_ A Review of the Quantitative Studies, 1900—1952. United States Air Force Personnel Training Research Center, 1954. 151 pp. National Association of State Directors of Teacher Education and Certification, and the American Association for the Advancement of Science, Guidelinesyrpr Science and Mathe- matics in the Preparation Prpgram of Elementary_School Teachers. Washington: ’NASDTEC-AAAS Studies, 1963. 15 pp. Odenweller, A. L., Predicting the Quality of Teaching. Teachers College Contributions to Education, No. 676. New York: Teachers College, Columbia University, 1936. 158 pp. 146 Ruddell, A. K., Dutton, W., and Reckzeh, J., "Background Mathematics for Elementary Teachers," Twenty—fifth Yearbook of the National Council of Teachers of Mathe— matics. Washington: The Council, 1960. Pp. 296-319. Ryans, D. G., "Prediction of Teacher Effectiveness,“ Encyclopedia of Educational Research, Third Edition. New York: Macmillan Co., 1960. Pp. 1486-1491. Somers, G. T., Pedagogical Prognosis. Teachers College Contributions to Education, No. 140. New York: Teachers College, Columbia University, 1923. 129 pp. Taylor, H. R., "The Influence of the Teacher on Relative Class Standing in Arithmetic Fundamentals and Reading Comprehension," The_Twenty-Seventhrrearbook of the National Society for the Study of Education, Part II. Bloomington: Public School Publishing Co., 1928, Pp. 97-110. Tiegs, E. W. and Clark, W. W., Manualy_California Achieve- ment TestsI Elementary Level, 1957 Edition. Los Angeles: California Test Bureau, 1957. 62 pp. Ullman, R. R., The Prognostic Value of Certain Factors Related to Teaching Success. Ashland, Ohio: A. L. Garber Co., 1931. 133 pp. Whitney, F. L., The Prediction of Teaching Success. Journal of Educational Research Monographs, No. 6. Bloomington: Public School Publishing Co., 1924. 85 pp. Wisner, R. J., "CUPM--Its Activities and Teacher Training Recommendations," Report No. 1 of the Committee on the Undergraduate Program in Mathematics, Five Confer- ences on the_Training ongathematics Teachere. Berkeley: The Mathematical Association of America, September, 1961. 90 pp. Young, K. G., "Science and Mathematics in the General Edu— cation of Teachers," The Edpcation of Teachers as Viewed by the Profession. Washington: National Com- mission on Teacher Education and Professional Standards, National Education Association, 1948. Pp. 146-150. 147 B. PERIODICAL ARTICLES Ackerman, W. I., "Teacher Competence and Pupil Change," Harvard Educational Review, 24:273-289, Fall, 1954. Anderson, H. M., ?A Study of Certain Criteria of Teaching Effectiveness," The Journal of Experimental Education, 23:41-71, September, 1954. Barnes, K., Cruickshank, R., and Foster, J., "Selected Educational and Experience Factors and Arithmetic Teaching," The Arithmetic Teacher, 7:418-420, December, 1960. Barr, A. S., "The Measurement and Prediction of Teacher Efficiency: A Summary of Investigations," The Journal of Experimental Education, 16:203-283, June, 1948. , "The Measurement of Teaching Ability," The Journal f Educational Research, 28:561-569, April, 1935. Bassham, H., "Teacher Understanding and Pupil Efficiency in Mathematics--A Study of Relationship," The Arithmetic Teacher, 9:383-387, November, 1962. Bathurst, J. E., "Do Teacherszmprove with Experience?" The Personnel Journal, 7:54-57, June, 1929. Bean, J. C., "Arithmetical Understandings of Elementary School Teachers," Theyglementary School Journal, 59: 447-450, May, 1959. Boyce, A. C., "Qualities of Merit in Secondary School Teachers," The Journal of Educational Psychology, 3:144-157, March, 1912. Breckenridge, E., "A Study of the Relation of Preparatory School Records and Intelligence Test Scores to Teach- ing Success," Educationaerdministration and Spper- vision, 17:649-660, December, 1931. Buswell, G. T., "Scholarship in Elementary-School Teaching,“ Theiglementarngehool Journal, 48:242—244, January, 1948. Domas, S. J. and Tiedeman, D. V., "Teacher Competence: An Annotated Bibliography," The Journal of Experlmental Education, 19:101-218, December, 1950. 148 Eccles, P. J., "The Relationship Between Subject Matter Competence of Teachers and the Quality of Science Instruction in the Elementary School," The Alberta Journal of Educational Research, 8:238-245, December, 1962. Fulkerson, E., "How Well Do 158 Prospective Elementary Teachers Know Arithmetic?" The Arithmetic Teacher, 7:141-146, March, 1960. Hardgrove, C. E. and Jacobson, B., "CUPM Report on the Training of Teachers of Elementary School Mathematics," The American Mathematical Monthl , 70:8704877, OCtober, 1963. Houston, W. R. and DeVault, M. V., "Mathematics In-Service Education: Teacher Growth Increases Pupil Growth," The Arithmetic Teacher, 10:243-247, May, 1963. Knight, F. B. and Franzen, R., "Pitfalls in Rating Schemes," The Journal of Educational Psychology, 13:204-215, April, 1922. Kranes, J. E., "The Child's Needs and Teacher Training,“ §chool and Societyj 88:1551156, March, 1960. LaDuke, C. V., "The Measurement of Teaching Ability, Study Number Three," The Journal of Experimental Education, 14:75-100, September, 1945. Lindstedt, S. A., "Teacher Qualification and Grade IX Mathe- matics Achievement," The Alberta Jourpal ofp§ducation, 6:76-85, June, 1960. Lins, L. J., "The Prediction of Teaching Efficiency," The Journal of Experimental Education, 15:2-60, September, 1946. Long, M., "A Synthesis of Recent Research Studies on Pre- dicting Teaching Efficiency," The Catholic Educational Review, 55:217-230, April, 1957. McCall, W. A. and Krause, G. R., "Measurement of Teacher Merit for Salary Purposes," The Journal of Educational Research, 53:73-75, October, 1959. Morsh, J. E., Burgess, G. C., and Smith, P. N., "Student Achievement as a Measure of Instructor Effectiveness," The Joprnal of Educational Psychology, 47:79-88, February, 1956. 149 Morton, R. L., "Mathematics in the Training of Arithmetic Teachers," The Mathematics Teacher, 32:106-110, March, 1939. The National Council of Teachers of Mathematics, "Guidance Report of the Commission on Post-War Plans," The Mathe- matics Teacher, 40:315-339, November, 1947. Orleans, J. 8., er al., "Some Preliminary Thoughts on the Criteria of Teacher Effectiveness," The Journal of Educational Research, 45:641-648, May, 1952. The Panel on Teacher Training, "Recommendations of the Mathematical Association of America for the Training of Mathematics Teachers," The American Mathematical Monthly, 67:982-991, December, 1960. Rabinowitz, W. and Travers, R. M. W., "Problems of Defining and Assessing Teacher Effectiveness," Educational Theory, 3:212-219, July, 1953. "Relation Between Teaching Efficiency and Amount of College Credit Earned While in Service," The Pennsylvania School Journal, 77:291, January, 1928. Remmers, H. H., "Report of the Committee on the Criteria of Teacher Effectiveness," The Review oprducational Research, 22:238-263, June, 1952. Rolfe, J. F., "The Measurement of Teaching Ability, Study Number Two," The Journal of Experimental Education, 14:52—74, September, 1945. Rostker, L. E., "The Measurement and Prediction of Teaching Ability," School and Society, 51:30-32, January 6, 1940. , "The Measurement of Teaching Ability, Study Number One," The Journal of Experimental Education, 14:6-51, September, 1945. Ruediger, W. C. and Strayer, C. D., "The Qualities of Merit in Teachers," The Journal of Educational Psychology, 1:272-278, 1910. Ryans, D. G., "The Criteria of Teaching Effectiveness," The Journalpr Educational Research, 42:690-699. May, 1949. , "A Study of the Extent of Association of Certain Professional and Personal Data with Judged Effective- ness of Teacher Behavior," The Journal oT_Experimental Education, 20:67-77, September, 1951. 150 Ryans, D. G., "Teacher Effectiveness Research: Problems and Status," The California Journal of Educational Research, 9:148-158, September, 1958. , "Teacher Personnel Research," The California Journal of Educational Research, 4:19-27, 73-83, January, 1953. Schunert, J. R., "The Association of Mathematical Achieve- ment with Certain Factors Resident in the Teacher, in. the Teaching, in the Pupil, and in the School," The Journal of Experimental Education, 19:219-238, March, 1951." Shim, Chung—Phing, "A Study of the Cumulative Effect of Four Teacher Characteristics on the Achievement of Elementary School Pupils," The Journal of Educational Research, 59:33-34, September, 1965. Standlee, L. S. and Popham, W. J., "Preparation and Per- formance of Teachersl'lndiana University School gr Education Bulletin, 34:1-48, November, 1958. , "Teacher Variables Related to Job Performance," Psychological Reports, 6:458, June, 1960. Stipanowich, J., "Mathematical Training of Prospective Elementary-School Teachers," The Arithmetic Teacher, 4:240-248, December, 1957. Swineford, E. J., "A Study of Factors That Affect Teaching Behavior," The California Journal of Educational Research, 14:214-224, November, 1963. Taylor, H. R., "Teacher Influence on Class Achievement," Genetic Psychology Monographs, 7:81-175, February, 1930. Thorndike, E. L., "A Constant Error in Psychological Ratings," The Journal of Applied Peychology. 4:25-29, March, 1920. Tomlinson, L. R., "Pioneer Studies in the Evaluation of Teaching," The Educational Researeh Bulletin, 34:63- 71, 1955. , "Recent Studies in the Evaluation of Teaching," The Educational Research Bulletin, 34:172-186, 196, 1955. 151 Von Haden, H. I., “An Evaluation of Certain Types of Personal Data Employed in the Prediction of Teaching Efficiency," The Journal of Experimental Education, 15:61-84, September, 1946. Watters, W. A., "Annotated Bibliography of Publications Related to Teacher Evaluation," The Journal of Experi- mental Education, 22:351-367, June, 1954. Webb, W. B. and Bowers, N. D., "The Utilization of Student Learning as a Criterion of Instructor Effectiveness," The Journal of Educational Research, 51:17-23, September, 1957. Yamamoto, K., "Evaluating TeaCher Effectiveness: A Review of Research," The Journal of School Psychology, C . THESES Garner, M. V., "A Study of the Educational Backgrounds and Attitudes of Teachers Toward Algebra as Related to the Attitudes and Achievements of their Anglo-American and Latin-American Pupils in First-Year Algebra Classes of Texas." Doctor's thesis, North Texas State University, 1963. 158 pp.: Dissertation Abstracts, 24:189, No. 1, 1963. Gathercole, F. J., "Predicting the Quality of Teaching: A Study of the Relation of High School Marks, Intelli- gence, Standardized Tests Scores, and Normal School Standing to Teaching Success," Master's thesis, Univer- sity of Manitoba, 1946. 129 pp. Gleason, G. T., "A Study of the Relationship Between Vari- ability in Physical Growth and Academic Achievement Among Third and Fifth Grade Children." Doctor's thesis, University of Wisconsin, 1956. 167 pp.; Dissertation Abstracts, 17:563, 564, No. 3, 1957. Glennon, V. J., “A Study of the Growth and Mastery of Certain Basic Mathematical Understandings on Seven Educational Levels," Doctor's thesis, Harvard University Graduate School of Education, 1948. 190 pp. Houston, W. R., "Selected Methods of In-Service Education and the Mathematics Achievement and Interest of Elementary School Pupils." Doctor's thesis, University of Texas, 1961. 215 pp.; Dissertation Abstracts, 23:157, No. 1, 1962. 152 Kriner, H. L., "Pre-Training Factors Predictive of Teacher Success." Doctor's thesis, Pennsylvania State College, 1931. 91 pp. Lang, A. R., "Teaching Ability as Related to Experience and Professional Training." Doctor's thesis, Stanford University. Leonhardt, E. A., "An Analysis of Selected Factors in Rela- tion to High and Low Achievement in Mathematics." Doctor's thesis, University of Nebraska, 1962. 307 pp.: Eissertation Abstracts, 23:368913690, No. 10, 1963. Moore, R. E., "The Mathematical Understanding of the Elemen- tary School Teacher as Related to Pupil Achievement in Intermediate-Grade Arithmetic." Doctor's thesis, Stanford University, 1965. 90 pp.; Eissertation i Abstracts, 26: 213, 214, No. 1, 1965. Neill, R. D., "The Effects of Selected Teacher Variables on the Mathematics Achievement of Academically Talented Junior High School Pupils." Doctor's thesis, Columbia University, 1966. 316 pp.; Eissertation Abstracts, 27:997-A, No. 4, 1966. Nelson, T. 8., "Factors Present in Effective Teaching of Secondary School Mathematics." Doctor's thesis, University of Nebraska Teachers College, 1959. 393 pp.; Dissertation Abstracts, 20:3207, 3208, No. 8, 1960. Peskin, A. 8., "Teacher Understanding and Attitude and Student Achievement and Attitude in Seventh Grade Mathe- matics." Doctor's thesis, New York University, 1964. 179 pp.; Elssertation Abstracts, 26:3983, 3984, No. 7, 1966. Schunert, J. R., "The Association of Mathematical Achieve- ment with Certain Factors Resident in the Teacher, in the Teaching, in the Pupil, and in the School." Doctor's thesis, University of Minnesota, 1951. Simmons, E., "Correlation of Administrative Ratings of Teachers and Pupil Achivement." Doctor's thesis, George Peabody College for Teachers, 1932. 97 pp. Smail, R. W., "Relationships Between Mean Gain in Arithmetic and Certain Attributes of Teachers." Doctor's thesis, State University of South Dakota, 1959. 151 pp.; Dissertation Abstracts, 20:3654, No. 9, 1960. 153 Soper, E. F., "A Study of the Relationships Between Certain Teacher-School Characteristics and Academic Progress, as Measured by Selected Standardized Tests, of Elemen- tary Pupils in Grades Four, Five, and Six of New York State Public Schools in Cities Under 10,000 Popula- tion." Doctor's thesis, Syracuse University, 1956. 135 pp.; Dissertation Abstracts, 17:570, 571, No. 3, 1957. Sparks, J. N., "A Comparison of Iowa High Schools Ranking High and Low in Mathematical Achievement." Doctor's thesis, State University of Iowa, 1960. 225 pp.; Dissertation Abstracts, 21:1481, 1482, No. 6, 1960. Stein, H. L., "Teacher Qualifications and Experience and Pupil Achievement." Master's thesis, University of Manitoba, 1935. 144 pp. Steinbrook, R. S., "A Study of Some Differences in Back- ground, Attitude, Experience and Professional Prepara- tion of Selected Elementary Teachers with Contrasting Local Success Record." Doctor's thesis, University of Indiana, 1955. 205 pp.; Dissertation Absrracts, 15:1013, No. 6, 1955. Taylor, H. R., "The Relationship of Estimated Teaching Ability to Pupil Achievement in Reading and Arithmetic." Doctor's thesis, Stanford University, 1928. APPEND ICE S 154 APPENDIX A DESCRIPTION OF STATISTICS I. STATISTICS RELATED TO DISTRIBUTIONS Xi and X5 represent independent variables. Xit represents the t-th observation of Xi. N represents the number of observations of a variable. 1. The arithmetic mean of X1 is a measure of central tendency of the values of Xi' It is denoted by Xi. X 1 it IIMZ .t Z 2. The standard deviation of Xi is a measure of dispersion of the values of Xi. 2 W 1/2 (x. 5:.) 1 1t 1 N-l IIMZ ._;E J. 3. The measure of skewness of Xi is a measure of the degree of symmetry of the graph of the distribution of the values of Xi. A value of zero indicates perfect sym- N N%'Z (X.t;§fl)s metry, a positive value indicates t=11 1 a. skewness to the right, and a Z _. 2 >3 (x.t-x.) , , , t=1 1 1 negative value indicates skew- ‘ ness to the left. 155 156 4. The measure of kurtosis of Xi is a measure of the degree 4 of flatness or peakedness of the (xit-xi) 2 p | n ' n ubaz "b42 |._\ x P. n I xl HV ID |——_. N graph of the distribution of the values of Xi. A value of 1 indi- cates perfect flatness. The greater the value above 1, the greater the peakedness of the graph. II. SIMPLE CORRELATION STATISTIC 1. The simple (Pearson product moment) correlation coeffi— N __ __ cient of Xi and X. indi— 2 (xit—xi)(x.t-x.) 3 t=1 3 3 cates the amount of N _, N ‘_ Z (X. -X.)2 Z (X. -X.)2 relationship between the t=1 it i t=1 jt 3 values of Xi and Xj. A value of 1 indicates a perfect negative correlation, a value of 0 indicates no correlation, and a value of 1 indicates a perfect positive correlation. III. MULTIPLE CORRELATION STATISTICS A. STATISTICS RELATED TO INDIVIDUAL INDEPENDENT VARIABLES Y represents the arithmetic mean of the dependent vari- able Y. Yt represents the t-th observation of Y. Yt represents YiJY. K represents the number of independent variables X1, X2,..., Xk. 1. 157 The multiple regression coefficients are the numbers N A 2 b0, b1, b2,..., bk such that 21(Yt-Yt) is a minimum t? and bo+b1X1t+b2X2t+...+bkxkt¥Yt. The equation may be A used for predicting Yt from Xit. th, ..., th. The standard partial regression coefficients (beta weights) are the normalized values of the multiple regression co- efficients. A comparison of the beta weights correspond- ing to independent variables provides a comparison of the relative importance of those variables, since the beta weights are in terms of a common scale, standard deviations. If Bi represents the beta weight for Xi, bi represents the multiple regression coefficient for Xi, STDxi represents the standard deviation for Xi, and STDY represents the standard deviation of the dependent vari- able Y, then Bi = bi STDxi/STDy. The partial correlation coefficient indicates the degree of correlation between one of the independent variables and the dependent variable after the effects of the remaining independent variables have been nullified. The delete for an independent variable Xi is the coeffi- cient of multiple determination for the dependent vari- able and the set of independent variables after Xi is deleted from consideration. 158 STATISTICS RELATED TO ALL VARIABLES COLLECTIVELY The coefficient of multiple determination R2 indicates N A 2 the percent of variation which 2 Z (Yt—Yt) R = 1 - t=1 is accounted for by the varia— N _. 2 Z (Y -Y ) tion of the independent varie t=1 t t ables. The coefficient of multiple correlation R indicates the .L R = (R2)2 degree of correlation between all of the independent variables and the dependent variables. The standard error of estimate indicates the amount of error to be expected in predictions based on the multiple regression equation. N __2 K N ”1% 2 (Y -Y) - 2 b- 2 x. -Y t=l t i=0 lt=1 it t N—K-1 - J APPENDIX B QUESTIONNAIRE AND COVERING LETTER 159 Name: :2. Below on the left, number the areas of study in order of your preference Address: . a: a EEEflSflE using 1 for greatest ‘ preference, 2 for next preference, ZIP: I etc.. below on the right, indicate - in a similar way your preference £2; Phone: Code No.: : teaching these areas: . Code No.: (as a student) (as a teacher) 1. Indicate the number of years and half-years that you studied each of language arts the following high school subjects: science general mathematics ‘ mathematics business mathematics art algebra music geometry social science trigonometry analytic geometry 3. How many years of teaching experience calculus did you have prior to 19 7 other: 4. List your academic subject majors and minors (not areas of Specialization 6. 7. such as elementary or secondary education): Majors: Minors: Classify each of your college and graduate courses in mathematics or in methods of teaching mathematics, according to its major emphasis. Usually, mathematics courses are taught by mathematics departments, and methods courses by education departments. Use the reverse side, if you need to. Brief, Number of T Year you approximate semester quarter took the course title hours hours course METHODS OF TEACHING MATHEMATICS: MATHEMATICS: 'Classify your in-service workshops in the same manner: Erief, approximate Number of Year you took workshop title clock hours the workshop METHODS OF TEACHING MATHEMATICS: MATHEMATICS: Thank you. Check here, if you would like a summary of the research. [3 161 LANSING SCHOOL DISTRICT LANSING. MICHIGAN WILLIAM R. MANNING SUPERINTCNDENT OFFICE OF DIRECTORS AND CONSULTANTS Your assistance is needed to supply some crucial information for a research project which is being conducted by the Office of Research of the Lansing School District and the College of Education of Michigan State University. This project was initiated in May, 1966, with the objective of determining the effect of the amount of mathematics preparation of teachers on the mathematics achievement of their students. The student population of the project consists of approximately 200 students who completed the eighth grade at Pattengill Junior High School in June, 1966, and who attended only the schools of the Lansing School District in grades K through 8. The teacher population consists of the approximately 275 teachers with whom these students studied in these grades. Since each student in this group had at least 9 teachers, it is meaningless to relate his mathematics achievement with any one of them, for this would imply that one teacher alone contributed to this achievement. It does seem reasonable, however, to relate a student's mathematics achievement with the average mathematics preparation of his complete set of teachers in grades K through 8. This the project will attempt to do. Student IQ, student reading ability, and average teacher mathematics preparation will be considered as factors contributing to the mathematics achievement of the student. The CDC 3600 computer of Michigan State University will be used to analyze these factors by means of multiple correlation techniques. Because we need information regarding the amount of mathematics you have completed in high school, college, and workshops, we are asking for a few moments of your time in completing the enclosed questionnaire. Each respondent's information will be treated confidentially and will be coded to preserve anonymity. Your assistance and cooperation in this research project are essential to its success. We will be happy to send you a summary of the results of the project. If you wish to receive such a summary, please check the appropriate box on the questionnaire. We would like to have the questionnaire returned by December 15. Sincerely, Edward Remw Wil liam Rouse Consultant in Research Instructor Lansing School District Michigan State University APPENDIX C HIGH SCHOOL MATHEMATICS CATEGORY VALUE SCALE This scale was used to assign values of high school mathematics preparation to the individual teachers. The appropriate portion of each returned questionnaire was carefully examined to determine the value which would represent the extent of the teacher's high school mathema- tics preparation. VALUE 0 EXTENT OF HIGH SCHOOL MATHEMATICS No information available No high school mathematics Had one or more of these: general mathematics, business mathematics, consumer mathematics Had one of these or one-half year of each: first year algebra, first year geometry Had two of these: first year algebra, first year geometry Had one of these: third semester algebra, third semester geometry, trigonometry Had two of these: third semester algebra, third semester geometry, trigonometry Had three of these: third semester algebra, third semester geometry, trigonometry Had analytic geometry Had calculus 162 APPENDIX D COLLEGE MATHEMATICS CATEGORY VALUE SCALE This scale was used to assign values of college mathematics preparation to the individual teachers. The appropriate portion of each returned questionnaire was care- fully examined to determine the value which would represent the extent of the teacher's college mathematics preparation. VALUE 0 1 EXTENT OF COLLEGE MATHEMATICS No information available No college mathematics Had one or more of these: general mathematics, business mathematics, functional mathematics, arithmetic for teachers Had one of these: college algebra, trigonometry, analytic geometry Had two of these: college algebra, trigonometry, analytic geometry Had three of these: college algebra, trigonometry, analytic geometry Had at least one of these: one term of calculus, concepts in algebra, concepts in geometry, concepts in calculus Had one of these: theory of equations, theory of‘ numbers, theory of polynomials, theory of matrices“ foundations of mathematics, foundations of analysis, college geometry, projective geometry Had two or more of these: (same courses as for 7) Had one or more of these: ordinary differential equations, advanced calculus, differential geometry, abstract algebra, topology 163 APPENDIX E TOTAL MATHEMATICS CATEGORY VALUE SCALE This scale was used to assign values of total mathe- matics preparation to the individual teachers. The appro- priate portion of each returned questionnaire was carefully examined to determine the value which would represent the extent of the teacher's total mathematics preparation. VALUE 0 EXTENT OF TOTAL MATHEMATICS No information available No mathematics Had one or more of these: general mathematics, business mathematics, consumer mathematics, func- tional mathematics, arithmetic for teachers, workshop in modern mathematics Had one of these or one-half year of each: first year algebra, first year geometry Had two of these: first year algebra, first year geometry Had one of these: third semester algebra, college algebra, third semester geometry, trigonometry, analytic geometry Had two of these: (same courses as for 5) Had three of these: (same courseswasffor 5) Had four of these: (same coursesras.for‘5) Had one or more of these: one term of calculus, concepts in algebra, concepts in geometry, concepts in calculus 164 VALUE 10 11 12 165 EXTENT OF TOTAL MATHEMATICS Had one of these: theory of equations, theory of numbers, theory of polynomials, theory of matrices, foundations of mathematics, foundations of analy- sis, college geometry, projective geometry Had two or more of these: (same courses as for 10) Had one or more of these: ordinary differential equations, advanced calculus, differential geometry, abstract algebra, topology 1. APPENDIX F A FLOW CHART OF THE TRANSFORMATION PROGRAM The program read and stored the following data: a. teacher identification numbers represented by IDTCH b. teacher characteristic values represented by TCHCHAR c. student characteristic values represented.by INFOSTUD These data were stored in the form of matrices. The program chose each student in order from 1 to 206. For each student the program chose each teacher charac- teristic from 1 to 4. For each teacher characteristic the program determined the arithmetic mean of the teacher characteristic values of the teachers for each semester throughout a specified sequence of semesters (e.g., K-4B, K-6B, K-8B). The program incorporated a means of determining the proper divisor for computing the arithmetic mean. The program computed the arithmetic mean as zero, if data did not exist for even one semester of the sequence, thus providing an indication of missing data. The program prepared data cards as output. These cards incorporated student data and the corresponding averaged teacher data. Separate sets of cards were prepared for the three grade level periods. 166 167 Temp=0 X = 0 IGR4(I.L) IGR6(I.L) IGR8(I,L) II II II 000 IDTCH(I,J IDTCH(I,J)> 0 TCHCHAR 0 + TCHCHAR(K.L) + J-15=O TCHCHAR(K,L J-1 J-Z "IIIIIIIIIIIIIIITES