TEE EF‘EECTS 0F EMERING ACHIEVEMENT LEVEL ARE} NEE SPENT EN CQURSE COMPLETRGH ON PERM EXAMKNATEON PEEFORREMCE ER A" EEMEDEAL ALGEBRA (2033351 FGR UNEVERSKTY STUDENTS .flmsia for the Degree 0‘3 99!. D. MICHEGAN STATE UNIVERSITY John Howard Jcaes 1971‘ rhubm «mm/m WW ~ u... 3 1293 00658 2583 '~—- , 'Mlchlgm s85? UH? Versrty This is to certify that the thesis entitled THE EQFECTS OF ENTERING ACEIEVENENT LEVEL AND TIRE SlE‘l‘T IN CLURFE CG I’LE‘IIUI LN FINAL ‘VAII ATILN .EIFLFIAICE IN A REl-m DIAL ALGEBRA CO REE FLR UNIV‘ RSITY SJUD..1.¢TS presented by JOHN HOLARD JONES has been accepted towards fulfillment of the requirements for Ph. D. Mathematics—Education degree in may 7 1971 Date ’ 0-7639 ABSTRACT THE EFFECTS OF ENTERING ACHIEVEMENT LEVEL AND TIME SPENT IN COURSE COMPLETION ON FINAL EXAMINATION PERFORMANCE IN A REMEDIAL ALGEBRA COURSE FOR UNIVERSITY STUDENTS BY John Howard Jones The population of the study consisted of University students enrolled in a non—credit, remedial algebra course. The purposes of the study were: (1) To investigate the effect of time allotted for study and instruction on final examination scores, and (2) To investigate the effect of entering achievement level on final examination scores, and (3) To investigate the interaction effect of time and entering achievement level on final examination scores, and (4) To investigate the success rates of these students in the succeeding college credit algebra course. The main effects and interaction effects were studied by randomly assigning one group of students to study the course in 6 weeks while a randomly equivalent group studied John Howard Jones the course in 10 weeks. The measure of entering achievement level was the score earned on a test designed for that pur— pose. All students took a final examination at the end of their respective instructional periods. The scores on the final examination served as the dependent variable for a two way full factorial analysis of variance. The independent variables were time (2 levels) and entering achievement score (4 levels). ‘ The proportion of the 6 week group who successfully completed the succeeding college credit algebra course was compared to a similar proportion for the 10 week group. This part of the investigation was conducted during the term immediately following the study of the remedial course. The analysis of variance was conducted on the CDC36OO computer at the Michigan State University Computer Center. The Tukey HSD Post Hoc comparisons were used to locate the significant differences. The prOportion comparison was conducted with a Chi-Square test of proportion differences. The analysis of variance failed to uncover significant effects which could be attributed to time group or to inter— action of time group and entering achievement level. The analysis of variance did uncover a significant effect due to entering achievement level. Post Hoc comparisons showed that students of lower entering achievement in the 10 week group performed as well John Howard Jones (in the sense of the HSD test) as students of higher enter— ing achievement in the 6 week group. There were no significant differences of the success rates in the succeeding college credit algebra course between those who studied the remedial course for 6 weeks and those who studied the remedial course for 10 weeks. THE EFFECTS OF ENTERING ACHIEVEMENT LEVEL AND TIME SPENT IN COURSE COMPLETION ON FINAL EXAMINATION PERFORMANCE IN A REMEDIAL ALGEBRA COURSE FOR UNIVERSITY STUDENTS BY John Howard Jones A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY College of Education 1971 DEDICATION Judy and Matt ii ACKNOWLEDGMENT S The writer wishes to acknowledge: The years of guidance and friendship which were freely offered by the members of his doctoral committee: John Wagner, Chairman Joseph Adney Maryellen McSweeney William Fitzgerald The financial support offered by the citizens of the Lansing Community College District through a Sabbatical leave which was granted by the Board of Trustees of Lansing Community College. The interest and cooperation of the Department of Mathematics, Michigan State University. iii TABLE OF CONTENTS CHAPTER Page LIST OF TABLES . . . . . . . . . . . . . . . . Vi I. INTRODUCTION . . . . . . . . . . . . . . . . l The Problem . . . . . . . . . . . . . 1 Statement of the Problem . . . . . . . . 1 Importance of the Study. . . . . . . . . 2 Design of the Study . . . . . . . . . . . . 6 Background . . . . . .‘. . . . . . . . . 6 The P0pulation . . . . . . . . . . . . . 9 The Sample . . . . . . . . . . . . . . . 9 Group B Described. . . . . . . . . . . . 9 Group A Described. . . . . . . . . . 12 Design Analysis and Data Matrix (ANOVA). 14 Design Analysis and Data Matrix (Chi— Square) . . . . . . . . . . . . . . . 15 Assumptions. . . . . . . . . . . . . . . 17 Limitations. . . . . . . . . . . . . . . l7 Succeeding Chapters. . . . . . . . . . . 18 II. BACKGROUND OF THE PROBLEM AND REVIEW OF RELATED RESEARCH. . . . . . . . . . . . . . 19 Background. . . . . . . . . . . . . . . . . 19 Review of Related Research. . . . . . . . . 32 III. STATISTICAL ANALYSIS OF THE STUDY. . . . . . . 44 Introduction. . . . . . . . . . . . . 44 The Analysis of Variance (Background) . . . 44 The Analysis of Variance (Results). . . . . 48 The Analysis of Variance (Hypothesis Test— ing) . . . . . . . . . . . . . . . 50 The Analysis of Variance (Post Hoc Tech— niques). . . . . . . . . . . . . . . . . 50 The Chi-Square Test of Proportion Differ- ences (Background) . . . . . . . . . . . 56 The Chi-Square Test of Proportion Differ— ences (Results and Hypothesis Testing) . 57 iv PLEASE NOTE: Some Pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS TABLE OF CONTENTS--C0ntinued CHAPTER Page IV. SUMMARY, CONCLUSIONS AND RECOMMENDATIONS. . . 60 Summary. . . . . . . . . . . . . . . . . . 60 Conclusions. . . . . . . . . . . . . . . . 61 Purpose 1 . . . . . . . . . . . . . . . 61 Purpose 2 . . . . . . . . . . . . . . . 62 Purpose 3 . . . . . . . . . . . . . . . 63 Purpose 4 . . . . . . . . . . . . . . . 63 Recommendations. . . . . . . . . . . . . . 64 Conclusion . . . . . . . . . . . . . . . . 65 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 67 APPENDIX . . . . . . . . . . . . . . . . . . . . . . 71 Michigan State University Mathematics Place— ment Test. 0 O O O O O O O O O O O O O O O 72 Information-Assignments—Time Schedule-Group B 73 Entering Achievement Test . . . . . . . . . . 77 Education 327-N-—Groups A and B . . . . . . . 80 Sample Quiz and Answer Key--Groups A and B. . 82 Sample Final Examination—-Mathematics 082 . . 87 Final Examination—-Mathematics O82° . . . . . 89 Information-Assignments—Time Schedule—Group A 93 Course Outline-~Mathematics O82 . . . . . . . 96 TABLE 8. 9. 10. LIST OF TABLES Data Matrix for the Analysis of Variance. . Data Matrix for the Test of Proportion Differ- ences . . . . . . . . . . . . . . Basic Statistics. . . . . . . . . Two-way Analysis of Variance. . . Post Hoc Pairwise Comparison of Eight Cell Means . . . . . . . . . . . . . . Cell Means I. . . . . . . . . . . Interpretation of Table 6 . . . . Cell Means II . . . . . . . . . . Success in Mathematics 108 Winter Chi—Square Table. . . . . . . . . vi Page 15 16 47 49 52 53 53 55 57 58 CHAPTER I INTRODUCTION The Problem §tatement of the Problem The population under study consisted of University students enrolled in a non-credit, remedial algebra course° The purposes of this study were: 1. To investigate the effect of tim§_allotted for study and instruction on final examination scores, and 2. To investigate the effect of entering achievement level on final examination scores, and 3. To investigate the interaction effect of timg_and entering achievement level on final examination scores, and 4. To investigate the success rates of these students in the succeeding college credit algebra course. The main effects and interaction effects of time and entering achievement level on final examination scores were studied by randomly assigning one group of students to study the course in 6 weeks while a randomly equivalent group studied the course in 10 weeks. The measure of entering achievement level was the score earned on a test designed for that purpose. All students took a final examination at the end of their respective instructional periods. The scores on the final examination served as the dependent variable for a two way, full factorial analysis of variance. iThe independent variables were time (2 levels) and entering achievement score (4 levels). The proportion of the six week group, who successfully completed the succeeding college credit algebra course was compared to a similar prOportion for the ten week group by employing the non parametric techniques of the Chi—Square test of proportion differences. This part of the investiga- tion was conducted during the term immediately following the study of the remedial course. Importance of the Study Most educational institutions fix, by course, the length of time which is permitted for the learning of a given number of topics in mathematics. Variable grades are assigned at the end of the course and many students fail: The persistent student may repeat the course but others abandon or alter their educational goals on the basis of failure. In any case there is a stigma associated with academic failure. Educational psychologist, Carroll, has advanced an hypothesis which I paraphrase: Almost all students can learn any topic to criterion if sufficient instructional time is provided.1 1John B. Carroll, A Model For School Learning, Teachers College Record, Vol. 64, 1963, pp. 722—733. The writer will refer to this hypothesis as the Carroll Conjecture although many others have recognized that learning rates vary greatly in any population of learners. Learning theorist, Bloom, has made the general observa- tion: Implicit in this formulation (Carroll's) is the assumption that, given enough time, all students can conceivably attain mastery of a learning task. If Carroll is right, then learning mastery is theoretically available to all, if we can find the means for helping each student. It is this writers belief that this formulation of Carroll's has the most fundamental impli- cations for education.2 Shulman paraphrased an analogy to modern course struc- ture which was originally given by Byers: Imagine the miIe run if it began with the firing of a gun and ended at the end of four minutes when another gun went off and everyone was to stop wherever they were. It would be even more startling if about five minutes later another gun went off for the next race and everyone began that race from the same point at which they had ended the previous one. By repeating such a process, we would guarantee the development of cumulative deficits for some runners as they fell progressively further behind after each successive race.3 Concern with learning rates in mathematics is not new. Metzler authored the keynote article in the first volume of 2Benjamin S. Bloom, Learning For Mastery, Evaluation Comment (Bulletin of the UCLA Center for the Study of Evalua- tion of Instructional Programs), May 1968, p. 3. Also quoted by Lee S. Shulman in Sixty-Ninth Yearbook of NSSE, p. 48. 3Lee S. Shulman, Psychology and Mathematics Education. The Sixty-Ninth Yearbook of the National Society for the Study of Education, Part I, 1970, University of Chicago Press, Chicago, Illinois, p. 49. The Mathematics Teacher. Metzler mentions the problem. Examinations will be ever with us, though it is to be hoped that they will not always continue to be quite so much of a bugbear as now. The bugbear will disappear as soon as students come to their examinations properly prepared. It is perhaps not usual but it is quite possible for students to enjoy examinations and they all will enjoy them when they are so prepared that they feel that they can pass any reasonable examination on the subject. Every healthy person likes once in a while to test their strength, mental as well as physical. Let those students who cannot thus be prepared well in the allotted time take longer for it. Some students have more ability than others and some are slower than others. In view of these facts, it seems strange that so many teachers expect every member of a class to doAthe same work almost equally well in the same time. More recently, Hocking has written We should most seriously reconsider the conse- quences of equating "fast" with "good". Because most of the faculty came through the standard sieve, it is quite natural that we make this equation. (I have even noticed implicit tendencies to equate "speed of learning" not only with intelligence and potential for success but with such totally independent matters as moral superiority and political sagacity.) Most students have goals which diverge from those of the faculty and this is as it should be: Should we not take some basic and very difficult decisions and set out upon new, more realistic paths?5 There exists little empirical evidence in support of the Carroll Conjecture as applied to mathematics education 4W. H. Metzler, "Where Shall We Place the Emphasis?" The Mathematics Teacher, Vol. I, September l908-June 1909, p. 3. 5J. G. Hocking, unpublished report consisting of Ten Lecture Outlines for Mathematics 082, a non-credit remedial algebra course offered by Michigan State University, 1970. but one impressive experiment has been conducted. Begle describes the experiment and the principal results obtained by Herriot for the School Mathematics Study Group. At about the time that Carroll made this sugges- tion the School Mathematics Group was organizing an experiment which, as it turned out, provided evidence in favor of this hypothesis. This experiment involved two groups of experimental students and two correspond- ing groups of control students. The first experimental group consisted of students entering seventh grade (and thus between 12 and 13 years of age) who were between the 25th and 50th percentile in ability, whether measured by a standard IQ test or by a standard mathematics achievement test. The other experimental group consisted of students in the same ability range who were entering the ninth grade. The control groups, which were selected a year later, consisted of students entering seventh or ninth grade who were between the 50th and 75th percentile in ability. Both the experimental and the control seventh grade students followed the same mathematics curriculum and used the same textbook, a seventh grade text pre- pared by SMSG. Similarly, the experimental and the control ninth grade students followed the same mathe- matics program and used the same SMSG algebra text. What was different was that the experimental groups were given two school years to study the material which the control groups studied for the usual one school year. A battery of tests was administered at the end of the experiment. Analysis of the test results showed that the seventh grade experimental students performed al- most, but not quite, as well as the control students on this battery. Analysis of covariance using scores from a battery of pretests strongly indicated that the experimental students had learned considerably more, given two years, than they would have if they had only the usual one year. At the 9th grade level the results were in the other direction. The experimental students outscored the control students on the final battery of tests. Here is a case then where students of below average ability were able to reach about the same level of achievement as students of above average ability as a result of an increase in the amount of instruction provided them.6 The paucity of research concerning the validity of the Carroll Conjecture in mathematics education moved the writer to conduct this investigation. If the Carroll Conjecture is valid, then many failures might be avoided by providing more time for learning and instruction to the student of lower entering achievement. It might ultimately be more realistic to view the criterion for completion of a course as fixed and to view the amount of time required by the individual to perform to criterion as variable. Design of the Study Background Students who enroll in first year mathematics courses at Michigan State University are required to take a 50 minute, 30 response, multiple-choice, placement examination (Al Appendix). If a student scores 24 or more correct responses, then he is permitted to take mathematics 112 which is the first course in a 4 term unified sequence of courses in analytic geometry and calculus. If the student scores 18 through 23 correct responses then he is permitted to take Mathematics 111 which is a one term course in college algebra 6E. G. Begle, "The Role of Research in the Improvement of Mathematics Education," Educational Studies in Mathematics, V01. 2, pp. 237-238. and trigonometry. If a student scores 14 through 17 correct responses then he is permitted to take Mathematics 108 which is the first term of a two term unified sequence of courses in college algebra and trigonometry. Exceptions to the fore- going parameters are often made on the basis of personal interview. If a student scores 0 through 13 on the place- ment examination then he must take Mathematics 082. Mathematics 082 is a beginning algebra course covering topics ranging from the arithmetic of real numbers through the study of the solution of quadratic equations with rational coefficients and real roots. No credit is given for successful completion of Mathematics 082, since the course is regarded as remedial. The course must, however, be successfully completed before the student can enroll in the first college credit course, Mathematics 108. Almost all students in Mathematics 082 wish to enroll in Mathematics 108. In recent years, Mathematics 082 had been conducted through individual study of a workbook-—A Survey of Basic Mathematics—-2nd Edition—-by Fred Sparks, a McGraw-Hill publication. The student was given the opportunity to attend recitation sections at which teaching interns presided. The program proved less than satisfactory. A committee of Mathematics Department Faculty was appointed in 1969 and was charged with developing a more efficient and effective method of offering Mathematics 082. Professor John G. Hocking, Department of Mathematics, Michigan State University, authored a series of ten lectures for Mathematics 082 during the summer of 1970. The lectures were designed to be offered in conjunction with the previ- ously mentioned work book. The committee decided that one lecture would be pre- sented on each Thursday afternoon during the Fall Term of that 1970-71 academic year. A quiz covering the content of this previous lecture was to be presented on the following Tuesday afternoon. Both lecture and quiz were to be pre- sented at 4:00 p.m. in large lecture meeting of all students enrolled in Mathematics 082. In addition to the large meetings there were to be 20 recitation sections established for the purpose of answer- ing questions and amplifying the lecture material in a more congenial setting. The 20 recitation sections were to be staffed by students enrolled in Education 327-N, Methods in The Teaching of Secondary Mathematics, a course for senior students who are candidates for the degree in secondary mathematics teaching. These recitation sections met for one hour at various times of the day on Tuesday and again on Thursday. A student enrolled in Mathematics 082 during Fall 1970—71 would therefore receive a lecture on Thursday and a quiz on Tuesday, both in large session meeting. In addi— tion he would attend a small recitation meeting once on Tuesday and once on Thursday. The writer conducted his investigation with these conditions in effect at the start of his work. The Population The population under study is all students who have enrolled, are presently enrolled, or who will enroll in Mathematics 082 at Michigan State University. The Sample The writer selected the students whose Tuesday-Thursday recitation sections met at 9 A.M. in Fall 1970, as the sample from which both experimental and control groups were drawn. This was done so that time of day was eliminated as a confounding variable in the experiment. Ten sections met at 9 A.M. They were sections 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Five of these sections, 3, 6, 8, 11 and 12, were selected at random and were assigned to complete the required topics of Mathematics 082 in 6 weeks. The writer will refer to sections 3, 6, 8, 11 and 12 as experimental group B in the sequel. The other five 9 A.M. sections, 4, 5, 7, 9 and 10 together with all other Fall term sections of mathematics 082 were designated as group A. Sections 4, 5, 7, 9 and 10 were designated control group A*. Group B Described The unique assignmemt for group B necessitated a differ- ent time schedule for lectures and quizzes than that which had been planned by the Mathematics 082 committee. The writer 10 secured a different room location for the meeting of the Tuesday-Thursday large sessions. A detailed paper entitled Mathematics 082--Information, Assignments and Time Schedule, Sections 3, 6, 8, 11 and 12 was prepared and delivered to the sections of group B during the second recitation meeting of the term (A2 Appendix). An entering achievement test was also administered during this recitation meeting (A3 Appendix). The time schedule detailed the offering of the 10 Hocking lectures on successive Tuesday and Thursday after— noons at 4 P.M. starting on September 24th and ending on October 27th. Ten quizzes were to be given during the 9 A.M. recitation sessions on both Tuesday and Thursday starting on September 29 and ending October 27th. The average enrollment per recitation section was 14 students. Each recitation section was staffed by 4 student interns from the Education 327-N class which was previously described. The interns were given a brief descriptive hand- out and also retained a copy of the group B information sheet (A4 Appendix). The interns partitioned the recitation sec- tion with 3 to 5 Mathematics 082 students per intern so that small group question sessions could be conducted. Each quiz given in the recitation section covered material presented during the previous large lecture meeting. Individual printed answer keys were given to each student immediately upon completion of each quiz (A5 Appendix). These keys were used by the student for self scoring and served to ll determine the type of problem the student would wish to con- sider with his teaching intern. The quiz usually required 20 to 25 minutes of time in the recitation section. The remaining half hour was spent in small groups with the intern. The quizzes were not used to determine student success or failure in the course. The 10 Hocking lectures were delivered by three Michigan State University Mathematics Department staff members. They were: Professor Patrick Doyle, Associate Professor William Fitzgerald and the writer. The writer was appointed to the staff by the department for the duration of this investiga- tion. The three lecturers divided the 10 Hocking lectures as evenly as possible and in such manner that if a lecturer delivered lecture "K" to group B then he would later deliver lecture ”K” to group A. This was done in an effort to eliminate "lecturer" as a confounding variable in the study. The last of the 10 Hocking lectures was presented to group B on October 27th. A sample final examination was presented to each student following the last lecture (A6 Appendix). This was done in an effort to guide student review and to allay fear of the actual final examination. The student was able to consider the sample final on an individual basis and then ask questions about the problems during the last recitation meetings on October 29th and November 3rd. The large session meeting of October 29th was devoted to review. 12 A 35 point final examination was given to group B on Tuesday, November 3rd during the large session meeting (A7 Appendix). Extraordinary security precautions were taken to assure that the integrity of the examination was main- tained since the same examination was used to evaluate group A on Thursday, December 3rd. The results of the final examination were presented to group B students during their recitation meeting on Thursday, November 5th. Those students who earned a score of 24 or more points were told that they had successfully completed Mathematics 082 and would be permitted to enroll in Mathe- matics 108 the following term. Those students who earned a score below 24 points were encouraged to join in the remain— ing 4 large lectures with group A and were told that they would have the Opportunity to retake the final examination with group A on December 3rd. In addition, the recitation sections continued for the students who did not pass the examination. Group A Described We recall that all Mathematics 082 sections other than those in Group B comprise group A. The control group A* consisting of 9 A.M. recitation sections 4, 5, 7, 9 and 10 is therefore a subclass of group A. A detailed paper entitled—-Mathematics 082--Information, Assignments, Time Schedule, Sections 1, 2, 4, 5, 7, 9, 10, 13, 14, 15, l6, l7, l8, l9 and 20 was prepared and delivered 13 to group A during the second recitation meeting of the term (A8 Appendix). The entering achievement test was also administered during this recitation meeting. The time schedule detailed the offering of the 10 Hocking lectures, one on each Thursday afternoon starting on September 24th and ending on November 24th.7 Group A quizzes were given during the large session meetings on Tuesdays starting September 29th and ending November 17th. Individual printed answer keys were given to each student immediately upon completion of the quiz. The group A quizzes were written in the style of parallel form to the quizzes in group B (A5 Appendix). Recitation sections of group A met with student interns on Tuesdays and Thursdays as described for group B. The recitation meetings for the control group A* met at 9 A.M. The meeting time for each other section within group A is detailed in the Fall term, 1970 schedule of classes. It is worthwhile to note that no quizzes were given during the group A recitation meetings. As a consequence there was twice as much time available for questions in the recitation meetings. Except as noted, the procedure for conducting the recitation meetings was exactly as described for group B. The large lectures were delivered as described under group B with the exception of the different time schedules. 7Lecture 10 was presented on a Tuesday due to the Thanksgiving recess. 14 The last of the 10 Hocking lectures was delivered on November 24th. Following the lecture, the students in group A received the sample final examination. The last recita— tion meetings on December lst were devoted to review and consideration of quiz 10 and the sample final examination. The last large lecture meeting on December lst was devoted to review. The 35 point final examination was given on Thursday, December 3rd at 4:00 P.M. during a large session meeting. Those students who earned a score of 24 or more points on this examination successfully completed Mathematics 082 and were allowed to enroll in Mathematics 108 the following term. Design Analysis and Data Matrix (ANOVAL The statistical hypotheses tested within the following design were: 1. There is no significant group difference in per— formance on the final examination due to time. 2. There are no significant differences in performance on the final examination due to entering achieve— ment level. 3. There are no significant differences in performance on the final examination due to interaction of the variables time and entering achievement level. Data for a two way, full factorial analysis of variance was tabulated in the following data matrix. 15 Table l.——Data Matrix for the Analysis of Variance B A* 01 xx-.. x... Q2 XXX°°° xx... Q3 x... xx... Q4 xx... xxx... The symbols are degined: A* is the 10 week control group consisting of sections 4, 5, 7, 9 and 10. B is the 6 week experimental group consist- ing of sections 3, 6, 8, 11 and 12. 01, Q2, Q3, and Q4 are the quartile classes on the entering achievement test. x is the individual score on the final examination. Design Analysis and Data Matrix (Chi-Square) The last purpose of this study was to investigate the success rates of the students in the sample in the succeeding college credit algebra course. This was done by employing the following design. 16 Table 2.-—Data Matrix for the Test of Proportion Differences Bl BZ Al* X Y Z The symbols are defined: El. consists of all students from group B who passed the Mathematics 082 final on November 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. consists of all students from group B who passed the Mathematics 082 final on December 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. it” Alj_consists of all students from group A* who passed the Mathematics 082 final on December 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. Z; X, and g_are ratios. The denominator of each ratio is the number of students from the associated category who enrolled in Mathema- tics 108 during Winter term of the 1970-71 academic year. The numerator of each ratio is the number of students from the associated category who received a final grade of 2.0 or higher in Mathematics 108 during the Winter term of the 1970-71 academic year. The statistical hypothesis tested within this design The design was analyzed by a Chi-Square test of prOpor- tion differences. l7 Assumptions 1. The usual assumptionsfor conducting analysis of variance are made. 2. The usual assumptions for conducting a Chi—Square test of proportion differences are made. 3. The population will not differ significantly from a similar pOpulation at another institution. A basic course outline is given in the Appendix (A9). 4. The integrity of the final examination was main- tained between the two dates at which the examina- tion was written. Limitations This study was not designed to improve the teaching or learning of Mathematics 082. The only purposes of the study are clearly stated in the introduction. The writer is compelled to offer an opinion which is neither nurtured nor starved by the formal conduct of this study. The writer has grave reservations concerning the large lecture method used in the teaching of Mathematics 082. The teaching methods in use during this study were probably superior to the methods in use prior to the fall of 1970 but not by large measure. Approximately fifteen years of the lecture method has left the student in a position requiring him to take a remedial algebra class in college. We then offer the stu- dent another lecture course which is significantly different from courses in his prior experience only in the ratio of number of students to lecturer. The student has been trained 18 to view mathematics as a spectator sport in which the number of spectators is ever increasing as the opportunity to perform is ever decreasing. A major change of method is due in the teaching of Mathematics 082 and such change should be in the direction of individualized instruction. Such methods as audio-visual- tutorial learning, programmed learning and computer assisted learning should be considered for immediate adoption to relieve the plight of the student in Mathematics 082. These comments are editorial in nature and as such, no justification is offered. Succeeding Chapters In Chapter II the background of the problem and a review of related studies is provided. Chapter III consists of the formal numerical analysis of the stated hypotheses. Chapter IV contains the results, an interpretation of data and recommendations based upon this study. CHAPTER II BACKGROUND OF THE PROBLEM AND REVIEW OF RELATED RESEARCH The way out of scholastic systems that make the past an end in itself is to make acquaintance with the past a means of understanding the present.8 Background Curriculum modification in mathematics has been largely a matter of installing new subject matter in an old frame- work. The framework consists of a fixed time structure which is called by the various names: semester, trimester, quarter or term. Courses are designed to fit within these time units and all students within a given institution budget their academic programs on the basis of the same unit of time. Mathematics courses may differ greatly in content from year to year but the fixed time framework is ever present. Massive content modification is observed in mathematics courses at every level. Two major groups have been instru— mental in changing content. 8John Dewey, as quoted by Sidney Hook in the preface to Education for Modern Man--A New Perspective (New en- larged edition; New YOrk: Alfred A. Knopf Publishing Co., 1966). 19 20 The work of the School Mathematics Study Group (SMSG) has influenced content at both the primary and secondary school levels. Wagner, was Assistant to the Director of SMSG when he wrote: History In the spring of 1958, after consulting with the presidents of the National Council of Teachers of Mathematics and the Mathematical Association of America, the president of the American Mathematical Society appointed a small committee of educators and univer- sity mathematicians to organize a School Mathematics Study Group whose objective would be the improvement of the teaching of mathematics in the schools. Professor E. G. Begle was appointed director of the Study Group, with headquarters at Yale University. In addition, the organizing committee appointed an Advisory Committee, consisting of college and university mathematicians, high school teachers of mathematics, experts in educa— tion, and representatives of science and technology, to work with the director. The National Science Foundation through a series of grants, has provided very substantial financial sup- port for the work of the Study Group. Objectives The world of today demands more mathematical knowl— edge on the part of more peOple than the world of yesterday, and the world of tomorrow will make still greater demands. Our society leans more and more heavily on science and technology. The number of our citizens skilled in mathematics must be greatly in- creased, and understanding of the role of mathematics in our society is now a prerequisite for intelligent citizenship. Since no one can predict with certainty his future profession; much less foretell which mathe- matical skills will be required in the future by a given profession, it is important that mathematics be so taught that students will be able in later life to learn the new mathematical skills which the future will surely demand of many of them. To achieve this objective in the teaching of school mathematics, three things are required. First, we need an improved curriculum which will offer students not only the basic mathematical skills but also a deeper understanding of the basic concepts and 21 structure of mathematics. Second, mathematics pro— grams must attract and train more of those students who are capable of studying mathematics with profit. Finally, all help possible must be provided for teachers who are preparing themselves to teach these challenging and interesting courses. Each project undertaken by the School Mathematics Study Group is concerned with one or more of these three needs. The School Mathematics Study Group is, of course, not the only organization concerned with the improve- ment of mathematics in our schools. Some of the others are the Secondary School Curriculum Committee of the National Council of Teachers of Mathematics; the Uni- versity of Illinois Committee on School Mathematics; The Commission on Mathematics of the College Entrance Examination Board, and the University of Maryland Mathematics Project. 'Liaison with these groups is provided by the simple expedient of including represen— tatives of these organizations on the Advisory Commit- tee and in the work of the various panels.9 More recently, Herriot of SMSG has observed: During the last decade the secondary school mathe- matics curriculum has been subject to critical scrutiny and has undergone a gigantic upheaval. The prime movers of this curriculum revision questioned the rationale of the consumer utilitarian philOSOphy with its major emphasis on repetitive drills and acquisition of skills. The speed with which "the new math" won a place in school curricula offerings was startling, even to the most dedicated proponents of a new approach to mathe- matics. The advent of an era of reform was hailed so quickly by sufficiently diverse and numerous groups of mathematicians in industry, research and teaching that it was evident that some reform was obviously long over- due. The time was ripe for a change. Through the efforts of the School Mathematics Study Group and other curriculum projects, the secondary school mathematics offerings have been modified or dramatically altered in these recent years. The initial emphasis on the quality of the mathematics diet of the college-bound student, and the lack of recommendations 9John Wagner, "The Objectives and Activities of the School Mathematics Study Group," The Mathematics Teacher, 22 and pilot studies for the average and below-average students, do not necessarily indicate absence of concern about this large group, but reflect only the university mathematician's genuine interest in and intimate knowledge about the college—bound group.‘0 The effect of the Committee on the Undergraduate Program in Mathematics (CUPM) has been observed in under- graduate mathematics courses across the land. Thompson and Poe provide background. The Mathematical Association of America.appointed its committee on the Undergraduate Programs in Mathe- matics (CUPM) in January, 1959. CUPM is charged with the responsibilities of recommending and influencing undergraduate mathematics curriculum changes. While carrying out its responsibilities, CUPM has exerted a dominant effect within a national mathematics curriculum reform movement. Many colleges and univer- sity mathematics departments have used CUPM recommenda- tions as guidelines in structuring their present under- graduate mathematics course offerings to provide an efficient program of studies for the students of today.11' Modification of course content has been accompanied by alteration of teaching objectives and by associated changes in teaching method. It is not entirely clear that all the recent changes in the mathematics curricula are the result of rational analysis and logical conclusion by the practitioners of mathematics education. Begle has made the following observation. Our major mistake in mathematics education has been our failure to recognize that we have not possessed the tools needed to do a good job in improving mathema- tics education, and that in the course of carrying out 1°Sarah T. Herriot, The Slow Learner Project——The Secondary School "Slow Learner" in Mathematics, SMSG Report No. 5, Stanford University, 1967, p. l. 11p. E. Thompson and R. L. Poe, "A Report on the CUPM Recommendations in the State of Texas," The American Mathema- tical Monthly, Vol. 75, No. 10 (Dec. 1968), p. 1107. 23 our normal activities as teachers and as mathematicians we are not likely to be provided with these tools. Let me hasten to say that I do not believe that this mistake has had disastrous results. On the con- trary, I am convinced that even though the guide—posts we followed and the tools we used in our attempts over the last decade to improve mathematics education were of dubious validity, we did move in the right direction and we have achieved positive results. All of us have received large amounts of anecdotal evidence both from students and teachers to the effect that what we have done has been good. I might say that we are very pleased at the results we are getting in our analyses of the data collected in our longitudinal study. The mathema— tics provided in the School Mathematics Study Group text— books seems to provide a better understanding of mathe- matics provided in the more classical textbooks. The time and effort we have devoted to reform during the last decade has not been wasted. Nevertheless, we cannot stop now. Further improve- ments are essential. Our children will live in an even more complicated and more quantified world than that of today. They need a better mathematics program than they now are getting. We still have many difficult problems to solve before we can make further improvements. In fact, I believe that so far we have attacked only the easier problems of mathematics education.12 A further warning against complacency is given by Allendoerfer. In spite of all the thunder and lightning, I be— lieve that we have not made any really fundamental change in school mathematics. ‘We have taken a structure which was well built to start with, have shifted its partitions, redecorated its rooms, put in new and faster elevators, dressed up the lobby, and cleaned the grime off the exterior. Certainly it is still intended for the use of the upper classes. We have not gone to the back streets and done anything with the (educational) slums, and we have hardly thought about new types of structures which are properly suited to the modern age. It is high time that we moved in to these new areas of activity. 12E. G. Begle, loc. cit., pp. 237-238. 13Carl B. Allendoerfer. "The Second Revolution in Mathematics," The Mathematics Teacher, Vol. 58, No. 8 (Dec. 1965), p. 692. 24 Begle feels that mathematics educators do not possess the "tools" required to effectively change mathematics education. The writer notes that there is a more funda- mental problem of the kind Allendoerfer mentions. We in mathematics education have failed to recognize and attack at least one important component of the curricula we would alter. We have failed to pay attention to the fixed time structure which is implicit in almost all current mathematics courses. Altering content and method while ignoring the variant learning rates of individuals is like offering a new instructional tape recording to a student without realizing that the student may be deaf. Why have we defended the fixed time structures for so many years when such structures are obviously in conflict with our knowledge that learning rates are quite variant in any population of learners? The answer is quite unpleasant. A part of this answer is given by Goodlad. A widely held expectation for schooling has been coverage of a set body of material denoting elementary, secondary, or higher education. It is implied in such often-heard statements as: this child is not ready for school; he has completed the fifth grade; or, he has had calculus. Such an expectation has led to prescrip- tion of a common body of topics for a class or grade, with little or In: predetermination of what is already known; to nonpromotion and grade repetition for learners who cover too little; to the distribution of identical textbooks to an entire class; to the preparation and use of tests stressing the possession of facts; to evaluative judgments based on group norms rather than prOgress on a learning continuum. These practices appear not to be compatible with our knowledge of indi- vidual differences or our growing concern for the individual. ~ 25 This notion that "a school is to cover” has spread a drab cloak of conformity over virtually every act of schooling, splashes of light breaking the surface only because a few lighthouse schools and enlightened teachers have dared to differ. Children from upper socio-economic class families have been admitted early to kindergarten because they already possessed what kindergarten was designed to provide. Teachers have en— gaged children in trivia because they hesitated to tread on ground reserved for the next grade. Adminis— trators have purchased identical books marked "4" or with four dots in order to match books by the thousand with fourth-graders by the thousand. High school teach- ers have criticized elementary school teachers for failing to provide adequately for children already advanced far beyond this restrictive expectation. And parents think they know what "fourth grade" or "A" in science means and happily report to grandparents that Susie passed. Brown provides a further reason for the retention of "grade" in school. An analogous argument may be applied to course within grade. The durable attractiveness of the grade lies in its administrative convenience. It serves as a comfort— able holding pool in which school administrators can and do throw youngsters for custodial purposes and forget them for a year.15 We are thus trapped in our fixed time structure by "tradition" and "administrative efficiency". Can we feel justified in maintaining our old ways without knowing why? There have been some interesting attempts at breaking down the fixed time structure but unfortunately few of these have taken place in the University. The non-graded plan 14John I. Goodlad, Diagnosis and Prescription in Edu- cational Practice. New Approaches to Educational Instruc- tion, A conference held on May 11, 1965 by Educational Testing Service, Princeton, N. J. 153. Frank Brown, "The Non—Graded High School," Phi Delta Kappan, Vol. 44 (Feb. 1962), p. 206. 26 for K-12 is an admirable venture. Brown observes: If the public schools in America are ever to achieve the ideal of having each youngster progress at his best rate of learning, then some form of non- grading must be instituted. This leads to the shattering implication that within the next five years every intellectually respectable high school will have some degree of non-gradedness. For the grade is a trapping of the outworn past. It was first conceived during the Middle Ages in a "gymnasium" at Stuttgart, Germany, and has grown sterile in the age of universal education and the hydrogen bomb.16 Brown also suggests: The first step in recovering from decades of intellectual disaster wrought by the grade must be to reclassify youngsters for learning on the basis of their achievement rather than the grade to which they have been chronologically promoted.l7 The most ambitious experiment in non—graded education yet attempted is the "Nova Plan" of south Florida. The Nova High School, which opened in September, 1963, is an initial unit of the South Florida Education Center, an educational complex imaginative in design and advanced in concept. Eventually this complex will house tax-supported schools encompassing kindergarten through junior college, plus a private university with a graduate school. This long-range program, when com— pleted, will present a continuous integrated process of learning unparalleled in education history. Known as "The Nova Plan," this new approach may well develop a model educational system for the county, state, and nation.18 Later in the same article, Kaufman and Bethune observe: 16B. Frank Brown, loc. cit., p. 206. 17Ibid., p. 207. 18Burt Kaufman and Paul Bethune, "Nova High—Space Age School," Phi Delta Kappan, Vol. 46 (Sept. 1964), p. 9. 27 Each discipline has identified the approach its faculty thinks best suited to remove extraneous retardation in the case of the gifted and relaxation of time pressures for the less able. The idea here is predicated on the fact that each student should go as far as his capabilities will carry him without pressures to complete material for the sake of arti— ficially set standards which do not take into considera— tion the individual differences among students. Instead of being promoted a grade level each year, students progress through a series of achievement levels called I'units" in each subject area. At the end of each unit a test is given which determines whether the student may continue to the next level. A student per— forming below minimum levels on the test must repeat that unit satisfactorily before being allowed to advance. Thus no student ever fails a whole year's work in any subject. At graduation time it is not expected that all students will have completed the same amount of the curriculum. However, a student who reports for a certain unit is expected to have a set of values, skills, and knowledge in common with others who are promoted to the same units. The identification of material within a unit allows for ladder—like steps upon which the student is constantly building and will use in later units.19 The non-graded concept has now been tried in varying degrees in many primary and secondary schools across the land. Brown has predicted the effect this will have on the Colleges and Universities: We realize that bringing about change in the college curriculum is somewhat like moving a graveyard, but the impact of students educated without the aca- demic bridle is already bringing a new respect for change in higher education. Spurred by the increased intellectual excitement of their students from non- graded high schools, colleges too will turn to the non- graded system.20 19Ibid., pp. 10-11. 20Brown, loc. cit., p. 207. 28 The writer is amazed that University sponsored research in learning theory is last applied in the University. It is as if there were an unwritten law which states that educa- tional experimentation and innovation have K—12 as their domain. It has been hypothesized that two humans are most alike at the moment of conception and that those character- istics which make peOple individuals diverge with time from conception to death. We note that efforts to individualize learning are primarily made at the elementary school level. The greatest provision for individual difference is thus made when that difference is least. We should maintain our efforts at the elementary level and increase our commitment to individualized instruction at the secondary and college levels. We have observed that the result of University spon- sored educational research is often implemented in the K-12 program rather than in the College or University program. There may be hidden advantage in this for those who would alter University teaching. We can learn from the mistakes we observe in K—12. Glaser developed a basic teaching model which affords us the opportunity to analyze the teaching function in terms 21 of four basic components. These components are: Instruc- tional Objectives, Student Entering Behavior, Instructional 21Robert Glaser, Psychology and Instructional Tech— nology, Training Research and Education, University of Pittsburgh Press, Pittsburgh, 1962, p. 6. 29 Procedures, and Performance Assessment. How is this model used in planning a University mathematics course today? The Objectives for a mathematics course are implicitly defined by the table of contents of the textbook. Explicit objectives are seldom identified. Goodlad said it best when he noted that "school is to cover". Teachers have accepted such objectives since there is false satisfaction associ- ated with completing the required chapters in the required time. The teacher is thus able to "meet the objective". There is something very wrong here. Objectives are not for teachers; they are for students. The student is most likely to succeed if he knows explicitly what terminal performance he is expected to display. Let us suppose that a set of explicit objectives for a mathematics course has been estab- lished. By this we mean that student attainment of each individual objective is measureable. Why not state that successful completion of the course means that the student has attained all of the course objectives? It is obvious that different individuals will require different amounts of time and different amounts of instruction to do this. If a student has not attained an objective then he is still in pursuit of that objective. He does not fail due to the artificial imposition of a time limit on his pursuit. Academic excellence could thus be regarded as a function of absolute attainment as well as comparative attainment. Entering Behavior is probably the most neglected com- ponent of the basic teaching model. The student is told that 30 he may enroll in course "K+l" if he has completed course "K". That is the extent of attention now paid entering behavior. The writer recalls a statement attributed to Ausubel which asserts that the most important single factor influencing learning is what the learner already knows. If we accept Ausubel's position then we should pay greater attention to measuring the capabilities the student displays upon entering a mathematics course. If we measure a stu- dents entering behavior then we are obligated to decide whether the student's capability is adequate to carry on the studies planned for our course. Course objectives should be attainable by the students studying the course. That is a simple fact which is basically ignored in mathematics teaching today. Iggtructional Procedures today are virtually the same for all. The students hear the same lectures and are assigned the same problems. All students follow the same time schedule. The system is administratively convenient but is grossly in- effective. Failure rates of forty and fifty percent are not at all uncommon in mathematics courses offered during the first undergraduate year. We have never made a genuine effort to suit instructional pace and instructional method to the individual. Contrary to popular belief, the presence of television receivers on campus does not reflect an effort to individualize instruction but rather reveals our true pur- pose which is to lower the instructional cost per student. 31 Performance Assessment consists of assigning a variable grade which is based upon two or three tests uniformly dis— tributed throughout the term and a final examination which is given at the end of the term. This is an unfortunate practice in view of our knowledge about the varied learning rates of students. A fast learner might have been able to pass all the tests in three weeks. A slow learner might only be ready for the first unit test on the day of the final examination. As it is now, the fast learner is bored by the course but he does pass. The slow learner fails and the failing grade is used to identify the slow learner as a bad student. What right do we have to equate fast with good and slow with bad? Student attitude toward testing is justifi— ably negative. There are only two ideal responses to the question; How did you do on the test? They are; I passed or I have not yet passed. There is no place for the negative response; I failed. It is unfortunate but true that college instruction in mathematics is not greatly improved since 1938 when Wood and Learned noted: Each individual has some level peculiar to him- self at which his education in any subject must begin. Instead of expecting the members of a college class to conform to an average, we might better arrange circum- stances so that each student could make full use of what he has learned and could advance from the point where he really stands.22 2?w. S. Learned and Ben D Wood, The Student and His Knowledge (Carnegie Foundation for the Advancement of Teaching, New York, 1938), p. 44. 32 Review of Related Research We recall that the population of this study consists of University students enrolled in a non-credit, remedial algebra course. One purpose of the study is to investigate the effect of entering achievement level on final examina— tion scores in the course. The writer is thus concerned with a highly restricted sub class of the class of all college students. It is apprOpriate that we acquaint our— selves with the general achievement variability of the universal class of students in which our population lies. Fortunately, a monumental survey of this larger class of students was conducted by Wood and Learned.23 A sketch of the findings of this research is given by Cook and Clymer. Learned and Wood's study of The Student and His Knowledge gives a graphic portrayal of the intellectual and achievement variation among high-school and college students. Their research serves as an excellent illus- tration of the major findings of variation in ability at all educational levels. This study, perhaps the most comprehensive and detailed of the variation of high-school and college students, involved eight to twelve hours of testing of over 26,000 high-school Seniors, nearly 6,000 college Sophomores, and slightly less than 4,000 college Seniors. Intelligence which was measured by the Otis Self- Administering Test of Mental Ability revealed that the broad range of intelligence found in the lower schools has not disappeared in either the high school or college. While the college distributions of intellect were skewed toward the higher levels, the total range of intelligence at the college Sophomore and college Senior levels was nearly as great as that at the high- school Senior level. In addition to the wide range, there was extensive overlap of distributions of the 23Ibid., p. 44. 33 students at the various levels. The college Senior median was only one—half of a standard deviation above the college Sophomore median. The high-school Senior median was approximately three—fourths of a standard deviation below the college Sophomore median. These facts indicate clearly the extensive overlap in the distributions of intellect. The General Culture Battery, which consists of tests in general science, foreign literature, fine arts and social studies, was administered to gain informa- tion about the educational achievement of the high- school and college students. The overlap of the high- school and college classes in achievement, as measured by these tests, was striking. More than 28 per cent of the college Seniors had achievement scores below the average college Sophomore. Nearly 10 per cent of the Seniors had scores below the average high-school Senior. Looking at the distributions from the stand— point of the high—school, Learned and Wood discovered that more than one-fifth of the high-school Seniors exceeded the score of the average college Sophomore. One high-school Senior in ten exceeded the average college Senior performance on these tests of educa— tional achievement. One college was selected for intensive study for the purpose of learning the variability of achievement of classes within the same institution. The General Culture Battery was administered to the student body of this college along with a two—hour test in English literature, vocabulary and usage, and a lengthy test in mathematics. On the basis of the performance on these achievement tests it was found that if the gradu— ating class that year had been selected from the total student body on the basis of achievement instead of from the Senior class on the basis of courses taken and credits earned, only 18 per cent of the Senior class would have graduated! The remainder of the graduates would have been made up of 21 per cent of the Juniors, 19 per cent of the Sophomores, and 15 per cent of the Freshman. The mean score of the gradu— ating class selected on the basis of achievement would have been one standard deviation above the average of the class that actually graduated, and its mean age would have been two years younger. It is difficult to imagine data which give a more dramatic portrayal of the variation of achievement than the information presented by Learned and Wood.24 24W. W. Cook and Theodore Clymer, Acceleration and Retardation in Individualizing Instruction, Sixty-First Yearbook of NSSE, University of Chicago Press, Chicago (1962), pp. 185-186. 34 The population of this study is therefore located within a collection of students whose general achievement is extremely variable. we might conjecture that the mathematical achievement variability within the sample would be small since all students in the sample were en— rolled in the same remedial course. The performance of the sample students on the entering achievement test (A3 Appendix) does not support the conjecture. Scores ranging from zero to twenty were possible on the test. The grand mean for 126 students for whom complete data was available was 14.46 with a variance of 9.29. Lowest and highest scores of 3 and 19 were achieved. It is perhaps more inter— esting to state that a significant number of students were unable to do even half of the basic arithmetic problems while an equally significant number of the students were quite competent on all but two or three problems of the test. It was painfully obvious to the writer that no con- ventional course could be offered which would meet the individual needs of these students. It was not the purpose of the study to design a new course but rather to study some of the many relationships which exist between entering achievement level and learning rates within the conventional course framework. The writer hOpes that others will imme- diately attack the problem of individualizing the methods of instruction in Mathematics 082. There have been few experiments in mathematics educa- tion in which the primary variables of interest were ability 35 level and time. One such experiment was conducted by Begle. The students were in the middle of the fourth grade. A very small topic in mathematics, completely new to the students, was used and.was taught for one, two, or three days. In the longer teaching sessions, no new ideas were introduced, but there was time for a wider variety of illustrations of the idea introduced and for more student discussion and questioning than was possible in the shorter sessions. On the basis of‘a test of arithmetic reasoning, the students were grouped into three ability levels-- low, medium and high. The average scores on a post- test were not significantly different along any of the three diagonals leading from lower left to upper right. 1 day 2 days 3 days Low Ability Medium Ability High Ability This then is another example in which students of lower ability reached the same achievement level as students of higher ability when they were provided with more instruction on the material.25 Begle's experiment, though small in sc0pe, was the primary source from which the idea for this study was derived. The writer has provided Begle's description of the massive research project of SMSG which was directed by Herriot. It is suggested that the reader reconsider that description as given in Chapter I, this volume. Herriot's study takes on added significance in that the large number 25Begle, loc. cit., p. 238. 36 of students in the sample were selected from different geo— graphic regions within the United States. Homogeneous grouping by ability was again used in the Herriot study as it was in the Begle study. Analysis of covariance was em- ployed as a statistical technique since experimental and control groups were not randomly equivalent at the outset. The writer is concerned with the use of homogeneous grouping by ability level in the experiment by Begle and in the SMSG research of Herriot. ‘Kirk comments on the use of analysis of covariance as applied to existing non—random "intact groups". The same comments apply to homogeneous grouping. A note of caution concerning the use of intact groups is needed here. Experiments of this type are always subject to interpretation difficulties that are not present when random assignment is used in forming the experimental groups. Even when analysis of covariance is skillfully used, we can never be certain that some variable that has been overlooked will not bias the evaluation of an experiment. This problem is absent in properly randomized experiments because the effects of all uncontrolled variables are distributed among the groups in such a way that they can be taken into account in the test of significance. The use of intact groups removes this safeguard.26 Block has studied mastery learning at the eighth grade level in a fully randomized experiment. An abstract follows: The purpose of this study was twofold. First, a rationale for setting objective, criterion-referenced performance standards for sequential learning tasks was 26Roger E. Kirk, Experimental Design: Procedures for the Behavioral Sciences (Belmont, Calif.: Brooks/Cole Pub. Co., 1968), p. 456. 37 proposed, applied, and validated. Second, the cognitive and affective consequences of requiring students to maintain particular mastery levels throughout the learn- ing of a sequential task were examined. Eighth graders (n = 91) were taught three sequen- tial units of elementary matrix algebra over a school week. The students were randomly assigned to five groups. The control group (n = 27) learned the algebra under no requirement that they maintain any per unit mastery level. Each of the remaining groups learned the units to a different, preestablished level——65, 75, 85 or 95 per cent mastery. Per unit performance was measured by formative tests administered at each unit's completion. Feedback/correction-review procedures helped students maintain their required mastery level throughout the learning. For each pupil the following measures were taken: pre— and post-achievement and transfer, retention, total learning time per unit (including time spent in correction/review), and interest in and attitudes toward the algebra at various stages during the instruction and two weeks after its completion. The pretest and Metro— politan scores on reading and arithmetic served as indices of individual differences in student algebra entry resources. There were several major findings. First, mainte— nance of the 95 per cent mastery level produced maximal cognitive learning (achievement, transfer, and retention) but had long run negative effects on student interest and attitudes. Maintenance of the 85 per cent level produced maximal interest and attitudes, but slightly less than optimal cognitive learning. The maintenance of both the 85 and the 95 per cent levels, however, produced significantly greater cognitive learning than the main- tenance of no per unit mastery level. These data suggest, therefore, that some care must be taken in the selection of the mastery levels students are asked to maintain throughout their learning. The maintenance of one level (e.g., the 95 per cent) may have Opposite effects on cognitive and affective develOpment. The data also imply that a mastery level can be selected which, when main- tained, will maximize positive development of the desired learning-~cognitive as well as affective. Second, the maintenance of a high level of per unit mastery can make student learning increasingly efficient. By Unit Three, those in the 95 per cent group learned substantially more material, even without the use of the unit's feedback/correction-review procedures, than the control group. But the two groups spent the same amount of unit learning time. The data trends indicate that if 38 there had been additional learning units, most students in the 95 per cent group would have been able to main- tain their high mastery level virtually without need for spending time with the feedback/correction procedures. Like a crutch, these procedures might have been eventually discarded. Finally, despite the individual differences in the entry resources (pretest and previous achievement measures) of students in the 85 and 95 per cent groups, these differences were not reflected in their final achievement. Most students in each group learned to approximately the same high level. Further, while their resources played a large role in the learning of the first unit, they played a decreasing role in the learning of subsequent units. For control students (who maintained no unit mastery level), however, the resources played a large role in their final achievement and in their learn- ing throughout the sequence. That is, entry resource measures were predictive of student learning under usual instructional procedures, but not under the mastery learning conditions used. These findings suggest that the use of feedback/ correction procedures to supplement the original instruc- tion is a key to the transformation of ordinary group- based instruction into instruction of optimal quality for each student in the class. The results also raise some pertinent questions about the role of individual differences in classroom learning. They suggest that individual differences need not condition student learn- ing and that perhaps individual differences have largely been used as a scapegoat for ineffective instruction.27 The writer searched the literature for related research concerned with college students studying mathematics. Few results of controlled experimentation were found. A notable exception is the work of Collins at Purdue University. Collins employed the techniques of mastery learning to which Block referred. 27James H. Block, "The Effects of Various Levels of Performance on Selected Cognitive, Affective and Time Vari- ables" (Unpublished Ph.D. Dissertation, University of Chicago, Benjamin Bloom, Advisor, 1970). 39 This study investigated the effectiveness of Bloom's mastery learning strategy for the teaching of freshman college mathematics. The research involved two modern algebra courses for liberal arts majors (n = 50 approximately) and two calculus courses for engineering and science majors (n = 40 approximately). These courses were broken into smaller units, and students were assigned to learn the units under either mastery or non—mastery conditions. The mastery students were given a list of the objectives to be covered in each unit, each class session, and each assignment. During each session, they had five to ten minutes to solve a problem based on the objectives covered in the preceding session and assignment. Then, the problem was discussed and questions answered. The non-mastery learners were given neither a list of objectives nor daily problems. Both mastery and non—mastery students used the same textbook, received the same assignments, covered the same material in class, and took the same unit tests. Grades were determined by averaging student scores on the unit tests. In the modern algebra classes, 75% of the mastery compared to only 30% of the non-mastery students achieved the mastery criterion of an A or B grade. The calculus classes' results were similar: 65% of the mastery com— pared to 40% of the non-mastery students achieved the criterion. In both the modern algebra and the calculus course, D and F grades were for all practical purposes eliminated for mastery students. The smaller difference in the percentages of students who attained the criter- ion under mastery and non-mastery learning conditions for the calculus courses may be attributed to those factors: a) the greater importance of the courses to all engineering and science students; b) the higher and more homogeneous mathematical ability of the calculus students; and c) the clearer relationship between the problems discussed in class and the unit test problems.28 Several interesting non-experimental projects are being conducted in an attempt to meet the needs of college mathe- matics students with varying learning rates. 28Kenneth M. Collins, "A Strategy for Mastery Learning in Freshman Mathematics" (Unpublished study, Purdue University, Division of Mathematical Sciences, 1969). 40 One such attempt is being made at the University of Maine at Orono. The following letter describes the plan. Dear Mr. Jones: I am quite late in answering your letter of inquiry regarding research dealing with different learning rates of college students studying Mathematics. The only projects that we have in progress are the follow- ing. We are about to propose a modular program, using various audio visual aids, to enable us to be of better service to our entering algebra—trigonometry students. This is very much at the planning stages. We have a program in progress, however, in a calculus course, that follows the following pattern. After the first five or six weeks of the semester, we take those students who seem certain to fail the beginning calculus course out of their classes, and form new sections of a remedial course. The people so relocated go back to the be— ginning of the course, and spend the remainder of the semester and the next semester on the content of the one semester course, with liberal doses of review material as it is needed. The idea here is that these students have had the prerequisite courses, and there seems little point in having them take these a second time. We, therefore, attempt to go over this review material with the motivation and demands of the calcu- lus course to help us out. In carrying out this program, we manage to clear a great deal of dead wood out of the calculus sections. Also, this tech- nique seems to do more for a student than having him fail calculus and repeat the course. Inevitably, there is a large number of students who fail regardless of what is done, because of lack of application, lack of ability, or lack of motivation. However, those students that go into the second semester of the remedial course seem to do quite well, and would seem to benefit from the program. The course itself is a four hour course, and the remedial course also meets four hours a week, with two credits per semester if both semesters are completed. I'll be happy to hear of your results, and will furnish further details if you wish. Sincerely yours, John C. Mairhuber, Chairman Department of Mathematics University of Maine 41 The writer finds the efforts in developing a modular program in algebra-trigonometry to be particularly inter— esting. One can envision a course containing 20 or 25 modules each with clearly defined objectives, methods and required terminal performances. Certain modules might be omitted for students who already display the required terminal performance of that module. A student would not be bored by studying a module he already understands. A student would not be forced by time to proceed to a new module before he understands the one he is currently studying. A student would receive a passing grade upon completing examinations to criterion in each of the course modules. It is obvious that the time required to do this will vary with individual students. Developing administrative technique for such a course will be a most challenging problem. The procedure Mairhuber describes for the calculus course is also interesting in that the student is offered some alternative to immediate failure. A different approach to the problem of variant learning rates is offered by the Department of Mathematics, Lansing Community College, Lansing, Michigan. The student may enroll in one of three remedial courses with initial placement based upon entrance examination scores. The courses are Basic Arithmetic, Beginning Algebra, and Intermediate Algebra. The student has the choice of a traditional lecture course or a programmed instructional course in both Basic Arithmetic 42 and Beginning Algebra. Intermediate Algebra is offered in a choice of traditional lecture or audio-tutorial mode of instruction. All of the courses are offered on the basis of a 10 week term but the 10 week time periods are used only for fee assessment and record keeping. The key idea is the time flexibility within the 3 courses. The following ex- amples will illustrate the operation. If a student enrolls for the programmed arithmetic course and requires only 3 weeks to complete the program then he may immediately start the beginning algebra program. It is possible for the stu— dent to complete two courses within one term and pay fees for only one course. If a student enrolls in Beginning Algebra and requires more than 10 weeks to complete the pro— gram then he may re-enroll in the course by paying the fee a second time. If a student enrolls in a traditional course and finds the pace either too fast or too slow then he may transfer at any time to the prOgrammed course at the same level. Student reaction to these procedures is positive and the administrative problems which accompany the increased flexibility have been solved. Basic research in the psychology of learning has long- been concerned with learning rates. Fleishman and Hempel conducted one such experiment. A practice task together with a number of refer- ence tests was administered. A factor analysis of scores obtained at different stages of proficiency on the practice task together with scores on the reference tests was carried out. The results confirmed earlier findings with a different psychomotor practice task 43 that considerable but systematic changes in factor structure occur as a function of practice. However, in contrast to the previous study, the present task did not become progressively less complex (in terms of the number of factors measured) as practice was con— tinued. As before, there was an increase in the contribution of a factor common only to the practice task, but this increase was not as marked as with the previous task.29 Interaction of ability level with amount of time re-‘ quired in paired associate learning was studied by Cieutat and Stockwell.3° It is fitting that we conclude with the comments of Carroll on individual differences versus method of instruc- tion. In the first place, I predict that the study of instructional methods and individual differences is going to be extremely difficult and frustrating, even- if it is "most interesting" psychologically. It is, then, possible that research will never be able to come up with a sufficiently solid set of con— clusions to justify being adopted in educational prac- tice. Or, it may turn out that even though differentia- tion of instructional method is possible in an actuarial sense, the net gains are not of impressive magnitude. In many cases, the cost of differentiating instruction may be too high to suit the practical school adminis- trator, particularly if it involves elaborate and ex— pensive equipment or eXtensive teacher retraining. "Reality testing" in this field may be painful.3 29Edwin A. Fleishman and Walter E. Hempel, "The Relation Between Abilities and Improvement with Practice in a Visual Discrimination Reaction Task," Journal of Experimental Psychnggy, Vol. 49, No. 5 (1955). p. 312. 30Victor J. Cieutat and Fredric E. Stockwell, "The Interaction of Ability and Amount of Practice with Stimulus Response Meaningfullness in Paired Associate Learning," Journal of Experimental Psychology, Vol. 56, No. 3 (1958), p. 193. 31John B. Carroll, "Instructional Methods and Individual Differences,“ Learning and Individual Differences (R. M. Gagne, Editor), Columbus, Merrill and Co., 1967, p. 41. CHAPTER I I I STATISTICAL ANALYSIS OF THE STUDY Introduction The statement of the problem was presented on pages 1 and 2 of Chapter I. The design of the study and the limitations implicit in the design were presented on pages 6 through 18 of Chapter I. The reader may wish to review these pages prior to considering the analysis contained in this chapter. The Analysis of Variance (Background) Data for a two way, full factorial analysis of variance was tabulated in the following data matrix. Table 1.--Data Matrix for the Analysis of Variance (Reprinted from Chapter I, this volume). B A* 01 xx--- x... 02 xxx... xx--- Q3 x--- xx--- Q4 xx--- xxx-co 44 45 The symbols are defined: A* is the 10 week control group consisting of sections 4, 5, 7, 9 and 10. B is the 6 week experimental group con— sisting of sections 3, 6, 8, 11 and 12. 01, Q2, QB and Q4 are the quartile classes on the entering achievement test. x is the individual score on the final examination. It was possible to earn a score between 0 and 20 on the entering achievement test. A student was a member of 01 if his entering score, y, satisfied 0 g.y g_12.1; 02 if 12.1 < y g_14.1; 03 if 14.1 < y g_l6.3; and Q4 if 16.3 < y g_20. The standard deviation of the test was 3.05. Group B had a mean score of 14.2 and group A* had a mean score of 14.5. A frequency distribution appears with the test in Appendix A3. The statistical hypotheses tested within this design were: 1. There is no significant group difference in per- formance on the final examination due to time. 2. There are.no significant differences in performance on the final examination due to entering achieve- ment level. 3. There are no significant differences in performance on the final examination due to interaction of the variablestime and entering achievement level. Several assumptions are made prior to conducting an analysis of variance. we must assume that our sample is drawn from a normally distributed pOpulation and that the sample was drawn at random. It is further assumed that the 46 cell variances are equal and that observations between cells and within cells are independent. The design and conduct of this study make assumptions concerning randomness and independence tenable. The writer decided to conduct a basic statistical analysis of each cell within the data matrix in an effort to provide support for the remaining assumptions. The Michigan State University CDC3600 computer was used to secure this information.* (BASTAT Routine 5) The results are recorded within the appropriate cells of the data matrix as follows (see Table 3 on the following page). The reader is no doubt well acquainted with all the reported numerical values with the possible exception of skewness and kurtosis. The following explanation is in order. Skewness is related to the third moment about the mean. The values for skewness may range from arbi- trarily negative to arbitrarily positive (-oo to +00). An idealized normally distributed variable has a skew— ness of zero. Kurtosis is related to the fourth moment about the mean. The values for kurtosis may range from 1.0 to arbitrarily large (1 to +00). An idealized normally distributed variable has a kurtosis of 3.0.32 *The writer is indebted to James Mullin, Applications Programmer, MSU Computer Center, for his aid in preparing the data for the BASTAT—Routine and the LS—Routine which is described later in this chapter. 32Reprinted from page 19 of BASTAT Description 5, Michigan State University Computer Laboratory, Publication, January 1969. 47 Table 3.--Basic Statistics B A* Min = 6 Max = 27 Min = 8 Max = 29 Mean = 16.7500 Mean = 19.1579 Q1 SD = 7.0210 SD = 6.8008 SK = —0.1097 . SK = -0.0297 KU = 2.9359 KU = 1.7848 Nll = 12' N12 = 19 Min = 5 Max = 31 Min = 12 Max = 28 Mean = 21.4615 Mean = 23.3636 Q2 SD = 7.6661 SD = 6.3343 SK = -0.2237 SK = -0.2071 KU = 2.5476 KU = 3.1151 N21 = 13 N22 = 11 Min = 9 Max = 31 Min = 12 Max = 33 Mean = 23.2778 Mean = 23.8750 Q3 SD = 5.7272 SD = 5.4391 SK = -0.3263 ' SK = —0.3841 KU = 3.1892 KU = 2.7195 N31 = 18 N32 = 16 Min :1: 16 Max 2 32 Min = 18 Max = 34 Mean = 26.3333 Mean = 26.9524 Q4 SD = 5.8057 SD = 5.2717 SK = -0.3670 SK = -0.4830 KU = 2.4590 KU = 2.5568 N41 = 15 N42 = 21 The'symbols are defined: Min, Max-—The minimum and maximum values of the cell data.‘ ' Mean--The mean of the cell data. SD--The standard deviation of the cell data. SK, KU-—The skewness and kurtosis measures of the cell data. Nij--The number of observations within the cell. 48 The fact that all cell measures of skewness fall within the range, -0.5 < SK‘< 0.0, and that all cell measures of kurtosis fall within the range, 1.7 < KU < 3.2, is offered in support of the assumptions of normality. The fact that all cell standard deviations fall within the range, 5.2 < SD < 7.7, is offered in support of the assump- tion of equality of cell variances. The Analysis of Variance (Results) It is always desirable to have an equal number of observations within each cell of a full factorial design. This is not always possible due to loss of data during the history of the experiment. Kirk comments on the problem of unequal cell observations in his recent book on experi— mental design. An experimenter may plan to have equal n's in each treatment combination; but, for reasons unrelated to the nature of the treatments, data for some subjects are not obtained. One common reason for losing data is equipment failure.‘ Unequal n's may also result from the unavailability of subjects during the experi— ment. Subjects may be unavailable because they have forgotten their appointments, are ill, or, in the case of animal subjects, have died. If unequal n's occur for reasons not related to the nature of the particular treatments used in the experiment, and the cell n's do not have‘the proportionality shown in Table 7.9-1, an unweighted means analysis may be used. Procedures for carrying out this analysis are shown in Table 7.9-2. The results of this analysis are summarized in Table 7.9—3. Note that the total sum of squares is not included in the table. It is not included because in an unweighted means analysis the 49 sum of squares for A, 8, AB, and within cell do not add up to the total sum of squares.33 It is most fortunate that the Michigan State University Computer Laboratory maintains a program routine for the CDC3600 computer which employs the unweighted means analysis and allows for unequal cell frequencies. The program is known as the LS—Routine and is explained in Statistics Series Description 18. ‘The writer employed only those parts of the LS-Routine which were required to generate the ANOVA-TABLE for the design of this study. The table follows. Table 4.--Two-way Analysis of Variance iF============e Source ‘ SS df MS F 1. Group (B, A*) 56.9621 1 56.9621 1.6439 2. Entering Achievement - (Ql, 2,3,4) 1231.0919 3 410.3640 11.8427* 3. Interaction 30.8116 3 10.2705 .2964 4. Error 4054.1994 117 34.6513 ————— * Significant with p < 0.0005 No sum of squares total (SST) is reported since the estimates implicit in the unweighted means analysis invali— date the equation SS SS + SSint+ SS = SST' group+ ach. error 7 33R. E. Kirk, Experimental Design: Procedures for the Behavioral Sciences (Belmont, Calif.: Brooks/Cole Publish- ing Co., 1969), p. 202. 50 The Analysis of Variance (Hypothesis Testing) The statistical hypotheses and the decisions based upon the appropriate F values in Table 4 are: 1. There is no significant group difference in per— formance on the final examination due to time. We fail to reject this hypothesis at the .05 level. There are no significant differences in perform- ance on the final examination due to entering achievement level. We reject this hypothesis at the .05 level. There are no significant differences in perform- ance on the final examination due to interaction of the variables time and entering achievement level. We fail to reject this hypothesis at the .05 level. The Analysis of Variance (Post Hoc Techniques) The reader notes that the only significant effect which has been uncovered by the analysis of variance is associated with ability level. The presence of any significant effect allows us access to the data with various post hoc tech- niques. Kirk comments on the procedures. Many experiments are designed to determine if any treatment effects are present. If an over-all test of significance leads to rejection of the null hypothesis, attention is directed to exploring the data in order to find the source of the effects. A number of test statistics have been developed for data snooping. Several of these statistics can be used to make all possible comparisons among means. 51 A multiple comparison test similar to the LSD test has been proposed by Tukey (1953). This test, which is called the HSD (honestly significant differ— ence) test or the w procedure, sets the experiment- wise error rate at a. The HSD test was designed for making all pairwise comparisons among means. The basic assumptions of normality, homogeneity of vari— ance, and so on, described in Section 2.1 in connection with a t radio are also required for the HSD test. In addition, the n's in each treatment level must be equal or approximately equal. A comparison involving two means is declared to be significant if it exceeds HSD, which is given by EMS HSD = qo v/ ———-—e:r°r The value of q is obtained from the distribution of the studentized range statistic. If the n's in the treatment levels for Tukey's test are not equal, an approximate HSD test can be computed by substituting K for n. E is given by ~ k n = [<1/n1>+<1/n2)+...+<1/nj>1 where k = number of treatment levels and n ,n2,...,n. refer to the respective n's for the treatmént levels?34 The writer intended no comparisons of cell means other than pairwise comparisons. The Tukey HSD Test is known to be among the most sensitive of the post hoc procedures when pairwise comparisons are under consideration.35 There are 28 possible pairwise comparisons of means among the 8 cells in the design. It was decided to conduct all of these pairwise comparisons. The appropriate value q was selected from the studentized range statistic table, D.7, p. 531 of Kirk's book. The .05 34Ibid., pp. 88—90. 35Ibid., p. 90. 52 level with 117 degrees of freedom was employed. ’K was computed by the cited formula and the entry for mean square error was taken from the ANOVA—TABLE. HSD was computed to be 6.64 accurate to 2 decimal places. The results appear in the following table. Table 5.--Post Hoc Pairwise Comparison of Eight Cell Means Pair A I Al Significance Pair A I A I Significanc:l 1. Ell-£12 2.41 15. SEN—3522 0.08 2. 3211-3222 6.61 16. 3231—?” 0.60 3. 3211-53; 7.13 17. 2.1—£42 3.67 4. Ell-£42 10.20 18. £315“ 3.05 5. 3211-331 4.71 19. 3241-?” 7.17 6. 3611-3?“ 6.53 20. SEN-£22 2.97 7. Ill-32.1 9.58 21. SEN-Ea, 2.45 8. £15312 2.30 22. 2.1-3?” 0.62 9. E2152; 1.90 23. E12522 1.90 10. £215.32 2.42 24. Elz-Esz 4.72 11. 3:31-3:12 5.49 25. SEN-32.2 7.79 12. £21531 1.82 26. 3722-3132 0.52 13. 3221-?“ 4.87 27. @2542 3.59 14. 2.1-3E1; 4.12 28. 3732-3242 3.07 = Mean of cell with row index i and column index j. Notation: 36. . 13 *A Is significant if | A l > HSD = 6.64. The writer contends that students of lower entering achieve— ment who were in the 10 week group A* performed almost as 53 well as students of higher entering achievement who were in the 6 week group B. Consider what this would mean in terms of post hoc comparisons within the design. Table 6.—-Cell Means I B A* Ql E1 1 £1 2 Q2 X2 1 X2 2 03 E3 1 E3 2 Q4 3:4 1 . X4 2 The ggnrsignificance of any of the following contrasts support our contention since each of these compare the mean of a cell representing a lower quartile of group A* with the mean of a cell representing a higher quartile of group B. Non-significance means that the cell means did not differ statistically according to the HSD test at the 0.05 level with 117 degrees of freedom. The numbers are the indices of the contrasts in the Table 5. Table 7.--Interpretation of Table 6 8 - A = E12 - E}; is non-significant 14 — A = E}. - 231 is non—significant 15 - A = §52 - E3, is non—significant 20 - A = I}. - x21 is non-significant 21 - A = E52 - §.1 is non-significant 54 The pr-significance of any of the following contrasts will fail to support our contention since each of these contrasts compare the mean of a cell representing a higher quartile from group A* with the mean of a cell representing a lower quartile from group B. Table 7.-—Interpretation of Table 6 (Cont'd) - A = £52 - ii; is non-significant 3 -' A = §32 - 2:11 is Significant 10 - A = x3; - R51 is non-significant ll - A = x2; - I}, is non-significant 17 - A = In - 3631 is non—significant No contrasts were considered if the quartile differences exceeded 2. Of the 10 contrasts which were considered, 5 support the writer’s contention. They are 8, 14, 15, 20 and 21. Of the 10 contrasts which were considered, 4 fail to support our contention. They are 2, 10, 11 and 17. Contrast 3 neither supports nor fails to support the con— tention. The reader notes that there is the numerical support of the majority for the stated contention. It is unfortunate that inferential statistics is not governed by majority rule. It is interesting to consider the cell means within the design without regard to statistical analysis. 55 Table 8.--Cell Means II B A* Q1 in = 16.75 3212 = 19.16 02 £21 $21.46 £22 :- 23.36 03 £31 = 23.28 E32 = 23.88 Q4 3"“ = 26.33 3?... =- 26.95 Within a given quartile we note that the cell means of group A* always exceed the cell means of group B. It is 'therefore reasonable to ask why the formal analysis failed to uncover a significant main effect due to group. Note that cell mean E52 exceeds cell mean §31. The presence of such an observation might lead one to expect a significant effect due to interaction of quartile with group in the formal analysis. Why did this fail to occur? The answer is quite simple and is related to the F-ratio itself. The numerator of the F-ratio is a measure of variability between groups while the denominator is a measure of variability within groups. A value of F is significant only if F is large, that is if the variability between groups is very large relative to the variability within groups. This experi— ment is complicated by the fact that the cell variances are quite large and the differences between cell means are relatively small. Significant F-ratios were therefore in the minority in the formal ANOVA. 56 An associated discussion of the value of HSD could be given. The size of HSD is directly proportional to within cell variance. A moderately smaller value of HSD would have made contrasts 2 and 11 significant and the associated lack of support would vanish. Unfortunately, HSD = 6.64 and the consequence must be accepted. The Chi-Square Test of PrOportion Differences (Background) Data for the test were tabulated in the following data matrix. (Reprinted from Chapter I, this volume.) Table 2.-—Data Matrix for the Test of Propor- tion Differences El 32 A1* X Y Z The symbols are defined: 31 consists of all students from group B who passed the Mathematics 082 final on November 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. ’ B; consists of all students from group B who passed the Mathematics 082 final on December 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. A1* consists of all students from group A* who passed the Mathematics 082 final on December 3, 1970 and who then enrolled in Mathematics 108 during the next academic term. 57 z, x, and g are ratios. The denominator of each ratio is the number of students from the associated category who enrolled in Mathematics 108 during the Winter term of the 1970-71 academic year. The numerator of each ratio is the number of students from the associated category who received a final grade of 2.0 or higher in Mathe— matics 108 during the Winter term of the 1970—71 academic year. The statistical hypothesis tested within this design The Chi-Square Test of Proportion Differences (Results and Hypothesis Testing) Data for the following table were found in the class lists which were distributed at the close of registration (for the denominators of X, Y, and Z), and at the close of. the Winter term (for the numerators of X, Y, and Z). Table 9.—-Success in Mathematics 108 Winter Term 1971 Number Number of Grades Success Enrolled 2.0 or Higher Percentage Group B1 20 13 65 Group 32 9 4 44 Group A1 * 24 " 13 54 Total 53 30 57 58 In order to conduct the test of proPortion differences, the following table was constructed. Table lO.--Chi-Square Table Group 81 Group B2 Group A1* Grade 22.0 (2.0 22.0 (2.0 22.0 <2.0 frequency f 13 7 4 5 13 ll expected f=e 11 9 5 4 14 10 The writer used the values in Table 10 to compute the value of Chi-Square for the data of the experiment. A table of values of Chi-Square was consulted using 2 degrees of freedom at the .05 level.36 The critical value of Chi-Square is 5.99. The statistical hypothesis and the decision based upon the computed values of Chi-Square is: We fail to reject this hypothesis at the .05 level, since the computed value fails to exceed the critical value. 36Kirk, loc. cit. p. 530. 59 The writer is indebted to Maryellen McSweeney, Associate Professor, College of Education, Michigan State University for her extensive consultation in the design and analysis of this study. CHAPTER IV SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary The purposes of this study were: 1. To investigate the effect of film; allotted for study and instruction on final examination scores in Mathematics 082 and 2. To investigate the effect of entering achievement lgyg; on final examination scores in Mathematics 082 and 3. To investigate the interaction effect of pig; and enteging achievement level on final examination scores in Mathematics 082 and 4. To investigate the success rate in Mathematics 108 of those students who successfully complete Mathematics 082. The study was conducted during the Fall and Winter terms of the 1970—71 Academic Year within the Department of Mathematics, Michigan State University. Administrative procedures, classroom materials and evaluation instruments were develoPed and prepared during the summer months of 1970. 60 61 An appropriate research design and an analysis of variance was employed to satisfy purposes 1, 2 and 3 of this study. A different design and a Chi-Square test of pr0portion differences was employed to satisfy purpose 4. Chapters I, II and III of this volume were written to provide the reader with an understanding of the problem, its background and its resolution. Conclusions Purpose 1 The analysis of variance failed to uncover any sig— nificant differences in~the mean final examination perform- ances of 6 week group B and 10 week group A*. One might easily conclude that Mathematics 082 could therefore be offered as a 6 week course since the additional 4 weeks of study apparently do not produce significantly better results. There is great danger in'such a conclusion. The writer pre- sented the results of post hoc testing in Table 7 of Chapter III. Students from lower quartiles of the 10 week group A* performed essentially as well (in the sense of the HSD test) as students from higher quartiles of the 6 week group B. This was shown to be true in’5 of the 10 contrasts of most interest. The relative gain by students from lower quartiles, of group A* is small but cannot be ignored. The reader will recall that those group B students who did not pass the final examination at the end of the 6 week 62 period were given the opportunity to retake a final exami- nation at the end of the 10 week period. At the end of the 6 week period 51% passed. At the close of the 10 week period an additional 23% passed. The overall total of 74% is impressive in view of previous success rates in Mathematics 082. The 10 week group A* compiled a success rate of 58% at the close of the 10 week period. The writer concludes that there is no significant over- all difference in final examination performance due to time. However: 1. Students of lower entering achievement within group A* did benefit from the longer period of study and instruction. As evidence of this the writer offers Table 7, Chapter III, entries 8, 14, 15, 20 and 21. 2. The group B students who did not pass the final examination at the end of the 6 week period derived benefit from the additional 4 weeks of study and lecture. As evidence of this the writer offers the fact that an additional 23% of the group B students were successful at the end of the 10 week period. Purpose 2 The writer concludes that there is a significant effect of entering achievement level on final examination scores in Mathematics 082. This effect was uncovered in the analy- sis of variance and the magnitude of the effect is to be ' found in Table 4 of Chapter III. It is more interesting to 63 find the entries in Table 6 of Chapter III which demonstrate the source of this significance. Contrast 7 is significant within group B. Contrast 25 is significant within group A*. In each of these contrasts two distinct quartiles within the given group were compared. Purpose 3 The analysis of variance failed to uncover any sig— nificant differences in final examination scores which could be attributed to the interaction of time and entering achievement level. It is clear that the most interesting and enlightening results of the study are found in the post hoc analysis which is considered in detail in Table 6 and 7 of Chapter III. Overall significant effects were difficult to detect due to the large within cell variance in the design. The pairwise comparison of cell means by the HSD test uncovered signifi- cant information which was masked in the overall analysis of variance. Most notable among the results of post hoc test— ing are those results located in Table 7 of Chapter III, and in the associated discussion of Tables 6 and 7 in Chapter III. Purpoge 4 The Chi-Square test failed to uncover any significant'” differences in Mathematics 108 success which could be attributed to group membership in Mathematics 082. 64 The writer concludes that the level of mastery which was attained in Mathematics 082 is relatively independent of the time spent in attaining that mastery. Recommendations The following recommendations for further study of the relationship between entering ability level and time allotted for study and instruction in mathematics courses offered at the college level are in order: 1. A replication of the study in Winter and Spring terms . 2. A replication of the study in a different course. The writer suggests using Mathematics 108 for the ability 1eve1——time study and Mathematics 109 for the follow—up study. 3. A replication of the study in a course in which the time factor may be more clearly differentiated. The writer would suggest that two randomly equivalent groups study the same course with one group spending two complete ten week terms while the control group spends the tradi- tional single ten week term. The effects of time as a learning variable might be more easily detected if the differences between the two respective time periods are greater. 4. The writer suggests that audio-visual-tutorial materials be prepared and that these materials become the 65 basis for teaching an entire sequence of mathematics courses at Michigan State University. The courses suggested for this experiment are Mathematics 082, 108 and 109. The emphasis should be on complete time flexibility within these three courses. Most teaching would be carried on in a laboratory setting and each student would progress at his own time rate. A similar plan is in Operation at Lansing Community College, Lansing, Michigan. The L.C.C. plan was described in the latter pages of Chapter II. Careful evaluation of the experiment should be con- ducted to answer such questions as: (a) Is this mode of individualized instruction successful for students at every level of entering achievement? (b) Is student attitude positive? (c) Are overall gains in performance realized when the experimental group is compared to an equivalent group studying the same courses in a traditional manner? Conclusion The writer recognizes that much pain is associated with removing the artificial time boundaries which we have imposed upon the teaching and learning of mathematics. We have become comfortable with our system of quarters, terms and semesters. It is convenient to structure a course and offer the same topics to all students on a given day. It is convenient to test students and assign grades at uniform intervals throughout a term. It is convenient to 66 alter the material content of a course and call such altera- tion fundamental educational change. It is not convenient to alter the framework within which we present mathematics. It is not convenient to alter the basic methods employed in the teaching of mathematics. The practitioners of mathematics education must con- front the problem of providing for individual differences with the same level of effort which was applied to altering course content during the late fifties and all of the sixties. A sense of national urgency prompted us to alter course content. It is unfortunate that no similar sense of urgency drives us to solve the problems associated with individual differences. BIBLIOGRAPHY B IBLIOGRAPHY A. Books Ausubel, David P. The Psychology of Meaningful Verbal Learning. New York: Grune and Stratton, Incorporated, 1963. Bloom, Benjamin S. Mastery Learning: Theory and Practice. (Edited by James H. Block). New York: Holt, Rinehart and Winston, Incorporated, 1971. Broudy, H. S., and Palmer, J. R. Exemplars of Teaching Method. Chicago: Rand McNally and Company, 1965. Campbell, Donald T., and Stanley, Julian C. Experimental‘ and Quasi—Experimental Design for Research. Chicago: Rand McNally and Company, 1967. DeCecco, John P. The Psychology of Learning and Instruction: Educat§ona1 Psychology. Englewood Cliffs: Prentice— Hall, Incorporated, 1968. Eisner, Elliot W. (editor) ConfrontinggCurriculum Reform. New York: Little, Brown and Company, Incorporated, 1971. ‘ Gage, Nathan L. (editor) Handbook of Research on Teaching Chicago: Rand McNally and Company, 1964. ” Gagne, Robert M. (editor) Learning and Individual Differences. Columbus: Merrill Publishing Company, 1967. Gagne, Robert M. The Conditions of Learning. New York: Holt, Rinehart and Winston, Incorporated, 1965. Hays, William L. Statistics for Psychologists. New York: Holt, Rinehart and Winston, Incorporated, 1963. Kirk, Roger E. Experimental Design: Procedures for the Behavioral Sciences. Belmont: Brooks/Cole Publishing Company, 1968. 67 68 Krumboltz, J. D. (editor) Learning and the Educational Process. Chicago: Rand McNally and Company, 1965. Learned, W. S., and Wood, Ben D. The Student and His Knowledge. New York: The Carnegie Foundation for the Advancement of Teaching, 1938. National Society for the Study of Education, Individualizing Instruction. Chicago: University of Chicago Press, Sixty-First Yearbook, 1962. National Society for the Study of Education, Mathematics Education. Chicago: University of Chicago Press, Sixty-Ninth Yearbook, Part I, 1970. B. Booklets Bloom, Benjamin S. LearninggFor Mastery, in Bulletin of the U.C.L.A. Center for the Study and Evaluation of Instructional Programs. Berkeley: University of California Press, 1968. DeVenney, William S. Final Report on an Experiment with” Junior High School Very Low Achievers in Mathematics, S.M.S.G. Report Number 7, Stanford: Stanford University Press, 1969. Goodlad, John I. Diagnosis and Prescription in Educational Practice, Conference Report on New Approaches to Educational Instruction, Princeton, Educational Test- ing Service, 1965. Herriot, Sarah T. The Slow Learner Project—-The Secondary School "Slow Learner" in Mathematics, S.M.S.G. Report Number 5, Stanford: Stanford University Press, 1967. C. Periodicals Allendoerfer, Carl B. "The Second Revolution in Mathema- tics." The Mathematics Teacher, Volume 58, Number 8, 1965. Begle, E. G. "The Role of Research in the Improvement of Mathematics Education." Educational Studies in Mathematics, Volume 2, 1970. 69 Brown, B. Frank. "The Non-Graded High School." Phi Delta Kappan, Volume 44, 1963. Carroll, John B. "A Model for School Learning." Teachers College Record, Volume 64, 1963. Cieutat, V. J., and Stockwell, F. E. “Interaction of Ability and Amount of Practice in Paired Associate Learning." Journalyof Experimental Psychology, Volume 56, Number 3, 1958. Fleishman, Edwin A., and Hempel, Walter E. "The Relation Between Abilities and Improvement with Practice in a Visual Discrimination Reaction Task." Journal of Experimental Psychology, Volume 49, Number 5, 1955. Kaufman, Burt and Bethune, Paul. "Nova High—Space Age School." Phi Delta Kappan, Volume 46, 1963. Long, Meltzer and Hilton. "Research in Mathematics Education." Educational Studies in Mathematics, Volume 2, 1970. Metzler, W. H. "Where Shall we Place the Emphasis?" The Mathematics Teacher, Volume 1, Number 1, 1908. Thompson, P. E. and Poe, R. L. "A Report on the C.U.P.M. Recommendations in the State of Texas." The American Mathematical Monthly, Volume 75, Number 10, 1968. Wagner, John. "The Objectives and Activities of the School Mathematics Study Group." The Mathematics Teacher, Volume 53, Number 10, 1960. D. Unpublished Material Block, James H. The Effects of Various Levels of Perform- ance on Selected Cognitive, Affective, and Time Vari— ables, Doctoral Dissertation, University of Chicago, Benjamin Bloom, Advisor, 1970. Collins, Ken. A Strategy for Mastery Learning in Modern Mathematics, A Study within the Division of Mathematical Sciences, Purdue University, 1970. Collins, Ken. A Strategy for Mastery Learning in Freshman Mathematics, A study within the Division of Mathematical Science, Purdue University, 1969. 7O Hickman, J. D. A Study of Various Factors Related to Success in First Semester College Calculus, Doctoral Dissertation, University of Southern Mississippi, 1969. Hocking, J. G. Ten Lecture Outlines for Mathematics 082, Department of Mathematics, Michigan State University, 1970. Rockhill, T. D. Programmed Instruction Versus Problem Sessions as the Supplement to Large Group Instruction in College Mathematics, Doctoral Dissertation, State University of New York at Buffalo, H. F. Montague, Advisor, 1969. Scannichio, T. H. Student Achievement in College Calculus, Doctoral Dissertation, Louisiana State University, Sam Adams, Advisor, 1969. A1. A2. A3. A4. A5. A6. A7. A8. A9. APPENDIX Michigan State University Mathematics Placement Test. Information-Assignments-Time Schedule--Group B. Entering Achievement Test. Education 327-N, Groups A and B, Information For Teaching Interns. ' Sample Quiz and Answer Key-—Groups A and B. Sample Final Examination—-Mathematics 082. Final Examination--Mathematics 082. Information—Assignments-Time Schedule--Group A. Course Outline--Mathematics 082. 71 72 MICHIGAN STATE UNIVERSITY MATHEMATICS PLACEMENT TEST It is not possible to reproduce the placement test in this volume. The utility of the test as a placement aid would be seriously jeopardized if copies of the test were to be released. The writer has examined the test and has found the following topics are emphasized: 1. Basic operations with signed numbers, integral and rational exponents and rational algebraic expres- sions. 2. Factoring linear and quadratic expressions. 3. Division of quadratic expressions by linear expressions. 4. Solving equations. Complex numbers occur in the solution of one of the quadratic equations. 5. Basic word problems. The test will entrap the student who would make such —l+B--l)-1 = A+B, or XZoX3 = X6, or even %’+ %'= %3 errors as: (A A student who is truly proficient in the content material of a one year secondary school algebra course will do very well on this test. Unfortunately, such students are virtually extinct. 73 MATHEMATICS 082 Information - Assignments — Time Schedule Section 3 - (Recitation 9:10 a.m. - 133 Akers Hall) Section 6 — (Recitation 9:10 a.m. - 102C Wells Hall) Section 8 - (Recitation 9:10 a.m. - 119A Berkey Hall) Section 11 - (Recitation 9:10 a.m. - 107C Wells Hall) Section 12 — (Recitation 9:10 a.m. - 218B Berkey Hall) All students in these sections have the opportunity to complete all the work in Math 082 by November 5, 1970. This early completion will entail a change of room assignment for your Tuesday—Thursday large lecture meeting From 108B Wells Hall.Tg_1028 Wells Hall. Please report to 102B Wells Hall this afternoon at 4:10 p.m. for your lecture meeting. All sections other than those listed above will meet throughout the term according to the Schedule of Courses. To receive a passing grade in Math 082 it is required that you pass a final examination in the course. The sections listed above will be given the final examination on Tuesday, November 3rd at 4:10 p.m. in the large lecture meet- ing. If you pass the examination at that time then you will have completed your obligation in this course and need not~ report for any further classes or examinations in Math 082. If you do not pass the examination on November 3rd then you will have the opportunity to attend additional lectures and recitation meetings for the remainder of the term and you may retake the final examination on Thursday, December 3rd at 4:10 p.m. This latter Option will be explained in detail after November 3rd to those students who need the additional time. A time schedule with daily assignments is attached for your use throughout the term. If you do not wish to participate in this early comple- tion plan then we suggest that you request a change of section to any Math 082 section not listed above. The procedure for section change is described on page 154 of the Schedule g§_Courses--Fa11 1970. 74 MATHEMATICS 082 Fall 1970 Time Schedule for Sections 3, 6, 8, 11, 12 Date Thursday 9/24 No Meeting Lecture I Tuesday 9/29 Questions on Lec. I + Lecture II self quiz Thursday 10/1 Questions on Lec. II + Lecture III quiz Tuesday 10/6 Questions on Lec. III + Lecture IV quiz Thursday 10/8 Questions on Lee. IV + Lecture V quiz Tuesday 10/13 Questions on Lec. V + Lecture VI quiz Thursday 10/15 Questions on Lee. VI + Lecture VII quiz Tuesday 10/20 Questions on Lee. VII + Lecture VIII quiz Thursday 10/22 Questions on Lec. VIII + Lecture IX quiz Tuesday 10/27 Questions on Lec. IX + Lecture X quiz Thursday 10/29 Questions on Lec. X + Brief Review quiz + hand out sample final Tuesday 11/3 Go over sample final FINAL EXAM Thursday 11/5 Hand out scores. If you pass then you are through. If you do not pass then we will explain in detail the additional Lectures and Help Sessions available to you, including another final on December 3rd. 75 MATH 082 Fall 1970 Assignments for sections 3, 6, 8, ll, 12. (A11 assignments refer to-—A Survey of Basic Mathematics, 2nd edition, Fred w. Sparks) LECTURE I §1.4 - 1.8 pp. 5 - evens 12-34; pp. 7,8 - 8,12,15,17,20,22,24 pp. 12,13 - 8,12,17,22,26,29 pp. 17,18 - 2,3,6,10,13,14 LECTURE II §1.9 - 1.15 pp. 25,26 - 10,12 + odds 13-51 pp. 28-29, odds start with 7. pp. 30-32, evens, pp. 33-38, 2,6,10,14,18,22, 26,20,34,38,42,46,50,54,57. pp. 4-43, every 4th starting with 1. pp. 45-48, 2,6,10,14,18,22,26. LECTURE III §2.1 - 2.8 pp. 52-54, 7,12,17,18,20,24,25,26,27,32,35,36 pp. 61-63 every 4th starting with 2. pp. 65-67 every 4th starting with 2. pp. 69-71 every 4th'starting with 2. pp. 72-73, 2,5,7,10,12,15,16,17,18 pp. 76-80, every 4th starting with 2. LECTURE IV §4.1 - 4.4 pp. 87-89 every 4th starting with 2. pp. 90-92 every 4th starting with 2. pp. 93-95 every 4th starting with 2. LECTURE v, §5.1 - 5.7 pp. 99-103, 1,3,5,7,9,11,13,15,23,26,28,32,36,40,44,48 pp. 104-108 every 4th starting with 2. pp. 109-111 every 4th starting with 2. pp. 112-118 every 4th starting with 2. pp. 119-122, 2,6,10,14,18,21. LECTURE VI §6.1 - 6.5 pp. 127-129 every 4th starting with 2. pp. 130-133 every 4th starting with 2. LECTURE VII §6.6 only PP- 76 135-144, 2,6,10,14,18,22,26,30 LECTURE VIII §7.2 - 7.6 PP; PP- PP; PP- 147-149 every 4th 150—152 every 4th 153-161 every 4th 163-165 every 4th LECTURE IX §8.1 - 8.11 PP- PP- PP- PP. PP- PP; LECTURE X 176-178 every 4th 180-182 every 4th 183—185 every 4th 187-191 every 4th starting starting starting vstarting starting starting starting starting with with with with with with with with NNNN 0000 14 20 2. 2. 192-194, 2,6,10,14,18,21 pp. 194-195, 2,6,10,12 197-199, 1,3,5,7,9 §9ol "' 906 pp. 201-205 every 4th starting with 2. pp. 208-213 every 4th starting with 10. pp. 214-218 every 4th starting with 14. pp. 223-224, PP- 220-222, 2,8,14,20 1,5,11,13. 77 MATH 082 - F311 1970 Diagnostic Quiz Name: Section: Record the letter corresponding to the proper response in the blank. 1. 27 - 12 + 531 = (a) 546 (b) 570 (c) 516 (d) none of these 2. 26.3 - .9 = (a) 17.3 (b) 27.2 (c) 25.4 (d) none of these 2 l 3. 3 + 2 = 6 l 3 (a) - (b) l- (c) - (d) none of these 7 6 5 l _ .2. 4. 6 5 _ -7 7 (a) 30' (b) -l (c) §—- (d) none of these 4 7 5. 5 X 6 = 2.2 a 14. (a) 30 (b) 5 (c) 15 (d) none of these \ 6 25 x 13 = (a) i—g (b) 385 (c) 325 (d) none of these 2 7. .7 = (a) 300 (b) 30 (c) 3 (d) none of these 10. 11. 12. 13. 14. 15. 16. 78 323%:-= (a) 96 (b) 12 (c) 9 (d) none of these 2,4- 5 3 " 3 8 .l (a) IO— (b) 33 (c) 4 (d) none of these .15 x .05 = (a) .75 (b) .0075 (c) 7.5 (d) none of these 2% of $6.50 is (a) 50.12 (b) $0.13 (c) $1.30 (d) none of these $5.00 is 80% of (a) 84.00 (b) $40.00 (C) $6.25 (d) none of these -2 - (-3) = (a) -6 (b) -5 (c) l (d) none of these 2(A+B) - (2A-B) = (a) -A (b) 0 (c) 38 (d) none of these 7 3 4 7D - 3C 7D 3C — I (a) CD 1» CD (c) C+D (d) none of these If A - 3 and B — 2 then -—§§:r 79 17. (3x + 2y)(x - By) s (a) 3x2 - 7xy - 6y2 (b) 3x2 — llxy — 6y2 (c) 3x2 - 6y2 (d) none of these 18. The completely factored form of 2x2 - 4x - 6 = (a) (x+1) (2x-6) (b) (x-3) (2x+2) (c) 2(x-3:) (x+1) (d) none of these 19. If K s g-- 15 and K = —1 then B = (a) %g- (b) 42 (C) 12 (d) none of these 20. If A > 0 and B > 0 then (A2B3)1/3 = (a) Ag/3B (b) (AB)2 (c) (AB)5/3 (d) none of these FREQUENCY DISTRIBUTION OF SCORES FOR 321 MATHEMATICS 082 STUDENTS FALL 1970 Number Correct 19—20 17-18 15—16 13-14 11-12 9-10 7-8 5-6 Freq. 27 60 78 78 35 32 10 l 80 ED 327N Information concerning Math 082 recitation sections: 3 - (Recitation - Tuesday + Thursday 9:10 a.m. - 133 Akers Hall) 6 — (Recitation - Tuesday + Thursday 9:10 a.m. - 102C Wells Hall) 8 - (Recitation - Tuesday + Thursday 9:10 a.m. - 119A Berkey Hall) 11 - (Recitation — Tuesday + Thursday 9:10 a.m. - 107C Wells Hall) 12 - (Recitation - Tuesday + Thursday 9:10 a.m. — 218B Berkey Hall) The first meeting with your Math 082 recitation section will occur on Tuesday, September 29 at 9:10 a.m. in one of the locations listed above. The students are to receive papers dealing with information, assignments and a time schedule at the first recitation meeting. A diagnostic test is also to be given during this first meeting. Please allow 40 minutes for the test. Be certain to emphasize the fact that the test grade does not determine the course grade in any way but will be used to help in designing future Math 082 courses. Be sure that each student records both his full name and section number on the diagnostic test. The completed tests are to be returned to your Ed 327 instructor at the next 327 meeting. You will note that QUIZ I is also scheduled to be given at the September 29 recitation meeting. This test is stapled to the papers of information and is not to be taken in class. All later self-quizzes are to be given in the scheduled recitation meetings. Answer keys are to be provided for each student as soon as he completes the quiz. The quizzes are designed to be Completed in 20-25 minutes. No record of quiz scores is needed. 81 ED 327N Information for Math 082 Recitation Sections 1, 2, 4, 5, 7, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20. The first meeting with your Math 082 recitation section will occur on Tuesday Sept. 29 at the hour and location designated in the Fall 1970 Schedule of Courses. The Math 082 students are to receive the papers dealing with general information, assignments and a time schedule at the first recitation meeting. A diagnostic test is to be given during the first meeting. Please allow 40 minutes for the test. Be certain to emphasize the fact that the test does not determine the course grade in any way but will be used to help in designing future Math 082 courses. Be sure that each student records his full name and section number on the diagnostic test. The completed tests are to be returned to your Ed 327 instructor at the next 327 meeting. 82 MATH 082 - Fall 1970 Quiz II--Group B Name Use scratch paper 1. Reduce to lowest terms (a) 13g: (b) 2315“ Convert 72' to an improper fraction. Evaluate and reduce the answers to lowest terms. (b) 141+ 23: (c) %-x g-_ (d) %+%= Gold sells for $32 per ounce on the legal world market and for $40 per ounce on the black market. A guard at Fort Knox steals }-pound on one day and %-pound the next. If he sells all the stolen gold on the legal market, how much will he get? (besides the prison term) how much on the black market? Legal market Black market A formula for making dynamite calls for a steady hand, a cool head and i-pound of nitroglycerine to 40 pounds of cellulose carrier. The manufacturer has only 8 pounds of cellulose and wishes to use all of it in the formula. How much nitro should he use in order to maintain the proportions set forth in the formula. 83 Convert each decimal to an equivalent percent. (a) 3.21 (b) .017 A one ton shipment of grain is 32% oats. How many pounds of oats are in the shipment? $152 is 41% of what amount? Convert 22 feet per second into miles per hour. 84 MATH 082 - Fall 1970 Key - Quiz II - Group B (a) % (b) {—0 241' (a) 5% (1:02—65- or 4-61- (013—4 $52 ; $65 §%-pound (a) 321% (b) 1.7% 640 pounds $370.73 to the nearest cent 15 mph (d) gun 85 MATH 082 - Fall 1970 Quiz VII-Group A Name Sec. No. 1. Ten gallons of pure alcohol is added to 80 gallons of a mixture containing 25% alcohol. What is the percentage of alcohol contained in the final mixture? One number, N, is three less than twice another number M. Write an equation exPressing the relationship of N and M. The tens digit of a certain number of two digits exceeds twice the units digit by 2. If we reverse the order of the digits, the sum of the new number and the original number is 121. What is the number? Fast Frank can paint a barn in 6 hours and slow Sam can paint the same barn in 9 hours. (a) What fraction of the barn can Sam paint in one hour? in Two hours? In T hours? (b) How long will it take Frank and Sam to paint the barn if they work together? A 24 hour road race employs a staggered start. A Lotus leaves the starting point at noon and travels at 80 MPH. A Ford Cobra leaves the starting point at 1 PM the same day and travels at 100 MPH. Assume that both cars travel the same road and function normally. At what time will the Ford catch the Lotus? How far from the starting point will this occur? Remember that Rate x Time = Distance. WATCH OUT ON THIS ONE. A man drives the first mile of a two trip at an average speed of 15 MPH. How fast must he drive the second mile in order to average 30 MPH for the entire trip? 86 MATH 082 - Fall 1970 Key - Quiz VII - Group A 334% N+3=2M or N=2M-3 83 1 2 T . (a) gu‘g. 5' (b) 3i-hours. or 3 hours 36 min. 5 PM, 400 Miles It can't be done. Ask your recitation instructor why. 87 MATHEMATICS 082 Problems similar in type and number will be on the final examination. Simplify and perform the indicated operations. M 4T2—49 _ 1. (a) A_289C (b) 2T+7 (c).J7 J12 + 73' (d)'i§%- (e) .1: (f) 2(3U-V)-7(U+V)-(-3V) 1+- X 2 . . <9) :- --i-- m --i?% m —.i‘. - .733- . 2 3 32 . 2 3 3 2 (J) (3 ) (k) “—3- (1) 3 -3 (m) 3 - 3 3 2 2 x2+2x-8 3x2+4x+1 2. Factor completely. (a) 82+28-15 (b) 4T2-l (c) 4C2-8c-l40 (d) 4A282-4A3B+4A28 3. True or False 2 (a)-JA2+B2 = A+B (b) fié$§;—- = l (C)-JA‘2+4 = J4A2+1 A (d) (AB)° = A°B° 4. Solve for x x 2 (a) EIE' = 3- (b) x+.Jx+3 = 9 (c) 6x2 - x = 1 (d) 2x2+x-3 = 0 88 Find x and y such that 2x-y = 7 4 the system is satisfied 7x+3y = 6 P154 £2 4 P137 £# 6 89 FINAL EXAMINATION MATHEMATICS 082 Fall 1970 INSTRUCTIONS: NAME: 1. Show all pertinent work Rec. Sec. on the test. 2. Record answer in the box. Score: Each correct answer is one point. Assume fraction denominators are non-zero. Write each expression in simplest possible form. (a) :QEYEY.= ’Y x2-4 (b) x-2 = -7. 2. 3 (c) A A8 B = B (d) J"-J§'= (e) A = A and B are positive ‘5 (f) §_:_§_.= '§ (9) 3(A-B)-(3A—B) = 90 II. Perform the indicated operation and simplify if possible. uflw _ 1.- 3 _ (a) (b) 13.6 '3— .02 = (C) 2x-l _ 2x+l 2:33.1_ (d) x+l 3 ‘ (e) 53 - 52 = as a power of 5 (f) (53)2 = as a power of 5 53 (9) -- = 52 L. as a power of 5 x2 -6 2x2 5 3 (h) __4?£__ . .___t_§i_._ 2x +3x x+3 (i) (3x+2) (2x-3) :- III. Factor each expression completely. (As described in your lectures.) (a) 9B2 - 49 = 91 (d) 6X2Y — 3XY2 = (e) 2KT + 3K + 4T + 6 = IV. Circle T for True or F for False. 75+ -i-= %+%- T (b) 03“)”1 = A T (c) (3-3‘+2‘1)‘1 = 5 T (d) 3x? = 1 T X is not zero V. Find all values of X which satisfy each of the following equations. (a) 2X+5=3X-2 (b) x2+2x-3 ll 0 (c) 6X2—X-2 ll 0 (d) VI. VII. VIII. 92 Find the values of S and T for which the system ZS - T = 1 4S - 3T = 7 is satisfied S *3 II II A new compact car gets 28 miles per gallon on regular gasoline when moving at 50 miles per hour. The car only gets 3/4 as good mileage at 70 miles per hour. What is the total cost of a 210 mile trip if the car is driven at 70 miles per hour and regular gasoline costs 37 cents per gallon? An enterprising student was selling wall posters for $3.00 each or 2 for $5.00. Sixty-five people made purchases and the student had $269.00 in total sales. How many of the peOple purchased 2 posters? 93 MATHEMATICS 082 Fall 1970 Information - Assignments — Time Schedule Sections: l,2,4,5,7,9,10,13,14,15,l6,l7,l8,19,20. In order to receive a passing grade in Math 082 it is required that you pass a final examination in the course. The sections listed above will be given the final examina- tion on Thursday, December 3, at 4:10 P.M. in room 108B Wells Hall. Ten lectures will be presented on Thursdays at 4:10 in 108B. These occur on Sept. 24, Oct. 1,8,15,22,29, Nov. 5,12,19 and on Tuesday, Nov. 24. Weekly quizzes will be given every Tuesday during the 4:10 P.M. meeting. These occur on Sept. 29, Oct. 6,13,20,27, Nov. 3,10,17. These quizzes are intended to help you to evaluate your progress and will not be used in determining your grade. The Tuesday - Thursday recitation sections are provided so that you will receive individual help not available in the large afternoon meetings. An assignment sheet is attached which will be of use during the entire term. 94 MATHEMATICS 082 Fall 1970 Assignments for Sections l,2,4,5,7,9,10,l3,14,15,16,17,l8 19,20. (All assignments refer to - A Survey of Basic Mathematics, 2nd ed., Fred W. Sparks.) LECTURE I §1.4 - 1.8 p. 5 all even-numbered exercises starting with 12. pp. 7-8; 6,9,10,12,13,15,17,18,20,23,25. pp. 12-13; 8,10,12,14,15,l7,20,22,23,26,28,29,31. pp. 17-18; 2,3,6,7,10,11,13,14. LECTURE II §1.9 - 1.15 pp. 25-26; 10,12,13-22, 31,34,35,37-42, 43-51. pp. 27-29; 7,9,10,12,13,17,l8,l9,22,24,25,26. pp. 30-32; all even—numbered problems. pp. 33-38; all even-numbered problems 2-36 plus 37,38,39,40,42,44,46,49,52,56. pp. 4-43; every fourth problem. pp. 45-48; all even-numbered problems. LECTURE III §2.1 - 2.8 pp. 52-54; 7,12,17,20,22,24,26,27,32,35,36. pp. 61-63; all even-numbered problems. pp. 65-67; all even-numbered problems. pp. 69—71; all even-numbered problems. pp. 72-73; 2,5,7,10,12,15,16,17,18. pp. 76-80; all even-numbered problems. LECTURE IV §4.1 - 4.4 pp. 87-89; all even-numbered problems. pp. 90-91; all even-numbered problems. pp. 93-95; all even-numbered problems. LECTURE V §5.1 — 5.7 pp. 99-103; 1-15,l9,23,26 (28-48) all even—numbered. pp. 104-108; every third problem 2,5,8,ll,...29. pp. 109-111; ditto 2,4,...,23. pp. 112-118; ditto 3,6,9,...,48. pp. 119-122; all even-numbered problems. 95 LECTURE VI §6.l - 6.5 pp. 127-129; do as many as possible in your head and the rest on paper. pp. 130-133; every 3rd problem 1,4,7,...20; then 22,23,24. LECTURE VII §6.6 only Homework assignment pp. 135-144. All except no. 6 should be carried at least through Step 4. LECTURE VIII §7.2 - 7.6 pp. 147-149; all even-numbered problems. pp. 150-152; all even-numbered problems. pp. 153-161; set up all problems but solve just the odd-numbered. pp. 163-165; 4,7,11,13,18,21,24. LECTURE IX §8.1 - 8.11 pp. 176-178; all even-numbered problems, starting with 14. pp. 180—182; all even—numbered problems. pp. 183-185; all even-numbered problems. pp. 187-191; 1-12 plus even-numbered problems there— after. pp. 192-194; all even-numbered problems. pp. 194-195; all even-numbered problems. pp. 197-199; 1-9. LECTURE x §9.1 — 9.6 pp. 201-205; 3,6,12,14-17,19,21,24,27,28,32,35,38, 42,44,49,50,51. pp. 208-213; every third problem starting with 9 plus 48-51. pp. 214-218; every third problem starting with 13 plus 43-46. pp. 220-222; 2,7,11,12,13,16,18,20. pp. 223-224; 1,4,5,7,ll,12,13. Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture Lecture II III IV V VI VII VIII IX X 96 TOPICAL OUTLINE-MATHEMATICS 082 LECTURES - Basic Arithmetic Operations with Whole Numbers and Decimal Fractions. - Basic Arithmetic Operations with Rational Numbers. Percent. - The Real Number System. Order. Prime Factorization of Integers. — Polynomials and Factoring. - Polynomial Fractions. - Equations. - Application Problems. - Systems of Linear Equations. - Exponents and Radicals. - Quadratic Equations. "Ilfiiililfljlllfllifllflllflllfi