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WM¥1$M¢£*\R x... \ehkafldn zagggg ‘ ' ‘3‘; n ur- » Jaw,” aim“! u LIBRARY ° mos 3 1293 10575 3374 This is to certifg that the thesis entitled MATHEMATICS FOR PROSPECTIVE ELEMENTARY TEACHERS IN A COMMUNITY COLLEGE: A COMPARISON OF AUDIO- TUTORIAL AND CONVENTIONAL TEACHING MATERIALS AND MODES presented by Harriett Elenor Emery has been accepted towards fulfillment of the requirements for All; degree in Mi On ”WE—g4 Major professor July 23, 1970 I)ate 0-169 ABSTRACT MATHEMATICS FOR PROSPECTIVE ELEMENTARY TEACHERS IN A COMMUNITY COLLEGE: A COMPARISON OF AUDIO—TUTORIAL AND CONVENTIONAL TEACHING MATERIALS AND MODES by Harriett Elenor Emery The purposes of this study were: (1) to compare the effectiveness of audio—tutorial material and material from a commercial textbook in teaching a semester course in "modern mathematics" to junior college students who are prospective elementary school teachers, and (2) to compare the attitudes of these students toward selected aspects of mathematics. For the comparison of the effectiveness of the audio— tutorial materials as opposed to the commercial textbook alone, multi—media materials which were appropriate to the junior college level and to the unit and course objectives were designed and prepared for use by the laboratory group. In addition supplementary aids, both teacher—prepared and commercially prepared materials, were incorporated into the laboratory learning situation. A rating scale was developed as the instrument to assess the student attitudes as measured by their reactions to comments illustrating attitudes toward aspects of Harriett Elenor Emery mathematics which are appropriate to the prospective elementary school teacher. The sample for the study was the students in Mathematics 103 at Schoolcraft College, Livonia, Michigan, in the winter semester of 1970. These seventy-seven students had pre- enrolled when the study was undertaken, and, therefore, one group was randomly selected as the experimental group. At the end of each unit of instruction a test on the unit was admin- istered to the classes. A two-hour final examination con- cluded the course. The attitudinal survey was administered on the last day of class (part I) and on the examination day (part II). The analyses for the study were computed on the 3600 computer at Michigan State University Computer Center, using the Missing Data Statistics Program (MDSTAT) and the Finn Multivariate Analysis (FINN). The analysis of covariance, using the Converted Rank in Class scores as the co-variable, was used to correct for the lack of random sampling in the original groups. The Scheffé Post Hoc comparisons were used to locate the significant differences. On the units where no confounding variables were intro- duced into the study by the history of events beyond the control of the experimenter, the audio-tutorial method was successful above the conventional groups-—either control group or the average of the groups. When the data were con- trolled for age, the older group was more successful on unit III, Operations and Algorithms. Harriett Elenor Emery The control groups had a significant correlation between attitude and achievement, but not the audio—tutorial group. On the question directly related to the audio— tutorial laboratory all groups were in favor of the laboratory. The audio—tutorial group of students (27 out of 27) responded so that the average of their ratings was 4.609 out of a 5.000 possible score; the control groups rated the laboratory H.063. In the correlation between achievement and attitude toward the laboratory, the control groups had negative cor- relations indicating the students who are not achieving their goals in the course wanted a source of additional assistance. Further study and experimentation with this mode of instruction in mathematics courses at the community college level was clearly indicated in the study. A resource labor— atory for basic mathematics classes would be a further exten— sion of the basic concept of the audio-tutorial mode of instruction. MATHEMATICS FOR PROSPECTIVE ELEMENTARY TEACHERS IN A COMMUNITY COLLEGE: A COMPARISON OF AUDIO—TUTORIAL AND CONVENTIONAL TEACHING MATERIALS AND MODES By Harriett Elenor Emery A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1970 Copyright by HARRIETT ELENOR EMERY 1971 DEDICATION Mom, Dad, Jane Who Made It Possible Renee, Michelle, and Jean—Jean Who Cooperated, Sometimes ii ACKNOWLEDGMENTS The Writer wishes to acknowledge: The inspiration, guidance, and help of my family-—my mother, my sisters, and my brother. The enlightened counsel and direction of her doctoral committee--Dr. T. Wayne Taylor, Chairman Dr. J. Sutherland Frame Dr. W. Robert Houston Dr. George Myers. The encouragement, advice, and assistance of-- the administrators the Board of Trustees the faculty and the staff at Schoolcraft College and especially Dr. Eric J. Bradner, President Dr. Robert Keene, Vice President of Instruction Mr. Robert A. Stenger, Dean of Instruction Miss Barbara A. Geil, Assistant Dean of Student Affairs and Director of Admissions Mr. Norman E. Dunn, Registrar Mr. Richard V. Chatham, Data Processing Manager Mr. Patrick Butler, Librarian Miss Gale Buchanan, Assistant Librarian/Audio-Visual Department. ‘ The cooperation and aid given to this project by Harriet 0. Morgan and the other members of the Biology Department. The interest and cooperation of the Mathematics Department. The thoughtfulness and the care displayed for the effective— ness and cost of the photography by Norma Rae Torr and her staff. And, especially, the forbearance and understanding of colleagues, friends, and relatives, who withdrew early and quietly from the site of activity. iv TABLE OF CONTENTS DEDICATION ACKNOWLEDGMENTS. LIST OF TABLES LIST OF FIGURES. Chapter I. INTRODUCTION The Problem Definition of Terms Design of the Study Hypotheses . Assumptions and Limitations .of the Study Related Studies Succeeding Chapters II. BACKGROUND OF THE PROBLEM AND REVIEW OF THE LITERATURE . . . . Introduction. The Problem Review of Literature Summary III. THE STUDY Introduction Mathematics for Prospective Elementary Teachers . . , . . . The Sample . The Audio— Tutorial Laboratory The Measures, The Design The Hypotheses The Analysis. . The Audio— Tutorial Mode Summary Page ii iii vii IO 12 13 l6 l8 Chapter Page IV. EVALUATION OF THE AUDIO-TUTORIAL MODE . . . . 90 Introduction. . . . . . . . . . . . 90 The Statistical Techniques . . . . . . . 90 The Analyses. . . . . . . . . . . . 93 The Findings. . . . . . . . . . . . l01 The Laboratory . . . . . . . . . . . l2l V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS. . . 127 Summary . . . . . . . . . . . . . 127 Discussion . . . . . . . . . . . . 135 Conclusions . . . . . . . . . . . . 1A0 Recommendations. . . . . . . . . . . 142 BIBLIOGRAPHY. . . . . . . . . . . . . . . 1““ APPENDICES - . . . . . . . . . . . . . . ISA Appendix A. SMCCMP . . . . . . . . . . . . . . 154 B. ELEMENTARY ARITHMETIC TEXTBOOKS USED IN 1967 IN SCHOOLCRAFT COLLEGE DISTRICT . . . . . . 173 C. ATTITUDINAL SURVEY . . . . . . . . . . 175 D. AUDIO—TUTORIAL [BIOLOGY 101] AT SCHOOLCRAFT COLLEGE . . . . . . . . . . . . . 180 E. MATH CARD . . . . . . . . . . . . . 187 F. MATERIALS FOR LABORATORY . . . . . . . . 189 G. SEQUENCING MATERIALS. . . . . . . . . . 701 H. STUDENT RESPONSES FOR OPEN—ENDED COMMENT . . . 206 vi Table 1“.) A. 10. 11. 12. LIST OF TABLES Mathematics Faculty at Delta College in Fall, 1961. . . . . . . Mathematics Faculty at Schoolcraft College in 196u—1965 . . . . . Mathematics Faculty at Schoolcraft College in 1969—1970 . . . . . . . . . . . Students Enrolled in Mathematics 103 and 112 During 1968—69, By Age (Given in Percent) Number of Students Enrolled in Mathematics 103 by Sections in Winter of 1970 at Schoolcraft College. . . . . . . . . . . Characteristics of Students Enrolled in Mathe- matics 103 Schoolcraft College, Winter Semester, 1970 (in Percents, to Nearest Tenth) Transfer Curricula Selected by Students Enrolled in Mathematics 103, Winter, 1970, (Given in Percent, to Nearest Tenth) Senior College Selected for Transfer by Students Enrolled in Mathematics 103, Winter, 1970, (Given in Percent, to Nearest Tenth) Experimental Design Comparison of Mathematics 103 Groups on Converted Rank in Class Scores (The Co—Variable) Group Correlations of Test Grades and Converted Rank in Class Scores Matrix of Correlations of Tests and Converted Rank in Class Scores (Groups 01,02, and ET). vii Page 56 59 59 69 71 73 75 83 98 99 TABLE Page 13. Matrix of Correlations with Covariable, Con— verted Rank in Class, Eliminated (C ,C , and T l 2 E ) . . . . . . . . 99 IA. Matrix of Correlations of Tests and Converted Rank in Class Scores (Groups C ,C , and E) 100 1 2 15. Matrix of Correlations of Tests with Co—variable, Converted Rank in Class, Eliminated (Groups 10 C1,C2, and E). l 16. Mean Scores for Groups in Mathematics 103 on Tests Examination, and Average, Winter Semester, 1970 . . . . . . . 103 17. Standard Deviations for Groups in Mathematics 103 on Tests and Final Examination, Winter Semester, 1970 103 18. One—Way Analysis of Variance for Test I (Groups Cl’C2’ and ET) 105 19. One—Way Analysis of Covariance for Test I (Groups Cl’C2’ and ET). 105 20. One—Way Analysis of Variance for Test I (Groups 01,02, and E). 106 21. One—Way Analysis of Covariance for Test I (Groups 01,02, and E) 106 22. One—Way Analysis of Variance (Groups C ,C , and -1 1 2 E ) . . . . 108 23. One—Way Analysis of Covariance (Groups ClC?, and 0+) . . T 108 2A. One—Way Analysis of Variance (Groups C ,C , and a 1 2 E) . . . 110 25. One—Way Analysis of Covariance (Groups Cl’C2’ and E) . . . 110 26. Summary of Decisions to Reject, 11A 27. Differences of Mean Scores on Tests,Groups Controlled for Factors. 115 28. Summary of Decisions to Reject. 116 viii Table Page 29. Attitude Toward Mathematics as Measured by the Means of Responses by Students on Attitude Scale . . . . . . . . . . . . . . 119 30. Summary of Decisions to Reject. . . . . . . 120 31. Mean Values of Student Responses to Audio— Tutorial Laboratory in Mathematics 103 . . . 122 32. Correlation of Achievement and Attitudes Toward Mathematics by the Groups. . . . . . . . 12A 33. Correlations of Achievement and Attitude Toward Audio—Tutorial Laboratory. . . . . . . . 125 3A. Summary of Decisions . . . . . . . . . . 126 ix LIST OF FIGURES Figure Page 1. Converted Rank in Class Scores for Groups 01,02, ET. . . . . . . . . . . . . 95 CHAPTER I INTRODUCTION The Problem Statement of the Problem The purposes of this study were: (1) to compare the effectiveness of multi—media material (audio—tutorial) and material from a commercial textbook in teaching a semester course in "modern mathematics" to junior college students who are prospective elementary school teachers, and (2) to com— pare the attitudes of these students toward selected aspects of mathematics. For the comparison of the effectiveness of the multi- media materials as opposed to the commercial textbook, multi— media materials (slides, audiotapes, posters, and problem work sheets) which were appropriate to the junior college level and to the unit and course objectives were designed and prepared for use by the laboratory group. In addition, supplementary aids, both teacher-prepared and commercially prepared materials, were incorporated into the laboratory learning situation. For comparison of the student attitudes toward selected aspects of mathematics, a rating scale was developed. This instrument was used to assess the student attitudes. The measurement scale was the student reaction to comments illus— trating attitudes toward aspects of mathematics. The aspects considered were those appropriate to the prospective elemen— tary school teacher, who will be teaching arithmetic. Importance of the Study In the last fifteen years the trend in mathematics education has been marked by a definite and very decided increase in the subject matter that is offered at each level in the educational structure. New developments in mathematics at the graduate level have created an increase in the math- ematics curricula available to the graduate and to the under— graduate. This increase in mathematics courses has resulted in the addition of material to secondary school mathematics; this addition has, in turn, led to the expansion of the ele— mentary school mathematics curricula. One result of this consistent trend of moving the sub- ject matter downward in mathematics education has been that (l) the mathematical training of a prospective elementary school teacher, and (2) the attitude of that teacher toward mathematics, have become more important to his successful teaching of mathematics than they ever have been. In 1961 (and again in 1966) the Committee on the Undergraduate Program in Mathematics (CUPM) stated that If present trends continue, the better elementary schools may soon be teaching the rudiments of algebra and also some informal geometry. Even in the teaching of arithmetic, sound mathematical training is needed because the teacher's understanding affects his views and attitudes; and in the classroom, the views and attitudes of the teacher are crucial. To an under— trained teacher, arithmetic is merely a collection of mechanical processes and is regarded with boredom, or dislike, or even fear. It is not surprising that, in such cases, students react to arithmetic in the same way. Children should be taught arithmetic for meaning and understanding as well as for skills. To teach in this way, a teacher needs to have a kind of training which conveys this understanding and also shows mathematics to be rewarding and worthwhile. The teacher cannot give something which he does not have.1 In Michigan the community college concept has been accepted and utilized more successfully each year. In 196A there were seventeen community colleges in Michigan; in 1969-1970 there were twenty—eight. Schoolcraft College is a community college located in Livonia, Michigan, and supported by the five communities of Clarenceville, Livonia, Plymouth, Northville, and Garden City. At the time that Schoolcraft College opened its doors (196A), the number of students enrolled in the semester course "mathematics for elementary teachers" and referred to as Math— ematics 103 was thirty students. The enrollments in this course increased to ninety students enrolled in three classes of Mathematics 103, each semester in the period from 1968— 1970. 1Mathematical Association of America, Recommendations for the Training of Teachers (January, 1961; Committee on the Undergraduate Program in Mathematics, Mathematical Association of America [Berkeley, Calif.: Distributed by the CUPM Central Office, 1961]), p. 5. All the community colleges in Michigan offer one semester of mathematics for prospective elementary school teachers, with a similar topical outline; some of these colleges have added to their curricula a second course in mathematics for such students, but this course covers algebra or geometry and not a stronger treatment of arithmetic which is the recommen— dation approved by the senior colleges (not yet implemented). Two course outlines and a bibliography for a two—semester sequence of courses in "mathematics for elementary teachers," a core curriculum, which had been developed by the subpanel selected by the Southeastern Michigan Community College Mathe— matics Project (SMCCMP) were accepted and approved at the annual meeting of the Southeastern Michigan Association of Community College Mathematics Instructors on May 11, 1970, at Macomb County Community College in Warren, Michigan. [See Appendix A, for the outlines and bibliography]. The implementation of these outlines in the participat— ing two—year colleges should lead toward a strengthening and unification of the mathematics topics offered to prospective elementary school teachers at these colleges. A survey of the commercial textbook series that have been adopted by the communities of the College District was Inade in 1967, including the "modern mathematics topics" and ‘the level of the concepts. Even then, mathematics super- ‘visors in the schools stated that they were anticipating the time when they could legally make a new selection-—that would be "more modern” than the series now in use. [See Appendix B for the texts.] The need for strong and effective courses in mathematics for elementary school teachers was stressed by the Committee on the Undergraduate Program in Mathematics in its 1969 report: Most prospective elementary teachers are not highly motivated toward scientific and mathematical studies and are apt to be less well prepared than many other students. But, if we are to offer to children in the elementary grades a good mathematics program, indeed, if we wish to have the current commercial elementary textbook series well taught, we must manage to persuade these students that mathematics is an important disci- pline which they can understand. It is imperative that they begin studying the recommended two years of mathe- matics early, before they have lost too much contact with their earlier training, and in time to strengthen their mathematical backgrounds for other courses in their programs.1 The addition of a second course in the mathematics pro- gram for the prospective elementary school teacher will be an improvement and a necessary factor in the strengthening of the mathematical training of these students. Before a student can enroll in the second course, he must have passed the first mathematics course. Among these students are many who, though they have succeeded in their other courses and though they showed a potential ability to become satisfactory ele— mentary teachers, could not manage to pass this mathematics course. Some pass it but have a poor attitude, which is due 1Mathematical Association of America, A Transfer Curric- ulum in Mathematics for Two—Year Colleges, A Report of the Committee on the Undergraduate Program in Mathematics (Berkeley, Calif.: Committee on the Undergraduate Program in Mathematics, Mathematical Association of America, 1969), p. 36. to their experiences and struggles to drag themselves through the course. Re—enrollment in the course has not been an acceptable solution and certainly is not a satisfactory solu- tion for the student or the instructor. In the educational field where good elementary school teachers are still needed, this loss of potentially successful candidates for elementary school teaching should not be ignored by the community college mathematics instructors and other educators. At Delta College, a community college near Bay City, Michigan, in the period of 1961—1964 and at Schoolcraft College in the period of 1964—1970 this problem has been apparent to the writer and to the mathematics instructors assigned to teach the course, Mathematics 103. Instructors at the other community colleges and at the four—year colleges of South- eastern Michigan Association of Community College Mathematics Instructors reported that a similar situation existed and even larger enrollments were involved.1 These instructors expressed concern for their students because many of the topics in the course have been extremely and consistently difficult for the students enrolled in the class. Extending time to be spent on a topic decreases the time available for some other topic. Re-teaching the topic has not increased the achievement suf— ficiently to justify boring other students. 1Latest reports were made on April 9, 1970, to the Southeastern Michigan Community College Mathematics Project Panel, and on May 11, 19703at the annual meeting of all mem- bers of Southeastern Michigan Association of Community College Mathematics Instructors. In a course which had no prerequisites, these students were required to learn and demonstrate the principles and concepts of arithmetic and, in addition, set theory and other topics of "modern mathematics.” The junior or community college students who are usually enrolled in the classes of mathematics for elementary school teachers are faced with a very difficult obstacle to their chosen vocation; an apprecia— tion of the barrier presented to these students can be fully realized only by the person who is aware of the typical com- munity college student who does enroll in this course. Definition of Terms Achievement This term referred to the scores or grades on the examinations which required responses of the students to items testing the performance of the behavioral objectives for the concepts involved. _ Test This term referred to a teacher-prepared one—hour examination over material and objectives covered in the com— pleted unit of the course. Final Examination This term denoted a teacher—prepared two-hour examina— tion over the objectives for the course. English Test This term referred to the English Expression Coopera- tive Test, Form 1 C, from Educational Testing Service. The scores were based on a possible score of 90. Arithmetic Test This term referred to a local test on arithmetic com— putation, developed by the mathematics department to screen for admission to the basic or beginning mathematics class for algebra, business arithmetic, mathematics for elementary teachers, or technical mathematics. The scores were based on a possible score of A5. Converted Rank in Class This term referred to the score which was obtained by equating the rank in class at graduation to a standard score on a distribution with a mean of 50 and a standard deviation of 10. This chart is available to qualified personnel from the Educational Testing Service. Semester This term referred to sixteen weeks of college work. glass Time This term was used to denote the time spent in class work: (1) for the control group the class time for each student was the three one—hour periods each week that the Class met, for a total of forty—eight hours; for the experi— HKNltal group, class time was defined for the individual student as two one—hour class periods plus the time spent that week in the laboratory, a total of 32 hours plus the total time spent in the laboratory. Modern Mathematics This term referred to the material found in set theory, operations in sets, numeration systems, and operations in different base systems. Finite systems also were included. Attitudinal Scale This term referred to the departmental survey submitted to the classes at the end of the course for their reactions to statements about attitudes toward aspects of mathematics. Admissions Battery This term referred to a selected set of tests which were administered to incoming freshmen at Schoolcraft College. The earlier set had been discarded and a new set was being evaluated. At that time two tests, English and Mathematics, were used. Since the College has an "open door policy," the tests were used for placement purposes. Objectives This term was used to mean objectives, stated in behavioral terms, which can be measured for successful com— pletion at the level of the community college students enrolled in the course. 10 SMCCMP This term refers to the Southeastern Michigan Community College Mathematics Project-—A National Science Foundation study grant, Dr. Arthur F. Coxford, Research Director. SMACCMI This term refers to the Southeastern Michigan Associa- tion of Community College Mathematics Instructors--organized at Schoolcraft College in 1965, under the leadership of Delbert Piller, Chairman of the Department of Mathematics. Design of the Study At Schoolcraft College, a community college in Livonia, Michigan, students who are enrolled in the course, Mathe- matics 103, "mathematics for elementary teachers,"are pro— spective elementary school teachers on an elementary educa- tion program. This course is required for the Associate degree, which is granted by the College, and it (the course) is also required for graduation from the senior institution to which the student may elect to transfer. From the population of all students who have enrolled in this class, since the College opened in the fall semester of 1964, those who are now enrolled in the course, or those who will be enrolled in the course in the future, the sample chosen to be used in this study was the students enrolled in the course, Mathematics 103, in the winter semester of 1970. 11 In the fall semester, 1969, the administrative personnel of Schoolcraft College were presented with the design proposed for the study; approval was granted for the study using college facilities and equipment. Arrangements were made for a multi—media laboratory to be set up. At Schoolcraft College the teaching assignments for the following semester are made very early in the preceding semester. As a result of this policy the writer was the instructor for all three of the classes in Mathematics 103 in the winter of 1970, since the printed schedule had been distributed before final arrangements for the study were completed. Alternate teacher assignments were planned, but they had to be rejected when it became apparent that the elimination of this one difficulty would introduce others which were undesirable. To assign a probationary teacher to teach an audio-tutorial class, without knowing his attitude toward this teaching method, would not be the only problem in the situation. Would the teacher spend the designated hours so that he was available in the laboratory? Would he make the audio—tape recordings so that his students could identify with him as they listened and worked? Would he be able to understand these students and their special needs? What would his attitude toward the class be if he should resent the change of assignment? The judgment was made to let the enrollments proceed just as any other regular semester, with no undue attention given to these classes. 12 By random sample the ten o'clock class was designated to be the experimental group and the eight and nine o'clock classes became the control group. (Selection was made before registration began.) The control classes met three days a week for the usual class hour, which is informal lecture, discussion, question and answer, or test session. The experimental class met two days a week for the usual informal lecture, discussion, or test session. The third class hour was replaced by an audio-tutorial laboratory. The Hypotheses There will be a definite increase in achievement as a result of the audio-tutorial method and materials. There will also be an improvement of the student attitude toward mathematics as a result of the audio-tutorial mode of instruc— tion. In order to test the significance of these hypotheses, the following null hypotheses were used for statistical analysis: 1. There are no significant differences between the effectiveness of the multi—media materials and the selected materials from a commercial textbook. 2. There are no significant differences in achieve- ment under the two modes of instruction, when the older students are compared with the younger in each group. 3. There are no significant differences in achieve— ment under the two modes of instruction, when the groups 13 are compared on the basis of entering college immediately upon graduation or after a lapse of time. A. There are no significant differences in achieve— ment under the two modes of instruction, when the groups are compared on the basis of attendance at an urban or a subur- ban high school. 5. There are no significant differences in achieve- ment under the two modes of instruction, when the groups are compared on the basis of attendance at a private or a public high school. 6. There are no significant differences in attitudes, the total attitude, or the eight categories, under the two modes of instruction. 7. There are no significant differences in the attitudes, the total or the eight categories, under the twc modes of instruction when the groups are compared on the basis of (a) age, (b) lapse/direct from high school, (c) private/ public high school, and (d) urban/suburban high school. Assumptions and Limitations of the Study Assumptions 1. The sample will not differ significantly from the population. Special mention should be made of the motivation problems involved in teaching this course. The students who are enrolled in this course have already acquired the stan— dards, aims, interests, goals, and reinforcement and reward systems of American high school graduates of average ability 1A (especially those of the high schools in the immediate vicin— ity). In their previous studies of mathematics, which vary considerably, these students have developed attitudes toward mathematics which range from tentative hope of success to a feeling of uncertainty, dislike, distrust, and sometimes even dread. The more mature students differ from the younger in that (1) they take more time to learn the same amount of material, (2) they persevere longer and strive harder, (3) they set higher standards for themselves, both on the individ— ual assignments and during the entire course, (A) they want to learn as much as they can so that they will be better pre— pared to teach arithmetic, (5) they develop much more tension, anxiety, or frustration than the younger students do, and (6) they usually have more and sometimes pressing family respon— sibilities or emergencies. 2. At the time that the experiment was conducted the classes covered the following material: (a) importance of mathematics in modern living; (b) patterns; (c) sets——nota— tion, universal set, null set, element of a set, union of sets, intersection of sets, subsets, proper subsets, car— dinality of a set, finite sets, infinite sets, whole numbers, natural numbers, even numbers, odd numbers, multiples, primes, bases,constants, variables, Cartesian products, ordered pairs, modulus; (d) whole numbers with bases that = 10 and bases that # 10; (e) changing bases; (f) operations with whole numbers in various base systems; (g) algorithms for four 15 fundamental operations in whole numbers; and (h) problem— solving in arithmetic. 3. The sample, which did not differ significantly from the population, will not differ significantly from similar classes in other community colleges. A. It was assumed that the examination materials were kept secure and that students had no opportunity to practice the questions on the tests. (Review sheets were given to each student a week before the test period.) It was further assumed that the tests were given at about the same time, that the directions for testing were followed in all groups, and that adequate test conditions were maintained. 5. It was assumed that the students were frank and sincere in their responses on the survey, thereby trusting the writer to follow the procedure as described. [See Appendix C for Attitude Scale.] Delimitations The study was designed to evaluate the effectiveness of the audio—tutorial method of teaching when used in a mathe— matics class, Mathematics 103. It was not designed to develop materials for teaching the course, an incidental result of the study. It was not designed to demonstrate a need for a second course to follow this course. The decisions on the subject matter to be included in the course and the course which should follow had already been made when this study was carried out. 16 The topics and concepts included in the course were the same for both groups. All students enrolled in Mathematics 103 at Schoolcraft College in the winter semester, 1970, were included in the study. Students who dropped the course were later excluded from the study. Limitations The study was limited by the size of the sample, the number of students enrolled in Mathematics 103 (N=77). A further limitation was the number of students who graduated from Detroit schools where rank in class was not computed. Rank in class was used as the co-variable in the analysis of the data. Related Studies A review of studies, which included approximately 70 reports submitted at the Audio—Tutorial System Conference at Purdue University on October 20-21, 1969, revealed that the method was used most frequently at the college level, two— year and four—year colleges (60 reports). The reports were in these areas: biology (20), botany (ll), chemistry (1), physics (2), geography (2), earth or soil science (2), nursing (3), medicine (1), foods (2), social studies (5), electronics (2), and others; none were in mathematics or mathematics education.l 1See Appendix D, for Audio—Tutorial Method in Biology at Schoolcraft College. 17 The reports mentioned above were descriptions of the results of adoptions and adaptations of the audio—tutorial method to a class, a department, or to a division of the school involved. None demonstrated a controlled study in the use of audio—tutorial method; even Husband and Postlethwait have stated that a small experiment in 1962 showed no improve— ment in achievement but did show a definite saving of stu— dent time in learning the concepts of the course (freshman botany).l This experiment used audio—tapes for the lectures that were to be delivered the following week. The tapes were on reserve in the library for student use. From this begin— ning study at Purdue University, with a sample of 36 students in one class, the audio—tutorial method was declared a success and was improved in the years that followed, without any further testing. Fitzgerald has stated that no controlled experiments have been done to evaluate the effect or achievement obtained by including the two—hour laboratory in the mathematics for elementary teachers course at Michigan State University.2 ‘ 1D. D. Husband and S. N. Postlethwait, "The Concept of lhidio—Tutorial Teaching" (unpublished Ph.D. dissertation, Dept. Cff Biological Science, Purdue University), p. 25. 2Interview with William Fitzgerald, Professor, Michigan State University, June, 1969, and address by William Fitzgerald at Michigan State University on November 8, 1969, for South- Eflistern Michigan Association of Community College Mathematics InStructors. 18 Boonstra found that there was no significant transfer of these laboratory materials into the teaching of classes by these prospective elementary teachers. A study of the volume, Research in Mathematics Education, from the National Council of Teachers of Mathematics revealed no studies of mathematics that would involve (l) prospective elementary school teachers, (2) mathematics, and (3) the audio— tutorial method. Studies on prospective elementary teachers were: (1) developing materials, (2) contrasting textbooks with lectures, (3) closed circuit television, (A) types of lectures, (5) types of exercises with concepts, (6) developing additional materials and courses.2 Succeeding7Chapters In Chapter II the background of the problem has provided a description of the evolving situation and a review of the literature has given an evaluation of completed studies and those proposed in the area of this study, as a basis for the completion of this experiment. In Chapter III is the problem described in detail. Included are the procedure and the method lPaul Henry Boonstra, ”A Pilot Project for the Investiga— ticon of the Effects of a Mathematics Laboratory Experience: A Case Study” (unpublished Ph.D. dissertation, Michigan State University, 1970). 2Boyd Holtan, ”Some Ongoing Research and Suggested Research Problems in Mathematics Education,” Research in Mathe— IEflggcs Education, ed. Joseph M. Scandura (Washington, D.C.: gfiftional Council of Teachers of Mathematics, 1967), pp. 109- 3. 19 of collection of the data. Chapter IV is a description of the analysis and interpretation of the data. In Chapter V are conclusions, results, and recommendations for further research. CHAPTER II BACKGROUND OF THE PROBLEM AND REVIEW OF THE LITERATURE Introduction The background of the historical development of the problem was presented as a perspective for the current study; the background of research and theory was the foundation for the study. For a study of the effectiveness of the audio-tutorial mode of teaching modern mathematics to community college students who are prospective elementary teachers a survey of the current literature reporting research studies was conducted. In order to provide a clear perspective for the audio—tutorial study the literature was reviewed and classi— fied in these categories: (1) prospective or in—service teachers of elementary school mathematics, (2) junior college students, (3) attitudes toward mathematics, and (A) audio— tutorial mode of teaching. The Problem Historical Background In the 1950's mathematics instructors meeting at the annual conventions of the National Council of Teachers of 20 21 Mathematics and the Michigan Council of Teachers of Mathematics were wont, during their late evening informal sessions, to gather in groups and to compare their experiences in teaching their current classes. The college instructors ascribed their teaching problems to failure at the high school level; and the high school instructors, in turn, attributed their problems to failure of the teachers at the elementary level.1 In the late 1950's two factors began to evolve that have been important in the development of the present-day (current) elementary school mathematics curricula. The first of these factors was ”modern mathematics." As early as 1957 instructors who utilized materials which had been written in 1953 and 1955 included these "modern topics in mathematics” in their graduate level mathematics courses. In 1959 these ”modern topics” were recommended for the secondary school curricula for the new subject matter and point of view are to be intertwined with much that is old and valuable in both subject matter and points of view, Land in this manner] we arrive at the conclusion that: (1) ”Modern 1In the 1950's the number of elementary school teachers who were interested in teaching mathematics and were attend- ing these annual meetings was very small in relation to the number of college and high school teachers in attendance. 2Phillip S. Jones, who is recognized nationalhyas an outstanding teacher, offered such a course at the University of Michigan, using the following "modern mathematics topics": (1) E. R. Stabler, An Introduction to Mathematical Thought (Cambridge, Massachusetts: Addison—Wesley Publishing Company, 1953), (2) W. L. Duren, Jr., and D. R. Morrison, Universal Mathematics, Part II, Structures in Sets (Preliminary Ed.; New Orleans, Louisiana: Tulane University Book Store, 1955), and (3) as a current events topic: a critical review of a proposed chapter of the then pending twenty-fourth yearbook for the National Council of the Teachers of Mathematics. '\) R) mathematics” does not mean a total abandonment of all-~or even much—-of what has been taught in the past.1 Agreement with this judgment was expressed by the School Mathematics Study Group in a newsletter of March, 1961, detailing subgroup plans as follows: II. Mathematics for Grades 9 through 12. This project is devoted to the production of a series of sample textbooks for grades 9 through 12. For the most part the topics discussed in these textbooks do not differ markedly from those included in the present— day high school courses for these grades. However, the organization and presentation of these topics is dif- ferent. Important mathematical skills and facts are stressed, but equal attention is paid to the basic con— cepts and mathematical structures which give meaning to these skills and provide a logical framework for these facts.2 From the high school and then into the elementary school, the need for a stronger mathematics program at the elementary school level was proposed in the same School Mathematics Study Group newsletter: In this project SMSG will undertake a critical study of the elementary school mathematics curriculum from the point of view of: increased emphasis on con— cepts and mathematical principles; the grade placement of topics in arithmetic; the introduction of new topics, particularly from geometry; and supplementary topics for the better students, for example from number theory.3 lPhillip S. Jones, "The Mathematics Teacher's Dilemma," The Universitygof Michigan School of Education Bulletin, XXX (January, 1959), pp. 65—72, reprinted in Notes for the Mathe— matics Teacher, Number 1 (New York: World Book Company), p. A. 2School Mathematics Study Group, Newsletter Number 6 (New Haven, Connecticut: Yale University, March, 1961), p. 5. 3 Ibid., pp. 7—8. 23 This need, recognized nationally, for a stronger and richer program in mathematics had been projected downward from the college level to the secondary school level, to the elementary school level, and then was brought back to the college level and into the teacher training programs. In 1961, the Panel on Teacher Training of the Committee on the Undergraduate Program in Mathematics announced its recom- mendations for an undergraduate mathematics curriculum for all students preparing to teach at what the Panel described as Level I: teachers confronted with the problem of presenting elements of arithmetic and the associated material commonly taught in grades K—6.1 During the time that the first factor, "modern mathe— matics," was gaining support, the second factor, "the mathematics curriculum for K—l2,” was starting to develop. This theme, which had been growing among educational groups, gained national recognition in 1959 with the publication of the Twenty—fourth Yearbook of the National Council of the Teachers of Mathematics, The Growth of Mathematical Ideas, K:12. Further recognition for the theme was forthcoming; in 1960 the Michigan Education Association used this K-l2 theme for all regional meetings-—especially for the subject matter conferences for each region. The theme for the one-day * 1Mathematical Association of America, Course Guides for the Training of Teachers of Elementary School Mathematics (Revised, 1968; A Report of the Panel on Teacher Training, Committee on the Undergraduate Program in Mathematics, Mathe— matical Association of America [Berkeley, Calif.: Distributed by the CUPM Central Office, 1968]), p. 1. a); (i program was the creation and communication of common goals and problems in mathematics for all grades, K—l2. At that time (1960) the National Council of Teachers of Mathematics and the Michigan Council of Teachers of Mathe— matics began to invite and to urge the attendance of ele- mentary school teachers at their meetings and their annual conventions. Each year these associations have scheduled meetings (sessions) so that they were led by elementary school teachers and have structured general sessions so that they were directed toward the interests and needs of these teachers. As a result, each year the number of elementary teachers attending these conferences has increased over the number that attended in previous years.1 Until 1969 the interest in elementary teacher training programs was directed toward the students enrolled in the four—year college or university. Earlier reports of the Panel on Teacher Training gave recommendations for a strong program of two years of additional mathematics in the train— ing of elementary school teachers. In 1969 the Panel on Teacher Training of the Committee on the Undergraduate Pro— gram in Mathematics recommended that "all two—year colleges offer the year course in the number system, Mathematics NS. It should be emphasized that, in the view of the Panel on 1In 1970 the National Council of Teachers of Mathe— matics voted a dues schedule of two dollars for elementary teachers and four dollars for all other educators. 25 Teacher Training, this is minimal training for the elementary school teacher."1 During the school year of 1969—70 a panel or committee from the Southeastern Michigan Association of Community College Mathematics Instructors (SMACCMI) has met monthly, under a grant from the College Science Improvement Program (COSIP) of the National Science Foundation (NSF) to consider the transfer curricula in mathematics from the community colleges, includ- ing mathematics for elementary school teachers.2 In September, 1969, at the invitation of the Panel on the Transfer Curricula of the Southeastern Michigan Association of Community College Mathematics Instructors, the Committee on the Undergraduate Program in Mathematics sponsored a symposium, at an invita- tional conference, for representatives from the mathematics departments of the institutions that are members of the Asso- ciation. The purpose of this conference was consideration of the new recommendations by the Panel on Teacher Training (CUPM) for the reactions of the conferees to these recommenda— tions. At the session devoted to mathematics for elementary 1Mathematical Association of America, A Transfer Curriculum . . . Two—Year Colleges, p. 36. 2The two-year colleges are: Delta College (Bay City), Flint Community Junior College, Henry Ford Community College, Highland Park College, Jackson Community College, Lansing Community College, Macomb County Community College, Monroe County Community College, Oakland Community College, School— craft College (Livonia), St. Clair County Community College, and Washtenaw Community College; and the four—year institu— tions are: Central Michigan University, Eastern Michigan University, Michigan State University, Northern Michigan University, Oakland University (Rochester), the University of Michigan, and Wayne State University. 26 school teachers, all conference participants accepted the recommendations as valid. It was pointed out that some two— year colleges offer two semesters of mathematics; only one of these is devoted to study of the number system (arithmetic). The decision to carry the recommendation to the curriculum committees of the two—year colleges involved was expressed by the members of this session. The attention of all con— ferees was called to the following paragraph: A subsequent survey by the Panel shows that the amount of mathematics required of prospective elementary school teachers had approximately doubled in the period from 1961—1966.' In spite of these gains, the full implementation of the recommendation has not yet been achieved. There are many reasons for this outside the control of the mathematics departments, primarily the tremen- dous demands on the time of the prospective ele- mentary school teacher.1 Background of Research and Theory There is an ever increasing interest in the develop— ment and use of audio—tutorial, or multi—media, instruction as a method of teaching. The advantages claimed for pro— grammed instruction, whether by machine or textbook, may apply equally well to this method of instruction. These advantages are: individualized instruction, behavioral objectives, sequencing, pacing, feedback, constructive eval— uation and improvement. To these advantages can be added the variety of media that may be utilized to involve the senses of sight, hearing, and touch; the senses of smell ‘ 1Mathematical Association of America, loc. cit., p. 36. 27 and taste, where appropriate to the objectives, could be used effectively. Most authorities include the following as the assets to be gained with the use of the audio—tutorial method: 1. the selection of the laboratory—learning period by the individual student. 2. the variety of learning activities included in each lesson. 3. the student's active participation in the learn— ing situation. A. the interaction of student and qualified instructor, at the time that the student feels the need for assistance. 5. the student's progress and achievement in each lesson is at his own pace or rate and is limited by his decision. Postlethwait said that with this method the student has the advantage of the following ingredients for an educational program: (1) repetition, (2) concentration, (3) association, (A) unit steps, (5) a communication vehicle appropriate to the objective, (6) a multiplicity of approaches, and (7) integrated experiences. 1S. N. Postlethwait, J. D. Novak, and H. T. Murray, Jr., The Audio—Tutorial Approach to Learning, Through Independent Study and Integrated Experiences (2nd. ed.; Minneapolis, Minnesota: Burgess Publishing Company, 1969), pp. 3-A. 28 At the 1969 Audio—Tutorial Systems Conference Postlethwait stated that he felt the method of teaching was successful, but that he had not conducted research to substantiate his conviction. At the same conference Novak made the following points on research that could be applied to the audio—tutorial technique: 1. The learning theories Of David Ausubel provide "a very adequate base for designing and interpreting research studies, especially those involving individualized instruction such as in audio— tutorial teaching."1 2. "You may wish to explore theoretical suggestions by Bruner, Gagne, Piaget, Smith and Smith, Skinner, or other psychologists but in the judgment of my graduate students Ausubel presents the most heuristic theory for proceeding in the design and analysis of research."2 3. "Distinguish rote reception learning from mean— ingful reception learning (explaining Ausubel); and there is need for subsumers (functional concepts) 1Joseph D. Novak, Relevant Research on Audio—Tutorial Methods, A Paper for the Audio—Tutorial Systems Conference at Purdue University (West Lafayette, Indiana: Purdue Univer— sity, October 20, 1969), p. 1. 2Ibid. 29 before meaningful learning, and advance organizers before the subsumers."l Galanter listed these advantages for programmed instruc— tion: "(1) all the advantages of a private tutor; (2) increase in the overall time efficiency of the learning process 3 to l; and (3) frees the teacher for creative and uniquely human tasks which only the teacher can perform."2 That there is a need for research in the use of programmed instruction is evidenced by professional literature, which points out this need explicitly. In a joint committee report on pro— grammed learning materials the committee recommended that (l) the effectiveness of a self-instructional program be assessed by finding out what students actually learn and remember from the program, and (2) active experimentation with self-instructional materials and devices in school systems is to be encouraged prior to large scale adoption.3 As Dr. Scannell said, "I believe that the state of the art (or science) called teaching elementary school mathe- matics can be advanced even if original, ice-breaking studies ‘— llbid. 2Eugene Galanter, "The Mechanization of Learning,” NEA Journal, (November, 1961), p. 18. 3"A Statement on Self—Instructional Materials and Devices by a Joint Committee of the American Educational Research Association (NEA), the Department of Audio-visual IDstruction (NEA), and the American Psychological Associa- tion," NEA Journal, (November, 1961), p. 19. 30 are conducted in limited settings, with limited generality."l He also added that relevant factors must be described in the fullest. Review of the Literature Elementary School Teachers In-Service Teachers In 1967 Suydam reported the research studies on ele— mentary school mathematics that had been published as journal articles, during the period from 1900 to 1965, as a part of her dissertation.2 Of these articles 158 were published in The Arithmetic Teacher; 132 in The Elementary School Journal; 36 in The Mathematics Teacher; and 30 in The Journal of Experimental Education. Other articles were published in journals, but only a few in any one of the journals. Suydam noted that The Arithmetic Teacher, although in publication only since 195A, published one fifth of all the reports.3 In 1970 Riedesel summarized the recent research in elementary school mathematics. The report covered three Categories of research: (1) approaches to content, (2) 1 Dale P. Scannell, "Obtaining Valid Research in Ele— mentary School Mathematics,” The Arithmetic Teacher, XVI (April, 1969), p. 295. 2Marilyn N. Suydam, "The Status of Research on Ele— mentary School Mathematics," Arithmetic Teacher, XIV (December, 1967), pp. 68A—689. 31bid., p. 685. 31 approaches to teaching, and (3) materials.1 A bibliography, with entries that were published from September, 19AA, to July, 1968, was included at the end of his article. These articles summarized by Riedesel reported research studies in the elementary school, with in—service teachers, and with elementary school pupils. In this category, a comparison of the dissertation abstracts and of the articles in the professional journals, including the new Journal for Research in Mathematics Educa— tion,2 and the first issue of The Two—Year College Mathematics Journal,3 revealed that studies completed before 1969 (dissert— ations) are usually found as articles in the professional journals. Ginther, Gibney, and Pigge have studied the ”mathematical understanding” of elementary school teachers classified accord— ing to (l) the size of community in which they taught, (2) the size of the community where they were graduated from high school, (3) the subject they preferred to teach, and (A) the lC. Alan Riedesel, ”Recent Research Contributions to Ele— mentary School Mathematics," Arithmetic Teacher, XVII (March, 1970), pp. 2A5—252. 2National Council of Teachers of Mathematics, Journal Egr Research in Mathematics Education, 1 (Washington, D.C.: National Council of Teachers of Mathematics, 1970), 3Mathematical Association of America, The Two—Year COllege Mathematics Journal, I (Boston, Massachusetts: PI‘indle, Weber and Schmidt, Incorporated, 1970). 32 subject they preferred not to teach.1 The summary of the analyses pointed out these results: (1) the mean score for teachers who graduated from medium—city high schools was significantly more than the mean score for the teachers from large—city high schools; (2) teachers who most preferred to teach mathematics did significantly better on the test than those who preferred to teach language arts, science, or social science; and (3) the teachers who least preferred teaching mathematics did significantly poorer on the test than those teachers who least liked to teach language arts, science, or social science.2 The studies referred to previously were devised to involve in—service elementary school teachers, as did the studies conducted by Clarkson and by Hunkler, rather than studies to involvepre—servicecm:prospective elementary school teachers. Clarkson recommended summer institutes to increase 1Thomas C. Gibney, John L. Ginther, and Fred L. Pigge, "What Influences the Mathematical Understanding of Elementary— School Teachers?” The Elementary School Journal, LXX (April, 1970), p. 368. 2This study supports the administrative decision of the elementary school principal who, by an informal arrangement with the teachers, allows the elementary school teacher who prefers to teach mathematics to do all or most of the arith- metic teaching in the school. This comment was made in an oral report to the Subpanel on Elementary School Mathematics at the Southeastern Michigan Association of Community College Mathematics Instructors Annual Conference on May 11, 1970, at Macomb County Community College, Warren, Michigan. 33 the effectiveness of in—service elementary school teachers.1 Hunkler drew the conclusions that (1) one course in college mathematics did not significantly improve the achievement of the pupils, and (2) that two or more courses had some positive effect on theachievementof their pupils, a verification of the recommendation for a full year of mathematics (Committee on the Undergraduate Program in Mathematics).2 These studies of research on in—service elementary school teachers were of significance as a background for the review of research that is concerned with the pre—service elementary school teacher. Pre—service and In—service Teachers In a recent report of research Gibney, Ginther, and Pigge compared the "mathematical understandings” of pre— service and in—service elementary school teachers.3 The sample consisted of (l) pre—service teachers, the students who were enrolled in elementary education courses and who had 1Donald Robert Clarkson, “The Effect of An In—Service Summer Institute on Mathematical Skills, Understandings, and Attitude Toward Mathematics of Elementary School Teachers,” (University of Connecticut, 1968), Dissertation Abstracts, XXIX, 9 (March, 1969), p. 3019—A. 2Richard Frederick Hunkler, ”Achievement of Sixth— Grade Pupils in Modern Mathematics as Related to Their Teachers' Mathematics Preparation,” (Texas A & M University, 1968), Dissertation Abstracts, xx1x, 11 (May, 1969y p. 3897—A. 3Thomas C. Gibney, John L. Ginther, and Fred L. Pigge, "The Mathematical Understandings of Preservice and In-service Teachers," Arithmetic Teacher, XVII (February, 1970), pp. 155— 162. 3A completed at least one three semester—hour course in mathe— matics, and (2) the in—service teachers who were enrolled in undergraduate or graduate mathematics education courses. The sample for the study was taken during 1968 and 1969 at Bowling Green State University, the University of Toledo, and Eastern Michigan University.1 The assumption was made that the sample had the characteristics of the population from which it was drawn, the population being the individuals who fit the cate- gory descriptions and who reside in the geographic location from which the three universities receive their students. The extension of the study is made to students enrolled in the courses described and at the universities, all students who have been enrolled or are enrolled. A test of selected modern mathematics topics in seven areas was administered to the subjects of the study. In Part I, the findings were: 1. Pre—service teachers had a higher mean score in each of the seven areas than did the in—service teachers, with five of these being significant, that is, (a) total, (b) geometry, (0) number theory, (d) structural properties, and (e) sets. These are listed as "modern," as opposed to fractions and operations which were regarded as traditional. 2. The mean score of the pre—service teachers was significantly higher than the mean score of each in-service group. ‘— Ibid., p. 156. 35 3. No significant differences were found among the mean scores of the four groups of in—service teachers who have taught 0—2, 3—5, 6—10, or more than 10 years. A. The in—service teachers who had taught more than 10 years had the lowest mean score and the largest standard deviation. The test used for this study had a .80 reliability, as computed by the Kuder—Richardson Formula 21. The data for this study were analyzed by using the means, the standard deviations, one—way analyses of variance, t—ratios of means, and a "t" ex post facto comparison for all possible pairs of means, and the Scheffé method, wherever apprOpriate.l The conclusion and summary of the study stated that the two groups of students are sufficiently different to warrant different treatments in the mathematics education courses for these groups. The preservice teachers should have a course which is designed and organized to meet their special needs and abilities. In Part II, the investigators compared the data on the basis of the grade level either to be taught or being taught. Significant differences,as indicated in parentheses,occurred for the pre—service teachers in the groups: kindergarten (1); first and second grades (A); third and fourth grades (5); fifth and sixth grades (1); and seventh and eighth grades (none). The results reported were: llbid., p. 156 and 158. 2 Ibid., p. 158. 36 1. Significant mean differences existed between pre— service and in—service teachers in favor of the pre—service teachers in grades 1 and 2, and grades 3 and A, and not in kindergarten, grades 5 and 6, and not in grades 7 and 8. 2. The pre—service teachers' scores, by the chosen grade level, did not yield significant mean differences in mathematical understandings in any levels, from the kinder— garten through the eighth grade. 3. A significant trend was found in in—service teachers scores in relation to the grade level taught——the higher the grade level taught, the higher the score.1 The previous study by Gibney, Ginther and Pigge had, as one part of the sample, students who (1) completed at least one mathematics course on the number system and (2) were enrolled in elementary education courses (no previous teach- ing experience). Pigge and Brune noted in their research, which involved students enrolled in a course on the methods of teaching mathematics, that no significant difference in achievementsof the groups resulted from their analysis of the data.2 They did, however, express a professional opinion that reviewing the teachers' manuals and the pupils' textbooks for elementary arithmetic did give that student group (manuals) 11bid., p. 162. 2Fred Pigge and Irvin H. Brune, "Lectures Versus Manuals in the Education of Elementary Teachers," Arithmetic Teacher, XVI (January, 1969), pp. A8—52. 37 the advantage of having worked with ”the tools of their future trade." The research articles, abstracts, and summaries described above have involved elementary school mathematics teaching, but none have applied to the student on an elementary education program and enrolled in his first (and required) mathematics course, which may be his only mathematics course. At the community college this means within the first two years of college credits and excludes the use of ”methods” materials in the course. The question that then arises is, ”What research has been done involving the community college or junior college?" Community College Students In a study at the University Community and Technical College of the University of Toledo, Morgan developed an equa- tion for predicting success in college mathematics classes, for the students enrolling at the college. Since the college administered the Cooperative Mathematics Test for Algebra II for purposes of student placement in the mathematics sequence (”Open door" policy of admission to community colleges), Morgan computed a discriminant equation which is reported to be accurate in 90 percent of its predictions when using the following predictor variables: "(1) score on a cooperative niathematics test; (2) years of high school mathematics; (3) mean grade in high school mathematics, and (A) age in months beyond the seventeenth birthday.”l \ 1William P. Morgan, "Prediction of Success in Junior CC>1lege Mathematics," Mathematics Teacher, LXIII (March, 1970), p- 261. 38 Although the students in the sample of Morgan's research are the same age as the subjects in this project, the students at the University of Toledo may differ in other character- istics; for instance, they are probably more mathematically trained and oriented than the subjects of this investigation, as students who are to be successful on an engineering tech— nology program need to be. In another study of junior college students Meyer studied the junior college student to evaluate the junior college as a means for status enhancement.1 The summary of his report included: 1. The low status, high skill, mobile group is the only group of the six that did use the community college for status enhancement. 2. Skill is believed to be the best single indicator of program preference and dropout. The major recommendation from the study included the admonition that consideration of all three factors be a part of any study of behavior patterns of junior college students.2 The research articles on community colleges, and on community college mathematics in particular, that are published in the current literature are scarce and hard to find. Much 1John David Meyer, "Junior College Students: Status Inconsistency,” (Stanford University, 1968), Dissertation Abstracts, XXIX, 11 (May, 1969), p. 3776-A. 2Ibid. 39 needs to be learned of the community colleges and the students at these colleges; for the increase in the number and size of the community colleges in Michigan alone shows the importance of gathering the data for decision—making at this level of education. How much of the research on four—year college students is applicable to the community college student, and especially to the students who are on transfer curricula, curricula which parallel the programs offered at the four— year schools? Will achievement of these two groups of students be equivalent? Will their attitudes be similar? Attitudes Toward Mathematics Neale has summarized some of the research on the attitudes and mathematics.1 He reported on the International Study of Achievement in Mathematics (Husen, 1967): The implication is clear. If certain attitudes are important objectives of mathematics instruction, then such attitudes must be given deliberate and separate attention, both in the development of mathematics curricula and in curriculum evaluation. Likewise, teachers need to give systematic attention to class- room activities that develop desirable attitudes.2 Neale presented sample items from the Dutton (1956) and the Aiken (1963) versions of a like—dislike and approach— avoidance scale to measure attitude toward learning mathematics. 1David C. Neale, ”The Role of Attitudes in Learning Mathematics," Arithmetic Teacher, XVI (December, 1969), pp. 2 Ibid., p. 632. A0 The sample items he included from the International Study1 also included items on perceived success in mathematics and aspirations to work in math—related occupations. Sample items illustrating a different approach, a semantic differential scale to measure attitude toward learn— ing mathematics, from research by Anttonen in 1967, are also included in Neale's article.2 On Anttonen's scale the sub— ject must record his attitude toward mathematics on a nine— point range between bipolar selections: (I) from distasteful to enjoyable; (2) from new to old; (3) from sharp to dull; (A) from weak to strong, or (5) from valuable to worthless. Neale classified the attitude toward learning mathe— matics as having a special status among attitudinal objectives in mathematics and he recommended that Mager's clear and engaging statement onthis belief would assist the instructor 3 of mathematics. Neale stated two results of his study of the articles on attitudes: (1) students develop an increas— ingly unfavorable attitude toward mathematics as they go through school and (2) at present these attitudes play only a slight part in learning mathematics. These results are from studies in which the subjects were elementary and secondary school students. 21bid. 3Robert F. Mager, Developing Attitude Toward Learning (Palo Alto, California: Fearon Publishers, 1968). Al Of interest here is the observation from the correlation and multiple regression studies for the proportion of mathe— matics achievement that is attributable to intelligence, to prior achievemenu and to attitude toward mathematics. With a variation or R2 of .776, the correlation for'achievement and attitude is .35; the correlationcfl‘achievementwith I. Q. and priorachievement,jointly, is .386, or 38.6% of variation in mathematics achievement. Neale discussed attitudes in children and attitudes toward learning which are interesting to an elementary teacher but are not as effective for students who are in community colleges, except as an explanation of the attitudes of these students as the result of their experiences in elementary and secondary school. However, Neale closed his article with an intriguing argument. After supporting the suggestion that students decide what should and shall be done in schools they attend, on the assumption that the principal motivation for learning is intrinsic interest, he then pointed out that commercial television has shown that reliance on intrinsic interest would not be likely to result in education, as he defined that term. He asked, rhetorically, "Who would go to work daily without pay and solely for the joy that comes from working?” In conclusion he expressed this value judgment: We are mistaken if we create in beginning teachers the impression that learning mathematics must spring from some unquenchable thirst for mathematical know— ledge. We are better off to tell the truth—-that A2 children learn mathematics for a combination of rea— sons, which include a desire to do their duty, be good children, and gain adult approval. The school should be a place where that work goes forward, where necessary tasks are made attractive and rewarding, where every motivation children and adults have is used to encourage learning. I applaud the efforts of mathematics educators who seek to appeal to curiosity and fun. I wish they would in addition work to change the institutions in which learning takes place, for I believe that such change attacks the problem of motivation for learning in a more fundamental way.1 Suydam and Riedesel have collected and listed the appli— cable generalized research findings which the classroom mathe— matics teacher may employ and test. Selection of sample items are: Individualizing instruction improves immediate achieve- ment, retention, and transfer. Modern mathematics programs tend to produce better rea— soning and retention but computational skills are not always better than in traditional programs. Teaching for transfer is necessary. Transfer is greatest when content is similar. Drill should be spaced and varied in type and amount. Periodic review increases retention. Immediate review of arithmetic test items increases . . 2 achievement and retention. lNeal, ibid. , pp. 639—6140. 2Marilyn N. Suydam and C. Alan Riedesel, ”Research Find— ifugs Applicable in the Classroom,“ Arithmetic Teacher, XVI “Racember, 1969), p. 6A1. A3 Anttonen has reported the research for his dissertation in an article on attitudes. His results are significant positive correlation (at the .05 level) between elementary attitude scores and secondary attitude scores. He also found significant positive correlations between all measures of attitude and achievement. The subjects of his research were 607 students from an above average socio—economic suburb of St. Paul, Minnesota.1 In another study Pitkin investigated the attitudes toward mathematics of three classes of teachers and found no significant differences in their attitudes. The subjects were eighty—four teachers classed as (1) those who had a content course and a methods course in modern mathematics, (2) those who had only a content course in modern mathematics, and (3) those who had neither a content nor a methods course in modern mathematics during the last five years.2 Reys and Delon stated the results of a study on the attitudes of students who were pre—service elementary school teachers. Included as subjects in this study were students in these courses: (1) mathematics content course, (2) methods of teaching course, and (3) problems of teaching arithmetic lRalph G. Anttonen, ”A Longitudinal Study in Mathematics Attitude," Journal of Educational Research, LXII, 10, pp. A67— A71. 2Tony Ray Pitkin, ”A Comparison of the Attitudes Toward Mathematics and Toward Pupils of Selected Groups of Elementary SChool Teachers Who had Different Types and Amounts of College Education in Modern Mathematics,” (University of South Dakota, 19638), Dissertation Abstracts, XXIX, 9 (March, 1969), p. 3025-A. AA course. The conclusions were that sixty percent expressed favorable attitudes toward arithmetic and that the college courses produced some of these changes. It is not surprising that the observed changes in attitude toward arithmetic in only a few months of instruction were small since most of these attitudes were conceived at least five years prior to entering college and perhaps even cultivated through the years. A large scale improvement of these deep—seated feel- ings might be expected from high quality instruction in a continuous mathematics program for a longer period of time. However, the problem will not be alleviated until favorable attitudes toward arith— metic are fostered throughout elementary, secondary, and collegiate levels.l A result of surveying research on attitudes toward mathematics was the conclusion that there is still a need to improve the attitude of the prospective elementary school teacher so that he may, in turn, improve the attitudes of his pupils. With this result clearly established, the fourth area of the survey is to be considered. Audio—Tutorial Mode Studies in mathematics using the multi—media technique of instruction, often referred to as audio-tutorial, are not found in the professional journals, not in the dissertation abstracts. The studies that approximate this mode of instruc- tion would be studies that use these techniques: (1) pro— grammed instruction; (2) audio—tapes; (3) films; (A) tele— vision; or (5) laboratory. 1Robert E. Reys and Floyd G. Delon, ”Attitudes of Pro- Spective Elementary School Teachers Towards Arithmetic," Agithmetic Teacher, XV (April, 1968), p. 360. A5 The Laboratory Three studies used a laboratory—learning situation in the research. The first study was a laboratory for high school students, a laboratory with multi—sensory aids and manipulative devices used to clarify concepts and arouse interests.1 The second article was a report of a mathematics laboratory for prospective teachers, a laboratory for a class which is a combination of the three courses usually taught to prospective elementary teachers: (I) the content course; (2) the methods course; and (3) the problems in teaching mathematics course.2 The third study was not reported as a 3 study. Houston referred to it as a pilot study. Fitzgerald described the results of his pilot study as a description of a successful innovation in individualizing instruction for large lecture groups by having some of the class sessions for small group laboratory activities.L1 As a part of this work lNicholas H. Borota and Gladys M. Veitch, "Mathematics for the Learning Laboratory to Teach Basic Skills to Tenth, Eleventh, and Twelfth Graders in a Culturally Deprived Area," Mathematics Teacher, LXIII (January, 1970), pp. 55—56. 2David M. Clarkson, "A Mathematics Laboratory for Pro- spective Teachers," Arithmetic Teacher, XVII (January, 1970), pp. 75—78. 3W. Robert Houston, ”Preparing Prospective Teachers of Elementary School Mathematics,” Arithmetic Teacher, XV (November, 1968), p. 6A5. “William M. Fitzgerald, "A Mathematics Laboratory for Elementary School Teachers," Arithmetic Teacher, XV (October, 1968), pp. 5A7-5A9. A6 with laboratory sessions Fitzgerald has a laboratory manual to be used in coordination with a text. A comparison of the two laboratory materials reported by Fitzgerald and by Clarkson showed that the materials are similar in nature. Boonstra used in his study selected students at Michigan State University who had taken the mathematics course under Fitzgerald. He reported that these students did not transfer their learning in the laboratory to their teaching experiences—— neither the use of materials nor the student-centered method of teaching.1 The studies by Fitzgerald and by Clarkson used the methods of teaching arithmetic to motivate learning. This approach to motivation, at least the direct use of methods, has been expressly rejected for community college courses by the curriculum planners and by the representatives from the four—year colleges who are responsible for approving the courses that will be accepted by the four—year institution for transfer credit. Programmed Materials A survey of the research on programmed instruction by textbook, machine, or computer has shown the interest in this mode of instruction. In 1966 May wrote an article on pro— gramming and automation in which he described the Skinner and lBoonstra, 10c_ cit. 117 the Crowder approaches to programming and also the programmed instruction by computer.1 Nagel used a programmed textbook for his research in the teaching of remedial college algebra classes (intermediate algebra).2 The results were not a signi— ficant improvement; however, the author set forth the conclu— sions that the failure rate was normal and, therefore, no harm was done to the subjects and (2) the amount of material covered was significantly more in the programmed group. Nagel also wrote of the student dissatisfaction and frustration for the group using the programmed material. Alton reported the use of programmed material in a remedial algebra course at the college level with a degree of success for the programmed method. She did comment on the students' desires and need for discussion with a teacher.3 Two studies used programmed instruction in the first course in algebra. Sneider described the use of programmed learning to study achievementiiia modern algebra class with the result that the programmed instruction was not better 1Kenneth O. May, ”Programming and Automation,” Mathe: matics Teacher, LIX (May, 1966), pp. AAA—A5A. 2Thomas S. Nagel, "Effects of Programmed Instruction in Remedial College Algebra Classes,” Mathematics Teacher, LX (November, 1967), pp. 7A8—752. 3Elaine V. Alton, “An Experiment Using Programmed Material in Teaching a Noncredit Course at the College Level [with] Supplement,” (unpublished Ph.D. dissertation, Michigan State University, 1965). A8 than the lecture demonstration instruction.l Devine did the research in student attitudes and achievement in teaching algebra 1.2 According to the results of the study (1) teachers reacted negatively to programmed materials and the inexperienced teachers more markedly than the experienced teachers, and (2) student attitudes were unaffected but achievement was signifi- icantly better in the traditional class. Devine recommended that ”the use of short units as homework assignments may be very helpful.”3 Hennemann and Geiselmann made use of programmed instruc— tion for the enrichment of a pre—calculus course. Students were satisfied with the programmed text but felt it took unnecessary fortitude and endurance to complete it. They expressed the feeling of a lack of interest and a need for contact with a ”human“ instructor. At Purdue University Bartz and Darby conducted a study on the effects of a programmed text in intermediate algebra 1Sister Mary Joetta Sneider, ”Achievement and Programmed Learning," Mathematics Teacher, LXI (February, 1968), pp. 162- 16A. 2Donald F. Devine, ”Student Attitudes and Achievement: A Comparison Between the Effects of Programmed Instruction and Conventional Classroom Approach in Teaching Algebra I,” Mathematics Teacher, LXI (March, 1968), pp. 296—301. 3Ibid., p. 301. I . . 4Willard W. Hennemann and Harrison A. Geiselmann, “Us1ng Programmed Learning in the College Classroom: A Case History,” Mathematics Teacher, LXII (January, 1969), pp. 27—32. A9 with one group to use the text in independent study. The control group, using a conventional text performed much better than did the experimental group.1 King studied group interaction with programmed instruc— tion. This study is not significant for a research project that is designed to individualize instruction.2 Pethtel has used closed circuit television to teach 3 college mathematics. The course enrollment of 810 students included the subjects of the study, 263 elementary—teaching majors. The course outline for this mathematics course, a survey, covered the following topics; numeration systems, sets, logic, algebra, probability, and descriptive statistics. The methods of instruction used were (1) only television, (2) television and discussion, and (3) traditional. Results of the study are (I) that students in television achieved and retained as well as the other groups, (2) that upper ability students learned more by television than the lower ability students. lWayne H. Bartz and Charles L. Darby, "The Effects of a Programmed Textbook on Achievement Under Three Techniques of Instruction," Journal of Experimental Education, XXXIV, 3, pp. A6—A9. 2Robert W. King, "Using Programmed Instruction to Investigate the Effects of Group Interaction on Learning Mathematics," Mathematics Teacher, LXII (May, 1969), pp. 393- 398. 3Richard D. Pethtel, "Closed Circuit Television Instruc— tion in College Mathematics," Mathematics Teacher, LXI (May, 1968), pp. 517-521. Audio Materials The two studies using audio materials in mathematics classes were reviewed. The first research project as described by Robinson gave tape—recorded instruction on arithmetic performance of seventh grade pupils.l In his description of the analysis Robinson specified that the application of the Kuder—Richardson Formula 21 to a teacher- made test resulted in a cumulative reliability of .73. Fisher's "t" technique was used for the evaluation. The list of results obtained in this eight—week study are: 1. The traditional arithmetic instruction group per— formed at a higher rate than the tape arithmetic instruction group. 2. Average ability students who received traditional arithmetic instruction performed at a higher rate than pupils who received tape arithmetic instruction. 3. Girls who received traditional arithmetic instruc— tion performed at a higher rate than the girls who received tape arithmetic instruction. A. Average ability girls who received traditional arithmetic instruction performed at a higher rate than aver- age ability girls who received tape arithmetic instruction. lFrank Edward Robinson, "An Analysis of the Effects of Tape-Recorded Instruction on Arithmetic Performance of Seventh Grade Pupils with Varying Abilities," (North Texas University, 1968), Dissertation Abstracts, XXIX, 11 (May, 1969)) p0 3782-A. 51 In the second study Byrkit used televised and aural materials for an in—service training program of junior high school mathematics teachers.1 The components of this study consisted of a programmed text, a televised lecture, post— tape exercises, homework exercises, and a summary. Fifty— four teachers were randomly assigned to six groups. Three groups studied the lesson on integers and the other three groups studied the number theory lesson, each lesson studied in the three modes. The television group showed greater superiority over the control group than the audio group over the control group. Strong possibility of type 11 error precluded acceptance of the hypothesis of no difference between the groups. The attitude survey showed more favorable attitude from the students who studied integers; the presentation mode made no significant difference in the attitudes. In 1968 Zoll presented a summary of research on pro— grammed instruction in mathematics.2 He classified the studies by the alternate method; such as, traditional instruc— tion, commercial programs compared to the conventional lDonald Raymond Byrkit, "A Comparative Study Concern— ing the Relative Effectiveness of Televised and Aural Mater— ials in the In—service Training of Junior High School Mathe- matics Teachers,” (Florida State University, 1968), Disserta— tion Abstracts, XXIX, 5 (November, 1968), p. lA63—A. 8Edward J. 2011, "Research in Programmed Instruction in Mathematics," Mathematics Teacher, LXII (February, 1969), pp. 103—109. 52 methods, characteristics of learners, attitudes toward pro— grammed instruction, changing the format of programs, teach— ing concepts, methods of teaching (guided discovery, expository). Zoll pointed out a need for still further research in programmed instruction. Summary In Chapter II the background of the problem has been presented in (l) the history and (2) the theory and the current literature on research, studies, and articles has been reviewed. The areas which were included were: ele— mentary school mathematics teachers, community college stu— dents, attitudes toward mathematics, and the audio-tutorial method (substituting other related media of instruction). The following conclusions from the literature are of impor— tance to this study: 1. Elementary teachers graduated from medium-city high schools understood mathematics more than teachers from large—city high schools. 2. Elementary teachers who prefer to teach mathematics understand mathematics better than other elementary teachers. 3. Elementary teachers who prefer not to teach mathe- matics understand less mathematics than teachers who prefer not to teach other classes. A. In accordance with Committee on the Undergraduate Program in Mathematics recommendations, two or more college 53 mathematics courses are needed by a teacher who wishes to have a positive effect on the arithmetic achievement of his students. 5. Pre—service elementary teachers need mathematics courses that are specially designed to meet their needs, attitudes, and desires; courses for in—service teachers will not suffice. 6. Utilizing in the college mathematics courses the materials and tools that the elementary school teacher may be able to use in his teaching career will create a positive readiness, or set, toward these materials and tools for the prospective elementary teacher. 7. There is not sufficient evidence for drawing a conclusion about community college students, not enough to demonstrate a trend toward a conclusion. There is need for further study in this area. 8. Achievement in mathematics is related to prior achievementand jointly to prior achievement and attitude toward mathematics. 9. After a course in college mathematics at the Univer— sity of Missouri, sixty percent of the pre-service elementary teachers have a favorable attitude toward mathematics. 10. The audio—tutorial mode of instruction as suggested by Postlethwait has not been tested for any of the research jIIInathematics and certainly not for mathematics for pro— SpeCtive elementary school teachers. CHAPTER III THE STUDY Introduction In this chapter are included (1) a detailed discussion of the problem, (2) the characteristics of the sample, (3) an evaluation of the measuring instruments, (A) the design of this comparative study of modes of instruction, (5) the hypotheses to be tested, (6) the methods to be used for that testing, and (7) a summary of the chapter. Mathematics for Prospective Elementary Teachers Delta College In the fall of 1961, Delta College, a community college located near Bay City, Michigan, offered its first classes. The college district was governed by an autonomous board of trustees. Delta College, which had been planned and designed with the assistance and cooperation of the administration, faculty, and staff of Bay City Junior College, absorbed the Junior College into its organization, offering positions to the Junior College employees who wished to assume duties at Delta College. Many faculty members of the Junior College decided to make this change; some of these preferred to change 5A 55 rather than locate in a new area. Of course, new faculty members were also offered positions, since the enrollment increased even more than had been anticipated by the planners. It [Bay City Junior College] had been in existence 30 years and was one of the oldest public two-year colleges in Michigan. It was housed on the upper floor of a high school, was established, was accepted by the community, was reasonably predictable in its growth, was progressive, and suffered only from a lack of elbow room. No one seriously questioned its value or its contribution to the community. The Mathematics Department At the opening of Delta College, the mathematics depart- ment was a department in the science division, but the mathe— matics department had duties and responsibilities that differed from those of the other science departments. Shortly after school opened the mathematics department prevailed on the administration for a change of classification and became an autonomous department, responsible through its chairman, Miss Meta Ewing, directly to Dean John Brinn, College of Community Services. The mathematics department of Delta College consisted of eight faculty members, four experienced in teaching in a community college in the area, one from a community college in California, and three from high schools in Michigan. [see Table l]. 1Eric J. Bradner, Report of the President, Schoolcraft College, 1960—1970, a printed report to the Board of Trustees. llivonia, Michigan: Schoolcraft College, 1970), p. 1. 56 TABLE l.——Mathematics Faculty at Delta College in the Fall, 1961. Number Area Men Women Aa Bay City Junior College 2 2 la Community College in California 1 I8 High School in Michigan 1 2b High School in Michigan 2 8 5 3 aCollege of Community Service bCollege of Letters Mathematics for Elementary Teachers As the classes in mathematics for elementary teachers, Mathematics 110 at Delta College, progressed through the course, during the years from 1961—6A it became apparent to the instructors, who were new faculty members, that these classes were different from those in the other courses of the transfer curricula. Different student problems would develop in these sections. After some discussion of the problems arising from the diverse needs, desires, attitudes toward mathematics, and tensions resulting from frustration, as demonstrated by the overt behavior of the students, the instructors consulted the ”older" mathematics faculty, or as they put it "Better aSLC a BCJC for help on that one!”1 1Bay City Junior College 57 The instructors were assured that this was the typical pattern for students enrolled in the course and that the anxiety-frustration rate, was the same as at other colleges, including the four year colleges and universities. Schoolcraft College In the fall of 196A, Schoolcraft College, a community college in Livonia, Michigan, offered its first classes, with a beginning enrollment of approximately 2000 students. School— craft College is governed by an autonomous board of trustees who are elected from the five communities of northwest Wayne County, which make up the college district: that is (l) Clarenceville; (2) Garden City; (3) Livonia; (A) Northville; and (5) Plymouth. Our history at Schoolcraft, although parallel in some ways with the example I have just cited, differs in one major respect. Here in northwestern Wayne County we have just gone through our first 10 years of living with the idea that we really do have an institution of higher education in our community. We are only now beginning to get used to it. As Schoolcraft College began to function in the fall of 196A, a small nucleus of administrators, instructors, and staff from Bay City Junior College and/or Delta College was on hand to provide experience in community colleges and/or in the area. The remainder of the administrators, faculty, and staff were from high schools, other community colleges, or colleges, in this state, or other states. lIbid. 58 Enrollment alone has increased 255 percent during the six years of the College's life. Faculty to teach the students has increased 335 percent. The amount of money needed to operate the College for one year has tripled. In a newly opened community college there was only one way for faculty members to judge the patterns of behavior and the needs of their students. That way was to compare the current observations with events and characteristics observed with the educational institutions, noting similar— ities and differences, in order to evaluate the current per— formance and to develop and apply corrective measures as they were called for. These evaluations were needed in the mathematics courses, especially in mathematics for elementary teachers. The Mathematics Department At Schoolcraft College, in 196A, the mathematics depart- ment was, as it has remained, directly responsible through its chairman to the Dean of Instruction. The department con— sisted of five instructors whose experience is shown in Table 2. The faculty at Schoolcraft College in the school year of 196A—l965 was similar to the faculty that taught at Delta College in 1961—196A. The current mathematics department for the school year of 1969 and 1970 consisted of a larger staff to meet the greater student mathematics enrollments, as shown in Table 3. lIbid., pp. 3—A. 59 TABLE 2.—-Mathematics Faculty at Schoolcraft College in 196A— 1965. Number Community College in Men Women 2 Wayne County I la 2 Michigan (Not Wayne County) 1 l 1 Out of Michigan 1b 5 3 2 aLeft for other position in Michigan bLeft for position out of Michigan TABLE 3.-—Mathematics Faculty at Schoolcraft College in 1969— 1970. Number Faculty Service a Men Women Begun Extent 3 196A- 6 years 2 l 2 1965— 5 years 2 3 1968- 2 yearsb 3b 2 1969— 1 yearb 2b 10 9 l 3Extent of service included 1969—1970 bProbationary status During the six years Schoolcraft College has been in operation the mathematics instructors have noticed and have remarked on the characteristics of the students who enroll in 60 mathematics for elementary teachers, designated as Mathematics 103 at Schoolcraft College. The Course, Mathematics for Elementary Teachers The enrollments in this course have increased in each semester so that, from 196A, with one section scheduled for each semester, this increase meant that in 1965 there were two sections each semester, and from 1968 on three sections were scheduled. For 1970—1971 there were scheduled three sections for the fall and four sections for the spring. At the suggestion of a mathematics instructor an evening class for Mathematics 103 was tentatively offered in the fall semester of 1965. This section, which had an enrollment of twenty students the first time it was offered (with limits on enrollment set at ten to thirty), has been offered each semester following that first offering. In 196A Mathematics 103, a three-semester hour course in modern mathematics at the arithmetic level, had no pre— requisites and was the required mathematics course for the elementary education transfer curricula. It was, therefore, a requirement for the granting of an Associate of Arts degree by Schoolcraft College. On occasion, students from another curriculum have elected this course, because it has been, and still is, the only arithmetic course at Schoolcraft College that carries transfer credit and is accepted by liberal arts colleges. 61 The Motivation The Mathematics 103 students at Schoolcraft College have displayed achievement behavioral patterns and attitudinal patterns similar to those "typical of the students at Delta College," according to the Bay City Junior College mathe— matics instructors. A special mention should be made of the motivation problems involved in teaching this type of course. The students who are usually enrolled in this course in mathe— matics have already acquired the standards, aims, interests, goals, and reinforcement and reward systems of the American high school graduates of average ability (especially of the schools in the immediate vicinity). In their previous study of mathematics, some students have had no mathematics since the eighth grade. Others have had three years of high school mathematics. These three years of mathematics may be a combination of any of the following: general mathematics; commercial mathematics; applied mathematics; business arith— metic; or the usual college preparatory mathematics. Many of these students have developed attitudes toward mathematics which range from tentative hope of success to a feeling of uncertainty, dislike, distrust, and sometimes even dread. At monthly meetings of the Southeastern Michigan Community College Mathematics Project and the annual meeting of Southeastern Michigan Association of Community College Mathematics Instructors, identified by SMCCMP and SMACCMI 62 respectively, mathematics instructors from community colleges and four—year colleges and universities have refuted sugges- tions and statements that the patterns and trends described above were in any way different from those observed at their institutions. They asserted positively that the observations were similar. At the Committee on the Undergraduate Program in Mathe— matics (CUPM) sponsored invitational Two—Year College Con— ference, held in September, 1969, at East Lansing, Michigan, and attended by mathematics faculty representatives of two— year colleges, and four-year colleges and universities through- out Michigan and the northwest part of Ohio, the members attend— ing the section meeting on the mathematics for elementary teachers (NS)l proposal of CUPM insisted that these students have similar patterns whatever the institution they attend. The professors of the universities stated that they are teach— ing the same kinds of students as the instructor at the two— year college—~the difference is the number of students in the section and not the kind of student. In these sections there are, noticeably, two classes of students which are characterized by the time lapse between high school graduation and the enrollment in this course. The time lapse between the periods of attending "school" and 1Committee on the Undergraduate Program in Mathematics Proposal of one year mathematics course in the number system. See Mathematical Association of America, A Transfer Curricu— lum . . ., pp. A0—A7. *— 63 the age difference of these two groups is much more pronounced in Mathematics 103 than it is in the other courses in the mathematics department (in the day sessions), as Table A will show. TABLE A.-—Students Enrolled in Mathematics 103 and 112 During 1968-1969, By Age (given in percent). 1968 1969 Winter Fall Winter Fall younger +1 85 8A 78 76 Math 103 a older +1 l5 16 22 2A younger +1 93 89 91 86 Math 112 a older +1 7 ll 9 1A aOlder is defined as over twenty years of age, or more than two years since graduation from high school. As the number of enrollments of the older students increased more rapidly than the enrollment of the younger students, adjustments had to be made in the teaching techniques in order to provide learning situations that would give all the students a maximum opportunity to learn the concepts of of the course. These older students differed from the younger students in one or more of the following character— istics; 1. They take more time to learn the same amount of material, especially as the course begins. 6A 2. They persevere longer and strive harder. 3. They set higher standards of achievement for themselves, on the individual assignments and during the entire course. A. They want to learn as much as they can so that they will be better prepared to teach arithmetic. 5. They develop more tension, anxiety, or frustration than the younger students do. 6. They usually have more and sometimes pressing family responsibilities or emergencies. Unless Mathematics 103 can be a rewarding learning experience many of these students will continue in their current unfavor— able attitudes toward mathematics. As prospective elementary school teachers these students will be expected to teach arithmetic in the future and they would communicate their true attitudes toward arithmetic to their pupils, even when they were attempting to conceal these antipathies. When their pupils sensed the real attitudes of these teachers they would become even more concerned about their arithmetic. In order to avoid this situation it was necessary to develop in the Mathematics 103 students a sense of security and satisfaction in mathematics and to increase the motivation in these students to the level at which they Could and would perform to the best of their abilities with— Out at the same time developing the extreme anxieties and frustrations which would interfere with learning and achieve— ment; this was, and is, one of the challenges of instructing 65 the mathematics class for prospective elementary school teachers. Attempts to Improve the Situation To date the efforts to provide a better learning situa- tion have been three-fold: 1. A pre-requisite for this course, Mathematics A5 or a satisfactory score on the Mathematics Admissions Test, was placed in the college catalogue two years ago. Until the fall semester of 1970 the test was used for data gathering purposes and voluntary placement. This pre-requisite will be required of all enrollees,beginning in the fall semester of 1970, since current study reinforced the basis for this decision. The inception of a three-hour course, Mathematics 10A, has been approved for the winter semester of 1971. This course was designed to give students the opportunity to have a full year of arithmetic, in accordance with the Committee on the Undergraduate Program in Mathematics recommendation. Course out- lines appropriate for such courses at community colleges have received approval from mathematics representatives of the four-year colleges and univer- sities at the April 9, 1970, meeting of the South- eastern Michigan Community College Mathematics Project at Oakland Community College at 66 Orchard Hills, Michigan. These outlines were approved by the Section on Mathematics for Ele- mentary Teachers at the May 11, 1970, meeting of the Southeastern Michigan Association of Community College Mathematics Instructors. This annual con- ference was held at Macomb County Community College in Warren, Michigan. [See Appendix A.] 3. By quietly demonstrating and reinforcing the attitudes of "Mathematics Is Fun," "Learning is Doing," and "Everyone (including teacher or student) has the right to make a mistake without being ridiculed," the students are encouraged to take an active part in the class sessions. Class discussions can cover any topics pertinent to the lesson or, time per- mitting, to mathematics in general. These discus- sions can provide daily feedback to students and to the instructor on the attitudes, anxieties, and achievements of the class. An indirect approach to the problem has been tried with a small degree of success, an improvement in the attitude. Additional projects, which are based on number theory, set theory, arithmetic, algebra, and geometry, some of which were originally teacher—made tests used in junior high classes, are designed to do the following for the students (plans for pro- jects proposed by students can be submitted for approval): 67 I. To decrease the extreme tension or anxiety that develops. 2. To give a sense of accomplishment to the student. 3. To offer enrichment topics and models for their future teaching. A. To provide further and advanced practice in basic fundamentals. To satisfy a need to create-~curve stitching, design- U'l ing, construction of solids, photographs of geo— metric figures in life situations. 6. To add a small score to the final average of the student. These attempts to correct the situation do not make a direct attack on the problem. Re—teaching and other methods cannot be used because of the time limitation and because each concept included in the course outline is essential to student learning in the course. The questions to be solved were, "How can we provide the student with the materials and means to study as much as he wants?" ”How can we provide him with assistance at the time that he feels the need for that aid?" "How can we help the student in Mathematics 103?” "How can we provide materials and tools to study the content of the Course and to make the student feel that he is learning or achieving?" "How can we arrange to provide the assistance when the student asks for it and to provide a flexible time schedule for studying?" 68 The problem, then, is (l) to provide the opportunity to succeed in learning modern mathematics, (2) to maximize the improvement in the attitude toward mathematics, and (3) to develop a mind set or readiness toward the audio—visual techniques of teaching through the student's personal expe— riences with the techniques. The Sample In November, 1969, approval for the study was obtained from the administration at Schoolcraft College, with a marked degree of interest in the idea of innovation of this technique (audio-tutorial) in some of the other mathematics classes. The Mathematics 103 classes for the winter semester of 1969— 1970 were, therefore, designated the sample for the study. This decision was made on the basis of the selection of this sample from the population of all students who have enrolled, were enrolling, or will enroll in this course at this college. At the time of this decision, the printed college time schedule for the winter semester was circulated to faculty, students, and community, with pre-registration already com— pleted. The classes had to be accepted as they developed at final registration, since they were scheduled for eight o'clock, nine o'clock, and ten o'clock. The experimental group, randomly selected, was the ten o'clock class. The determination was made before the registration process was completed but was not announced until the first day of class—— when laboratory arrangements were announced. Class lists 69 were not yet available and, therefore, there was no fore- knowledge of the characteristics of the groups. The class enrollments are limited to thirty students, with the exception that the instructor may give approval for admitting a student to his class, even if it has been announced as closed. Information on the Mathematics 103 classes was obtained at the second class sessioniimm1a math card [the current card is an adaptation of a math card used at Delta College, which was in turn a revision of the original version used at Bay City Junior College]. [See Appendix E]. These classes had the enrollment pattern that is shown in Table 5. TABLE 5.—-Number of Students Enrolled in Mathematics 103 by Sections in Winter of 1970 at Schoolcraft College. Class Initial Addsa Dropsb Final Enrollment 8:00 23 1 3 ' 21 9200 27 5 3 29 10:00 25 6 AC 27 aAdds must be completed within one week. bDrops must be completed within four weeks not to become a part of the student's record. COne drop was much later than the fourth week (WP was grade given); drop was due to difficulties unrelated to this study or to school. 70 The final enrollment of Mathematics 103 was seventy— seven students, with two groups of twenty—one and twenty—nine students to be the control groups and the section with twenty— seven students was the group for the experimental or audio— tutorial mode of instruction. The students who dropped the course in the first four weeks transferred to the non—credit course, Mathematics A5, or to courses in other departments, with the exception of the one student. That student, in the experimental group, was achieving his goals and his average was well above passing when he was forced by an emergency situation to drop out of the class, or to miss so many classes that his work would become unsatisfactory. The data on the sample characteristics were compiled into a table for easier reference and comparison of the group characteristics. [See Table 6.] From the data in Table 6, the largest groups of students would be: (1) in a ratio of 17 to 20, from a public high school and single; (2) in a ratio of 3 to A, under twenty years of age and directly out of high school, and (3) in a ratio of 3 to 5, college freshmen and from suburban high schools. This information from Table 6 made evident the individual differences of the students within each group that are apparent to educators. The students, or groups of students, that have one or more of the following character- istics are in a minority group in Mathematics 103: (1) male; (2) married; (3) older; (A) sophomore; (5) from an urban 71. :.mm c.e5 m.sa 5.mm m.mo m.am H.mm a.:o 5.2m m.m5 p.mH 3.2m 0.35 o.wm 55 w 0.2m 0.05 o.mH o.wm o.wm o.mm 0.2m o.oo 0.:m o.w5 0.:H o.wm o.w5 c.mm om mo+Ho m.mm. $.55 m.wa m.am 3.05 w.mm o.5m o.mo m.mm H.25 m.mH m.aw 5.wm m.mm 5m m o.Hm o.mm m.ma m.mm H.mm m.5m :.H: m.wm o.Hm o.mm 5.0m m.m5 m.m5 H.2m mm mo m.:a 5.mm m.m m.om m.o5 m.mm w.mm m.©5 m.:H 5.mw w.: m.mm o.Hm o.mH Hm Ho owned poosfia opo>flpm oflapsm oeznsm cons: oLoEoceom essences Loofio somcsow ceases: onCHm masses was: 2 dzoso Hoosom swam maze zpflo mmmHo ow< mspmpm xom Some owoaaoo CH Hopflpwz ca ooaaoscm Hoosom has: ooocoooa .Azpcop umopmoc one on mucoopod :HV .o5mH .popmoEom newsflz .owoaaoo pumeoaoosom .moa wOHBMEocpmz CH ooaaopcm mucoospm mo moflpmfipopompmnoul.m mqmflwv Q5ma .popcflz .moH moflmeoflme CH Umaaoecm mQCoUSPW he Umpooaom MHSOHLLSQ Lm%m£dhell.5 mqm Uw 320 Q mo D 2 go D sz DEM :23 am: Z QSOLU .Agpcop pmosooc on speooeod SH co>fimv Q5mH CH ooaaopcm meoUSpm an poemcmee pom oobooflom .sopcsa .moa mosoososeoz owoaaoo Loecomnl.w mqmcooll.H enemas .oH eo eoapmfl>oo osmocmpm ocm om mo some w Sufi: me>LOpCH CH mopoom mmmao CH xcmm ooppo>coo _ _ i _ di- .11— C _ _ 05 so or or am 05 Lo om a: om 05 we gm 0: om _ _ _ _ _ m _ o squepnis jo aeqmnN 96 control groups had seven in the 60—6A interval and the audio— tutorial group had three. The scores are standard scores. The control groups, therefore had ten students who were more than one standard deviation above the mean, whereas the audio—tutorial group had only three in that interval. A review of the characteristics of these groups, shown in Table 6, increased the basis for comparison of the two modes of instruction. A summary of the characteristics of the groups was established from the data collected: 1. The largest group of students was female, single, less than twenty years of age, and admitted directly from a public, suburban high school. 2. The characteristics of the other groups would yield the intersections of the sets of students who would be members of the sets above and of the complements of those sets. 3. The Converted Rank in Class score (CRC) mean scores for the groups showed that the ranges of the control groups were wider, with higher and lower values than those of the experimental group. A. In the Converted Rank in Class scores (CRC) the mean score of the experimental group (E+) and the single mean for the students in the two control groups are very close in value. 5. The sample mean score of the Converted Rank in Class scores and the mean score of the E+ group are the same (52.86A). The mean score of the Cl group is above and the 97 mean score of the 02 group is below this average value (52.86A). In addition to the characteristics of the groups, the correlations of the Tests and the Converted Rank in Class scores (the co-variable) were important to a clearer under- standing of the analyses that were to be performed. These correlations were placed in Table 11. Garrett suggested testing for significance of correlations in two different ways. His second method of testing for significance of the correlation of two variables was used: (I) the number of subjects and the number of variables were determined; (2) the Table J was entered for two variables and (n—2) degrees of freedom; and (3) significance was determined whenever the r—value in the Table J was less than the r-value being tested. All r—values are significant in the table, at .05 level, except in Test 1, C and E+, in Test II, C2, and in 1 Test III, C2. Some of the r-values are significant at the .01 level, as marked in the table. These values indicate the rejection of the null hypothesis—-no significance in the correlation of the Tests and the Converted Rank in Class SCOI‘SS . 1’ C2, and E+, the correlations of the scores on the Tests and the Converted Rank in Class (the For groups C co-variable) scores were placed in Table 12 in order to evaluate the resultant effect on the Test correlations when 98 TABLE ll.--Group Correlations of Test Grades and Converted Rank in Class Scores. Group N df Test 1 Test 2 Test 3 Exam 96 95 96 Cl 21 19 0.3379 0.6A88 0.5717 0.6A81 02 29 27 0.3897** 0.1895 0.3057 0.38l2** E 27 25 0.AIA7** 0.A987* 0.5Au3* 0.5A81* C1+C2 50 A8 0.3715* 0.370l** 0.378396 0.A52A* El 23 21 0.3722 0.A51A** O.A16A** 0.5359* 96* 99* * 'X' T 77 75 0.2801 0.2700 0.3A98 0.3701 TI 73 71 0.2909** 0.2866** 0.3671* 0.373A* * Significant at .01 level.1 ** Significant at .05 level. the data were controlled for the co-variable. The correla— tions in Table 12 are significant according to the chart supplied in Garrett's book.2 The resultant correlations, when data were controlled for the co—variable, were put in Table 13. The entries in this table are somewhat smaller than the entries in Table 12, showing the effect of the co—variable was not as great as had been anticipated from the pilot study. 1Henry E. Garrett, Statistics in Psychology and Education, Ath ed. (New York: Longmans, Green, and Company, 1955), pp. 397, 437-439. 2Garrett, ibid. 99 TABLE l2.-—Matrix of Correlations of Tests and Converted Rank in Class Scores (Groups Cl’ C2, and ET).1 T Test 1 Test 2 Test 3 Exam CRC Test 1 1.0000 Test 2 0.5066* 1.0000 Test 3 0.6827* 0.6618* 1.0000 9(- 96 99 Exam 0.6863 0.6279 0.7707 1.0000 969(- 9696 96 9E CRC 0.2801 0.2700 0.3A98 0.3701 1.0000 Degrees of freedom = 70 X Significance at the .01 level (r>0.302)l XX Significance at the .05 level (r>0.232)l TABLE l3.——Matrix of Correlations witIICovariable,Converted Rank in Class, Eliminated (Groupsc: C and ET).2 l’ 2’ Test 1 Test 2 Test 3 Exam Test 1 1.0000 X Test 2 0.A662 1.0000 X X Test 3 0.6502 0.6289 1.0000 X X X Exam 0.6533 0.5903 0.7368 1.0000 Degrees of freedom = 69 X All significant at .01 level (r>0.302)2 lGarrett,ibid. 2Garrett,ibid. 100 For groups C1’ 02, and E, the correlations of the scores on the Tests and the Converted Rank in Class (the co-variable) scores were placed in Table 1A in order to evaluate the resultant effect on the Test correlations when the data were controlled for the co-variable, for this group also. The correlations in the Table 1A are significant, as marked, according to Garrett. The resultant correlations were put in Table 15. Here again, the entries are less than those in the original Test correlations, before controlling for the co-variable. TABLE lA.-—Matrix of Correlations of Tests and Converted Rank in Class Scores (Groups CI’C2’ and E). Test 1 Test 2 Test 3 Exam CRC Test 1 1.0000 X Test 2 0.51A9 1.0000 X X Test 3 0.66A8 0.6961 1.0000 X X X Exam 0.6822 0.6A23 0.78A0 1.0000 XX X X X CRC 0.2909 0.2866 0.3671 0.373A 1.0000 Degrees of freedom = 7A 96 Significant at .01 level (r>0.302)l 96 Significant at .05 level (r>0.232)l lGarrett,ibid. 101 TABLE l5.--Matrix of Correlations of Tests with Co-variable, Converted Rank in Class, Eliminated (Groups CI’C2’ and E). Test 1 Test 2 Test 3 Exam Test 1 1.0000 X Test 2 0.A707 1.0000 X X Test 3 0.6270 0.6630 1.0000 X X X Exam 0.6A63 0.6023 0.7A98 1.0000 Degrees of freedom - 73. X All significant at .01 level (r>0.302)l With the methods of analysis decided and with knowledge of the groups involved in perspective, the results of the analyses were evaluated. Comments on these evaluations were reserved for the end of the chapter. The Findings Hypothesis 1 No difference was found in the achievement as measured by the mean scores of the groups on Test I, at the .05 level. The mean scores of the groups were computed on the missing data statistics program (MDSTAT) and were compared for values. From Table 16 it was obvious that the mean scores in order of magnitude, from the largest to the smallest, were: lGarrett, ibid. 102 ET, E, Tf, T, C , l(C + C ), and C . The range of the mean 1 2 l 2 2 score values was lA.9 points. The audio—tutorial group (both ET and E) mean score of performance exceeded the mean score of performance of all the sample (TJr or T) and every other group. The standard deviations for these groups were studied to estimate the variation of performance of each group. The standard deviations are given in Table 17. T For the groups C1’ C2, and E , the analysis of variance showed that there was significance in the difference of the mean score of achievement of the students on Test 1 at the .0107 level and therefore the null hypothesis was rejected for Test 1. The data for this decision were put into the one—way analysis of variance, as printed by the computer. Appropriate data, which were selected from the computer print- out (FINN), were augmented from the missing data statistics program (MDSTAT) to form special tables for Test 1, presented here for an easier interpretation of the hypothesis testing. The augmented data are marked off in Table 18 within the dotted lines. The Scheffé Post Hoc comparisons technique was used to evaluate the contrasts already set up, (1) E+—%(Cl + C2) and (2) Cl - C2. The result obtained was that significance was in the E+—%-(Cl + C2) comparison at .025 level, since zero is not a value included in the interval of 13.009il2.303. The comparison is significant at the .05 level but not at the .01 level. 103 TABLE l6.——Mean Scores for Groups in Mathematics 103 on Tests, Examination, and Average, Winter Semester, 1970. Final Group N Unit I Unit II Unit III Exam Averagea Cl 21 69.257 61.900 52.800 7A.0A8 66.A11 C2 29 6A.690 59.186 55.200 63.655 61.277 E 27 78.193 56.363 71.378 69.1148 68.8% CA 50 66.608 60.326 5A.192 68.020 63.A33 E“r 23 79.617 ' 60.352 76.617 73.913 72.882 T 77 70.670 58.937 60.218 68.Al6 65.333 T+ 73 70.707 60.33A 61.258 69.877 66.All a . . . . Average grade is defined as sum of examination grades [final exam is weighted as two tests] divided by five. TABLE l7.——Standard Deviations for Groups in Mathematics 103 on Tests and Final Examination, Winter Semester, 1970. Group N Unit 1 Unit II Unit III Exam Cl 21 15.617 17.532 1A.015 16.008 C2 29 21.122 19.82A 21.276 25.9AO E 27 12.905 20.387 19.812 17.8A7 CA 50 18.965 18.758 l8.AA6 22.71A El 23 12.935 18.1A8 16.268 13.561 T 77 17.12A 19.A37 19.033 20.875 TT 73 17.3A2 18.669 17.898 20.000 +A students deleted - did not complete course requirements. CA = (C1 + CZ) 10A The analysis of covariance, the co—variable of Converted Rank in Class scores being controlled as the independent variable, showed that there was significance in the difference of the mean score of achievement of the groups on Test I at the .00Al level. The null hypothesis was rejected, again, for Test I. For the analysis of covariance, with augmented data, see the Table 19. .1. The Scheffé Post Hoc technique yielded the E —32L—(cl + C2) contrasts as the significant difference in teaching techniques. .1. The only other comparison to yield significance was the E —C2. The achievement of the audio—tutorial group was more effective on Test 1 than the achievement of the average of the control groups as measured by the mean scores of these groups: C1’ + C2, and E . For the groups, 01’ C2, and E, the hypothesis of no difference in the achievement as measured by the group mean scores was tested by the analysis of variance and resulted in significance in the difference of the mean scores on Test 1 at the .0151 level. The data for the computation were pre- sented in Table 20 which has been augmented in same manner as in Table 18. The analysis of covariance showed significance in the differences of the mean scores at the .00Al level and, there— fore, the null hypothesis was rejected for the difference in achievement on Test I. The Scheffé technique resulted in 105 TABLE l8.——One-Way Analysis of Variance for Test I (Groups C C and El). l) 2, 7‘ _”—SUR°_'_' Source df I of l Mean F p Squares l Squares Ratio less than 1 —e Between 2 l 2920 2752 : 1A60.1376 A.8553 .0107a l l I I 21051.1A90 300.7307 ‘ 'Total 72 23971.A2A2 I ——__-——_—_——_~_——___—_ -— L-:lAugmented data. areject null hypothesis. TABLE l9.—-One—Way Analysis of Covariance for Test I (Groups C C and EI). 1’ 2’ iWISEIII Source df : of Mean F p l Squares Squares Ratio less than [ t l 1 Between 2 3357.7994 ; 1678.8997 5.9716 .OOAIa Within 69 19680.3180 281.1A7A 7 _. l I ; _______ ___.- l —— — ———-—- _..—- __ _ :Total 71 23038.117A ._ __ __ _— _. — _ —.- _ _.—.— .— __-—————— _—- tiiflAugmented data areject null hypothesis Covariable (CRC) eliminated. the comparison, E—%(Cl+C2), being the significant difference at .025 level for the interval of 11.585i11.52A. Zero was not included in the interval. The data for the analysis of covariance were placed in Table 21, with augmented data in the dotted line portion of the table. 106 TABLE 20.—-One—Way Analysis of Variance for Test I (Groups 0 C and E). 1’ 2 I“ “"sum‘ I Source df , of i Mean F p . Squares f Squares Ratio less than L. I I Between 2 I 2607.00AA I 1303.5022 A.AA52 .0151a I “““““ I Within 7A | 21699.6268 293.2382 I _‘-_——5.—-——— “Total 76 2A306.6312 —m—_————_—o.~—_-——-_ [IIIlAugmented data. areject null hypothesis. TABLE 21.--One—Way Analysis of Covariance for Test I (Groups C C and E). 1’ 2’ I""SuH‘—'—I Source df I of 1 Mean F p I Squares I Squares Ratio less than J J I . Between 2 : 3233.A6l2 :1616.7306 5.9A18 .00Ala Within 73 l2013A.9856 272.09AA I f—_ — M fl_— .7. | ITotal 75 23368.AA68 I -—-——~-—-—---—- _———— ——. - L-_3Augmented data. areject null hypothesis. Covariable (CRC) eliminated. Hypothesis 1: The Decision The hypothesis of no difference in achievement of the groups as measured by the mean scores on Test I was rejected. The achievement of the audio—tutorial group was significantly better on Test I than the average achievement of the control 107 groups, but not significantly better than each group. (E+ did perform better than C2). Hypothesis 2 No difference was found in the achievement of the groups as measured by the group mean scores on Test II, at the .05 level. The group mean scores showed a reversal of position from that shown in Test I; for the rank order, in test 11, I I from the largest value to the smallest, is: 01’ E , T , C C T, and E. The range of the mean score values was A, 2’ 5.537 points for Test II. The mean score values for all groups on all tests were placed in Table 16. C and E1L the analysis of variance failed For groups C1’ 2, to yield results to reject the null hypothesis of no signi— ficant difference of group mean scores at the .05 level since the rating for p was .8795 on Test II. The analysis of variance for Test 11 was included in the analysis of variance of all tests on the computer print—out. The FINN computer print-out for all tests was placed in Table 22. A compari- son of this table with the unmarked portion of Table 18 showed the data for the decision—making process, for each of the succeeding decisions, in turn. The FINN print—out was augmented with data from the (MDSTAT) program, shown by dotted lines. I For groups C1’ C2, and E the analysis of covariance results did not indicate rejection of the null hypothesis for 108 the difference of the group mean scores on Test II at .05 level, as p was less than .8768, as shown in Table 23. TABLE 22.-—One-Way Analysis ofTVariance (Groups 01,02, and E ). Mean Squares F p Variable Between Within Ratio less than IM_-_-I Test I 1A60.l376 i300.7307 I 4.8553 0.0107a Test II AA.8563 I3A8.5338 I .1287 0.8795 | I Test III 3996.2627 l320.3520 I l2.A7A6 0.0001a I Exam 931.2801 IA00.15A7 . 2.3273 0.1051 Degrees of freedom for hypothesis = 2. Degrees of freedom for error = 70. areject null hypothesis. TABLE 23.--One—Way Analysis of Covariance (Groups CI’C2’ and BI). Mean Squares F p Variable Between Within Ratio less than Test I 1678.8997 [28I.1A7A7: 5.9716 0.00A1a Test II A3.l850 £327.90A3 I .1317 0.8768 Test III A52A.5682 I285.2350 I 15.8626 0.0001a I I Exam 10A0.5328 I 350.3595_J 2.9699 .0580 ————~ -— Co—variable (CRC) eliminated. Degrees of freedom for hypothesis = 2. Degrees of freedom for error = 69. areject null hypothesis. 109 For groups 01’ C2, and E the results obtained for Test II with the analysis of variance at .05 level was failure to reject the null hypothesis at .05 level, with p less than .6188. The data from the (FINN) print-out were presented in Table 2A. The analysis of covariance results did not indicate rejection of the null hypothesis at .05 level (.835A). The data from the analysis of covariance print—out (FINN) were presented in Table 25. Hypothesis 2: The Decision The hypothesis of "no difference was found in the achievement of the groups as measured by the group mean scores on Test II" cannot be rejected at this time nor from these data. The decision at the present is failure to rejgct. The result of this failure to reject would be mislead- ing, if left without comment. The events occurring during the teaching of Unit II have had a greater effect on the audio-tutorial group than was anticipated. It was important to note that the lack of significant difference which resulted in the failure to reject the null hypothesis was not caused by the audio—tutorial laboratory mode. The variables that were interjected into the study at that time were, almost certainly, the factors responsible for the shift in the achievement in all the groups. One of these factors was the sequencing and programming of Unit 11. It was not as thorough as that for Unit I and Unit 111. Units I and III have been 110 TABLE 2A.—-One-Way Analysis of Variance (Groups C C2, and E). 1’ Mean Squares F p Variable Between Within Ratio less than Test I 1303.5022 [29312382] A.AA52 0.0151a I Test II 182.530A I377.8315: .A83l 0.6188 Test III 262A.1939 :362.2626i 7.2A39 0.001Aa Exam 668.89A9 LA35LZ62li 1.5350 0.2223 Degrees of freedom for hypothesis = 2. Degrees of freedom for error = 7A. areject null hypothesis. TABLE 25.—~One—Way Analysis of Covariance (Groups 0 C and E). l, 2, Mean Squares F p Variable Between Within Ratio less than Test I 1616.7306 {272709AA—I 5.9A18 0.00Ala Test II 63.3790 I351.5197 E .1803 0.835A Test III 3329 0536 :317.7366 I 10.A77A 0.0001a Exam 692.3521 I380.1A17 I 1.8213 0.1691 ___—..___—_— Co—variable (CRC) eliminated. degrees of freedom for hypothesis = 2. degrees of freedom for error = 73. areject null hypothesis. used previously and revised in another form. [See Appendix G]. The amount of material covered in Unit 11 (concepts) and the 111 complexity of the material was even more difficult for the students than had been anticipated. The second factor was that the Mathematics 103 classes had to be cancelled for one week and this cancellation occurred during Unit 11. All of the classes were affected differently by this loss of class time. Hypothesis 3 No difference was found in the achievement of the groups as measured by the group mean scores on Test 111, at the .05 level. The mean scores of the groups, when arranged in rank order from the largest to the smallest value, were: ET, E, T+, T, C2, CA’ and Cl: the range of these mean scores was 23.817 points. The mean scores were placed in Table 16. For the groups C1’ C2, and Er the analysis of variance showed significance at the .0001 level in Table 22 and, when the Scheffé Post Hoc comparisons method was applied, the contrasts that showed significance in the difference of group mean scores were: I l l. E - 2(01 + 02) with the interval, 22.A25i1A.29l. 2. E+ — G1 with the interval, 23.817il6.A37. 3. E+ — C2 with the interval, 21.Al7il5.760. The C1 - 02 comparison did not prove significant, since zero was in the interval, which meant that when Cl — 02 equalled zero, then C would equal C and the null hypothesis should 1 not have been rejected. 23 112 The results of the analysis of covariance showed signif— icance for Test III at the .0001 level and the comparisons above were again significant, though the values were not the same for the intervals. The data were placed in Table 23. For the groups C1’ C2, and E the results of the analysis of variance showed significance at the .00Al level on Test 111 in the Table 2A. The Scheffé Post Hoc comparisons gave the E — %(C1 + 02), E — Cl’ and E — 02 as significant, but again Cl - C2 was not a significant comparison. The results of the analysis of covariance showed a significant variance at the .0001 level. The comparisons listed above were the signifiant comparisons for Test 111. The data for the comparisons were shown in Table 25. Hypothesis 3: The Decision The null hypothesis of "no difference in the achieve— ment of the groups as measured by their mean scores on Test III, at the .05 level," was, therefore,rejected (.0001). Hypothesis A No difference was found in the achievement of the groups as measured by the group mean scores on the final examination. The group mean scores, when arranged in rank order, from the largest to the smallest, were: Cl’ E , T , E, T, CA’ and 02. The range of mean score values was 10.393 points. 113 The results of the analysis of variance for groups 01’ 02, and Er showed no significant differences for the final examination with a value of p less than .1051 in Table 22. The results of the analysis of covariance for groups .1. C1’ C2, and E final examination with a p at the .0580 level, in Table 23. also showed no significant variation for the For the C1’ C2, and E groups the results of the analysis of variance and the analysis of covariance showed no signif— icant variations with values of .2223 and .1691, respectively, in Table 2A and 25, respectively. The group mean score of Cl resulted from a reaction to the scores the students received on Test III. Those scores had increased the student's need to raise his average, in order to pass the course. Hypothesis A: The Decision The decision was failure to reject the null hypothesis of no difference in achievement of the groups as measured by the group mean scores on the final examination. The data would not at this time permit rejection. Hypothesis 5 No difference was found in the achievement of the groups as measured by the group mean scores on the Tests (including examination), crossed with age, at the .05 level. From the table below it was apparent that age was a factor of the variation in achievement of the groups and from 11A the mean scores it was apparent the group mean score for the older students was significantly better than the group mean score of the younger students. In every test category the Converted Rank in Class mean score for the older students was less than the mean Converted Rank in Class score for the younger students in the groups C1’ C2, and EI. From Table 26 it was seen that the older students performed better on Test III and for the groups C1’ C2, and E the performance on the examination was also significant to these groups, older and younger students. TABLE 26.——Summary of Decisions to Reject. Group Test Means p less than Contrast Difference (O—Y) Anova Ancova C C E+ III 11 00 0 0 A a 0 0A6Aa 0 Y 1:) 23 07 7 o 5 9 o "' Cl,C2,E III 12.8285 0.0336a 0.0298a 0 — Y 01,02,E Exam 11.8016 0.0569 0.0AAAa 0 — Y areject at .05 level. The mean scores of the older students were consistently higher on the tests than the mean scores of the younger students, as was shown in comparing the mean scores; the results were placed in Table 27 below. 115 TABLE 27.-—Differences of Mean Scores on Tests Groups Con— trolled for Factors.a Hb—‘u __ 1_ 1--.“. -—~ 1- —' --_. _-_., u ..—.— —-.____.——i Ageb Lapse—Direct Urban—Suburban Test Older- Younger I . ET E E E ET E Test I + 3C + 2C + 2 3 3 3 d d Test II + 7 + 9 +13 +12 A 7 Test III +12C +13C + 5 + 5 5 8 Exam +11d +12d + 7 + 8 l 3 aPrivate, public high schools differed by only one point. bGrouping in 5 levels by age resulted in Group 11 mean score being the highest consistently (23 to 27 years of age). c Mean spore above mean score of control groups; below mean score (E'). d . Mean score above all mean scores in Test, groups uncrossed. Hypothesis 5: The Decision The null hypothesis of no difference in achievement of groups as measured by the mean scores of the groups on the Tests, crossed with age,was rejected for Test 111 for both sets of groups 01’ 02, and E+ and Cl’ C2, fore, for Test 111 the fourteen older students achieved a and E; there— significantly higher mean score than did the sixty—three younger students. The older students were defined as more than twenty years of age and/or more than two years from high school graduation. 116 Hypothesis 6 No difference was found in achievement of the groups as measured by the group mean scores on the Tests, crossed with "lapse/direct from high school" classification. The differences of the mean scores for these two groups (L — D) on the tests was placed in Table 27. The group mean Converted Rank in Class score was less on each of the tests for the "lapse" group than for the "direct" group. However, from Table 28 it was seen that the "lapse" group achieved a higher mean score than did the "direct" group in both the Cl,C2,E+ and Cl’C2’ E groups. TABLE 28.——Summary of Decisions to Reject. Group Test p less than Comparison Anova Ancova Cl,C2,E+ II 0.0111a 0.0085a L — D Cl,C2,E II 0.02A7a 0.0156a L — D areject null hypothesis. Hypothesis 6: The Decision The null hypothesis was rejected for only Test II for both sets of groups with the students classified as the "lapse" group achieving a significantly higher mean score. Hypothesis 7 No difference was found in the achievement of the groups as measured by the group mean scores on the Tests, crossed with the public/private school classification. 117 The mean scores differed by only one point and the results of the analysis of variance and the analysis of covariance did not reveal significance. Hypothesis 7: The Decision The data did not show significance and the results of the analyses led to the decision to fail to reject the hypo— thesis, for this set of data. Hypothesis 8 No difference was found in the achievement of the groups as measured by the group mean scores on the Tests, crossed with the classification of urban/suburban. The differences of the group mean scores were put in Table 27. The data were used to test the null hypothesis by the analysis of variance and the analysis of covariance. No significance was revealed. Hypothesis 8: The Decision The data analyses resulted in the decision to fail to reject the hypothesis for the data collected. Hypothesis 9 No difference in the group mean rating of the students was found in the attitude of the students toward the seven (or eight) aspects of mathematics, as measured by their rating on the attitude scale, at the .05 level; or on the attitudes toward mathematics when crossed with each of the above cate— gories of characteristics of the sample, at the .05 level. 118 The missing data statistics program (MDSTAT) on the computer gave the mean scores for these seven categories. The mean scores were put in Table 29. The E group had a higher mean in all categories, except method, which referred to the textbook and class work and not to the laboratory. In all categories except the school, the ratings are above three, which means a more favorable attitude toward mathematics has replaced a less favorable attitude——the result of elementary and high school mathematics. The difference, course minus school, was a value of +.500 for the total group and +.566 for the E1” group (.592 for 02). All the differences were positive, which indicated an improvement of attitudes at the end of the course. The results of the analyses of variance and the analyses of covariance are presented in Table 30. Investigation of the analyses was extended to a different grouping arrangement to ascertain the effect of the mode of instruction as a possible cause for failure to complete the course require- ments and of the effect their deletion had on the statistical analyses. The deletion had decreased the number of rejections of the null hypotheses on attitude and, therefore, it was apparent that those deleted from the study did not dislike only the audio—tutorial laboratory mode of instruction, but still disliked mathematics so much that they could not, or would not, attend the classes for discussion, or the laboratory. Three of these students were below the mean, more than one standard deviation below on the Converted Rank in Class score. This low score meant that high school achievement had been low. 119 .Ham pm peoamoe o.eofio I psomom 039 .efl.pocs3o mason mosses bogmflsflec: ego .Empwoea sonsano B 22 .0189 —.0050 —.1835 .0053 Cl+C2 32(49) 30 —.l389 .0621 —.0819 —.0982 E 27(27) 25 .2385 .1822 .2889 .3079 .l. E 23(23) 21 .2294 .2315 .3077 .3277 None are significant at .05 level.1 [small n's] These correlations indicated that the student who needed the laboratory for assistance did spend time in the laboratory learning mathematics. In order to collect and summarize the decisions presented in this chapter as effectively as possible, the decisions have been arranged in tabulated form and placed in Table 3A. lGarrett, ibid. 126 somnmb on Hume cospos powwop 0p Hflmm Hoonom pomnop 0p afimm Cpmcoe pomhop on Hflmm wCHcowop Amo+aovmlm **mo. **pomnog pr0p Amo+HQVMum **mo. **pomwms mapsoo Amo+Hovmlm **mo. **pomnop Hmpocow mmmmmmm¢ sombflelmmamfi **mo. **pomhms sombfle\mmama x HH pmme Lowcsozupobao **:o. **poowop somcso>\pooflo x mem somcsozlpobao **mo. **poowop gmwczo>\pooao x HHH pmme poonop on Hamm Lomcso>\gooao x HH pmoe poonop co afimw Lowcsoz\poofio x H pmoe poonmp oo Hflmm mem mosm.ao-m.gmo+fiovm-m **Ho. **pomnmb HHH same summon Op Hflmc HH pmme moum.Amo+Hovm-m **mmo. **pommmb H same pcmEo>mH£o< condom Ho>oq COHmHooQ so mQSOLw coozumo monopommww oz .mQOHmHoma do spmsszmuu.zm mamas CHAPTER V SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary As early as 1961 (and again in 1966) the Committee on the Undergraduate Program in Mathematics (CUPM) stated that If present trends continue, the better ele— mentary schools may soon be teaching the rudi— ments of algebra and also some informal geom— etry. Even in the teaching of arithmetic, sound mathematical training is needed because the teacher's understanding affects his views and attitudes; and in the classroom, the views and attitudes of the teacher are crucial. To an undertrained teacher, arithmetic is merely a collection of mechanical processes and is re— garded with boredom, or dislike, or even fear. It is not surprising that, in such cases, stu— dents react to arithmetic in the same way. Children should be taught arithmetic for mean— ing and understanding as well as for skills. To teach in this way, a teacher needs to have a kind of training which conveys this under— standing and also shows mathematics to be re— warding and worthwhile. The teacher cannot give something which he does not have.1 Mathematics through its leaders, literature, studies, conferences, workshops, and education has, within recent years, focused attention on the need to provide prospec- tive elementary school teachers with effective training l . . . . I . Mathematical Assoc1ation of America, Recommenda— tions . . . Training of Teachers, p. 5. I27 128 so that the teachers will be able to develop important mathematical skills and concepts in their pupils. These teachers must also have a favorable attitude toward mathe— matics, if they are not to influence their pupils in an adverse way, toward a subject which is needed so much now, and will be needed even more in the future. The need for strong effective courses in mathematics for elementary school teachers was stressed by the Com— mittee on the Undergraduate Program in Mathematics in its 1969 report: Most prospective elementary teachers are not highly motivated toward scientific and mathemati— cal studies and are apt to be less well prepared than many other students. But if we are to offer to children in the elementary grades a good mathematics program, indeed, if we wish to have the current commercial elementary textbook series well taught, we must manage to persuade these students that mathematics is an important disci— pline which they can understand. It is impera— tive that they begin studying the recommended two years of mathematics early, before they have lost too much contact with their earlier training, and in time to strengthen their mathematical back— grounds for other courses in their programs.1 In the winter semester of 1970, at Schoolcraft Col— lege, a community college in Livonia, Michigan, the seventy—seven students who enrolled in Mathematics 103 became the sample for this study to evaluate the effect of an audio—tutorial laboratory for this course. By random selection the ten o'clock class became the 1Mathematical Association of America, A Transfer Curriculum . . . Two Year Colleges, p. 36. I29 experimental group and the eight and nine o'clock sections became the control groups. As prospective elementary school teachers these stu— dents would be expected to teach arithmetic. They would also communicate their true attitudes toward arithmetic to their pupils, even when they were attempting to conceal antipathies toward arithmetic. When their pupils sensed the real attitudes of these teachers they would become even more concerned about their arithmetic. In order to avoid this situation it was necessary to develop in the Mathematics 103 students a sense of security and satis- faction in mathematics and to increase the motivation in these students to a level at which they could and would perform to the best of their abilities without, at the same time, developing the extreme anxieties and frustra— tions which would interfere with learning and achievement. This was, and is, one of the challenges of instructing the mathematics classes for prospective elementary school teachers. Two methods for improving the mathematical training of these students were recommended: 1. The addition of another course, a Mathematics 10A, to follow the current Mathematics 103 in order to increase the breadth of knowledge. 2. 130 The improvement of the learning and achievement in Mathematics 103 in order to increase the depth of knowledge. Mathematics 10A will be offered in the winter semester of 1971. The remaining source of improving the mathematical training of these students, at the community college level, therefore, the improvement of the learning and achievement in the current course. The questions that we, then, had to solve were: 1. How can we provide the student with the mate— rials and means to study as much as he wants? How can we provide him with assistance at the time that he feels the need for that aid? How can How can student to feel How can we help the student in Mathematics 103? we provide materials and tools for the to study the content of the course and that he is learning and achieving? we arrange to provide the assistance when the student asks for it and to provide a flexible time schedule for studying? The selection of the audio—tutorial mode of instruc— tion was an attempt to adapt a technique which has been considered successful in the field of biology by leading educators, such as Postlethwait, Novak, and Coladarci, to the requirements of another discipline, mathematics, and especially to Mathematics 103. There has been and is a 131 great need to improve the amount of learning achieved in Mathematics 103; in the last ten years the amount of mathematics required by the prospective elementary school teacher has more than doubled. The materials (the audio— tapes and slides, and problem sheets) were selected for the laboratory. The students were, thereby, allowed to practice with these tools of instruction before they had their teaching experiences. The cost of the audio— tutorial materials was reasonable and the laboratory was set up in the library listening room. Audio—tapes, slides and problem sheets were easiest to revise, and maintenance of these materials would be relatively simple. The problem, then, was (1) to provide the oppor- tunity to succeed in learning modern mathematics, (2) to maximize the improvement in the attitude toward mathe— matics, and (3) to develop a favorable mind set, or readiness, toward the audio-visual techniques of teaching through the student's personal experiences with the tech— niques. The purposes of the study were: 1. To compare the effectiveness of the audio— tutorial materials and laboratory with the con- ventional materials and commercial textbook for the Mathematics 103, the mathematics for pros— pective elementary school teachers. 132 2. To compare the effect of the study on the atti— tudes of these students toward selected aspects of mathematics. The two control groups attended the usual informal lecture, question and answer, problem—solving and testing sessions, three days a week. The experimental group had the same schedule for two days a week. The third class session was cancelled and the students signed up for at least one hour a week for a laboratory session, where they used audio—tapes, slides, posters, and problem work— sheets designed for the laboratory. All materials were interrelated and were prepared to meet the behavioral objectives for the concepts covered in that unit. [Appendix F]. Supplementary aids and devices were also available. The data for the study were collected in three ways: 1. At the second class session the students filled out the math card which was handed out at the beginning of the hour and collected about fifteen minutes later. [Appendix E]. 2. At the end of each unit, in which the three sec— tions covered the same concepts, a test was given on the concepts which had been listed on the review sheet and available one week before the day for the test. The evaluation of these tests was returned to the student at the next session, following completion of the grading 133 process. At the end of that class hour the tests were col— lected again for further study and analysis. 3. The attitudinal survey was completed by the stu— dents in two parts; the first part was handed out the last day of class and was returned to the desk before the stu— dent left the classroom, and the second part, the open ended comments, was handed out at the examination period and was returned with the examination paper. [Appendix C and H1. Analysis of the Data Analysis of the data has revealed that the audio— tutorial mode of instruction did improve the achievement of the Mathematics 103 students in two of the four tests, and the improvement was enough to raise the average grade for this class above the average grade for either of the control groups. On these two tests the audio-tutorial group achieved a significantly higher mean score than did the combination of the two control groups. When the tests for achievement were crossed with other factors, the fac- tors that showed effect were: (1) age, the older students were more successful on Test III and on the examination, than were the younger students, the E — l/2(C1 + C2) group of older students; and (2) lapse, the ”lapse” students who had not entered college directly from high school were the more successful group on Test II. 13“ The attitude survey showed a small improvement in the direction of a more favorable attitude toward mathe— matics, as defined the difference between attitude in school and attitude at the end of the course, so that the present attitude is more favorable toward mathematics than it was at the beginning of the course. The significant correlations of attitude and achieve— ment, in the categories of general, course, and total were in the control groups; apparently the experimental group was so involved with the audio—tutorial method that it was not significantly concerned with other aspects of mathematics, or possibly so satisfied to be achieving that the other aspects did not affect them. All groups favored the audio—tutorial laboratory, and did so in the following order: 1. The experimental group (27 of the 27 students responded), having used the laboratory, favored the audio— tutorial method the most heartily with an average of 4.609 out of 5.000 points. 2. The second control group (2A of the 29 students responded) was decidedly in favor of the audio—tutorial method. The students having more difficulty in learning were apparently of the opinion that this approach would have helped them to improve their achievement and learn- ing. (Some of these students want to take Mathematics 10“.) 135 3. The first control group (8 of the 21 students responded) favored the audio—tutorial method, but more of these students had learned how to study in high school (as can be determined from the Converted Rank in Class scores in Table 10 and the mean score of the group in Table 16). These students preferred an optional labora— tory set up so that they could go to the laboratory only if and when they were in need of assistance. Discussion The audio—tutorial mode of instruction is an ex- cellent technique for individualizing instruction in order to meet the needs of the student. The adaptation of this method for future use in mathematics classes should be an effective means of improving instruction for many reasons. The first and strongest reason, from the pedagogical stand- point, is the almost unanimous acceptance with which the audio—tutorial method was received by the students. The students who used the instruction presented in this way rated it highly on the survey and recommended it highly, suggesting (l) the future use of the audio—tutorial method in this and other mathematics classes and (2) the extension of time for the laboratory. The students in the control groups expressed their desire for and need of the audio—tutorial method by their ratings and comments in the survey and from the negative correlations between 136 their achievement and their attitudes toward the labora— tory. The students with the poorer grades wanted the laboratory more and rated the laboratory higher. The second reason for the effectiveness of this method, which is important for these students, is that the improvement of their attitude toward mathematics was ef— fected by the use of the audio—tutorial mode of instruc— tion. That the significantly more favorable attitude toward mathematics of the experimental group was the re— sult of the method of instruction is apparent from the time spent in the laboratory (average time, 2 hours, and maximum time, 1“ hours per week) and from the student comments. [See Appendix H]. The audio—tutorial mode of instruction provided flexibility of scheduling and was very effective for those students who wanted to learn and were given the opportunity to use the assigned laboratory. 0f the three sections the experimental group seemed to be the class that had most of the unusual problems and emergencies to contend with. The flexibility of the laboratory hours allowed many of these students to make up their regular laboratory hours and to learn the mathematical concepts involved without the need for additional or unusual con- sideration by the instructor. Learning was achieved despite absences due to such traumatic situations as (l) a serious auto accident, (2) being kicked in the head by a 137 horse, (3) illness of a son, or (A) a funeral. These situations created tensions for the students involved and the availability of laboratory time and assistance made the adjustments for these problems somewhat easier. The course did not add to their burdens (nor the instructor's) in the way it would have done if they had been in a con— trol group. The audio—tutorial mode of instruction was instru— mental in the significant improvement of achievement by the experimental group in both Test 1 and Test III. The improvement of achievement was only slightly less than significant in the other two tests as a p of .0580 on the examination would indicate. However, this lack of significant effectiveness was not the result of the audio—tutorial mode. 0n Unit II two uncontrolled vari— ables were introduced into the study. First the sequence and program for the unit needed further analysis and evaluation. Second, and even more influential to student achievement in this unit, was the necessity to cancel classes for one week. On the final examination one uncontrolled variable was introduced into the study. The low scores the control group had received on Unit III greatly increased the normal need to achieve, which the students viewed as a challenge. The other two groups had no such motivation. 138 The audio~tutorial method provided the ”older" stu- dents an opportunity to spend time in the laboratory, re- freshing their previous knowledge of mathematics and in— creasing that knowledge to the level they desired to achieve. (College students set their goals and achieve— ment patterns, that is the grades they will settle for.) These students want to achieve but are hesitant about ask- ing for assistance. With this method they do not feel that they are creating any unusual problems for the in- structor. Teachers and administrators will find this audio- tutorial mode of instruction an effective technique for teaching. Student motivation and interest in the subject matter is much improved and therefore of less concern to the instructor (and administrator). With this technique the teacher can assist the individual student at the time the student requests aid. The student can repeat the concepts involved, until he feels confident that he knows the concepts. In the use of the audio—tutorial materials, with this teaching technique, the instructor can insure the student of l. the opportunity to concentrate on the lesson, without distractions. 2. the repetition of a lesson, or parts of the lesson, as he desires. 139 3. a multiplicity of materials with which to study. A. the adjustment of the concepts in the lesson for association, assimilation, and transfer. One of the advantages of using the audio—tutorial materials is that audio—tutorial materials can be revised and edited item by item, or part by part, or whole units at a time without disrupting the entire program. Another advantage is that the cost of editing the parts of the program can be kept minimal. Also, from time to time new materials can be incorporated into the program. In addi— tion, manipulative devices and inexpensive aids which are appropriate to the behavioral objectives can be utilized in the laboratory. Further, the slides can be revised and easily up—dated, singly or in groups, as needed. Finally, the audio—tapes can be erased and new recordings made on the tapes in order to improve the instruction. These "tools of the trade” are easy for the students to use. Indeed, the students should become acquainted with these tools, as educational techniques, since their pupils will certainly be aware of them, at least as a means of entertainment. 140 Conclusions In General Terms In the major portion of this study the analyses of the data collected have shown that the following hypo- theses are supported: 1. The audio—tutorial mode of instruction will (and did) increase the achievement of the experimental group significantly, as measured by the mean scores of the tests and examination for each group. 2. The audio-tutorial mode of instruction will (and did) improve the student's attitude toward mathe— matics significantly as measured by their ratings on the attitude scale. [For their comments, see Appendix H]. In Specific Terms l. The audio-tutorial method of instruction was used effectively to improve the achievement of students in mathematics for elementary school teachers. 2. The audio-tutorial method of instruction was more successful than the conventional teaching techniques for the mathematics for elementary school teachers, as measured by Test III. 3. The audio—tutorial method was significantly better than the average of the conventional techniques as measured by the achievement of these groups of stu- dents on Test 1. 141 A. The older students of each group performed better on the achievement tests for Test III and the final examination than did the younger students. After a period of adjustment to studying and to college, the older stu- dents apply themselves to their courses and they struggle to achieve. 5. The older students exhibited less tension and frustration in the audio—tutorial group. 6. The group using the audio—tutorial method was better than the average of all three groups and better than the average of the two control groups, when the un— controlled variables and Test 11 were removed from the study. 7. Sections in the same course and in the same semester, with characteristics distinctly different from one another are not unusual in education, whether in school or in college. The audio—tutorial method would allow a student to devote as much time as he felt that he needed to study the concepts included in the lesson and the unit. The teacher could devote more time to the stu— dent who needed the extra assistance. A range from one hour to fourteen and three—quarter hours per week were spent by the individual student in the laboratory during this study. 8. The student in the laboratory felt that he could progress at his own pace, repeating segments of 1A2 the program as he wished. He had assistance available when he wanted it. Recommendations The following recommendations for the further test- ing of the audio-tutorial method of instruction are in order: 1. A replication of the study with a random assign- ment of the students to the groups to verify the findings of this study. 2. The development of an audio-tutorial laboratory for mathematics courses at the basic level: arithmetic, business arithmetic, algebra, geometry, Mathematics 103 and 10“, with a study of the effectiveness of that lab- oratory. 3. The use of the audio-tutorial method to teach students how to study mathematics, with evaluation of the effectiveness of the laboratory for this use. A. For practical use, the audio—tutorial method shouldtxaused for a laboratory, with the instructor as the resource person. Selected students who have passed the Mathematics 103 and 10A course(s) could have teaching experience by serving as laboratory assistants. 5. If the laboratory of the previous recommenda— tions were set up to include other courses, the student assistant should work with the students for the courses 1113 for which he was qualified. Certainly, the work with a basic arithmetic course would increase his arithmetic efficiency, would show him through personal experience the need for knowing the basic arithmetic facts. 6. The inclusion of other inexpensive audio— visual aids and manipulative devices should be used in further study of this technique. BIBLIOGRAPHY 11M BIBLIOGRAPHY A. Books Ausubel, David P. The Psychology of Meaningful Verbal Learn— 4 ing. New York: Grune and Stratton, Incorporated, 1963. Borg, Walter R. Educational Research, An Introduction. New York: David McKay Company, Incorporated, 1963. Bruner, Jerome S.; Goodnow, Jacqueline J.; and Austin, George A. A Study of Thinking. New York: John Wiley and Sons, Incorporated, 1962. Bryant, Edward 0. Statistical Analysis. New York: McGraw— Hill Book Company, Incorporated, 1960. Campbell, Donald T., and Stanley, Julian C. Experimental and Quasi-Experimental Designs for Research. Chicago: Rand McNally and Company, 1967. Cox, D. R. Planning of Experiments. 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Mathematics Teacher, LXI (May, 1968), pp. 517-521. 151 Pigge, Fred, and Brune, Irvin H. "Lectures versus Manuals in the Education of Elementary Teachers." Arithmetic Teacher, XVI (January, 1969), pp. 48—52. Pitkin, Tony Ray. "A Comparison of the Attitudes Toward Mathematics and Toward Pupils of Selected Groups of Elementary School Teachers Who Had Different Types and Amounts of College Education in Modern Mathematics." Dissertation Abstracts, XXIX (March, 1969), p. 3025—A. Reys, Robert E., and Delon, Floyd D. "Attitudes of Prospec- tive Elementary School Teachers Toward Arithmetic." Arithmetic Teacher, XV (April, 1968), pp. 363—366. Riedesel, C. Alan. "Recent Research Contributions to Ele- mentary School Mathematics." Arithmetic Teacher, XVII (March, 1970), pp. 245—252. "Some Comments on Developing Proper Instrumentation for Research Studies in Mathematics." Arithmetic Teacher, XV (February, 1968), pp. 165—168. . "Topics for Research Studies in Elementary School Mathematics." Arithmetic Teacher, XIV (December, 1967), pp. 679~683. ., and Sparks, Jack N. "Designing Research Studies in Elementary School Mathematics Education." Arithmetic Teacher, XV (January, 1968), pp. 60-63. ., and Suydam, Marilyn N. "Research on Mathematics Education, Grades K—8, for 1968." Arithmetic Teacher, XVI (October, 1969), pp. 467—478. "Research on Mathematics Education, Grades K—8, for 1967." Arithmetic Teacher, XV (October, 1968), pp. 531—544. Roberts, Fannie. "Attitudes of College Freshmen Towards Mathematics." Mathematics Teacher, LXII (January, 1969), Pp. 25-27. Robinson, Frank Edward. ”An Analysis of the Effects of Tape— recorded Instruction on Arithmetic Performance of Seventh Grade Pupils with Varying Abilities." Dissertation Abstracts, XXIX (May, 1969), p. 3782—A. 152 Scandura, Joseph M. "Research in Mathematics Education--An Overview and a Perspective." Research in Mathematics Education. Washington, D.C.: National Council of Teachers of Mathematics, 1967, pp. 115-125. Scannell, Dale P. "Obtaining Valid Research in Elementary School Mathematics." Arithmetic Teacher, XVI (April, 1969), pp. 292-295. Schlinsog, George W. "The Effects of Supplementing Sixth— grade Instruction with a Study of Nondecimal Numbers." Arithmetic Teacher, XV (March, 1968), pp. 254—260. Sneider, Sister Mary Joetta. "Achievement and Programmed Learning." Mathematics Teacher, LXI (February, 1968), pp. 162—164. Spitzer, Herbert F. "A Proposal for the Improvement of the Mathematics Training of Elementary School Teachers." Arithmetic Teacher, XVI (February, 1969), pp. 137—139. "Statement on Self Instructional Materials and Devices by a Joint Committee of the American Educational Research Association (NEA), the Department of Audio—Visual Instruction (NEA), and the American Psychological Association." National Education Association Journal, L (November, 1961), p. 19. Suydam, Marilyn N. "The Status of Research on Elementary School Mathematics." Arithmetic Teacher, XIV (December, 1967), pp. 684—689. ., and Riedesel, C. Alan. "Research Findings Appli— cable in the Classroom." Arithmetic Teacher, XVI (December, 1969), pp. 640—642. . "Reports of Research and Development Activities, 1957-1968." Arithmetic Teacher, XVI (November, 1969), pp. 557-562. Tornyay, de, Rheba. "Instructional Technology and Nursing Education." Journal of Nursing Education, IX (April, 1970)) pp. 3‘8. Weaver, J. Fred. "Using Theories of Learning and Instruction in Elementary School Mathematics Research." Arithmetic Teacher, XVI (May, 1969), pp. 379—383. 153 White, Frances Jayne. "Observational Learning of Indirect Verbal Behavior Through the Medium of Audio—Tapes." Dissertation Abstracts, XXIX (March, 1969), p. 3030-A. Zoll, Edward J. "Research in Programmed Instruction in Mathematics." Mathematics Teacher, LXII (February, 1968), pp. 103—109. D. Newspaper Hechinger, Fred M. "Time to Teach Those Teaching Machines." New York Times, February 8, 1970. E. Unpublished Materials Alton, Elaine Vivian. "An Experiment Using Programmed Mater— ial in Teaching a Noncredit Algebra Course at the College Level [with] Supplement." Unpublished Ph.D. dissertation, Michigan State University, 1965. Boonstra, Paul Henry. "A Pilot Project for the Investiga— tion of the Effects of a Mathematics Laboratory Experience: A Case Study." Unpublished Ph.D. disserta— tion, Michigan State University, 1970. Hedley, Robert Lloyd. "Student Attitude and Achievement in Science Courses in Manitoba Secondary Schools." Unpublished Ed.D. dissertation, Michigan State Univer— sity, 1966. Husband, D. D., and Postlethwait, S. N. "The Concept of Audio—Tutorial Teaching." Unpublished Ph.D. disserta— tion, Purdue University, 1970. APPENDIX A SMCCMP MATHEMATICS FOR ELEMENTARY TEACHERS 1. Course Outlines 2. Bibliography 3. Course Descriptions and Textbooks 154 SMCCMP Carolyn Re, Chmn. Harriett Emery Edward Rathmell Curriculum for Mathematics for Elementary School Teachers A. Prerequisite: No algebra prerequisite should be set for the course(s). B. Philosophy: The tenor of the course should be intuitive, in fact, this intuition should be used to direct the stu— dents to their own discoveries of mathematical ideas and principles. The course should constantly refer itself to the topics currently being taught in the elementary school. Various models for the mathematical principles being taught should be presented so that the prospective teacher has several avenues by which she can reach students. Problem solving and the use of mathematical sentences should per— meate the course. The course should incorporate discus- sions of existing supportive materials and the mathematical principles behind each, such as movies, geo—boards, Dienes blocks, Cuisenaire rods and blocks, the abacus, drill sets, computer assisted instruction, and programmed materials. This can be augmented by visitations to elementary schools in the area; by correspondence with schools, or by exam— ination of textual materials being used by the schools. C. Objectives: 1. The student should develop an awareness, through the experience in this course, that mathematics can be a pleasant experience. 2. The student should learn the basic concepts of mathe— matics which are in this course outline. 3. The student should become familiar with the structural development of mathematics, such as the development from sets to natural numbers, to integers to rationals. 155 156 The student should experience and develop an awareness of the processes of inductive, and deductive reasoning and their place in mathematics, especially in problem solving. The student should be able to communicate using current mathematical language, i.e., mathematical symbols and vocabulary. The student should be familiar with the goals, content and sequencing of the elementary mathematics curriculum. The student should be familiar with supportive materials for teaching elementary school mathematics, the mathe— matical concepts they illustrate, and their effective— ness . 157 Course Outlines Mathematics for Elementary School Teachers This list of concepts is intended to be the content of the Mathematics for Elementary School Teachers' courses. The following order of presentation is suggested. I. Patterns — This concept is to be introduced and then to be used recurrently throughout the COUPSG . A. Arithmetic Geometric C. Special 1. Fibonacci numbers 2. Pascal triangle D. Sequences l. 2. Geometric Arithmetic E. Series 1. Arithmetic 2. Geometric 3. ”Square” numbers, numbers which can be put in square array, such as. .. ... 4. "Triangular" numbers, numbers which can be put in a triangular array, such as . . . F. Number line *11. Sets A. Well—defined sets B. Members of sets C. Set builder notation (use of variables) D. Empty set and universal set E. Infinite and finite sets *absolute must *III. 158 Relationship between sets 1. 2. 3. 4. Equal Subset Disjoint sets Overlapping sets Partitions of sets Operations on sets 1. 2. 3. 4 5 Unions Intersection Complement and relative complement Cartesian products Properties of these operations Cardinality of sets 1. 2. One-to—one correspondence Equivalent sets Venn Diagrams Whole Numbers A. Numeration systems 1. Systems without place value, e.g., Egyptians, Roman Systems with place value a. Base 10 b. Other bases Number theory oowmmL-wmw Divisibility Primes, composites Evens, Odds, naturals Multiples Divisors Factoring Greatest common divisor Least common multiples 159 Whole numbers as a system 1. Addition as the cardinality of the union of disjoint sets 2. Multiplication as the cardinality of cartesian product, and as repeated addition 3. Properties of whole numbers with addition and multiplication a. Closure b. Associative c. Commutative d. Inverses e. Identity f. Cancellation g. Distributive law 4. Subtraction and division as inverses 5. Graph of wholes x wholes Relations 1. Equalities 2. Inequalities and order of numbers *IV. Algorithms A. B. Definition Proof of algorithms 1. Addition 2. Subtraction 3. Multiplication 4. Division Algorithms — examples: 1. Scratch methods of 4 operations 2 Lattice method 3. Lightning method 4 Shortcuts based on: a. (a + b)(a - b) b. (a + b)2 c. (10a + 5)2 5. Distributive law for multiplying a(b + c) = ab + ac References: Ward and Hardgroves, Modern Elementary Megh, Second Print- ing, 1964. pp.155- 160. *V. 160 D. Operations in any non—decimal bases 1. Add 2. Subtract 4 Charts for an Operation are 3. Multiply Supplied to students 4 Divide 1 Positive Rationals A. Definition as {%Ia,b€W; b#0} or {glaew; ch} B. Vocabulary of rationals l. Improper 2. Proper 3. Mixed C. Ordered pairs Partitions of discrete sets, regions, and number line E. Renaming equal fractions F. Ordering by using g related to as ad is related to bc b G. Operations and algorithms on: H. Renaming and operating on numerators as whole numbers 1. Properties of operations and inverse operations - division; multiplicative inverses now exist J. Complex fractions K. Subsets l. B as a subset of F b a 2. Bb = {—Ela,er; ch} 3. B1 = W 4. B10 = decimal fractions L. Decimal fractions 1. Relating decimals and fractions, including converSIons 2. Estimation, rounding off, and order 3. Algorithms for decimals 4. Per cent 5. Problem solving *VI. VII. VIII. 161 Integers A. Definition {. . . , —3, —2, —1, 0, 1, 2, 3, . . .} B. Number line extended to I C. Operations on integers 1. Algorithms, using those on W 2. Properties, using those on W D. Models for operations on integers l. Vectors 2. Patterns 3. Madison Project Postman Stories and High in the Sky, in Arithmetic Teacher; PLUS (a book on techniques that work.) Reals and Rationals A. B. Define as all numbers with an infinite decimal expansion Find a pattern that would establish a non—repeating decimal Prove /2 is irrational by the usual method and by using the Fundamental Theorem of Arithmetic, i.e., Unique Factorization Theorem Operations with radicals Properties of the real number system Estimating radicals by interpolation Systems and Their Structure Definition Operations 1. Tables 2. Charts 3. Properties Examples 1. Modular arithmetic 2. Rectangle symmetry 3. Symmetry of equi—lateral triangle 4 Peg Board permutations 162 IX. Problem Solving and Mathematical Models Word problems Mathematical sentences Linear equations Order of operations WUOUJID Use of formulas, examples: 1 Interest 2 Perimeter 3 Area 4. Volume 5 Work 6 Distance F. Ratio and proportion G. Rate pairs Second Course X. Logic A. Inductive and deductive reasoning B. Mathematical meaning of l. Implication statement 2. Negation 3. Contrapositive statements 4. All, every 5. Some 6. None *XI. Geometry and Measure A. The concepts and definitions of geometric figures B. Euclidean and tOpological properties of the common geometric figures C. Classification of polygons Axioms and proofs of some simple incidence properties *XII. WXC—(HCEQ’II 163 Measurement 1. The concept of measurement as an approximation 2. Relate measurement to congruence 3. Length, area, volume, and angle measurement with standard English and metric units Reflections, translations, rotations, symmetries Congruence Constructions Size transformations and similarity Parallels and perpendiculars Angles and circles Optional Vector geometry Parallelogram law Algebra of vectors Problem solving Spherical geometry Differences in Euclidean and spherical geometry Finite geometry mflmmtwmp Duality Algebra and Analytic Geometry A. B. C. Addition of polynomials Multiplication of polynomials by a constant Cartesian plane 1. Vocabulary a. Quadrants b. Origin c. Coordinates d. Axes 2. Graphs of cartesian products 3. Relations and functions a. Definition b. Domain and range c. Graphs of relations and functions XIII. XIV. 164 4. Distances a. Directed b. Absolute value c. Between two points D. Linear equations 1. Solution in one variable 2. Graph of solution on number line 3. Graph on linear equations in two variables by plotting 4. Intuitive meaning of m and b in y = mx + b 5. Solution of linear systems by plotting E. Linear inequalities 1. Solution in one variable by cut—point method 2 Graph on number line 3. Graph of inequality in two variables 4 Linear programming as an example of inequalities in two variables F. Graph of higher order equations by plotting 1. Quadratic equations 2 Graphing y = x3 3 x2 + y2 = r2 for r = constant 4. y = bX for b = constant, b # 1, 0 5 Interpreting graph Probability A. Define probability of an event for discrete cases B. Counting problems C. Permutations and combinations D. Independent and mutually exclusive events E. Problem Solving Statistics A. Compiling and describing data B. Graphs C. Mean, median, mode D. Intuitive presentation of the normal and binomial distributions 165 Bibliography1 Articles Bellman, Richard. "Control Theory.” Scientific American, Vol. 211, No. 3 (September, 1964), pp. 186:200. Courant, Richard. "Mathematics in the Modern World.” Scientific American, Vol. 221, No. 3 (September, 1964), pp. 40:49. Davis, Philip J. "Number." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 50—59. Dyson, Freeman J. "Mathematics in Physical Science." Scientific American, Vol. 211, No. 3 (September, 1964), Gardner, Martin. ”Mathematical Games.” Scientific American, Vol. 211, No. 3 (September, 1964), pp. 218-224 (word games). "Mathematical Games." Scientific American, Vol. 215, No. 3 (September, 1966), pp. 264—272 (Mrs. Perkin's Quilt). "Mathematical Games." Scientific American, Vol. 219, No: 5 (November, 1968), pp. 140-146 (dice). "Mathematical Games." Scientific American, Vol. 221, No. 6 (December, 1969), pp. 122—127 (dominoes). Gregory, Richard L. "Visual Illusions." Scientific American, Vol. 219, No. 5 (November, 1968), pp. 66L79. Kac, Mark. "Probability." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 92-1083 Kline, Morris. "Geometry.” Scientific American, Vol. 211, No. 3 (September, 1964), pp. 60—69. Lane, William. "Abstract Mathematics for Upside—Down Readers. Enrichment Mathematics for High School, Washington, D.C.: National Council of Teachers of Mathematics, 1963, pp. 141—149. lStudent Bibliography of Study was submitted to subpanel in March, 1970. 166 Moore, Edward F. "Mathematics in Biological Sciences." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 1148-1674. Quine, W. V. ”Foundations of Mathematics." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 112-127. Raimi, Ralph A. "The Peculiar Distribution of First Digits." Scientific American, Vol. 221, No. 6 (December, 1969), pp. 109—113. Sawyer, W. W. ”Algebra." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 70—78. Stone, Richard. ”Mathematics in the Social Sciences.” Scien— tific American, Vol. 211, No. 3 (September, 1964), pp. 168—182. Suppes, Patrick. "Use of Computers in Education." Scientific American, Vol. 215, No. 3 (September, 1966), pp. 206: 223. Ulam, Stanislaw M. "Computers." Scientific American, Vol. 211, No. 3 (September, 1964), pp. 202—216. Booklets Fugii, John N. Puzzles and Graphs. Washington, D.C.: National Council of Teachers of Mathematics, 1966. Glenn, William H., and Johnson, Donovan A. Number Patterns, in Exploring Mathematics on Your Own, St. Louis, Mo.: Webster Publishing Co., 1960. Hartley, Miles C. Patterns of Polyhedrons. Ann Arbor, Michigan: Edwards Brothers, Inc., 1941. . Penetrating Polyhedrons. Chicago: University of Illinois, 1960. Johnson, Donovan A. Paper Folding for the Mathematics Class. Washington, D.C.: National Council of Teachers of Mathe- matics, 1957. , and Glenn, William H. Topology. New York: Webster Division of McGraw—Hill Book Co., 1960. May, Lola J. Elementary Mathematics: Enrichment. New York: Harcourt, Brace and World, (booklet as advertisement). 167 National Council of Teachers of Mathematics. Arrangements and Selections. booklet #5 of Experiences in Mathe— matical Discovery, Washington, D.C.: National Council of Teachers of Mathematics, 1966, pp. 21-28. National Council of Teachers of Mathematics. Formulas Graphs and Patterns. booklet #1 in Experiences in Mathematical Discovery, Washington, D.C.: National Council of Teachers of Mathematics, 1966. National Council of Teachers of Mathematics. Geometry. Unit #4 in Experiences in Mathematical Discovery, Washington, D.C.: National Council of Teachers of Mathematics, 1966. National Council of Teachers of Mathematics. Sets, booklet #1 of Topics in Mathematics for Elementary School Teachers, Washington, D.C.: National Council of Teachers of Mathematics, 1964. Nichols, Kalin, and Garland. Introduction to Sets. New York: Holt, Rinehart and Winston, Inc., 1962. Norton, M. Scott. Geometric Constructions. New York: Webster Division of McGraw—Hill Book Co., 1963. Wenninger, Magnus J. Polyhedron Models for the Classroom. Washington, D.C.: National Council of Teachers of Mathe— matics, 1966. Books Brumfiel and Krause. Elementary Mathematics for Teachers. Reading, Massachusetts: Addison—Wesley Publishing Company, 1969. Boehm, George A. W. The New World of Mathematics. New York: Dial Press, 1959. Dinesman, Howard P. Superior Mathematical Puzzles. New York: Simon and Schuster, 1968. Dudeney, H. E. Amusements in Mathematics. New York: Dover Publications, 1958. Gardner, Martin. Mathematics, Magic and Mystery. New York: Dover Publications, 1956} Gray, James F. Sets, Relations, and Functions. New York: Holt, Rinehart and Winston, 1962. 168 Heath, Royal Vale. Mathemagic. New York: Dover Publications, 1953. McFarland and Lewis. Introduction to Modern Mathematics. Indianapolis: D. 0. Heath Company, 1966. Merrill, Helen A. Mathematical Excursions. New York: Dover Publications, 1957. National Council of Teachers of Mathematics. Enrichment Mathe- matics for the Grades. Washington, D.C.: National Council of Teachers of Mathematics, 1963. Enrichment Mathematics for High School. Washington, D.C.: National Council of Teachers of Mathematics, 1963. Multi-Sensory Aids in the Teaching of Mathematics. Washington, D.C.: National Council of Teachers of Mathe— matics, 1945. Topics in Mathematics for Elementary School Teachers. Washington, D.C.: National Council of Teachers cxf Mathe— matics, 1964. More Topics in Mathematics for Elementary School Teachers. Washington, D.C.: National Council of Teachers of Mathematics, 1969. Historical Topics for the Mathematics Classroom. Washington, D.C.: National Council of Teachers of Mathematics, 1970. N. S. S. E. Mathematics Education. NSSE Yearbook LXIX, Part 1. Chicago: University of Chicago Press, 1970. Peterson and Hashisaki. Theory of Arithmetic. 2nd. Ed. New York: John Wiley and Sons, 1967. Schaaf, William L. Recreational Mathematics. Washington, D.C.: National Council of Teachers of Mathematics, 1958. (a bibliography). Vilenkin, N. Ya. Stories about Sets. translated by Scripta Technica. New York: Academic Press, 1968. Ward, M., and Hardgrove, C. E. Modern Elementary Mathematics. Reading, Massachusetts: Addison—Wesley Publishing Company, 1964. Wheeler, Ruric E. Modern Mathematics,eAn Elementary Approach. Belmont, California: Brooks/Cole Publishing Company, 1966. 169 Willerding, Margaret F. Elementary Mathematics, 2nd. Ed. New York: John Wiley and Sons, 1970. Periodicals The Arithmetic Teacher. Washington, D. C.: National Council of Teachers of Mathematics. The Mathematics Teacher. Washington, D. 0.: National Council of Teachers of Mathematics. Junior College Journal. Phi Delta Kappan. Scientific American. 170 COURSE DESCRIPTIONS AND TEXTBOOKS Southeastern Michigan Community College Project Transfer Curriculum in Mathematics PROGRAM DESCRIPTIONS Mathematics for Elementary Teachers Delta: Math 110: Mathematics for Elementary Teachers, 3 hr. credit, 4 hr. contact. Text: Peterson and Hashisaki (Wiley). Syllabus covers all of Chapters 1-7 and work on real numbers in Chapter 8 is discussed quickly. Henry Ford: Math 30: Mathematics for Elementary Teachers ' I, 3 hr. credit. Text: Peterson and Hashisaki (Wiley). In addition to the content in Chapters 1—7, the course includes some work on introductory logic. Math 31: Mathematics for Elementary Teachers 11, 3 hr. credit. Text: McFarland and Lewis, (Heath). The syllabus includes introductions to number theory, real numbers, non metric and metric geometry, mathe— matical systems, and solving linear equalities and inequalities. Highland Park: Math 231: Mathematics for Elementary Teachers, 3 hr. credit. Text: Ward and Hardgrove (Addison Wesley). The syllabus includes history of numerals, bases, sets, the whole and fractional numbers, fundamental operations and their properties, algorithms, and number theory. Math 232: Mathematics for Elementary Teachers, 3 hr. credit. Text: Ward and Hardgrove, (Addison Wesley). The syllabus includes integers, real numbers and their properties, algebra, graphing, informal geometry and informal topology. 171 Jackson: Math 111: Foundations of Mathematics, 3 hr. credit. Text: Peterson and Hashisaki (Wiley). The syllabus includes the text material through the rational numbers (Chapter 7) and selected topics con— cerning the real numbers. Macomb: Math 115: (No course title available), 3 hr. credit. Text: Wheeler (Brooks/Cole). The syllabus includes elementary logic, sets, natural and whole numbers, operations of these numbers, equalities and inequalities, integers and operations, numeration systems, and the rational numbers. Math 116: (No course title available), 3 hr. credit. Text: Wheeler, (Brooks/Cole). The syllabus includes decimal numeration, real numbers, number theory, modular arithmetic, elementary algebra, graphing, geometry both metric and non metric. Oakland: Math 251: Mathematics for Elementary Teachers 1, 3 hr. credit. Text: Wheeler (Brooks/Cole —— Fundamental College Mathematics). The syllabus includes sets, whole numbers, numeration systems, integers, number theory, fractions and rational numbers, and decimals and real numbers. Non metric geometry is optional. A supplementary booklet exists which contains performance objectives for the course and also study assignments, hints and time limits. St. Clair: Math 110: Foundations of Mathematics, 3 hr. credit. Text: Brumfiel (Addison Wesley). Schoolcraft: Math 103: Mathematics for Elementary Teachers, 3 hr. credit. Text: Ward and Hardgrove (Addison Wesley). The syllabus includes sets, whole numbers, numeration, binary operations, algarithms, informal geometry, fractions, algebra and problem solving, patterns in mathematics. (Note: This syllabus may change in 1970-71). Washtenaw: Math 107: Principles of Elementary Mathematics, 3 hr. credit. Text: Course not now taught. 172 Flint: Math 261: Mathematics for Elementary School Teachers, 3 hr. credit. Text: Peterson and Hashisaki. The syllabus includes numeration, sets, relations, and the systems of the whole numbers, the integers, the rationals,and the reals. Chapters 1-8 are covered. APPENDIX B ELEMENTARY ARITHMETIC TEXTBOOKS USED IN 1967 IN SCHOOLCRAFT COLLEGE DISTRICT 173 174 maco pxop mopmfioomm< goswomom mosoflom cpsosmam pxop pxop moHLom smabflmq moamochomflbo< oHHH>Sppoz m I m as: maoq Locooam ApcoEQOHpsov m I a ramp hoflmozIQOmH©©< meOmeoz Azoamv m I s maco>fiq psoEoHQQSm Locowop mopsfloomm< gopsomom mocoflom paw» sopmsflz pxop s osccocam .raom soaaochoaaooa Ahmaswopv m I s maco pxop maso pxop ll xcom smoflpmsoocoswao moowsw assesoEon ooasomaa omoaaoo reasoaoocom gasps: arostomao Hoocom oaaooa o>aa oer ca mommmao capoegpfls< mswpcosofim 2H smma CH com: mxooopxoe APPENDIX C ATTITUDINAL SURVEY 175 SURVEY Name The Mathematics Department of Schoolcraft College asks your cooperation in filling out this survey so that we may improve this course. This survey will be kept confidential and the data collected will be compiled and studied only after the semester grades have been turned in to the registrar. (Your name is needed for research purposes only.) Directions: Please write your name in the upper right hand corner. Each of the statements on this survey expresses a feeling which a person has toward mathematics. You are to express on a five—point scale, the extent of agreement between the feeling expressed in each statement and your own feeling. The five points of the scale are: Always (A), Often (0), Sometimes (S), Rarely (R), and Never (N). You are to put an X in the block for the letter that best indicates how closely you agree with the feeling expressed in the statement as it concerns you. 1. I am eager to study my mathematics [AIOJS [R IN] lesson because there is such a feel— ing of satisfaction in completing each assignment and knowing it is done correctly. *2. In mathematics classes at school I I I I I I (not college) I tried and tried, but . I needed help to complete my mathe— matics assignments. rating 9? 1+5 no * 5+1 176 *3. *4. *5. *6. *10. 11. *12. *13. *14. 15. 177 . . . A O S R N I dislike mathematics and I resent the 11 time I have had to devote to this mathematics course. Of all my classes in grade school and I I I I I ] high school, I dreaded mathematics - the most. The class hour seemed to drag on forever. Mathematics makes me feel depressed and I I, I :1 _[ uncertain; I put off studying it as long I I as I can. I needed more hel than I could et. p g Lilli! My parents didn't like mathematics I I I I I and I don't like mathematics; I ' guess it runs in the family. When I apply for my teaching assign— ment, I will request a position which L_1L_J l I -1 includes teaching arithmetic. Having the answers to the problems . ,j .1 I —1_ I increased my understanding and helped L . in finding errors. I will be glad to get through this L, I I I I I required course. e. . s I liked the textbook; 1 could study I I I at home whenever I wanted. . fir --n L__J A mathematics problem is always an incomprehensible jumble of words and numbers which I cannot translate into symbols and numerals. Learning "modern" mathematics has been L_1I I I J more difficult for me than learning s - the other mathematics. It was hard to keep up with the class; as . I needed more time to assimilate the L L141 1 L,:] required concepts. In high school I enjoyed my mathematics I L 1, 1 I I classes; I was sorry to hear the bell r . ring. 178 16. The teaching techniques (lecture, A 0 discussion, test, etc.) used in this [—1 ] 4L7 I J class made the material meaningful and clear. 17. I have enjoyed this course, but I still feel the need for more mathematics to be a a good elementary teacher. P—I ”T —-]h H school and this course has been a repetition and a waste of time; I have learned nothing new. *18. I had "modern" mathematics in high I j“ J, ] r] 19. I like mathematics; therefore, I I I 1 I 1 I finish my other studies first, in . order to devote my full attention to mathematics. 20. I have enjoyed this mathematics L 1J_ I _51] class; the time seemed to fly by -- . when I was studying my next assign— ment. *21. I am frustrated by mathematics; I will not accept any position which tel j—~l l I includes teaching arithmetic. 22. I like mathematics; I feel I will be es I I——I I I able to teach arithmetic in elementary school. 23. I like to explain solutions to mathe— I— I— I I lI matics problems to my classmates. - 24. I like mathematics more now than I did L I I~ I II before taking this course. 25. **I enjoyed the laboratory; I couldn't I I I J; .LJ have learned as much by studying at .1 home. 26. ** I liked the laborator sessions: t y ’ [IJIIJ I could work at my own rate and I could get help when I needed it. *27. **The laboratory contibuted greatly [I] «I 1 II to my frustration with mathematics. ~ ' It gave me a lot more to learn. 179 Name In addition to the above reactions, in the first part of the survey, I would like to have you know that: (Give suggestions, criticism, or comments.) APPENDIX D AUDIO—TUTORIAL [BIOLOGY 101] AT SCHOOLCRAFT COLLEGE by Gordon G. Snyder 180 History Audio-Tutorial (A.T.) at Schoolcraft College began in the fall of 1966 with an examination of Dr. S. N. Postlethwait's A.T. botany laboratory at Purdue University. Following that examination and after hours of discussions and visits to other A.T. laboratories in the area, the biology division decided to devise an A.T. approach for our general biology course. The division spent the remainder of that year in the general development of the procedures and materials for a pilot program. During the summer of 1967, two instructors were employed and charged with the specific development and production of materials for the pilot program. This included the writing of scripts, a student guide, and the construction of four booths and development of the laboratory area, which was a corner of a small greenhouse adjoining one of our traditional laboratories. The pilot programs involved two classes (one each semester) of 24 students. Weekly divisional meetings resulted in a constant evaluation of the program and revisions of it as it progressed. A final evaluation of the pilot program lead us to the conclusion that all the general biology students would benefit from being taught by an A.T. approach, and that a total A.T. 181 182 program should be developed. Two instructors were again employed for the summer of 1968 to complete the development of the total program. Audio—Tutorial Today Presently, our Biology 101 course has three components: The combined lecture, the open laboratory, and the seminar quiz. The combined lecture is a scheduled one hour session for approximately 90 students. There are four combined lectures each given by a different instructor. These lec— tures provide the opportunity for orientation, guest speakers, long films, exams and other aspects of a biology course that would be best suited to large group meetings. Major exams are also given during this time. The combined lectures are held on Mondays. The open laboratory is the only non-scheduled portion of the course. The laboratory is equipped with 29 individual booths each containing a tape-recorder, microscope and other appropriate equipment. Materials that do not lend themselves for booth activities are placed in peripheral demonstration areas. The student checks into the open lab with the help of a student receptionist. The open lab is available from 7:00 a.m. to 5:00 p.m. Monday through Thursday, and students may visit it as frequently and stay as long as they desire. Our experience has shown that the hours of greatest utiliza— tion occur from 9:00 a.m. to 3:00 p.m. We also know that 183 the first week involves a settling process; thus, students will at first encounter some difficulties in getting into the open lab when they so desire. After the first week, however, routines are well established and further difficul- ties are practically nonexistent. Information on booth availability is given to the students each week which helps them to determine their own schedules for the open lab. The learning sequence is on tape and includes lectures, laboratory exercises and experiments, readings from scientific journals, and textbook assignments. Periodically, the stu— dent will be directed to demonstration areas for observation of super 8 mm film loops, charts, living specimens, and to carry out exercises and perform experiments. The Burgess A.T. Systems tape recorders are under the individual control of the student, and he may repeat any portion of the learning sequence. One booth is available for make—up work in case a student should (because of illness or death in the family) miss a portion of a weekly learning sequence. A full—time biology instructor (any of 9 in our depart— ment) is always available in the open lab for tutoring students on a one—to—one basis. Students are also able to interact with each other in demonstration areas, booths, and in the A.T. reading room, which is available directly off the open lab. It contains numerous reference materials and is suitable for informal interactions between students and instructors. 184 Students are guided through the weekly learning sequence by focusing their attention on objectives. Each learning sequence has been divided into several objectives which are given to the students. The objectives state the goals of the weekly learning sequence by clarifying what the student should be able to do and how he is to do it. The following is an example: Objective 1. Gain an understanding of the problems of the regulation of body fluids. By listening to the tape, reading pages 446—447 of your textbook, viewing the film loop, and asking the instru— ctors for assistance, you should be able to: 1. List 7 problems in the regulation of body fluids. 2. Define plasmolysis. Evidence of your knowledge of this objective will be obtained from the seminar quiz and major exams given by your instructor. Other materials are also provided including a list of all the terms used on the tape. This word list is placed in the booth for easy reference while the student uses the tape. A student guide is also utilized which contains drawings, charts, and outlines of exercises and experiments. All tapes are made by one instructor. Seminar quizes are held during an hour's scheduled session on Friday and/or Monday. An instructor's Monday combined lecture sessions are divided into five quiz sessions containing 14—18 students. Instructors gear these sessions 185 to meet the needs of their students. Forty percent of the student's final grade will come from the instructor's eval— uation of the student during the seminar quiz. In conclusion, I have prepared a chart (fig. 1) which reflects the progression of ideas which were employed in the development of our present day Audio—Tutorial approach. .uHcs page Ho mo>Huoonno ozp co Acmputh HHHMszv Eon» mmuHsd omHm LouosppmcH one .muHcs szwm3 mchpooCOO wCOHummsd xmw ocm mEoHDOHQ mmzomHo mucocsum one .m .mo>Hpoonno Hmeoz mo owvazocx HHonu mumpumcosov zHHmpo mucoczpw one .2 .QMH some on» CH pcmam wH mEHu Ho coproa meow cwsocpHm .HMOHOHD Hapocow cmzp .B.< on» :H mH Lao Ho Losses opmszmop p0: moon .xmoz\mL: mm Lou .omgsoo HonOHn Hmpocow omOH ompsoo scams omen: Huwpm oco Locuwp pan .xcmn OHEoUmom porosppmcH mEHp HHsm Egon one .m pmH Como on» chuchE o» mEHu Lmnuo mH bmoH Lone omozz Losses Hanan a mopmcom .m LHozu couscop :onH>Hc L50 :H mL0poshpmcH HH< .H HH .mHmv NHzc pmcHEmm moLSBUoH woCHnEOQ : machozppmcH mEHp pumm m LoBCHz : : = = : : : : : L50: H conoocom .Ls H msouosppmcH oEHp HHsm : osmH .LOposppmcH oewm on» ma co>Hw ohm oncz mmNHSU LmCHEom m oucH omoH>Ho mum wpchSBm omega .mogspooH noCHo NHso gmcHEcm mwcspocH vochEoo : machossumcH.oEHp puma m Hme IEoo on» 90 H HHco mo>Hw hobosppmcH c< Loos H poHsomcom .Lc H mLOuosppmcH oEHu HHsm : mmmH .wmumcHEHHo NHsd LacHEomm mossoooH smchEoo m mLOposHbm:H oEHu whom o LoocHx COHpLoa HmHo .Lsoc H on bommmhocH NHSO L30: H UmHssozom .Lc H LOuojgumcH wEHp HHsm H momH .mLOuoscumcH HHm wcoem conH>Ho moNHSU ImouspooH UwcHnEoo HHm wm>Hm LocossumcH mco NHsv LMCHEmm machoshpmcH mEHp chum o Hme .mommeo HHm I Emnwopo Hqu Ho LopmoEom pmH Hmpo .L: m\H ooHsoocom .L: H mLOBosppmcH mEHu HHzm m momH . .mmNHsu con COHumpcoEHLquo NHso LMCHEmm mLOBOJprcH oEHu ppmm m LoBCH3 oHnmpooncoo .zHco mmmHo H .EmeOLm uoHHm Hmho .hc m\H UmHssocom .L: H LouozppmcH oEHu HHsm H womH NHsv meHEmm machoshumcH mEHp phwmm m HHmm .cho mmmHo H .EdeOLm BOHHm Hmpo: .9: m\H chsoocom .p: H hobozpumcH oEHp HHsmm H NomH muomdm< pcmoHchme NHso oLSBooH macaw oHsoocom szcm3 as am cam toucHs osmH .sm mm omm HHmm mme H: mm mmH soocHz momH 5m mm sHm HHmm wmmH mm a :m LoucH3 mmmH mmH a 2m HHmm sme nmq Como no mCOHumum mucoospm mo .02 coHBMLon mo major ku09 Ho .02 Hapoa .xopaa< Lopmoswm cam pom» momqqoo Bmmm mmB APPENDIX E MATH CARD 187 MATH CARD Last Name First Middle ::} Male :1 Female Address Street City __ ;__ Single Student No. Phone No. Date of Birth :: Married :1 Freshman High School City Year Graduated __ ;; Soph. Senior College Curriculum Math Placement Test H.S. Math Credit Grades __ Alg. 1 .1 Yes Geom. —— No Alg. II __ S. Geom. Trig. Gen. Math Bus. Math Other 188 APPENDIX F MATERIALS FOR LABORATORY Directions for Tape Recorder Laboratory Card Objectives--Lessons One to Four Student Bibliography 189 II. III. IV. VI. VII. VIII. IX. Math 103 Directions for Use of Tape Recorder Threading recorder. A. Take reel - place on left spindle, anchor into grooves of reel. B. With shiny side of tape facing you - pull out 12" of tape from reel. C. Holding end of the tape in your right hand and supporting tape in left hand place tape in slot of playing mechanism. D. Place end of tape on right hand reel - wrapping the tape around reel at least once to prevent slipping. Turn on tape recorder. Check that speed indicator is set for 3-3/4 rpm. Put on headset. Turn indicator to play. Adjust volume to your preference. Inst. stop may be used for short pauses; for longer span of time stop the machine. If you wish to replay a portion of the tape, turn indi— cator to rewind and allow sufficient time for tape to return to material desired. Turn to stop and then to play. To leave booth — for short time — (1) turn recorder to stop and turn off motor, (2) take off headset. If difficulty — call instructor. 190 191 .:H H QMH .0 .NH .m .HH .2 .0H .m .m .m .w .H p50 CH opmo Spoom commoH p50 CH mama npoom commoH oSHE oEHB a ommH Hochz MOH meHCD . « cmmpmmnsm Ho Hooszz Hmpoe ”oHQHosHLm v .pom m so SOHHmHoQo HpmsHm “oHQHosHHm HHH ego .HH .H moans Sosa moHQHocHLm Ucm mpdoosoo UopooHom 203 amc a smaoa ma oars .mnoa HH ”msH>Hom soHoosa Hm.Ha u a ”coaeaooz oom OHHHoon QHmsmsoo “v . >Hpmwoz mcocmpwsH Hmposow >HpHmom AMHHHHHMV m” mw OHBmpoz o A. rooHoHoHHoo n ”u HHHH “w“ »\\\\\\\\\\\\I pomcsm coaoeooz oHoHoHHoe a\\\\\\. All/Illivpomnzm Homopm "pdoocoo pom HHSZ \ V oHonmom mpombsm Lemons voHnHmmOQ mpmmpdm Ho Loossz Ho Loos: mICm cm HHH uhmgo HmQOHpmHophochv mpdoosoo Ho mcHsHmso .Hm .pom m Co COHmeoQo HCmCHm H .Q aoCmmw ‘ ”mHQHoCHCm L I|§|l pCoEon omHo>CH pCoEon soHoccoH gév/ é .xHCpmz C x C Cm CH mpom Co COHmeoQo Cm mo Epom pCmCo “oHQHoCHCH pCoEon pCoEon omCo>CH mpHpCooH \*l/ I In \ I \ V »\ / n\ /x x / »/\\ \\ /\ //\\ COHmmoH m Com OHCOB oIdoL 205 Task Analysis Chart V there a Show lack of Identity Identity Element Element (e)? Yes List Identity Element e =“__ \\\\\I there an Show lack Inverse No of inverse Element ’ elements (a')? Yes List inverse Elements inpairs t Exit to Next IIII‘IT““‘*~—~HH~iiggee Chart “*--~—Hii_hl or ' The End APPENDIX H STUDENT RESPONSES FOR OPEN—ENDED COMMENT 206 STUDENT COMMENTS by Groups Group I: Control 1. First of all, I thought a lab would have suited this class well. It would have been a great help. This comment resulted from the fact that sometimes if you missed class or did not understand a certain concept, it might not have been enough to explain it a second time in class. Or some students might have wanted to persue a certain technique, but it was not a requirement of the class to know that technique, and not enough time to ex— plain it for those who understood it. Second, I thought the class went too slow, but that's only an opinion. 2. Math does nothing more than frustrate me, I would rather just say thanks for the reminder and goodby. 3. I think that we spent too much time on the first few chapters because we had to go pretty fast through the last ones. Other than that I enjoyed the class and learned some new concepts. (Shortcut mult., especially!) 4. I have learned a great deal from you intent teachings and now have a better understanding of math. I enjoyed the class. 5. You have a very interesting class. You present the material in a very interesting way. Math has always been my downfall in school. I guess it's just not one of my gifts, but I do try. I appreciate everything you've taught me: which is quite a lot. 6. On the whole this class was very informative and worthwhile. I learned a lot by the way you taught. Also very helpful were the ideas you would present on ways to teach these new methods to children and the ways teaching math can be made into a "game" which would be 207 208 much more advantageous to the child, since children do love games. I don't feel that the text was of that much value to me (except for checking a couple of things). I learned a lot more by listening to the lectures in class. My honest opinion is that basically I don't think you should change any of your teaching habits. If students would learn to apply themselves (which I am guilty of sometimes too) instead of griping about classes, I'm sure they would see more value in the way you are teaching. 7. I was, at first, very frightened about taking math, because I had so much trouble with it in high school. I did, though think I learned more from this one course with you than I think I have from all the other ones I have ever taken. You explain everything very well, and I really enjoyed your teaching. I hope that someday I can get it across to my elementary children as well as you reached me. Your examples were great and they made it much easier to see and to understand. You are a very good teacher and I can see how children would have loved you. I think I would have enjoyed it more if it wasn't at 8:00 in the morning! The book was also very good and it helped explain the information while I was at home. The home tests you gave were good, because I sure needed the review if I'm ever to be able to explain it to children. You helped me to learn a lot, even though some of my grades didn't show it. Thank you! 8. Sometimes, Aiss Emery, it wasn't very clear what you were trying to get through to us. I know you know what you were talking about, but I just couldn't get it. Having you as a teacher is a lot easier than trying to understand the book, though. You sure know what you're talking about. 9. Some of the material that was presented in class was explained in such a way that it was confusing and difficult to understand. 209 Also directions could have been clearer on tests and take home tests to avoid misunderstanding. 10. When you give both take home tests and class tests there should be more directions as to what kind of answers you wanted. I found that you were not consistent, for instance: on one take home test you marked .60 wrong because it should have been just .6 but yet on an extra credit sheet I got 6/10 marked wrong because it should have been 60/100. One place I should have dropped the zero and the other I shouldn't. How am I supposed to know when I am and when I'm not? I never knew how you wanted numbers expressed (mixed, decimals, fractions) and since you gave no directions I got a perfectly good answer marked wrong because it should have been expressed another way. Be more specific, give directions or go over your set of rules at the beginning (which you never did in our class). llzwould be a great help! 11. My own lack of interest in the class can be the only contributing factor to the poor grades I have been receiving. Being away from math for so long does not prove beneficial, either. Math was never my best subject, but with sufficient effort I realize I am capable of doing well. I only regret that I hadn't looked at it as a 223 so simple course, when I first enrolled. I am acquainted with several people who take the course with that false and dangerous attitude right from the start! If they could be made to realize that it is not as easy as it looks, perhaps they would take it more seriously and avoid making the mistake that I unfortunately made. 12. I have enjoyed the class, but found the tests kind of hard. I feel the way you taught the class a book wasn't really necessary. I think we should have more quizes over the new things we learned before we have a big test to make sure we know what we are doing. 13. Needs more exchange between teacher and stu— dents——this would benefit all the class and teacher in understanding where the problems are. 14. No comment. 15. I have learned much that I hadn't known before, even though I had some knowledge about most of the mate- rial. What I had already known I found easy and what was new to me (bases, modulus, charts) were not too hard to pick up. I do feel however that I don't know all of this material well enough to teach it. 210 16. For me, I feel this course has done some good. Math, being my downfall—~as you can probably see, is a struggle. The way in which things have been explained has been good. I feel I understand much more now than when I first started the course. 17. I had a good deal of this material in high school and before——but it's having this class that allows us to use it again before it's completely forgotten about. Unfortunately there were quite a few things I had forgot. It's too bad there is not a continuation of this course. ——I think I would take it. 18. I enjoyed the class very much and hope to take part II when given. The only problem I found, which was confusing, was that the instructor and the book sometimes used different methods. 19. I feel that more time is needed so that the whole book can be covered more thoroughly, also more should to relate it closer to the principles by the teacher in the elementary classroom. I liked the class it is very interesting but I feel I in particular needed more time to grasp some of the concepts presented. 20. I feel we should have had more participation on behalf of the class, like small groups, etc. to dis- cuss difficulties rather than straight lecturing. Group 11: Control 1. Math has never been a good subject for me. I guess I made myself hate it when I was little because I always put it off till last. I still do it now. I thought that if I took another math course, it would make me like the class more, but it didn't help. I gig learn more about math though. But I think the lab idea is a good one. 2. I feel I've really learned from this class; however, I am sorry there is no continuating class to follow. Until this year math was just another subject but somehow you've made me realize how much fun it can be. I hope some day I can do the same for my students. 3. Part of my "essay" is on the other sheets. I like the idea of a lab——it's ideal for people like me. 211 [From the other paper] I've always had a hard time with math in grade school and high school. I've always enjoyed geometry (it seems to make more sense). At the present time I feel my previous background has been poor and I will not teach it until I've become more skilled in it. I feel I would cheat my students if I did. I know the exact feeling some of the nuns I had in grade school knew less than some of the brighter kids. Thank you. 4. I enjoyed this math class and did better than I expected to do. Everything was clearly explained and I understood most of the lessons right away. Thank you for making math more enjoyable. 5. I enjoyed your class and learned and refreshed much. There were a few times such as in bases that you went a little too fast for me as bases were entirely new to me. A lab course would have been very beneficial in my case. 6. (1) Be more demanding on requiring homework. (2) Allow no one in after attendance is taken. (a) Class taught well. 7. I was miserable in high school math and it amazes me how well I am doing here. I really and truly enjoy the class and I hope that if I ever have to take another math course it will be one of your classes. You've made math more interesting to me and I would really like to go into it more, if I could get teachers like you. I hope to have you for Math 104, if I can take it when it's given. 8. I was told my best bet would be to take this math course over Biology. I had to have one or the other. However I do not resent the fact that I made this choice. I did learn something, however minimal it may have been (and the little it is, is through only my own fault.) Science and math are not my strong points so I'm glad this course is over, but I did enjoy the class while I had it. 9. The class was taught very well, I'm just sorry I didn't benefit from it like I should have. My attitude 212 towards math is my problem. But the class is one from which a lot is to be learned and you seemed to have taught it in a way which makes the hardest math seem a little easier. Thanks. 10. In past years I have disliked math but this was different. I didn't love it, but I didn't dislike it as much as other math classes. 11. I have a very difficult time in understanding math. It takes me quite a bit of studying to comprehend math. If we had a lab like the Biology department, I think I could have understood the math better. If I have something visual I do better--abstracts escape me. I enjoyed the class but I still feel uncertain, de— pressed and undecided what I've learned or accomplished. I don't feel I've learned anything that I can apply to teaching. When I was told that it was elementary math I ex- pected I guess a lot of children's math, like from lst grade to 6th. 12. The course was O.K. 13. —— I enjoyed the class extremely—-not totally because I like math, but because it was well taught. —- A student can't help but love math, with a teacher who loves her field, and devotes more time to the class and to each student. P.S. I really learned a lot. 14. More hand out sheets for practice because I needed extra practice on certain parts. 15. I have learned quite a lot more from this class about the basic principles of math. I liked the text and I preferred the class without any use of laboratory. 16. I think you should have spent a little more time on certain things. I think the idea of a laboratory was a good one. It would give a person like myself whose not very good in mathematics someplace to go for addi— tional help. I feel you should try to make the class more interesting. I found it terribly boring (but then I find math terribly boring). 213 17. No comment. 18. I think a new textbook should be used to replace the one we now use. 19. I felt that this course was very interesting & that it was fairly easy to understand. With the high school background I have had in math I was able to under— stand most of the assignments using my background in math along with the new ways being taught. To me, this course was beneficial & I would like to teach a math course using the techniques we used in Math 103. If I had a chance to teach math I wouldn't turn it down, but the courses I am taking now I hope to go into Art Education. The course was well taught, the work was easy to understand, & I really enjoyed the class. 20. For the most part the material covered was already taught to me when I was in high school. I was never good in math at all, and this was proven by the fact that I did poorly on all the home tests, which were a review of what we learned years ago. I benefited a lot from this course but am still too weak to ever attempt to teach it to elementary school children. I have little confidence when it comes to math classes, & I therefore cannot picture myself ever excelling in it to the point where I would enjoy teaching it to others. After being away from math for nearly 2 years I was surprised to find myself taking an interest in it. 21. The class has too few exercises in showing how each operation works. I found the ditto sheets very help— ful. The class moved too fast on algorithms. 22. I think extra problems in each type of opera— tion would be helpful. Perhaps a workbook could be used along with the text, especially if one finds extra prac— tice is needed. Also, I believe testing more often would help clear up any misunderstandings one might have about the work. 23. I think a math laboratory would really help the students, especially if they were having trouble with the work. 24. My biggest problem was lack of background in math, it's been 10 years since I had any courses in math. 214 Something like a lab would have been quite helpful I think to brush up and go over my problems. 25. I would highly recommend the math work shop; working on similar problems would have been a great help. 26. I feel I could have benefited from a math lab. 27. I would suggest a whole year course of math. That would better prepare one for teaching. Group III: Experimental (Audio—Tutorial) 1. I have gained a lot of new elementary mathe- matical knowledge since taking this course. I found the lab work almost essential in learning the work, because I thought the book didn't explain everything well enough or provide useful exercises. Even though the lab work was sometimes long, it was very resourceful. I also liked the take home quizes because they helped to brush me up in things I needed to be refreshed on. While doing lab work I'm grateful that you were so patient and always ready to help. (I'm going to remember that when I teach.) 2. The lab is beneficial but the time should be cut down. 3. I thought the textbook was hard to understand. The lab helped greatly. I wish I could have spent more time in the lab. 4. I thought the lab was a great help and it made the class much better. 5. I would like a Q. 6. I would just like to say it was an enjoyable class & I think I really learned something. 7. I do think the lab did help, but I was unable to use it more than an hour each week because of the hours I have to work. Therefore, maybe I could have gained more by having the regular class hour. 8. I enjoyed this math class more than any class I had yet taken. However even after studying for a 215 considerable amount of time I find that I was not pre— pared for the final. I do not feel at all sure of what I had thought I knew. The test did not seem to be re- stricted with what we were studying at length. I found several sections, especially parts IX & IV, confusing. I feel that there was much we did not cover thoroughly enough in class that was part of this exam. 9. As I said in the first section of the survey, I believe that the class time was of more help to me than was the lab. 10. I believe that the lab was very helpful. I think it would be a good idea to install more tape re- corders and slide viewers so that instead of having only one hour a week to come in. Each pupil could feel free to come in whatever day is convenient and spend as much time as he needs. Most of the exercises could not be done in an hour and with more freedom I believe it would enable everyone to learn at a speed at which he is capable. It also would have been helpful in studying for tests if we had our own papers and tests to refer back to. 11. I enjoyed lab time very much—-it helped me quite a bit. Many concepts finally became clear to me in your classes in lab. Thank you. 12. I wish I could have devoted more time to this class but I had many personal problems this past semester and didn't have the time to devote completely to this course. 13. In the lab, you seemed very understanding and willing to help. But, in class it seemed the opposite. Maybe you assumed we knew what you were talking about. But I was left in the dark some times. 14. I think the laboratory work was extremely bene- ficial but in studying for the final exam, I think it would have helped more if we could have studied our pre— vious tests. That way you can see where you were more liable to make mistakes. With only a few minutes to look over tests before they are handed back in, you forget too much of it. I also felt the book was more confusing than helpful. It was difficult in many cases to figure out what they were trying to get across. 216 15. To me this class has been very rewarding and I've enjoyed it very much. I have learned a lot about the new modern math that I thought I never [would] be able to handle. The only criticism [I] have was that the hourly test were too long for the amount of time given. Another thing that I thought was very good and beneficial was the lab. I hope you continue using it. 16. I enjoyed the lab work very much. Out of any math class I have ever had, this one with the lab was the most "unique." I learned more from the tapes and slides than I have ever learned from any teacher. Learning math by slides and tapes sounds funny (awkward) but it sure was helpful. 17. Excellent lab usage, good idea and feeling from you that you really cared about the students and that you did understand their problems. Too often teachers, math especially, place themselves above the problems of a few students and move on to the next idea or problem. I enjoyed the course quite a bit. 18. I truly enjoyed your class; sometimes I wonder if I would have made it without the lab, tho. Have a nice vacation and good luck working on your Ph.D. By the way, I felt the final was much too hard, compared to our class tests. 19. I feel your math lab was a big success and I know I couldn't have done as well as I have without the help it gave me. This has been one math course that I really didn't dread coming to. 20. Actually, though, I found this math hard, (as I've been away from all Math close to fifteen years), it is a very fascinating field which is very vast. As some of this began to dawn on me, I do believe, if I had devoted more time, step by step to this par- ticular subject, I could've gotten more out of it. (All my studying must be done away from home-—during school hours only—-) and with seven people at home three days and weekends, evenings, etc. must be devoted to keeping things running there. 217 Yes, this is a very worthwhile field, these modern concepts and better ways of thinking and doing. 21. This was a very enjoyable class, and I was able to understand math more, because of the different ap- proaches you gave us. Your suggestions for teaching will be valuable for my teaching career. The math lab was very helpful. 22. You are a very pleasant person to be around. I enjoyed your class very much & I hope that some day I will be as acquainted with my material as you are in order so I can make a good math teacher some day. 23. The labs were very helpful and made all the material much easier to understand. 24. You're a very persistent teacher. Good luck in the future. 25. Going to the lab was very helpful, but I didn't work as hard as I should of in this course. I found that in high school the math was much easier for me, maybe because I put more time in. 26. Your teaching of math for elementary teachers was very helpful to me. When I had a question you were always concerned that I learned how to find the answer myself. When I did not catch on right away, you didn't get impatient but reviewed once again. You have a way of making us want to learn and be able to teach. You are a very understanding and patient. Too bad all my teachers aren't like you. Maybe my grades don't show it but I learn more in this class than in any other. Sure, the grade counts but to me what I learned is more important. IIIIIIHHHHIIIIHIWHIWIIHHIIIIHIIIHHHHIHIIHHI