A STUDY or THE RELATIONSHIPS mwem CREDIT IN CERTAIN HIGH SCHOOL MATHEMATICS AND SCIENCE counsas AND VARIOUS ASPECTS 0% success AT THE MICHIGAN COLLEGE or MINING AND TECHNOLOGY Thosis for TIT. Dogm of Ed. D. MICHIGAN STATE UNIVERSITY Donaid Hart Baker 1957 1“. 099,. ..._- ‘ LIBRARY Michigan State University ‘ This is to certify that the thesis entitled A STUDY OF THE RELATIONSHIPS BE'IWEEN CREDIT IN CERTAIN HIGH SCHOOL MATHEMATICS AND SCIENCE COURSES AND VARIOUS ASPECTS OF SUCCESS AT THE MICHIGAN COLLEGE OF MINING AND TECHNOLOGY. presented by Donald Hart Baker has been accepted towards fulfillment of the requirements for Ed. D! degree in Edllcation I Major profelssor / Date July 30: 1957 dw.‘ 0.1:. ' j J u ' 3 We "‘1‘. _. .ann-M'." . _ I ”Law. ~- 1 ~' v A STUDY OF THE REMTIQVSHIPS BETWEEN CREDIT IN CERTAIN HIGH SCHOOL MATHEMATICS AND SCIENCE COURSES AND VARIOUS APSECTS OF SUCCESS AT THE MICHIGAN COLLEGE OF MINING AND TECHNOLOGY By Donald Hart Baker A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF EWCATION Department of Foundations of Education 1957 A STUDY OF THE RELATIONSHIPS BETWEEN CREDIT IN CERTAIN HIGH SCHOOL MATHEMATICS AND SCIENCE COURSES AND VARIOUS ASPECTS OF SUCCESS AT THE MICHIGAN COLLEGE OF MINING AND TECHNOLOGY AN ABSTRACT OF THE THESIS This study was designed to discover and to make ex- plicit the relationships which may exist between the amount of credit a student possessed in certain high school mathe- matics and science courses and various aspects of success at the Michigan College of Mining and Technology. The high school courses involved in the study were algebra, solid geometry, trigonometry, biOIOgy, chemistry, and physics. The number of semesters of credit earned in each of the above courses by a student was taken from his high school transcript. The various aspects of college success investigated in the study were graduation from this college and receiving a good grade (A, B, or C) in the following courses: first and second term algebra, trigonometry, analytic geometry, first year chemistry, and each of the three terms of first year physics. College records were obtained from.the regis- trar's offices The study population was limited to students who en- tered the college after 1949 as freshmen without prior college experiences and who were either graduated or dismissed for academic reasons. A total of 1647 students were involved. (27? graduated and 170 dismissed) The majority of the stu- dents were male and majoring in a science or branch of engin- eering. The main part of the study involved establishing di- chotomies with regard to the amount of high school credit in a course or combination of courses and with regard to success or non-success in one of the college experiences. The num- bers of students in the resulting four categories were enter- ed in a 2 x 2 contingency table, and a Chi-squared value was computed to determine the probability of a non-chance rela- tionship between the high school and college achievements. If the value of Chi-squared was above that of the 0.01 level of significance, the tetrachorio correlatim coefficient was computed. All significant comparisons were reported. Three principal reasons were postulated to account for the appearance of particular high school courses in the sig- nificant comparisons: the advantage of having studied the same subject in high school, the opportunity to improve the grasp of skills and concepts, and the demonstration of high level ability and interest. 0f the approximately 800 statistical comparisons made, almost 590 yielded Chi-squared values above that of the 0.01 significance level. The following table indicates the high school credit found most useful for predictive purposes: College Success H.S. Courses Chi-squared rt Graduation 6‘ 51.. 72 0.75 First TermflAlgebra A,G,P 41.A8 0.48 Second TermflAlgebra A,G,T,P 22.27 0.58 Trigonometry A,G,P 22.12 0.58 Analytic Geometry A,G,T,P 7.04 0.25 First Year Chemistry A,0,P 42.54 0.54 First Term.Physics P 19.87 0.55 Second Term.Physics P 12.55 0.45 Third Term Physics A,P 7.69 0.51 . e A - Three or more semesters of algebra G - One or more semesters of solid geometry T - One or more semesters of trigonometry 0 - Two or more semesters of chemistry P - Two or more semesters of physics IA non-credit college course in solid geometry was shown to be equivalent to high school solid geometry. A non- credit college course in elementary algebra was shown to be inferior to three or more semesters of high school algebra. It must be remembered that the demonstration of a sta- tistical relationship is not, by itself, a proof of a cause— and-effect relationship. Even in the case of two variables, one of which is an event that takes place before the second, it is necessary to investigate the effects of all other var- iables in the situation before cause—and-effect is proved. The author wishes to express his gratitude to the fol- lowing people, who rendered exceptional assistance in this study: Dr. Hilosh Muntyan, guidance committee chairman, who had the unenviable task of converting a terse physicist into a fluent educator; Dr. Willard Harrington, who was often can- sulted on statistical questions; Dr. Walker Hill, who gave valued advice on the conduct of the study; Prof. Thomas Ser- mon, Registrar of the Michigan College of Mining and Tech- ’ nolOgy, and his staff, who made the necessary records avail- able for the study; and Rebecca L. Baker, who helped substi— tute for the equipment of the IBM Corporation. TABLE OF CONTENTS CHAPTER PAGE I.INTRODUCTION.................. 1 II. THE HIGH SCHOOL PREPARATION OR THE STUDY POPULATIQI..................15 III. THE COLLEGE SUCCESS ASPECTS OF THE STUDY POPULATIGI..................15 Iv. THE STATISTICAL TECHNIQUES USED IN THE STUDY . . 24 v. COLLEGE GRADUATION AS RELATED TO HIGH SCHOOL PREPARATION..................53 III. PERFORMANCE IN FIRST YEAR MTHEMATICS AS RELATED TO HIGH SCHOCL PREPARATION . . . . . . 51 VII. PERFORMANCE IN FIRST YEAR CHDIISTRI AS RELATED To HIGH SCHOOL PREPARATION . . . . . . 76 VIII. PERFORMANCE IN rum YEAR PHYSICS AS RELATED TO HIGH SCHOOL PREPAPATION . . . . . . 82 1x. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . 92 APPENDIXss'ssessesesesoseeesosee0105 LIST OF TABLES Chi-squared Values and Tetrachoric Correlation Coefficients for Various Combinations of High School Courses Compared with TABLE I. II. III. IV. V. VI. VII. VIII. Graduation from.College . . . . . . Graduation from.College . . . . . . Graduation from College Good Grades Good Grades Good Grades Good Grades Good Grades Good Grades Good Grades in in in in in in in First TermMAlgebra First Term Algebra First TermIAlgebra . Second Term Algebra . Trigonometry and Analytic Chemistryssessssessess Physics.............. Geometry PAGE 56 45 A6 54 59 62 66 71 78 84 Tables showing the observed cell frequencies for the significant comparisons appear in the appendix in the same order as the tables listed above. CHAPTER I INTRODUCTION This studykwas designed to discover and to make ex- plicit the relationships which may exist between certain aspects of students' high school preparation and their success at the Michigan College of Mining and TechnclOgy. The approach to the study was statistical in nature. A Chi-squared value was computed to test for the significance of an apparent association between the high school and col- lege performanfes being investigated. If the Chi—squared value equaled or exceeded that at the 0.01 level of signif- icance, a tetrachoric correlation coefficient was computed to determine the degree of association. .A more detailed discussion of the statistical techniques used in this study is given in a later chapter. The high school courses considered in this study were limited to mathematics and science courses. More specifi- cally, cognizance was taken of the number of semesters of algebra, solid geometry, trigonometry, bi010gy, chemistry, and physics taken by each student in the study population. Only these courses for which a student received a passing grade in high school were counted. The second chapter treats the subject of high school preparation more completely. There are many possible criteria of college success. 2 The most immediate and readily discernible of these is grad- uation or academic dismissal from college. The study is also concerned with whether or not a good grade, i.e., an.A, B, or C, was received in various introductory mathematics, chemistry, and physics courses. The meaning of success in college is con- sidered in more detail in the third chapter. The study populatim was limited to those students who entered the Michigan College of Mining and Technology as freshman without prior college experiences and who were either graduated or dismissed for academic reasons from that college. Since college curricula do change, this study was made reason- ably current by considering only those students who entered during the academic year l950~51 or later. This made it pos- sible to include three graduating classes in the study, a number considered necessary to furnish enough cases for sta- tistical reliability. A total of 4&7 students were involved in the study. Of these, 277 were graduated, and the remaining 170 were dismissed for academic reasons. At this institution, grad- uation follows the completion of a particular curriculum with at least an average grade of C. Academic dismissal occurs automatically when a student fails four or more courses within three consecutive quarters. A larger number of graduated students might have been considered except that students who entered this college after attending any other college were excluded from the study. ‘Apparently about fifty percent of the recent grad. uating classes consists of such transfer students. These students were not included in the study because of the dif- ficulty involved in comparing their introductory college courses with those being investigated. The number of students dismissed during the period of the study may also be misleading. As is the case in most colleges and universities, a large percentage of the entering freshmen voluntarily leave this college for various reasons. Undoubtedly a major reason is the imminence of dismissal and the desire to leave with a relatively unblemished record. No estimation could be made of the number of students in this category. Consequently, no student who did not graduate was included in the study unless his record clearly indicated dismissal for academic reasons. It would undoubtedly be desirable to have a larger study population. This, however, would require including students who entered college before 1950. It is felt that the advantage gained by this step might well be more than canceled by the probability that conclusions derived from a longer study period might be less pertinent to the present situation. The period of time of the study was chosen so as to minimize the number of students who entered the college more 4 than the usual three months after graduating from high school. Almost all of the world war II veterans had left the college, and few veterans of the Korean conflict had arrived. This decreased the possibility that the students had experiences of a vocational or military nature between high school and college that would obscure the relationships sought in the study. An analysis of high school graduation dates and col- lege entrance dates indicates that only a very small per- centage of the students did not graduate from.high school and enter college the same year. A tabulation of the high schools from which these students graduated indicates a rather surprising geographical spread, considering the location and relative smallness of the college. Moreover, the students dismissed from college were by no means all from the generally smaller high schools of Michigan's upper peninsula, as is seen from the following percentages. Of those in the study who graduated, forty-nine per- cent went to Michigan upper peninsula high schools, thirty- five percent went to Michigan lower peninsula high schools, and the remaining sixteen percent came from out of state. Of those in the study population who were dismissed, forty- four percent went to upper peninsula high schools, thirty- six percent went to lower peninsula high schools, and the other twenty percent were from out of state. Moreover, the high schools ranged in size from those in the metropolitan area of Detroit to the country schools of rural Michigan. .As nearly as could be determined by a brief survey of the high school data, there were students in appreciable numbers from high schools in all categeries of size. It should also be of interest to note something of the college these students attended. The Michigan College of Mining and Technology is a state-supported institution estab- lished in 1885. The main campus is located in Houghton, in the upper peninsula of Michigan. A branch of the college, established in 1946, is located at Sault Ste. Marie. Only freshman and sephomere courses are offered at the branch of the college. Since courses cannot be identical, with the branch 265 miles from the main campus, no student who atten- ded the Sault Ste. Marie branch of the college was included in the study. The original function of the college was to furnish men trained in mining and metallurgy, especially for the mineral industries of Michigan's upper peninsula. In 1891 this college graduated more mining engineers than did any other college in the country. In 1927 the college increased the scope of its curricula to include most of the major fields of engineering and science. Later, curricula in for— estry and engineering administration were added. 6 At present, the college offers courses leading to the Bachelor of Science degree in nine branches of engineering and six branches of science, as well as in forestry and en- gineering administration. Courses are also offered leading to the Master of Science degree in most of these fields. The majority of students on the main campus are enrolled in some branch of engineering. It is of some interest to note that, although the college is coeducational, the overwhelming majority of the students in the study are male. This is accounted for by the fact that most of the curricula offered are in the fields of engineering and science, with no liberalarts type of program available. Most of the girls in the study had ma- jored in chemistry with emphasis on medical technology. The enrollment on the Houghton campus during the aca- demic year 1955-56 was about two thousand, predominantly undergraduates. In recent years the enrollment has been increasing at the rate of about nineteen percent each year. There are two major questions which, it is hoped, this study may assist in answering. First, does the posses- sion of high school credit in any course or combination of courses among those under consideration correlate highly with college graduation? That is, could the possession of credit in certain high school courses be used in predicting graduation from college? Second, does the possession of high school credit in any course or combination of courses among those under consideration correlate highly with the achievement of a good grade in certain introductory college courses? That is, are there apparently certain high school courses that are taken by those students who receive good grades in their introductory college courses? Questions of this general nature have, of course, been asked many times by many people. The Encyclopedia 2§H§gggg: ‘tigggl,Research1 reports numerous investigations, especially to find a suitable predictor of college graduation. Many possibilities have been considered, including intelligence tests, aptitude tests, high school grades, ranking in high school graduating class, and college grades in previous se- mesters. It seems to be agreed that the degree of intel- ligence needed for success in college cannot be stated cate- gorically.2 Many investigators report that high school marks generally provide a more accurate basis for the prediction of college scholarship than do intelligence tests.5 For example, Segel claims that a comparison of average high school marks and college scholarship yields a median correlation coefficient 1 Encyclopedia 2£_Educational Research, (New York: The Macmillan Company, revised edition, 1950). 2 Ibide, pe 88’s 5 Ibide, pe 885e 8 of 0.55, which is Ooll higher than the median coefficient be- tween intelligence tests and college scholarship.4 Smith,5 using previous records to estimate college success, found correlati ms between predicted grade-points and grade-points actually earned for the third, fourth, and fifth semesters to be 0.6), 0.71, and 0.70 respectively. Smith noted that there was a tendency for any record to lose prognostic value after a year or two and that the best single predictor of scholastic success in any given semester is the previous semester's work. Smith made use of all of a student's high school grades in order to have ”grasp objectivity“. Numerous studies have demonstrated that a combination of several factors may be considerably more valuable in pre- dicting general scholarship than any single factor. Edds and McCall,6 employing a combination of average high school marks, Otis Group Intelligence Test, and Cross English Test scores, obtained a mltiple correlation of 0.81 with general scholarship. 4 Encyclopedia of Educational Research, op. cit., p. 883, citing David. Segel, Prediction oi Success in College (U. 8. Office of Education, Bulletin No. .15, 1952;):- 5 F. F. smith, "The Use of Previous Records in Be— timting College Success', ,1. Ed. Psychol., 56: 167-76, 1945. 6 Encyclopedia g; Educational Research, 0p. cit., p. 885, citing J. H. Edds and W. M. McCall, “Predicting the Scholastic Success of College Freshmen“, i. _E_d_. Psychol., 50: 251-65, 1959. 9 Perhaps the most extensive bibliography of studies in this area appears in an article by Cosand.7 It also includes tables of correlation coefficients, obtained by a number of investigators, relating various aspects of college success to high school data and/or psychological test scores. The period between 1928 and 1950 is covered, with the work done between 1940 and 1950 receiving major emphasis. Cosand points out that the current trend in college admissions work is toward multiple predictors including some high school data and various psychological tests. The literature previously cited in this chapter appears in Cosand‘s article and contains correlations representative of those cited by him. The only investigation known to consider the same sort of high school data, i.e., semesters of credit in high school courses, as this_study is reported in a doctoral thesis by Leasman.8 He studied the records of 1024 graduates of the Illinois public high schools who entered four Illinois uni- versities in the fall of 1950. The college grade point average (C.G.P.A.) was taken as the criterion of college success, and the following categories of high school subjects 7 J. P. Cosand, "Admissions Criteria', College and University, 28, April, 1955. 8 R. E. Leasman, "The Relation of the Pattern of High School Courses to College Success', Ed. D. thesis, The University of Illinois, 1954. 10 were considered: English, foreign languages, mathematics, social sciences, natural sciences, and vocational subjects. No distinction was made among the various high school courses in any category; for example, all mathematics courses were treated simply as mathematics, without distinguishing among algebra, geometry, or trigonometry. Leasman used several statistical treatments in seek- ing relationships between his criterion of college success and each of the categories of high school credit. For the most part, he used analysis of variance methods, which an- abled him to held constant the factors of ability and apti- tude. In addition, contingency tables were employed in those cases where ability and aptitude were not controlled. The measure of ability was the percentile rank in the high school class. The measure of aptitu'de was the percen- tile score on the A.C.E. Psychological Examination. Most of Leasman's investigation dealt with the 665 students in his study population who had completed at least two years of college. Separate studies were made with that part of the study population attending each of the four Illinois universities involved in the investigation. ‘ Leasman established dichotomies in terms of the amount of high school credit in each category of courses, as was done in this study. For example, in the part of his study involving high school mathematics, he divided the appropriate 11 part of his study papulation into two groups: those who had three or more units (years) of mathematics and those who had less than this amount of credit. The means of the college grade point averages of the students in the two groups were compared statistically to detenmine if a significant differ- ence existed between them. If the factors of ability and aptitude were held con- stant, Leasman found no significant difference, at the 5% level, between the mean C.G.P.A.'s of the two groups in any of the six categories of high school courses. If the factors of ability and aptitude were not held constant, Leasman found that students with three or more units of credit in high school foreign languages or in mathe- matics achieved significantly higher grade point averages than did students with less than this amount of credit. No such results were obtained wiflh the other four categories of high school subjects. Leasman concluded that the relation between college grade point average and ability and aptitude is higher than between college grade point average and high school course .pattern. It should be pointed out that the approach to this study differs in several ways from that of Leasman. Fewer categories of high school subjects are involved, and distinc- tions are made among the various courses in any category. In 12 addition, this study is concerned with graduation or the type of grade received in a single college course rather than with the college grade point average, as an indication of college success. However, Leasman's study was the only one discover- ed, among investigations carried out during;the last twenty- five years, that considered the college student's high school preparation from the same standpoint as does this study. It is of interest to note that the apparently slight difference in the approaches of these two studies yields somewhat dif- ferent conclusions, as evidenced by Leasman's conclusions mentioned earlier and those of this study discussed in the final chapter. There have, of course, been many more investigations of the relationships between college success and high school preparation than the few mentioned in this chapter. With the exception of the study by Leasman, these investigations have been cited to indicate the diversity of approach to this area of study and to indicate the highest of the correlations ob- tained. It will be noticed that, in general, the correlation coefficients thus far reported are not entirely satisfactory for the prediction of college success. CHAPTER II THE HIGH SCHOOL PREPARATION OF THE STUDY POPULATION As was mentioned in the introductory chapter, the high school preparation data on the 447 students making up the study population consist essentially of the number of semesters of credit each student received in mathematics and science courses. The actual data are as follows: 1. The student's name 2. The location of the high school from which he graduated 5. The year of graduation 4. The number of semesters of credit in: a. algebra b. solid geometry c. trigonometry d. biology e. chemdstry f. physics This data was obtained directly from official high school transcripts made available by the registrar's office of the Michigan College of Mining and Technology. Although some of the transcripts had been issued before the student's actual graduation from high.school, letters from the high school authorities were available to indicate the credits 14 received between the date of issue of the transcript and the date of graduation. It may be observed that no records were made of the semesters of plane geometry taken by each student. Among the entrance requirements of this college is an inflexible one that each student must have credit in two semesters of high school plane geometry. Since this is apparently the maximum number of semesters of plane geometry offered by all, or practically all, high schools, there are no comparisons that can be made involving this subject, and consequently no record of it was kept. A very pertinent question may be asked at this point as to why the marks received in these high school courses were not used in this study. It is apparent that most in- vestigations reported in the literature involve comparisons of high school marks with some aspect of college success. However, after considering his own educational experiences and after discussing the question with a number of college faculty members, this investigator concluded that there must be a significant loss of validity whenever course grades from different institutions are combined and treated as es- sentially equivalent. There is certainly the possibility of some sort of internal consistency in grading procedures in any one high school, although even this would seem to be more a goal than a probability. With each high school supplying 15 only a few students for the study population, it seemed a dubious procedure to assume that the grades received in any particular course at different high schools had a reasonable degree of comparability. Of course, in ignoring the grades received in these courses, one is rejecting most of the available indications of differences of ability, retaining only the possession or lack of credit. Inasmuch as college success is partly a func- tion of ability, the rejection of these indications might conceivably result in such a homogeneity that there remain no distinctions identifiable with college success. At least, there must be some loss in the sensitivity of the statistical results. After a consideration of these arguments, it seemed that the procedure of grouping students according,to the num- ber of semesters of credit received in a particular high school course is a more defensible one. It is true that the same question of equivalence is present, but to a much lesser extent. There is no doubt that a certain level of perform- ance might earn course credit for a student in one school but not in a different school, or perhaps not even in the same school under a different teacher or at a different time. Here, however, one is concerned only with consistency in the pass-or-fail judgments of the teachers rather than in the which—of-five-grades type of decision. While the question of 16 comparability still exists, the counting of course credit seems to reduce its importance about as much as possible. Consequently, the approach of this study is seen to be a reasonable compromise involving some gain in comparability of student achievement and some loss in sensitivity of the statistical results. A problem arose during the processing of the data on high school science courses. In general, it was found that students possessed either no credit or credit for two semes- ters of a particular science course. However, a relatively few students had credit for only one semester or for more than two semesters in some science course. The difficulty arose in determining how the unusual amount of credit should be considered in making comparisons involving high school science and the various aspects of college success. Since most students with any credit in a particular science had credit for two senesters, it was felt that all students would have to be considered as having either no credit or credit for two semesters of each science. Then it was arbitrarily decided that students having credit in only one semester would be treated as having no credit and that students having credit in more than two semesters would be treated as having only two semesters. This decision is mainly justified by the small number of students who did not follow the usual pattern with regard 17 to the amount of credit in a high school science. There were not enough to these students to yield statistical re- liability if they were grouped in separate categories, and yet it was felt that their science courses should be involved in the comparisons, along with those of the other students. The decision to count one semester of credit as no credit is based largely on the feeling that less than half as much is likely to be learned in one semester of a science compared to that learned in two semesters. CHAPTER III THE COLLEGE SUCCESS ASPECTS OF THE STUDY POPULATION The consideration of college success in this study is limited to two types: graduation from the Michigan College of Mining and Technology and the receiving of a good grade (A, B, or C) in introductory mathematics, chemistry, and physics courses. The following college informatim was obtained for each student: 1. The year of matriculation 2. The existence of any entrance conditions and when they were satisfied 5. The grades received in: a. the first and second terms of algebra b. trigonometry c. analytic geometry d. the first year of chemistry (one grade given) e. the first year of physics (three grades given, one for each of three terms) 4. Whether the student was graduated or dismissed for aca- demic reasons. There are only two non-credit courses, designed to satisfy entrance requirement deficiencies, which are of in- terest in this study. A course indicated as A00 is taken by 19 those students who, on a placement test given during their first term in college, show poor preparation in mathematics. A survey shows that the studmts most likely to have to take the £99 course are those with credit in less than three semes- ters of high school algebra. This course is taken before the first term of regular college algebra. In order to observe the effect of £99 on college success, various combinations of high school algebra and/or Agg_were used in making statistical comparisons. The results of these are described in later chapters. The other non-credit course, indicated as 52, is one in solid geometry for those students deficient in it. Stu- dents are nominally required to have had a semester of solid geometry in high school or to take thejig course. In con- trast to A09, which records indicate as always taken in the first term.of residence, £9 seems to be taken at any time before graduation, often daring the last term of residence. Consequently, caution had to be exercised in counting the number of students who had taken.AQ_before any one of the college courses taken as a criterion of college success. Moreover, a significant number of students somehow escaped taking:AQ even though they did not have high school credit in solid geometry. The majority of the students at this college take four introductory mathematics courses in the freshman year. The 20 first term of algebra is usually taken during the first term of residence. The second term of algebra and a course in trigonometry are taken during the second term. A course in analytic geometry follows in the third term. Successful completion of the mathematics courses taken the previous term is the only prerequisite for entering the later courses of this group. That is, the courses must be taken and passed in the order given above, being contingent upon no other col- lege courses. Several of the college curricula do not re- quire the course in analytic geometry. Consequently, this added to the effect of dismissals in reducing the number of students available for comparisons involving analytic geom- etry. The students taking,the medical technology, pre-dental, or pro-medical curricula are allowed to take a different series of three introductory mathematics courses, only roughly comparable to the four described earlier. Because of the lack of equivalence between the individual courses of the two series, data concerning the three-course series were not used in this study. Only a very small fraction of the study population took the three-course series. The introductory course in chemistry, considered in this study, is normally taken by all students in the freshman year. There are no college course prerequisites. The course is described in the college bulletin as "A study of the 21 principles of chemistry and of the properties, preparation, and uses of the more common elements and their compounds. Principles of, and practice in, the separation and identifi- cation of the more common cations are taken up in the latter part of the course.“ Both lecture and laboratory work are included in the course. During,the period of the study all college curricula required the successful completion of this first-year course in chemistry. At that time one grade was given for the entire year's work. There are two introductory three-term sequences in physics. Since they appear to be equivalent except for a small difference in the mathematical approach to the subject matter, data on these courses were combined, and no distinc- tion was made between them at any point in the study. The first term's work is concerned with mechanics and sound; the second with heat and light; the third with electricity and magnetism. Both lecture and laboratory work are in- cluded in these courses. The prerequisites for the engin- eering physics sequence, taken by almost all of the students in the study population, consist of the four introductory mathematics courses described earlier. A three-term sequence in calculus must be taken at least concurrently with the three terms of this physics course. The prerequisite to the other introductory physics course is either the three-term series of introductory mathematics or the first three terms 22 of the four-term series. A few students in the engineering physics course take what is usually the third term of the course between the first and ordinarily second terms. Since no distinction between these and the normal cases was made in recording the data, it will be impossible to draw any con- clusions involving the order in which the second and third terms of the course are taken. As was described in the first chapter, graduation requires the successful completion of one of the four-year curricula with a 0 average both for the entire four years and for the last year. Dismissal occurs automatically if a student fails four or more courses within three consecutive terms. There are no other grounds for academic dismissal. A dismissed student may apply to a committee composed of faculty and administra- tive representatives for readmission. An appreciable per- centage of dismissed students are readmitted. Strenuous efforts were made to insure that no student was included in the study if he was dismissed during the period of the study but later readmitted without finally graduating. That is, it is believed that all of the students in the study popu- lation were either graduated or dismissed to return no more. It was necessary to decide how to treat the quite appreciable number of courses that were failed and repeated by some of the students in the study population. The decision 25 was made to count only the first grade received in any of the courses being considered and to ignore grades received on re- petitions of any of them. No allowance for course repetition was made in any of the computations. This procedure will have to be remembered if any conclusions are drawn concerning the effect of one college course on a presumably later one. No comparisons of this nature are considered to be within the scope of this study. CHAPTER IV THE STATISTICAL TECHNIQUES USED IN THE STUDY It is, of course, possible to gain an impression of the association between credit in a given high school course and college success by computing several numerical propor- tions, using the numbers of those who did and did not achieve college success and of those who did and did not have the high school credit. While this procedure is relatively quick and perhaps satisfactory under some circumstances, it has several shortcomings. If only the proportions mentioned above are used to judge, perhaps, the relative merits of high school courses as they appear to be related to college success, two partic- ular difficulties arise. This procedure gives no indication of the probability that the distribution of numbers in the categories mentioned is not merely a chance distribution, indicating an association that does not exist. In addition, it is extremely difficult with this procedure to compare the apparent association between credit in one high school course and college success with the apparent association between credit in a different high school course and college success. For these reasons, a more elaborate statistical treatment of the data is indicated. At this point, it might be well to mention that the ll! ‘11“ ,. 1‘5: I. ii \ll-I... ll‘.’ Isl-I'll 25 following discussion of the statistical techniques employed in this study presupposes a basic knowledge of statistics. Treatments of the various concepts used in this study may be found in most statistics textbooks.1'2'5 It will be observed that both the high school and college data described earlier represent rather natural di- chotomies. For example, a student did or did not receive credit in a particular high school course; he did or did not achieve college success. Taking advantage of these two di- chotomies, one can divide the study papulation into four categories, and the number of students (that is, the fre- quency) in each category can be entered in a two-by-two con- tingency table. From the data presented in such a table, it is possible to obtain several statistical indications. See Figure 1 on the following page for an example of the 2 x 2 contingency table as it was used in this study. l O. 0. Peters and I. R. Van Vocrhis, Statistical Procedures an§,their Mathematical Bases (New York: McGraw Hill Company, 19505. 2 M. J. Hagood and D. 0. Price, Statigtics for So- ciologists (New York: Henry Holt and Company, revised, 1952). 3 Quinn McNemar, Psychological Statistics (New York: John Iiley and Sons, second edition, 1955;. College Dismissed H Graduated totals Have credit b a t1 in solid 108 220 528 geometry High (125 ) (205) School Do not d c t2 have credit 62 57 119 in solid ‘geometry (45) (24) t5 t4 . N totals } 170 277 1 1:47 Figure 1 AN EXAMPLE OF THE THO-BY-TUO CONTINGENCY TABLE USED IN THIS STUDY The numbers 108, 220, 62, and 57 are'the observed frequencies obtained from.the’data. The numbers 528 and 119 are the row totals; 170 and 277 are the column totals. The total number of cases involved in the comparison is 447. The numbers 125, 205, 45, and 74 are the expected frequencies, explained in any of the references given on the previous page. The first computation based on the data in the contin- gency table is the testing cf’the null hypothesis: that there is no association between the classification along the ver- tical marginal and the classification along the horizontal marginal. If this hypothesis can be rejected, there is small probability that the apparent association between classifi- cations is due to chance. Consequently, it will be assumed that the association actually exists. 27 The null hypothesis was tested in this study by deter- mining a Chi-squared value for the observed distribution in the two—by-two contingency table. The following equation can be used to compute Chi-squared: 2-- Z W x - fexp. However, according to many authors, a correction for contin- uity should be applied to Chi-squared if the expected cell frequency in any cell is less than a certain minim value. Althougi there is not general agreemt regarding the exact mininum value, those mentioned in the literature would in- dicate that the Yates correctim for continuity is neces- sary in this study. Consequently, that correction was applied to every value of Chi-aqua red. With reference to the equation given above, the cor- rection consists of reducing by one-half unit the magnitude of each difference between the observed and expected frequen- cies in a cell. For example, referring to Figure l, in cell a the difference (the deviation) is 17. This was reduced to l6.5 before squaring and dividing by the expscted frequency 205. The correction reduces the value of Chi-squared com- pared to that obtained without using the correction. . It is also suggested by authorities on statistics that the Chi-squared test of sigiificance is not valid if the expected frequency of any cell is less than five. A fsw 28 of the comparisons in the study were rejected for this rea- son. The value of Chi-squared can be used to test the null hypothesis mentioned earlier. First, however, the investi- gator met decide upon the confidence level above which he wishes to operate. This level and the number of degrees of freedom in the distribution of frequencies determine the critical value of Chi-squared. If a comparison yields a value of Chi-squared less than this critical value, the null hypothesis cannot be rejected. The choice of confidence level, called by some the level of significance, is to some extent an arbitrary matter. The lower the chosen confidence level, the greater certainty one has that the apparent association is not due to chance. On the other hand, in not rejecting the null hypothesis when the Chi-squared value is below the critical value, the inves- tigator runs the risk of discarding distributions in which a dependmcy or association actually exists. There is always this risk, and it must be recognized that some comparisons, rejected in this study because of a low Chi-squared value, may involve some degree of association between the high school credit and college success. The 0.01 confidence level represents a compromise, acceptable to this investigator, between accepting a chance distribution and rejecting a valid association. Consequently 29 the critical value of Chi-squared is 6.655, which represents that confidence level with the one degree of freedom inherent in a two-by-two contingency table. Comparisons were rejected if they yielded Chi-squared values less than this. After determining that a given distribution is very likely due to a valid relationship, one can compute the degree of association by one of several statistical concepts. Among these are the contingency coefficient and the tetrachoric correlation coefficient, rt. The tetrachoric correlation coefficient has the advantage of ranging in possible values between positive and negative one; thus comparing in range with~the more commonly seen Pearsonian correlation coefficient, which is used with other types of distributions. The validity of the tetrachoric correlation coefficient depends upon several conditions. It must be demonstrated, or at least assumed, that the dichotomized characteristics are actually continuously distributed, not discretely distributed, and that they are also normally distributed. Furthermore, the statistical regression must be linear. It may be observed that these conditions are often very difficult to demonstrate and can only be assumed to exist, on the basis of a carefully reasoned judgment. There seems to be sufficient cause in this study to assume that these conditions have been reasonably well met; or at least there is no evidence that they have not. 50 The exact equation for computing the tetrachoric cor- relation coefficient is somewhat difficult to use, and other methods are commonly employed. Perhaps the most convenient one, and the one finally used in this study, involves refer- ence to a set of curves4 from which the correlation coefe ficient can be read directly, with the aid of certain pro- portions from the distribution. Since, in a study of this nature, the task of sorting data becomes of overwhelming magnitude when all data must be scanned for each comparison, it is essential that some me- chanical process be employed. No mechanical sorter of the IBM type was available; however it did prove effective to transcribe the data to special cards, one for each student, by notching or leaving unnotchsd a series of holes around the edge of each card. A semi—mechanical sorting was pos- sible by lining up a group of cards and inserting a long slender rod through any particular hole. Lifting the rod removed the unnotchsd cards. The card holes were coded to represent the various high school and college data. The cards thus sorted were counted visually to obtain the frequencies for the contingency tables. Visual counting is, of course, more susceptible to 4 L. Chesire, M. Saffir, and L. L. Thurstone, Com- utin Diagrams for the Tetrachoric Correlation Coefficient Chicago: University of Chicago Bookstore, 1955). 51 error than is mechanical counting. All reasonable precau- tions were observed in order to obtain accurate frequency values. Continual cross-checking and recheeking of the com- putations were carried out to reduce the possibility of errors. At first glance, it might appear that the six high school courses considered in this study would permit only six comparisons with any one aspect of college success. However, the possibility of using credit in two or more high school courses in one comparison makes many more com- parisons available. The procedure adopted in this study and described in.the next chapter allows for at least sixty- three comparisons, enumerated below. There are six comparisons taking one high school course at a time, fifteen taking two courses at a time, twenty taking three courses at a time, fifteen taking four courses at a time,six taking five courses at a time, and finally one taking all six courses at once. These numbers can be easily verified on the basis of combination and per— mutation theory. It must be remembered that the demonstration of a statistical relationship is not, by itself, a proof of a cause-and-effect relationship. Even in the case of two variables, one of which is an event that takes place before the second, it is necessary to investigate the effects of 52 all other variables in the situation before cause-and-effect is demonstrated. Of course, cause-and-effect is a possible reason for a statistical relationship and_as such will be considered in those cases where a statistical relationship is obtained. CHAPTER V COLLEGE GRADUATION AS RELATED TO HIGH SCHOOL PREPARATION Perhaps the most interesting question with which this study is concerned is whether or not the knowledge of the courses a student took in high school can be used in the pre- diction of graduation from college. Even after deciding to restrict the consideration of the study to the number of semesters of credit in certain high school courses, disregarding grades and other courses, the investigator still had several alternatives in treating the high school data. The first alternative might be called the positive type of comparison. In this treatment the study population would be divided into two categories: the first would include all students who did possess credit in a particular high school course or combination of courses: the second would include the remaining students, who did not possess credit in all of the courses being considered at that time. For example, in comparing graduation from college with credit in high school solid geometry and chemistry, the first category would include students who did have credit in solid geometry and chemistry: the second would include those who did not have credit in one or both of these two subjects. In the chapter on statistical procedures it was shown 54 that there are sixty-three different combinations of the six high school courses considered in the study, taking them first one at a time, then two at a time, and so forth. The second alternative might be called the positive and negative type of comparison. In this treatment the study population would again be divided into two categories. This time the first category would include students who did have credit in some of the high school courses and who did not have credit in others. The second category would include the remainder of the study population. For example, the first category might include those students who had credit in solid geometry and chemistry but who did not have credit in trigonometry. The second category would include the stu- dents who did not fit into this pattern. It can be imagined that there are a great many more possible ways of combinaing the high school data using this type of comparison than exist using the first alternative. There is no doubt that the second alternative would provide a more exhaustive analysis of the relationships be- tween credit in the high school courses and graduation from college. Nevertheless, it is believed that the end results of the two approaches would be somewhat similar; inasmuch as both alternatives involve roughly the same range of course combinations. If this premise is granted, a practical as- pect of the study sways the argument in favor of the first 55 alternative. The unfortunate, but common, limitation of the time available for the study made the first alternative the desirable one. The lack of completely mechanized card sort- ing and counting equipment contributed to the very appreciable difference in the time required to carry out the complete set of comparisons by the two procedures. Consequently, in view of the approximate equivalence of the two methods, the first alternative was taken. In searching for the best basis for predicting grad- uation from college, three main series of comparisons were made. In the first series no attention was paid to the non- credit college courses, AQ_and 599, described in an earlier chapter. The second and third series, which take these two courses into account in different ways, demonstrate the necessity of considering this type of course, designed to remedy entrance deficiencies, in a study of this sort. 0f the sixty-three comparisons in the first series, only twenty-seven yielded Chi-squared values greater than 6.655, which represents the 0.01 significance level. The high school course combinations with their Chi-squared and tetrachoric correlation coefficient values for these twenty- seven comparisons are shown in Table I on page thirty-six. The comparisons were first separated into groups with re- spect to the number of high school courses involved; then the comparisons in each group were ranked according to the TABLE I 56 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COHBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GRADUATION FROM COLLEGE Courses Chi-squared rt P 16.20 0.59 G 12.74 0.50 A 10.55 0.52 G,P 26.05 0.40 A.P 22.94 0.40 0,? £5.72 0.55 A,G 15.28 0.55 A,0 10.42 0.27 0.0 10.56 0.25 A,G,P 29.67 0.42 G,C,P 20.02 0.55 A,C,P 19.62 0.35 A,G,C 11.57 0.27 0,3,? 10.40 0.25 G,T,P 9.72 0.24 A,G,T 0.54 0.25 A,G,B 7.77 0.22 A,T,P 7.70 0.22 A,G,C,P 21.95 0.57 A,G,T,P 12.77 0.28 A,G,B,P 12.59 0.27 G,B,C,P 9°57 0'24 A,B,C,P 7.55 0.22 A,G,B,C 7.57 0.21 G,T,C,P 7.02 0.21 A,G,B 0,? 10.97 0.26 A,G,T C,P 8.85 0.25 'CCDCDF9533' - Three or more semesters of algebra - One - One - Two - Two - Two or more or MOIO or more or more or more semesters semesters semesters semesters semesters of solid geometry of trigonometry of b1010gy of chemi stry of physics 57 values of Chi-squared. Table 1a in the appendix shows the cell frequencies observed in these twenty-seven comparisons. In analyzing the data presented in Table I, attention was first drawn to the highest values of Chi-squared and rt in each group of course combinations. Considering only these top-ranking values in each group, one may notice that both Chi-squared and the correlation coefficient increase to a maximum in the case of the three-course combination, then decrease. The following extract from Table I points out this trend: Courses Chi-squared rt P 16.20 0.59 G,P 26.05 0.40 A,G,P 29.67 0.42 A.G.0.P 21.95 0.57 A,G,B,C,P 10.97 0.26 where the symbols representing the courses are the same as those used in Table 1. Thus it appears that the pattern of credit in three or more semesters of algebra, one or more semesters of solid geometry, and two or more semesters of physics shows the highest degree of relationship to graduation from this col- lege. 0n the basis of the first series of comparisons, this combination of high school credit is apparently the best pre- dictor of graduation from college. Another sort of observation of Table I involves the frequency with which each high school course appears, out of 58 the thirty-two combinations in which it might appear. Solid geometry has the highest frequency, nineteen, followed close- ly by physics with eighteen and algebra with seventeen. Chemistry appears thirteen times; biology seven; and trig- onometry six. This observation of frequencies is rather thought provoking, in view of the courses students commonly take early in their work at this college. Courses in chemistry, algebra, and trigonometry are taken in the freshman year; physics is taken in the sophomore year. Solid geometry is taken only as a non-credit course to overcome an entrance deficiency. Relatively few students take college courses directly related to biolOgy. Discussions with faculty members of the appropriate departments have yielded the information that the introduc- tory mathematics and science courses are taught at this college essentially as if the student had little or no high school background in those subjects. One might expect that the high school courses that are essentially duplicated in the early college years would not yield the higher correlations when compared to graduation from college. Those high school courses not duplicated in college would be expected to correlate highly with graduation if they provided a valuable preparation for college. It is readily apparent that this does not occur in 59 these comparisons involving graduation from this college. Physics, chemistry, and algebra are among the high school courses that are duplicated; yet they appear to be important. The picture is further confused when the remaining high school courses are considered. Trigonometry is duplicated; it does not seem to be important. Solid geometry is not duplicated; it seems to be quite important. Biology is not duplicated; yet it does not seem to be very important. The unexpected pattern of this information must be due to a combination of several factors. It is certainly reasonable to expect that certain high school courses do provide subject matter background useful in college. It is also quite probable that the possession of credit in a course does not necessarily imply a retained knowledge of the course subject matter. Differences in motivation un- doubtedly exist. Finally, there is very likely a sort of hierarchy of courses, common to most high schools, which is taken by those students who are seriously concerned with preparing themselves for college. The following paragraphs treat these factors as they may affect the statistical results of this study. First, it must be re-emphasized that a statistical comparison cannot be used, alone, to demonstrate a cause-and- effect relationship. Consequently, the discussion which fol- lows is based on conjecture reinforced by a limited knowledge 40 of students and of high school and college curricula. Any conclusions must be considered as tentative and subject to modification in the light of further investigation. If one assumes that high school courses provide a de- sirable subject matter preparation for college, these sta- tistics are difficult to believe. There is every reason to expect that the student with the most complete high school background would be the one most likely to be successful in college. Yet this appears to be refuted by the fact that th° highest Chi-squared and rt values were found to be as- sociated with a combination of three high school courses rather than with a combination of all six of the courses in- volved in the study. With the possible exception of biology, all of the courses in mathematics and science in high school should provide a useful background for students enrolled in science or engineering curricula in college. Yet one might interpret these statistics to indicate that a student with credit in only algebra, solid geometry, and physics is more likely to graduate from college than one with additional credit in mathematics and science. This skepticism is well founded. Remember that the entire study population is forced into two categories with respect to the students' high school records. This dichotomy is a natural one when credit or no credit in only one course is involved, but it leaves something to be desired when 41 credit in more than one course is considered. For example, consider the comparison involving a combination of three high school courses A, B, and C. The first category in the di- chotomy would include only those with credit in all three courses. The second category, however, would include those with credit in.A but not B or 0, credit in B but not A or 0, credit in C but not in A or B, credit in A and B but not C, and so forth. In this example, the second category would actually consist of seven groups, only one of which includes those with no credit in A, B, or 0. It requires only intui- tion to imagine that the contrast between the two categories becomes progressively weakened as more high school courses are added. Consequently, one might expect that the statis- tical symptoms of the contrast would also suffer. Unfortunately, this is difficult to demonstrate. This investigator was unable to find any other statistical treat- ment that would satisfactorily indicate the benefit or detri- ment of a more extensive high school background. Some anal- yses were carried out with the aid of contingency coefficients for tables containing as many cells as there were different groups. Thus the example cited in the previous paragraph would call for a two-by-eight contingency table to treat graduation or dismissal from college and the eight different categories of high school credit. The drawback to this ap- proach is that it is impossible to compare the contingency 42 coefficients obtained from contingency tables of different sizes, which cannot be avoided in this situation. Hence, one is forced to conclude that the statistical data obtained in this study cannot be used to justify the exclusion of any high school course for those students intend- ing to go to college. The second factor mentioned at the beginning of this discussion is the doubtfulness of assuming that credit in a given high school course implies a retained knowledge of the course content. Any teacher will verify the impression that some students, in spite of a record of previous courses, react to a course with the innocence of the academic new-born. Again, this study was handicapped by an inability to evaluate this factor. It is certain only that it must be influential to some extent. The third factor mentioned earlier is motivation. There is no reason to believe that any student has the same degree of motivation during two consecutive days, to say nothing of a period of years. There are numerous cases of students with mediocre high school records who somewhere find the incentive to excel in their college endeavors. Un- doubtedly a vocational decision can mark the turning point. This frequently comes in the early college years. It is the question of motivation that undermines any prediction of college success based on the high school record. 1+5 The last factor, concerning a pr0posed hierarchy of courses, allows for interesting speculation. Specifically, it is suggested that a pyramid-like arrangement of high school courses may be constructed on.the basis of the se- quence of courses taken by the student who plans to go to college. Algebra and biology, of the courses considered in this study, probably constitute the bottom, or earliest, layer of the pyramid. Trigonometry and chemistry likely have an intermediate position. .Among those with whom this idea has been discussed, there are conflicting Opinions as to the relative levels of physics and solid geometry. In any event, one or the other probably represents the highest level of achievement in most high schools. If this idea has any basis in fact, it is possible to explain the statistical results of this chapter by stating that the students most likely to graduate from this college are those who have reached the highest level of the hierarchy of high school courses. The above discussion has been carried out with the ex- press purpose of demonstrating the difficulty involved in se- tablishing reasons for the differences in Chi-squared and the tetrachoric correlation coefficient observed in different combinations of high school courses when compared to gradua- tion from college. This study can claim only to point out the existence of the relationships, not the reasons for them. 44 When the two non-credit courses _A__C_)_ and 599 are con- sidered, it is possible to obtain much higher correlations between high school credit and graduation from college than were obtained in the first series of comparisons. The sta- tistical results of the comparisons in the second series are shown in Table II on page forty-five. The statistical re- sults of the comparisons in the third series are shown in Table III on page forty-six. As usual, no comparisons yielding Chi-squared values less than 6.655 were entered. Furthermore, these two series consist solely of comparisons which include algebra or solid geometry or both. The other comparisons, those not including algebra or solid geometry, would be duplicates of the cor- responding ones in the first series. The procedure in the second series involves elimi- nating certain students from the study population in some comparisons if they took A9 or £99. Specifically, no student who took 59 was included in any comparison which included high school solid geometry, and no student who took 599 was included in any comparison which included high school algebra. Thus the number of students involved in the comparisons in this series varied depending upon the high school subjects included in each comparison. Table 11a in the appendix lists the cell frequencies and the total number of students in each. 0f the sixty-three comparisons possible using all six high 45 TABLE II CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GRADUATION FROM COLLEGE’ Courses Chi-squared rt G 51.55 0.78 G,P 62.12 0.70 A,G 59.22 0.75 0,0 25.95 0.44 GOT 15-81 0052 G,B 12.52 0.51 A,P 11.55 0.55 A,G,P 46.97 0.66 6.0.? 59.44 0.55 G,T,P 18.45 0.57 G,B,P 17.87 0.55 A,G,C 14.59 0.57 G,B,C 11.42 0.28 A,G,B 10.57 0.29 A,G,T 10.00 0.28 A,G,P 9.59 0.26 G,T,C 8.07 0.24 A,G,0,P 25.40 0.45 G,B,0,P 15.56 0.54 AsGsTsP 14°17 0‘53 G,T,C,P 12.91 0.51 A,G,B,P 11.76 0.51 A,G,B,C 7.52 0.25 A,G,T,C,P 9.16 0.27 A,G,B,C,P 9.05 0.27 A - Three or more semesters of algebra G - One or more semesters of solid geometry T - One or more semesters of trigonometry B - Two or more semesters of biology C - Two or more semesters of chemistry P - Two or more semesters of physics ’ wherever algebra, A, appears in a course combination, no student who took A00 was counted. Wherever solid geometry, G, appears, no student who took 59 was counted. TABLE III CHI-SQUARED VALUES AND ETWACHORIC CORRELATICN COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GRADUATION FROM COLLEGE‘I Courses Chi-squared rt G 54.72 0.75 G,P 61.72 0.6) A,G 57.99 0.58 0,0 26.65 0.45 A,P 16.50 0.57 G,B 9.99 0.26 G,T 8.49 0.25 A,G,P 52.17 0057 6900? 58.80 0.48 A,G,C 22.05 0.58 G,B)P 17e50 0'55 A,G,P 15.40 0.52 G,T,P 12.95 0.28 G,B,C 9.68 0.24 A,G,B 9054 O. 24 A,G,T 8.46 0.2) A,G,C,P 55.55 0.45 A,G,B,P 15.99 0.51 G,B,C,P 14.88 0.50 A,G,T,P 12060 0.28 A,GpBgo 8.58 0.24 G,T,C,P 8.54 0.2} A,G,B,C,P 15.71 ' 0.28 A,G,T,C,P 8.55 0025 A - Three or more semesters of algebra G - One or more semesters of solid geometry T - One or more semesters of trigonometry B - Two or more semesters of biology C - Two or more semesters of chemistry P - Two or more semesters of physics * Credit in A00 is considered equivalent to credit in three or more semesters of algebra. Credit in.AQ is con- sidered equivalent to credit in solid geometry. 47 school courses, it can be shown that only forty-eight contain algebra and/or solid geometry. In twenty-five of these, Chi- squared values above the 0.01 significance level were obtain- ed. The third series involves a somewhat different way of treating Ag and £02. In this series a student who took £99 was considered the same as a student who had credit in three or more semesters of high school algebra, and a student who took 59 was considered the same as a student who had credit in high school solid geometry. Thus all students in the study population were included in each comparison. Table IIIa in the appendix contains the cell frequencies observed in these comparisons. There were twenty-four, out of the pos- sible forty-eight, with Chi-squared values above the 0.01 significance level in this series. The second and third series have several characteris- tics that are similar. They both yielded values of Chi- squared and rt that are considerably higher than those values for the corresponding comparisons in the first series. This would indicate that A2 and A92 exert an appreciable influence on the students who take these non-credit courses. This ob- servation is somewhat confirmed by the results of a fourth, brief, series to be described later in this chapter. If there were no appreciable effect of taking .A_0_ or A92, one would expect that the results of the second and third series 1,5 would be practically identical with those of the first series. In both the second and third series the highest value of Chi-squared occurs for the comparison between solid geomp etry and physics and graduation from college, while the high- est value of rt occurs for the comparison involving solid geometry alone. Given values of Chi-squared so much greater than that at the 0.01 significance level, it would sesm.that the comparison with the highest value of rt would have the most predictive value. That is, with such high probabilities of an association in both of the top comparisons in each series, the comparison with the higher degree of association should be the more useful. Therefore, it appears that the best predictor of graduation from this college is credit in high school solid geometry or in Ag. 1 It seems likely that the discussion earlier in this chapter concerning the reasons for differences in the statis- tical values in the first series also applies to the results of the second and third series. Certainly it is ridiculous to suggest that the student most likely'to graduate from college is the one who takes only solid geometry in high school. More likely, credit in solid geometry represents about the highest level of achievement in high school. Several more comparisons involving 50 or £90 were made in an effort to establish the apparent value of these courses. 49 The first compares high school credit for three or more se- mesters of algebra, but no AQQ, with credit in A99 - in place of the usual dichotomy of high school credit or no credit. College graduation or dismissal, as usual, provided the other dichotomy for the comparisons in this series. This comparison yielded a Chi-squared value of 20.08 and an rt of 0.42, in- dicating that three or more semesters of high school algebra are preferable to taking A29. The second comparison in this group involved credit in 599 compared to less than three as- mesters of high school algebra and no 599. 7A Chi-squared value of 0.01 was obtained in this case, and rt was not de- termined. In view of the low Chi-squared value, it is not likely that a relationship exists. The third comparism involved credit in Lg against no credit in high school or college solid geometry. This yielded a Chi-squared value of 40.19 and an rt value of 0.85, indicating that students who took:AQ were more likely to grad- uate from this college than those who did not take a course in solid geometry. The fourth comparison was between high school credit in solid geometry and credit in £9. The Chi- squared value for this was 0.16, too low for further con- sideration. Again, in view of the low Chi-squared value, it may be assumed that no relationship exists. In summary, the investigation of graduation from col- lege as related to credit in certain high school courses has 50 shown that the student most likely to graduate from the Mich- igan College of Mining and Technology is one who has either taken solid geometry in high school or taken the college non— credit ccurse in solid geometry, A9. A Chi-squared value of 54.72 and a tetrachoric correlation coefficient of 0.75 were obtained in comparing credit or no credit in solid geometry with graduation or dismissal from this college. It has been pointed out that these statistics, because of the several factors that influence them, cannot be used alone to justify any particular high school curriculum for those students who plan to go to college. A statistical study of the two non-credit courses, Ag and £99, indicated that a student who has credit for three or more semesters of high school algebra is somewhat more likely to graduate from this college than is one who must take £99 to remedy a deficiency in algebra and that a student who takes 59 to remedy a deficiency in solid geometry is much more likely to graduate than one who has never taken a course in solid geometry. CHAPTER VI PERFORMANCE IN FIRST YEAR MATHEMATICS AS RELATED TO HIGH SCHOOL PREPARATION As was described in the introductory chapter, most students at the Michigan College of Mining and Technology take four elementary mathematics courses during their fresh- man year. These are first term algebra, normally taken dur- ing the first term; second term algebra and trigonometry, two separate courses normally taken during the second term; and analytic geometry, normally taken during the third term. The only college prerequisite for any one of these courses is credit in those courses which precede it in the sequence described above. The criterion of college success throughout this part of the study was a good grade, i.e., an A, B, or C, in the particular mathematics course being considered. The study population was divided into two groups: those who received a good grade the first time they took the coirse and those who received a poor grade, i.e., a D or P, the first time they took the course. The high school data was treated the same as described in the preceding chapter. The 2 x 2 contingency table was, as before, used to obtain a Chi-squared value and if that was significant at the 52 0.01 level, a tetrachoric correlation coefficient was calcu- lated. The remainder of this chapter is divided into five sections, one for each of the four college mathematics courses under consideration, and a summary. FIRST TERM ALGEBRA Again, as in the case of college graduation, three main series of comparisons were made. In the first series no attention was paid to the non-credit college courses, 59 and A99, described in an earlier chapter. The second and third series, which take these two courses into account in different ways, yielded results quite different from those obtained in the comparable series in the study of college graduation. The fourth series was also included in this part of the study. Of the sixty-three comparisons in the first series, fifty-sight yielded Chi-squared values greater than 6.655, which represents the 0.01 significance level. In addition, an extra comparison involving only high school algebra was made, in which the study population was divided into two groups: those who had four or more (instead of the usual three or more) semesters of high school algebra and those who had less than four semesters. This comparison was included with those of the first series. The high school course combinations with their values 55 of Chi-squared and tetrachoric correlation coefficients for these fifty-nine comparisons are shown in Table IV on pages fifty-four and fifty-five. Table IVa in the appendix shows the cell frequencies observed in these comparisons. In attempting to analyze the data presented in Thble IV by considering the highest values of Chi-squared in each group of course combinations, as was done in the previous chapter, one finds a rather surprising pattern. This is shown in the following extract from Table IV: Courses Chi-squared rt T 51.82 0.45 G,P 58.27 0.46 A,G,P 41.48 0.48 A,G,T,P 58.48 0.47 A,G,T,C,P 29.95 0.45 A,G,T,B,C,P 15.02 0.51 where the symbols representing the courses are the same as those used in Table IV. The first unexpected observation is that trigonometry, not algebra, yields the highest Chi-squared value among the comparisons involving single high school courses. The second is that trigonometry does not appear again until the four- course_combination, in contrast to the orderly growth of the pattern in the study of graduation from college (see page 57). These observations will be discussed later in this section. It is much easier to appreciate the pattern built up by considering the highest tetrachoric correlation in each 54 TABLE IV CHI-SQUARED VALUES AND TEI'RACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN FIRST TERM ALGEBRA A W Courses Chi-squared rt T 51.82 0.45 0 27.60 0.42 A (4/) 25.09 0. 59 A 25.07 0.46 P 12.46 0.54 G,P 58.27 0.46 A,T 54.90 0.44 0,1 54.44 0.44 1,0 51.45 0.45 A,P 51.44 0.46 T,P 50.48 0.41 0,0 25.87 0.57 T,0 21.59 0.56 A,C 19.27 0.56 T,B 18.44 0.55 G,B 10.75 0.26 G,P 8.81 0.24 A,B 7.21 0.21 A,G,P 41.48 0.48 A8631. 59-17 0.48 AaTaP 55076 0.44 G,T,P 52.06 0.45 A,G,C 28.21 0.40 0,0,? 27.48 0.59 A,T,C 26.08 0.59 A,G,P 25.96 0.58 G,T,C 25.76 0.58 T.G.P 21.79 0.58 A.T.B 17-96 0.55 G,B,P 17.50 0.53 G.T.B 16.14 0.55 T,B,P 14.62 0.50 A,B,P 12.22 0.27 A,G,B 12.08 0.28 G,B,C 11.68 0.28 T,B,C 11.51 0.28 A,B,C 9.14 0.24 table continued on next page TABLE IV (continued) 55 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN FIRST TERM ALGEBRA Courses Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q QQQQQQ Q Q Q Q Q Q Q Q Q Q QQQQQ Q Q Q WHmHQFBmPBQQHD-JGQQ Q owowwmomawooaoe Q 'UO’UOO’O'U’UW’U‘U'UO'U’U 80>>>Q0>>>§3>>>> Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q c:>->->->->- ~3~acaEacn¢a wWHO-JWH (scannincoca Q Q Q A,G,T,B Chi-squared rt 58.48 0.47 52.65 0.44 50.07 0.45 25.67 0.59 25-97 0-58 19.15 0.55 18.48 0.56 16.11 0.55 15.57 0.51 15.86 0.51 15.46 0.50 15.42 0.50 11.55 0.27 11.58 0.28 10.05 0. 27 ,P 29.95 0.45 ,P 18.48 0.55 ,P 16.08 0.54 ,0 14.01 0.52 ,P 11.86 0.29 ,P 10.47 0.28 ,C,P 15.02 0.51 'Ucacnuacaas - Three or more semesters of algebra of solid geometry of trigmometry of biology of chemistry - One - One - Two - Two - Two 01' more 01' MOI‘O or more or more 01‘ more semesters semesters semesters semesters semesters of physics 56 group of course combinations. This is shown in the following extract from Table IV. Courses Chi-squared rt A 25.07 0.46 1,? 51.44 0.46 or G,P 58027 Oeh6 A,G,P 41.48 0.48 or A,G,T 59017 0048 A,G,T,P 58.48 0.47 A,G,T,C,P 29.95 0.45 A,G,T,B,C,P 15.02 0.51 where the symbols are again the same as those used in Table IV. This pattern, built up without undue rearranging of the data in Table IV, gives one a more reasonable picture of the apparent relative importance of the high school subjects under consideration. As courses appear in the growing pat- tern, it is easy to visualize the opportunities that these courses present for increasing skill in the various algebraic manipulations. Nor is it surprising that biology is the last to appear. In both of these extractions from Table IV it is the combination of algebra, solid geometry, and physics that yielded both the highest Chi-squared value and highest tetra- choric correlation coefficient of all the comparisons in this series. The high school credit represented by this combina- tion would therefore provide the best predictor of a good grade in first term algebra at this college, although the 57 combination of algebra, solid geometry, and trigonometry is practically as good. Again, as in the case of graduation from college, it is quite unreasonable to assume that the student suffers by taking more than the three courses noted in the previous paragraph. The reader is referred back to pages forty and forty-one for a discussion of the possible reason for the decreasing in Chi-squared and rt observed with combinations of more than three high school courses. It must be restated that this statistical handicap impairs the value of the data when it is used to justify a particular combination of high school courses as a college entrance requirement. Perhaps the best generalization concerning this series of comparisons arises from simply counting the number of the comparisons that exhibit significant Chi-squared values. It seems apparent that the capable student can hardly avoid be- coming proficient in the various aspects of elementary alge- bra. This would help explain the existence of the several patterns, shown earlier in this section, derived from the data obtained in the first series. It was expected that somewhat higher correlation coef- ficients would again be obtained with the second and third series of comparisons, in which the two non-credit courses 9.9 and _A_9_Q were considered. This was observed in the part of the study dealing with graduation from college, discussed 58 in Chapter Five. However, the correlations obtained in these two series were slightly lower than the corresponding ones of the first series, and after making sample computations to de- termine that this was very likely to occur in all comparisons where the criterion of college success was a good grade in a particular course, the second and third series were discon- tinued. There were forty-eight comparisons in the second series, of which forty-three yielded Chi-squared values high- er than 6.655, the critical value for the 0.01 significance level. These are shown in Table V on pages fifty-nine and sixty, with the observed cell frequencies shown in Table Va in the appendix. In this series no student who took 59 was included in any comparison which included solid geometry, and no student who took 599 was included in any comparison that included algebra. Thus the number of students involved in the comparisons in this series varied depending upon the high school subjects included in each comparison. This series contains only comparisons which include algebra or solid geometry or both. The third series consists of the same forty-eight com- parisons as the second series, with an additional six for a more detailed examination of the effects.of ég_andrégg. In the first forty-eight comparisons, students who took1§Q*were treated as if they had credit for high school solid geometry, 59 TABLE v CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATICNS OF HIGH SCHWL COURSES COMPARED WITH GOOD GRADES IN FIRST TERMLALGEBRA * Courses Chi-squared rt 0 22.46 0.40 A 8.66 0.57 G,P 52.19 0.44 G,T 29.76 0.45 A,T 22.26 0.59 0,0 19.50 0.55 A,G 18.95 0.40 A,P 14.86 0.57 0,8 8.29 0.24 0,1,? 28.87 0.42 A,G,T 27.95 0.45 A,G,P 26.64 0.44 6,0,? 25.45 0.58 A,T,P 22.66 0.40 G.T.C 20.40 0.55 ACTJO 16002 Os 5“ A,G,C 15.28 0.55 G,B,P 14.64 0.51 G,T,B 15.86 0.51 A,T,B 10.48 0.29 A,G,P 9.61 0.29 0,8,0 9.57 0.25 A,G,B 6.95 0.25 A,G,T,P 28.64 0.44 G,T,C,P 20071 0'57 A,G,T,C 20007 0.58 A,G,C,P 19016 0057 A,G,B,C 17.50 0.25 A,T,C,P 16.66 0.55 3.3.0.? 12-77 0-29 G,T,B,P 11.98 0.29 A,G,T,B 11.79 0.52 1.0.8.? 10.68 0.28 A,T,B,P 9.77 0.28 G,T,B,C 9051 0026 A,T,B,G 7.35 0.24 table continued on ggxt page 60 TABLE V (continued) CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN FIRST TERM ALGEBRA * Courses Chi-squared rt A,G,T,C 21.65 0.59 A,G,T,B 10.82 0.29 AsbaBsC 9012 0026 G’T'B,C 8077 00% A'GDTIB 8057 0.26 A,T,B,C 6.85 0.25 A,G,T,B,C,P 8.07 0026 - One - One - Two - Two - Two ’UCDCDFBCDF' or or or or or more more more more more semesters semesters semesters semesters semesters Three or more semesters of algebra of solid geometry of trigonometry of biolOgy of chemistry of physics * Whenever A appears in a course combination, students with credit in A00 were not counted. Whenever G appears in a course combination, students with credit in 59 were not counted. 61 and students who tookiggg were treated as if they had credit for three or more semesters of high school algebra. In the additional six comparisons either 119 or A29 was ignored in several comparisons involving both algebra and solid geom- etry. These six are clearly indicated in the table. 0f the first forty-eight, there were again forty-three comparisons that yielded Chi—squared values higher than 6.655, although not all the same as in the second series. The forty-nine comparisons in the third series are shown in Table VI on pages sixty-two and sixty-three, with the observed cell fre- quencies given in Table VIa in the appendix. Every compar- ison in the third series involved the 459 students who took the first term algebra course. The results of the second and third series are very similar, and the values are somewhat lower than those ob- tained in the first series. As might be expected, the six additional comparisons of the third series yielded values intermediate between those of the first series and the others of the third series. It is rather surprising that controlling A_Q and A_(_)_C_)_ does not lead to higher values of Chi-squared and the cor- relation coefficient. No very profound reason, statistical or educational, can be given to explain this. Apparently these two courses are not very effective as preparation for the first term of algebra at this college. TABLE VI 62 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN FIRST TERM ALGEBRA ‘ Courses ’Q>Q>>?’QQ CD 9 QQ QQQQQQ- a. QQQQQQ-sa to 60-) Q Q Q wmoweaafioeaoeoou OW’U’UOOIUO'OO'U'U’Ut-ib Q Q Q Q G>>Q>>G>GQ>>Q>>P Q Q Q >> Q - Q- 062 QQ Fit-i QQ A0801's AoGoT: G,T,C. A,G,O AoTooo "O’U'UO'U table continued on next page '0'!) Chi-squared 19.51 55.07 50-96 50.20 28.05 28.01 18.91 16.44 16.17 7.97 6.82 59.21 54.58 59-72 50.90 50.60 29.22 25.14 21.48 21.29 16.58 16.20 16.10 15.95 11.55 8.28 8.10 56.78 55.21’ 52.06 25.55 22.57 20.85 20.49 ’t 0.56 0.45 0.45 0.42 0.41 0.41 0.55 0.52 0.55 0.25 0.25 0.47 0.44 0.44 0.45 0.45 0.41 0.58 0.55 0.55 0.54 0-55 0.52 0.29 0.28 0.25 0.25 0.47 0.44 0.45 0.58 0.57 0.55 0.55 TABLE VI (continued) 5) CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN FIRST TERM ALGEBRA ‘ Courses A,G,T,B A,G,B,P A,T,B,P G,T,B,P G,B,C,P G,T,B,C A,T,B,C A,G,B,C A G,T,C A G,T,B A G,T,B A 6,8,0 0 T,B,C A T,B,C A Chi-squared 17.02 14.58 14.56 15.86 12.58 11.91 11.45 0.45 25.22 14.44 12.50 12.02 10.47 10.00 10.96 rt 0.54 0.50 0.51 0.51 0.28 0.29 0.28 0.24 0.58 0.52 0.50 0.28 0.28 0.27 0028 ‘ 'UOUHG? A One or Two or - Two or - Two or includes One or 30 1'0 IO 1'. mar O DOTO nor 0 those students with credit in.A00; G in- Three or more semesters of algebra semesters semesters semesters semesters semesters of solid geometry of trigonometry of biology of chemistry of physics eludes those students with credit in A2. 0' does not include those students with credit in 29.“ A' does not include those students with credit in A00; 64 Again, as in the study of graduation from college, a fourth, short series of comparisons was made to determine the apparent value of .A_Q and A993 (See page forty-nine for the description of these comparisons.) In this case, of course, the criterion of college success was the achievement of a good grade in the first term algebra course. Only one of the comparisons yielded a Chi-squared value higher than that of the 0.01 significance level. That compared high school cre- dit for three or more semesters of algebra, but no A29, with credit in 599,- in place of the usual dichotomy of high school credit. A Chi-squared value of 22.94 and a tetra- choric correlation coefficient of 0.45 were obtained, indi- cating again that three or more semesters of high school algebra are preferable to £99. In summary, it seems that there are more than a few patterns of high school course combinations that correlate fairly well with receiving a good grade in the first term algebra course at the Michigan College of Mining and Tech- nology. The most important consideration seems to be that the student take, in addition to three or more semesters of high school algebra, several more high school courses in which algebra is used. The best predictor of a good grade in first term al- gebra, with rt equal to 0.48, is credit in a combination of high school algebra, solid geometry, and either physics or 65 trigonometry. There were, however, a number of other combi- nations which yielded correlations only slightly less. SECOND TERM ALGEBRA Only one main series of comparisons was made in the study of second term college algebra. In this series there was no attempt to control the effect of .A_O_ and .A_O_Q inasnuch as lower values of Chi-squared and the correlation coeffi- cient were obtained when this was done. Thirty-six of the regular sixtybthree comparisons yielded Chi-squared values above that of the 0.01 level of significance. In addition, the four-or-more semesters of algebra dichotomy of the previous section was investigated in this part of the study (see page fifty-two). These thirty-seven comparisons are shown in Table VII on pages sixty-six and sixty-seven. The observed cell frequencies for these comparisons are shown in Table VIIa in the appendix. A cursory inspection of Table VII leads to two obser- vations. There are not nearly as many significant compari- sons, and the Chi-squared and correlation values are quite a bit smaller, compared to those involving first term algebra shown in Table IV. These observations are not entirely unexpected. It is reasonable that the effect of the high school background on a specific college course should dimtnish as related college 66 TABLE VII CHI-SQUARED VALUES AND TETRACHDRIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN SEOND TERM ALGEBRA Courses Chiosquared rt 1 12.58 0.29 A 11.49 0.54 0 8.84 0. 27 A (4/) 6.66 0.20 11,? 18.28 0. 54 6’11. 15087 0. )5 G,P 15.48 0.55 1,? 14.57 0.51 A,P 14.55 0.54 A,G 10.95 0.29 T,C 8.06 0.25 0,0 7.67 0.25 , , 20.41 0.56 . . 19-90 0-56 , , 18.54 0.55 18.10 0.54 12.72 0.29 11.80 0.28 10.41 0.26 QQ?>>o-3.Q>QOP>F mfleOficfit-iOHGGv-J mummomoommmew Q Q s 0 9'7? 0‘5 , , 9.08 0.25 . . 8049 0025 , , 7.86 0.25 , , 7.19 0.22 , , 6.77 0.21 table continued on next page TABLE VII (continued) 67 CHI—SQUAR- VALUES AND IETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES 1N SWOND TERM ALGEBRA moon-“71> — One One - Two - Two - Two Courses Chi-squared rt A,G,T,P 22.27 0.58 A,G,C,P 14.50 0.51 A,T,C,P 15.75 0.50 G,T,C,P 15.15 0.29 A,G,T,0 12.91 0.29 A,G,T,B 9.89 0.26 A,G,B,P 8.85 0.24 A,T,B,P 7.54 0.24 G,T,B,P 6.68 , 0.22 A,G,T,C,P 15.81 0.5) A,G,T,B,P 9.54 0.25 A,G,B,C,P 7.56 0.22 Three or more semesters of algebra or more semesters of solid geometry or more semesters of trigonometry or more semesters of bioIOgy or more semesters of chemistry or more semesters of physics i131 68 courses form a part of the more immediate background. In this case, one would engot that the first term algebra would become a dominant factor in determining a student's success in second term algebra. Furthermore, the results of the study by Smith, cited on page eight of the introductory chap- ter, indicated that records lose their prognostic value as time elapses. An analysis of the data shown in Table VII, in terms of the highest values in each group of high school course combinations, indicates the same sort of confusion as observ- ed in the study of first term algebra, with only one differ- ence beyond those already mentioned. High school trigonom- etry has replaced solid geometry in the apparent order of importance of these subjects. Following are extracts from Table VII, indicating the highest values in each group of course combinations: Courses Chi-squared rt T 12.58 . 0.29 A,T 18.28 0.54 A'T’P 20041 0056 A,G,T,P 22.27 0.58 A,G,T,C,P 15.81 0.55 A 11.49 0.54 11.? 18.28 0.54 or A,P 14s§5 Osih A,T,P 20.41 0.56 or A,G,T 19.90 0.56 A,G,T,P 22.27 0.58 A,G,T,C,P 15.81 0.55 The first part above ranks the course combinations by the 69 highest Chi-squared values; the second by the highest corre— lation coefficients. It will be observed that the best predictor of a good grade in second term college algebra, with rt equal to 0.58, is the high school credit represented by the combination of algebra, solid geometry, trigonometry, and physics. The short series of comparisons investigating the particular effect of Q and A__9_0_ again yielded only one with a value of Chi-squared above that of the 0.01 significance level. That was again the comparison of high school credit for three or more semesters of algebra, but no 599, with cre- dit in.AQQ,- in place of the usual dichotomy of high school credit. A Chi—squared value of 25.65 and a tetrachoric cor- relation of 0.55 were obtained, indicating that apparently AQQ.is an even poorer substitute for high school_algebra in the case of second term algebra than in the case of first term.algebra. TRIGONOMETRY Again only the first of the main series of comparisons was made in studying the relationship of high school credit and, this time, a good grade in college trigonometry. There were twenty-seven, out of the sixty-three, com- parisons that yielded Chi-squared values above that of the 0.01 significance level. The comparison involving four or 70 more semesters of high school algebra did not yield a value high enough for inclusion in the table. These twenty—seven comparisons are shown in Table VIII on page seventy-one, with the observed cell frequencies shown in Table VIIIa in the ap- pendix. Following is the usual extract from the table, showing the course combinations with the highest Chi-squared values in each group of combinations: Courses Chi-squared rt G 17.5} 0.55 G,P 20.17 0.56 A,G,P 22.12 0.58 A,G,T,P 18.18 0.55 A,G,T,C,P 11.76 0.28 Inasmuch as trigonometry and second term algebra are both normally taken during the second term.af the freshman year, it is not surprising that their Chi-squared and cor- relation values are quits comparable in magnitude. The surprising observation is that high school trig- onometry is apparently not important enough to appear in the t0p~ranking comparisons until the four-course combination. This is quite strange in view of the place occupied by trig- onometry in the study of the two college algebra courses. Perhaps the most tenable explanation is that proficiency in trigonometry depends largely upon the level of high school achievement as evidenced by credit in solid geometry and physics. This idea was discussed in some detail on page TABLE VIII 71 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN TRIGONOMETRY AND ANALYTIC GEOMETRY Trigonometry Analytic Geometry A G T B C P - Three or more semesters of algebra semesters of solid geometry semesters of trigonometry One One Two Two Two Courses b’raca QQQ O’OO'Ot-BHQ'U H>Q§>Q>Q Q Q Q Q H>Q>P>Q>> OHHQPHHQQ 'OOOO'O’U'UD-B’O QQQ Q QQQ Q Q Q or more 01' more 01' more or more or more P P C 12.26 P P B 17.55 11.91 8.70 20.17 19.12 16s95 14.75 10.92 10.67 9-77 8.74 22.12 19.68 15.47 15.52 12°55 11.28 10.96 10.55 7072 18.18 14.07 10.54 9.25 7-66 Chi-squared rt 0.55 0.28 0.51 0.56 0-57 0.54 0.52 0.27 0.27 0.28 0.25 semesters of biology semesters of chemdstry semesters of physics 72 forty-three, in connection with graduation from college. The best Predictor of’a good grade in college trigo- nometry is high school credit in the combination of algebra, solid geometry, and physics - with rt equal to 0.58. The fourth, short series involving _A_g and 5953 again yielded a significant value of Chi-squared only in the case of the comparison concerning AQQ. This time the dichotomy gave a Chi-squared value of 17.87 and a tetrachoric correla- tion coefficient of 0.44, indicating that AQQ_is also not a very good substitute for high school algebra as preparation for college trigonometry. ANALYTIC GEQMETRY The size of the study population was considerably re- Aduced in that part of the study dealing with success in the college analytic geometry course. Apparently, students begin to drop out of this college in sharply increasing numbers at the end of the second term.of the freshman year; also students majoring in forestry are not required to take the course in analytic geometry. This resulted in a decrease in the study population from over four hundred for the second term courses to three.hundred thirty-two for this course. The usual sixty-four comparisons of the regular series, including the two involving only high school algebra, yielded only one with a Chi-squared value above that of the 0.01 level 75 of significance. This is shown at the bottom of Table VIII on page seventy-one, with the observed cell frequencies shown similarly in Table VIIIa in the appendix. Analytic geometry differs markedly in one respect from the other mathematics courses discussed in this chapter in that no comparable course is commonly offered in high school, in contrast to high school courses in advanced algebra and trigonometry. Thus, in this case, the high school courses could only serve as lower level preparation for this course. This, and the greater time lapse between high school and taking analytic geometry, probably account for the lack of comparisons yielding Chi-squared values above the chosen level of significance. In view of the low correlation coefficient obtained in the one comparison, 0.25, it is doubtful that it is use- ful for predictive purposes. Again the short series yielded a significant value of Chi-squared only in the case of the comparison concerning 993. This time the Chi-squared value was 7.28 and the tetrachoric correlation coefficient was 0.26, indicating a very weak pre- ference for high school algebra over A00. SUMMARY The part of the investigation concerned with the four elementary college mathematics courses yielded a number of 74 unanticipated, as well as anticipated, results. In view of Smith's study of previous records, sited on page eight, it is reasonable to find the number of com» parisons yielding Chi-squared values above that of the 0.01 significance level decreasing with the later college courses. After observing the similar effect in the part of the study concerned with graduation from college, it was not sur- prising to find the maximum values of Chi-squared and of the correlation coefficient occurring with the three- or four- course combinations of high school credit. As was suggested in the preceding chapter, statistical reasons make it un- likely for the maximum values to occur with combinations of more high school courses. Consequently, it is unwise to use the statistical results of this study to justify the exclu- sion, particularly, of any course from among the entrance requirements of the college. Following is a table of the best course combinations to use for predictive purposes for the four college courses in this part of the study: College Course H.8. Courses Chi-squared rt First Term Algebra A,G,P 41.48 0.48 or A,G,T 59.17 0.48 Second Term Algebra A,G,T,P 22. 27 0.58 Trigonometry A,G,P 22.12 0.58 Analytic Geometry 1,03,?" 7.04 0.25 ' The only significant comparison obtained The first unexpected observation in this part of the 75 study was that controlling the two non-credit college courses, Ag,and:AQQ, did not lead to higher values of Chi-squared and of the correlation coefficient than when the two courses were ignored. Apparently, the value of these two courses can be discounted as far as preparation for'the four college courses is concerned. Another surprising observation was the somewhat con- fused pattern formed by the top-ranking comparisons in each group of high school course combinations. One might expect that credit in high school algebra would be the best single predictor of success in college algebra; instead, high school trigonometry yielded a higher Chi-squared value for both terms of college algebra. For college trigonometry, credit in high school solid geometry turned out to be the best single-course predictor. This phenomenon might be at least partially explained in terms of the proposed hierarchy of high school courses discussed in Chapter Five. In an attempt to determine the relative merits of 29. and high school solid geometry and of A99 and three or more semesters of high school algebra, the only comparison yield- ing statistically significant results was in each case that one involving A29. It was concluded that A__0_(_), the non-credit college course in algebra, was not a particularly effective substitute for three or more semesters othigh school algebra in any of the four college mathematics courses. I‘ll'l‘llll’IerI V’.i.}isl[ CHAPTER VII PERFORMANCE IN FIRST YEAR CHEMISTRY AS RELA'ED TO HIGH SCHOOL PEEPABATICN During the interval of time covered by this study all students at the Michigan College of Mining and Technology were required to take the same three-term course in elemen- tary chemistry, described on pages twenty and twenty-one. At that time a student did not receive a grade for the course until he had completed the three terms of work; at which time he received one grade. Inasmch as there are no college pre— requisites, this courss in chemistry is normally taken in the freshman year. The criterion of college success was again the achieve- ment of a good grade (A, B, or C) in the college course. As before, that part of the study population that completed the course was divided into two groups: those who received a good grade the first time they took the course and those who re- ceived a poor grade (D or F) the first time they took the course. There were 579 students involved in this part of the investigation. The difference between this number and the 447 involved in the study of graduation from college is en- tirely due to dismissals before completion of the course in Chem 8 try s 77 The high school data was treated in the same way as described in the preceding chapters. The 2 x 2 contingency table was, as before, used to ob- tain a Chi-squared value and, if that was significant at the 0.01 level, the correlation coefficient was calculated. Only one main series of comparisons was made, that series in which A_0_ and A510, are ignored. There were forty- five comparisons, out of the total of sixty-three, that gave significant Chi-squared values. These are shown in Table IX on pages seventy-eight and seventy-nine, with the observed cell frequencies given in Table IXa in the appendix. The usual analysis of the statistical results, in terms of the top-ranking values is each group of high school course combinations yields again some surprising observations, as well as some to be expected. Following is the extract from Table II indicating these comparisons: Courses Chi-squared rt P 28.10 0.57 A,P 55.79 0.55 A,G,P 42.54 0.54 A,G,C,P 58.29 0.52 A,G,T,C,P 27.58 0.47 A,G,T,B,C,P 11.59 0.55 Once again the pattern shows an increase to a maximum then a decrease in the values of Chi-squared, for probably the same statistical reason that was discussed in Chapter Five. It is therefore unwise to use these statistics alone 78 TABLE 1x CHI-SQUARED VALUES AND TETRACHDRIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN CHEMISTRY Courses Chiesquared rt ? 28. 10 0.57 0 10.91 0.56 A 9.75 0.56 0 7.90 0.28 A,P 55.79 0.55 0.? 55.10 0.52 G,P 27.55 0.45 A,C 20.74 0.45 0,0 15.62 0.56 A,G 12.91 0.54 1,0 12.90 0.55 1,? 10.54 0.50 0,1 9.07 0.28 1,1 8.57 0.27 A,G,P 42.54 0.54 A,G,P 55.16 0.50 0,0,? 51.87 0.48 1,0,0 20.54 0.40 Tpopp 19038 0.40 A,T,C 17.54 0.57 1,1,? 15.85 0.56 G,T,P 15050 0056 A,G,T 14.45 0.54 0,1,0 15.84 0.54 8,0,? 9.45 0.28 A,B,C 8.65 0.28 1,8,0 7.55 0.26 table continued on next page TABLE IX (continued) 79 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN CHEMISTRY Courses Chi-aqua red A,G,C,P 58.29 A.T.C.P 24.76 A,G,T,P 22058 G,T,C,P 21.12 A,G,T,C 18.91 A,B,C,P 15.68 G,B,C,P 12.04 A,T,B,O 8.98 T,B,C,P 8.56 4.0.3.? 7-99 0,1,8,0 7.74 A,G,B,C 7.47 A,G,T,C 27.58 A,G,B,C 15.64 A,T,B,C 10.58 A,G,T,B 9.69 0,1,8,0 9.69 A,G,T,B,C,P 11.59 rt 0.52 0.44 0.45 0.42 0.40 0.54 0.52 0.29 0.28 0.26 0.27 0.25 0.47 0.54 0.51 0.29 0.29 0.55 "UOWD-am> - One One - 1V0 - Two - Two Three or more senesters of algebra or more semesters or more semesters or more semesters or more semesters or more semesters of solid geometry of trigonometry of biology of chemistry of physics 80 to justify the exclusion of any high school courses from among the college entrance requirements. The pattern shown on page seventyeseven is built up in an orderly fashion, in contrast to those obtained in the in- vestigation of college mathematics. Physics, the top-ranking single course, appears in all of the other top-ranking com- binations; algebra, which first appears in the two-course combination, appears in all of the remaining ones. It seems reasonable then to use this pattern in estimating the relative importance of the high school courses as far as the first year course in college chemistry is concerned. With this premise, it is surprising to note the posi- tion of high school chemistry in the pattern on page seventy- seven and in Table IX. Noting that the combination of algebra and physics has a Chi—squared.and rt only slightly greater in negnitude than has the combinatim of chemistry and physics, one would then conclude that three or more semesters of high school algebra are about equivalent to a year of high school chemistry as preparation for college chemistry. This would seem to indicate an appreciable difference in the levels of the usual high school chemistry course and the chemistry course at this college. Numerous comments by students at this college tend to reinforce that conclusion. It also appears that a year of high school biology is relatively unimportant to success in college chemistry. 81 The high correlation found between credit for a year of high school physics and success in college chemistry is possibly due to two reasons. Physics, a science usually taught with a rigorous approach, might be good preparation for dealing with the complex aspects of college chemistry. Also, credit for physics, representing perhaps the peak of achievement in high school, might indicate the ability and seriousness of the student. For predictive purposes high school credit in the com— bination of algebra, chemistry, and physics is probably the best - with rt equal to 0.54; inasmuch as that combination yielded a Chi-squared value appreciably higher than that of any other combination. Although physics alone yielded a slightly higher correlation coefficient, the difference be- tween the Chi-squared values of physics and the algebra— chemistry-physics combination makes the latter somewhat pre- ferable. The short series of comparisons involving 5p and £92 again yielded a significant Chi-squared value only in the comparison withalgebra and 2.9.9: AChi-squared value of 18.16 and a correlation coefficient of 0.48 reaffirm the conclusion that three or more semesters of high school algebra are pre— ferable to A00. CHAPTER VIII PERFORMANCE IN FIRST YEAR PHYSICS AS RELATED TO HIGH SCHOOL PREPARATION During the interval of time covered by this study all students at the Michigan College of Mining and Technology were required to take one of the two three-term.courses in elementary physics, described on pages twenty-one and twenty- two. Students received grades at the end of each of the three terms. College mathematics prerequisites resulted in physics being taken normally during the sophomore year. The criterion of college success was again the achieve- ment of a good grade (A, B, or C) in the college course. As before, that part of the study population that completed a portion of the course for which a grade was given was divided into two groups: those who received a good grade the first time they took that part of the course and those who received a poor grade (D or F) the first time they took that part of the course. The high school data was treated in the same way as described in the preceding chapters. The 2 x 2 contingency table was, as before, used to obtain a Chi-squared value and, if that was significant, a tetrachoric correlation coefficient was calculated. Inasmch as controlling _A_Q and A00 in the comparisons 85 of high school credit and college success did not lead to higher values of Chi-squared and ’t' only the first of the three main series of comparisons was made for each of the three terms of physics. In this series 52 and A_QQ were ig- nored. Also, for each term, the effect of these two non- credit college courses was investigated in the usual short series. However, no comparison of the short series yielded a significant Chi-squared value for any of the three terms of physics. The remainder of this chapter is divided into four sections, one for each of the three terms of the elementary physics sequence, and a summary. FIRST TERM PHYSICS There were 547 students in the study population who received a grade for the first term of physics, which is con- cerned with a study of mechanics and sound. The other 100 students of the original study population were all dismissed before receiving a grade in this course. There were only eleven, of the sixty-three, compari- sons that yielded Chi-squared values above that of the 0.01 significance level. These are shown in the firstpart of Table X on page eighty-four, with the observed cell frequen- cies given in Table Xa in the appendix. Following is the usual extract from the table giving TABLE X 84 CHI-SQUARED VALUES AND TETRACHORIC CORRELATION COEFFICIENTS FOR.VARIOU8 COMBINATIONS OF HIGH SCHOOL COURSES COMPARED WITH GOOD GRADES IN PHYSICS First Term Physics Second Term Physics Courses P >00? to» Q o Q'U'U'U Q Q o 0000 Q O'U'U’U Q >->- >r¢a>»>' Q (3'6 >900 nub OG'U’U Q Q Q Q Q Q Q QQ>>PQ Q Fifi-3.0.9530 O'U’UO’U'U Q Q Q Q Q Q Q Q Q >.>QQ.>?Q> 05380-553985} on.rao:oo:aouoo 'UOO'U’UO’U'U Q Q Q table continued on next page Chi-squared 19.87 18.56 15.84 12.51 6-79 16.55 L6.15 9.45 7.01 12.58 7.00 rt 0.55 0.45 0.58 0.55 0.26 0.57 0.57 0.28 0.26 0.55 0.34 0029 0.28 0051 00§2 0.27 0.27 TABLE X (continued) CHI-SQUARED VALUES‘AND TETRAOHORIO CORRELATION COEFFICIENTS FOR VARIOUS COMBINATIONS OF HIGH SCHOOL COURSES COMBARED WITH GOOD GRADES IN PHYSICS Courses Chi-squared rt A ,G,T,C 1P 10' 17 0° 55 G,T,B,C,P 9°15 0'52 A’G'B’C'P 8057 0.50 A,G,T,B,P 7.22 0.50 A’G’T'B,C 7008 0.50 AsGsTsBsCsP 7'65 0‘51 Third Term. A,P 7.69 0.51 Physics A,T,P 7.49 0.28 A,G,T,P 8.06 0.29 A,T,C,P 7.05 0.27 A,G,T,C,P 8.18 0.50 A - Three or more semesters of algebra G - One or more semesters of solid geometry T - One or more semesters of trigonometry B - Two or more semesters of b101ogy C - Two or more semesters of chemistry P - Two or more semesters of physics 86 the top-ranking course combinations in each group: Courses Chi-squared rt P 19.87 0.55 A,P 18.56 0.45 A,G,P 16.55 0.57 A,G,C,P 12.58 O.§§ The small number of significant comparisons is not at all surprising, in view of the trend observed in that part of the study dealing with college mathematics and also in view of Smith's study cited on page eight. The relatively low values of Chi-squared are also likely due, especially, to the background in college mathematics and science gained during the freshman year. Unusual is the observation that, in this series, the highest Chi-squared value and correlation coefficient are associated with credit in a single high school course, rather than with a combination of three or more courses. Perhaps this also is the result of college mathematics courses in- tervening between high school mathematics and college physics, in addition to the statistical l'dilution" effect discussed on pages forty and forty-one. It is not surprising to see the preponderance of ma- thematical courses appearing in the significant comparisons, since mathematical logic is commonly used throughout the first term of college physics. - h It is apparent that the best predictor of a good grade 87 in first term college physics is credit for one year of high school physics, with rt equal to 0.55. SECOND TERM PHYSICS There were 516 students in the study population who received a grade in the second term of physics, which is can- cerned with a study of heat and light. The decrease of 51 students, compared to the number receiving a grade in the first term of physics, is representative of the difficulty of the first two years of engineering curricula. It must be remembered that the phrase “second term" applied to this part of the year of college physics does not necessarily signify that the students took this part of the course consecutively with the first term. As stated on page twenty-two, a small fraction of the study population reversed the usual order of the second and third terms. There were twenty-six, of the sixty-three, comparisons that yielded Chi-squared values above that of the 0.01 level of significance. These are also shown in Table X on pages eighty-four and eighty-five, with the observed cell frequen- cies given in Thble Xa in the appendix. Following is the extract from the table giving the top-ranking course combinations in each group: 88 Courses Chi-squared rt P 12.65 O.#5 G,P 15.08 0.56 0.0,? 14.80 0.58 A,G,C,P 12.69 0.55 A,G,T,C,P 10.17 0.55 1.63.3.0; 7.65 0.51 There is a marked difference between the patterns of tsp-ranking course combinations in the cases of first and second terms of physics. In that of the first term, algebra appears to be more important than solid geometry and chemis- try.v The reverse is apparent in the pattern of the second term. It is believed that this is symptomatic of the appre- ciable difference between the approaches to the subject mat- ter of these two terms. In the first term a relatively high level of mathematical competence is perhaps a major require- ment for the successful completion of the work, with compre- hension of the subject matter unfortunately of secondary importance. In the second term mathematical manipulations are of secondary importance to the ability to visualize pro- cesses and to understand abstract concepts. . With this premise it is possible to eXplain the appear— ance of the high school courses in the pattern above in terms of the hierarchy of courses proposed in Chapter Five. Aside from the importance of high school physics as a specific part of the background, it is apparently most important that the 89 student have demonstrated the ability and interest to take the higher level high school courses. I Inasmuch as there is little difference between the Chi-squared values for the combination of solid geometry, chemistry, and physics - the highest - and physics alone, the higher correlation coefficient obtained in the case of physics alone - rt equal to 0.45 - makes credit for one year of high school physics the best predictor of a good grade in the second term of college physics. THIRD TERM PHYSICS There were 515 students in the study population who received a grade in the third term of physics, which is con— cerned mainly with a study of electricity and magnetism. This is almost the same number of students as the number who completed the second term of the course. There were only five, of the sixty-three, comparisons that yielded Chi-squared values above that of the 0.01 level of significance in the study of third term.physics. These are shown in Table X on page eighty-five, with the observed cell frequencies given in Table Xa in the appendix. The size of this series does not warrant the usual extraction of top-ranking combinations; consequently the reader is referred directly to Table X in the discussion of the relative importance of the various high school courses. 90 The size of this series is undoubtedly a reflection of the increased effect of thetaccumulated college background, rather than an indication of the relative simplicity of this part of the three-term sequence. On the contrary, it is generally conceded that this is the most difficult term of the three. A study of electricity and magnetism probably in- volves abstract concepts of the highest level among those en- countered in elementary physics. In common with the first term, it is often possible to substitute skill in mathematical manipulation for comprehen- sion in the third term's work.' This very possibly accounts ‘for the closer resemblance of the significant course combina- tions in this series to Shoes of the first term rather than to those of the second term. For predictive purposes, credit in algebra and physics, rt equal to 0.51, is slightly preferable, although all Chi- squared and correlation values are so uniformly low that none of them is probably worth much for this purpose. SUMMARY The patterns of the significant comparisons in the in- vestigation of elementary physics differed to some extent from those of the other college courses involved in the study. They can, however, be explained by the same influences that were postulated for the other courses. The course content III-l.lui!" xllllll‘in‘ls \Il lilti.i.t 91 effect of high school physics on performance in college phys- ics is probably more pronounced than, for example, the effect of high school chemistry on performance in college chemistry. .In the first and third terms of physics, high school courses that emphasize mathematical skills seem to be important, com- pared to those high school courses that might be regarded as indicating the highest level of achievement. In the case of second term physics, the level of achievement in high school seems to be the more important. As far as the use of high school credit in predicting performance in college physics is concerned, credit for one year of high school physics seems to be the best criterion for the first and second terms of college physics. There is no very good criterion, of those studied, for the third term. CHAPTER IX SUMRY AND CONCLUS IONS This study has established the existence of a large number of statistical relationships between credit in certain high school mathematics and science courses and various as- pects of success at the Michigan College of Mining and Tech— nology. These relationships are in the form of Chi-squared values and tetrachoric correlation coefficients, obtained from.2 x 2 contingency tables made up by dichotomizing the high school credit and the college achievement of between 447 and 515 students. More than 800 statistical comparisons were made, with almost 590 of these yielding a Chi-squared value above that of the 0.01 significance level. If a comparison showed a significant relationship, according to the Chi-squared cri- terion, a tetrachoric correlation coefficient was computed in order to determine the degree of association. The statis- tical results are shown in Tables I through I, with the con- tingency tables? observed cell frequencies for these compari- sons given in Tables Ia through Xa in the appendix. After the existence of these significant relationships had been established, it seemed desirable to suggest reasons to account for the apparent importance of particular high school courses, credit for which correlated with success in 95 one of the aspects of college work under consideration. Three principal reasons have been proposed to explain the existence of the significant relationships. The first, and most obvious, is tint familiarity with the course content of a particular high school course is required, or at least advantageous, for success in a college course that is con- cerned with the same area of study. For example, success in college physics might depend in part upon a study of the same subject matter at the high school level. Another reason is that a particular high school course may give the student an opportunity to improve certain skills that were learned in an earlier course and useful in some college course. An example of this might be the various a1- gebraic procedures learned perhaps in the first high school algebra courses, with skill in using them improved in high school trigonometry and physics and essential to success in college chemistry. In this example, credit in high school trigonometry and physics would correlate with success in col- lege chemistry. The third reason is that success in a college course may depend, at least in part, upon a level of ability and in- terest demonstrated by credit in certain high school courses. It is proposed, in this regard, that a hierarchy of courses exists in high school - with algebra and biolOgy at a lower level, trigonometry and chemdstry at an intermediate level, lllll Illlllillllll 1AA..J. 94 and solid geometry and physics at the highest level. Thus credit in high school solid geometry and physics might corre- late highly with success in college analytic geometry, in spite of relatively little similarity in the subject matter of these three courses. There are numerous reasons why a significant statis- tical relationship between credit in a high school course or courses and college success is not obtained. Perhaps the most obvious of these is that no aspect of the high school course is a factor in achieving a particular kind of college success. An example might be the comparison of credit tn high school Spanish with success in college chemistry. It would be surprising if a significant relationship were ob- tained, in view of the complete lack of similarity between the two courses. Had a greater variety of high school or col- lege courses been considered in this study, that reason might have been quite important. Within the limitations of the study, only biology seems to be but weakly related to college success. I The fact that the student may not retain the skills or subject matter implied by the possession of credit in a particular high school course can certainly mask a relation- ship that might otherwise be demonstrated. Change in the direction or degree of the student's motivation is undoubtedly a major factor. 95 There is also the question of the caliber of the high school and college courses. If any course were relatively undemanding of the students' intellectual efforts, compared to the generally accepted standard at that educational level, one would expect to find few significant relationships - whether the course was considered as high school preparation or as a criterion of college success. With many high schools contributing students to the study population, such an effect would be relatively unlikely. However, it is a distinct pos- sibility in the case of a college course taken by all of the students. The nature of both high school and college courses would have to be examined more closely before deciding that any course or set of courses is to be condemned. The four reasons mentioned above must be faced by any investigator who attempts to relate any details of high school achievement to any aspect of college success. There is, in addition, a handicap inherent in the statistical approach of this study that tends to reduce the statistical indications in a large group of the comparisons that were made. I This handicap is the result of dichotomizing the study population with regard to high school credit, no matter how many high school courses may be involved in a particular com- parison. If only one high school course is involved, the study population divides naturally into two parts: the first including only those students who possess at least a certain 96 minimum amount of credit in that subject and the second in- cluding those students who do not fall into the first group. Thus the 2 x 2 contingency table is an appropriate tool for seeking a relationship between high school credit in one course and some aspect of college success. If, on the other hand, credit in two or more high school courses is to be investigated, the study population actually consists of more than two parts. There are students who possess at least the minimum amount of credit in all of the high school courses involved in the comparison, students who have at least the minimum credit in all but one of the high school courses, students who have at least the minimum credit in all but two of the high school courses, and so forth. If the 2 x 2 contingency table is also used with these multi-group study populations, all but the first group of those mentioned in the previous paragraph would be combined into one, the group containing those who did not possess at least the minimum credit in all of the high school courses involved in that particular comparison. ’As the number of high school courses in the comparison is increased, the num- ber of groups squeezed into one classification increases seven more rapidly. If there are two high school courses in- volved, the Ihave not credit'I group actually contains three groups; if there are three high school courses involved, the "have not credit" group actually contains seven groups. .’I\I. 1’. 97 Not very much background in statistics is required to enable one to see that it becomes progressively more diffi- cult to obtain statistical indications of relationships that may actually exist, as the number of high school courses is increased in the comparisons. The reader might well ask why, in view of this dif- ficulty, some other statistical approach wasnot employed. The following line of reasoning was used: there were too few different amounts of credit involved to use the common Pear- sonian correlation coefficient, hence the advantage of the contingency table; Chi-squared values and correlation coef- ficients for different-sized contingency tables are not easy to compare, hence the use of only 2 x 2 contingency tables. This rather lengthy discussion has as its main purpose calling attention to the fact that relatively low statistical results arising from combinations of several high school courses cannot be interpreted as indicating as weak a rela- tionship as would the same results for a single course. That is, the positive aspect of the evidence should be emphasized rather than the negative aspect. This is particularly important to remember if the re- sults of this study are used to justify the modification of college entrance requirements. The following paragraphs give very brief summaries and interpretations of the statistical results. 98 Graduation from the Michigan College of Mining and Technology is best predicted by the possession of credit in solid geometry, either from high school or from the college course 59. Chi-squared is 54.72, and rt is 0.75. The order of importance of the high school courses can be explained on the basis of a hierarchy of courses, with credit in physics and solid geometry indicating the highest level of interest and ability. A good grade (A, B, or C) in first term college alge- bra is best predicted by high school credit for three or more semesters of algebra, one or more semesters of solid geometry, and a year of physics. Chi-squared is 41.48, and rt is 0.48. The most important courses seem to be those that give an op- portunity for improving mathematical skills. A good grade in second term.college algebra is best predicted by high school credit for three or more semesters of algebra, one or more semesters of solid geometry, one or more semesters of trigonometry, and a year of physics. Chi- squared is 22.27, and rt is 0.58. Here also the improvement of mathematical skill seems to be the dominant factor. A good grade in trigonometry is best predicted by high school credit for three or more semesters of algebra, one or more semesters of solid geometry, and a year of physics. Chi- squared is 22.12, and rt is 0.58. In this case the prOposed hierarchy of courses would help explain the apparent order of 99 importance of the high school courses. A good grade in analytic geometry is predicted by the same high school credit that was described for second term algebra. Chi-squared is 7.04, and r1" is 0.25. This was the only combination of high school courses that led to a Chi- squared value greater than that of the 0.01 significance level. The fact that this course is preceded by three other college mathematics courses would help account for the scar- city of significant comparisons. A good grade for the first year of chemistry at this college is best predicted by high school credit for three or more semesters of algebra, one year of chemistry, and one year of physics. Chi-squared is 42.54, and r1" is 0.54. Cre- dit for physics alone yielded a slightly higher rt but a much lower Chi—squared value. The interpretation of the statis- tics is somewhat more difficult in this case. Perhaps a com- bination of all three reasons,-mentioned earlier, best serves to explain the apparent order of importance of the high school courses. Good grades for each of the first two terms of college physics are best predicted by high school credit for one year of physics. vFor the first term, Chi-squared is 19.87, and r1" is 0.55. For the second term, Chi-squared is 12.6), and rt is 0.45. A good grade for the third term of college physics is best predicted by high school credit for three or more lOO semesters of algebra and one year of physics. Chi-squared is 7.69, and rt is 0.51. This pair of values is not appreciably better than the others obtained in the study of third term physics. Again, a combination of all three reasons gives perhaps the best explanation for the order of importance of the high school courses. Throughout the study there has been concern for the effect of taking the two non-credit college courses 52 and A29. A_(_)_ is a one term course in solid geometry for those students without high school credit in that subject. 1_A__(_)_O_ is a one term course in elementary algebra for those who show a lack of ability in algebra on an entrance examination. Taking g9 correlates significantly with college suc- cess only in the case of graduation from college, in which it is apparently equivalent to high school solid geometry and much better than no formal course inths subject. Three or more semesters of high school algebra were shown to be preferable to QQQ_in the cases of graduation and all freshman courses. No significant results were obtained concerning the value of £99 in the study of the three terms of college physics. This study is potentially of value to four groups of people: high school students, high school faculties, college admissions officers, and college faculties concerned with the specific college courses involved in the study. 101 High school students with college ambitions, especial- ly in the broad area of engineering, can see the advantages of taking as many as possible of the high school courses in- volved in this study, with the possible exception of biology. It may be surprising to some that a course, for example, in physics is apparently beneficial in a later course not neces— sarily concerned directly with physics. High school faculties, particularly advisers and those responsible for curricula, should be interested in making available - and advising students to take - those courses that seem to be related to future success in college. College admissions officers are apparently still searching for methods of predicting the graduation of pros- pective students and for bases of realistic entrance require- ments. The correlation coefficients obtained in this study are high enough to warrant consideration. The use of credit in solid geometry to predict graduation from this college, with an rt of 0.75, represents about as good a procedure as any given in the literature. It has been shown that solid geometry is more important than perhaps has been realized. A deficiency in algebra cannot be completely overcome by brief college courses such as 529. Even though trigonometry is taken early in college, it is apparently worthwhile to study the same subject in high school. .A year of high school physics is not a luxury. A year of biology does not seem to 102 be nearly as important as a year of physics or chemistry.' Those on college faculties who are concerned with the content and approach of elementary mathematics and science courses should be interested to note that high school courses do contribute to the background of college students, personal opinions to the contrary. Students with little high school credit in mathematics and science can be expected to have difficulty with the elementary college courses in the same fields. The fact that this study is not concerned with other high school courses should not be interpreted as a judgment of any other course. It is quite possible that these courses might yield as interesting information with the same statis- tical approach used in this study. APPENDIX OBSERVED CELL FREQUENCIES TABLE Ia FOR THE COMPARISONS SHOWN IN TABLE I (ENTIRE STUDY POPULATION USED IN EACH COMPARISON) 104 Courses Cell 5: Cellib Cell 3 Cell d >GJ’U Q>>Q>Q \- Q Q s s OOQ'U'U'U Q Q Q Q >>>QQ>>Q> Q HQQH.WQOOCJ "OWI-B’U’UO’U'U'U Q Q Q Q Q Q Q Q Q Q Q - Balm-CDQQ‘D QWOOWHO Q ’UO'U’U'U'U’U QQ QQ QQ ’3’ G>>G>>> Q Q Hm Q 00 8 ‘39 u to Q 'U'U QQ 260 220 257 210 241 255 217 226 194 207 189 216 191 144 151 155 150 166 186 149 142 152 151 154 156 H \N .b' 158 108 140 as 115 115 105 115 95 85 79 98 89 61 66 7o 68 78 75 61 57 55 69 59 61 52 57 17 57 20 67 56 41. 60 51 85 70 88 61 86 155 126 122 127 111 91 128 155 145 126 145 141 147 145 52 62 50 82 55 55 67 55 77 87 91 72 81 109 104 100 102 92 95 109 115 115 101 111 109 118 115 Cell g contains those who graduated and who possessed credit in all high school courses shown. Cell 3 contains those who were dismissed and who possessed credit in all high school courses shown. Cell g_contains those who graduated and who did not possess credit in all high school courses shown. Cell d contains those who were dismissed.and who did not possess Eradit in all high school courses shown. FOR THE COMPARISONS SHOWN IN ThBLE II TABLE IIa OBSERVED CELL FREQUENCIES 105 >QPQ>Q Courses 0.11 5" 0.11 g 0.11 g 0.11 g Total G QQ Q Q QQQ. c. Q WON-90mm QQ QQQ. Q re>>Q>QQG> Q Q Q Q Q Q Q Q one» Isiaracaaao: Q Q e cameo-goo Q we (1:: (3'070'0-s~0 Q Q ” >>Q’Q> s Q Q tot-J Vs Q “0'13 218 208 209 192 157 151 255 200 187 151 145 105 155 145 155 210 159 179 151 149 156 157 129 154 125 108 88 92 95 75 72 106 77 79 66 61 81 62 59 66 9O 69 70 55 58 61 53 55 54 49 5 15 5 29 64 7O 24 12 54 7o 78 29 86 67 57 49 82 55 90 65 85 75 55 78 87 58 53 27 55 71 74 50 42 67 80 85 5s 54 60 55 45 77 49 91 61 as 66 66 65 7O 56? 551 551 551 551 0 credit Cell §.contains those who graduated and who possessed in all high school courses shown. Cell 3 contains those who were dismissed and who possessed credit in all high school courses shown. Cell g_contains those who graduated and who did not possess credit in all high school courses shown. Cell‘d contains those who were dismissed and who did not possess credit in all high school courses shown. OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TABLE III (ENTIRE STUDY POPULATION USED IN EACH COMPARISON) TABLE IIIa 106 Courses 0.11 g’ 0.11 p 0.11 g 0.11 g QQ QQ QQ>9>Q Q QQ QQQQ Q QQ QQ .- a. s-JUJO'U'U‘UO’O‘U >->-c>c>>-c:>'mw> Q Q 0sla:a:n:afa Intacsrsfaoacacncn ~303m1c161m1 Q Q G>>G>> 5 him emu-300:0 >->» Q one: Q QQ QQQQQQ 'UO'U’U'U'U u 00 Q ’U'U 274 257 266 241 252 190 174 250 251 254 178 225 167 167 152 106 150 112 150 89 82 104 95 110 74 109 72 76 as 81 93 75 66 71 75 67 65 66 5 20 11 )6 25 87 105 27 46 45 99 52 110 110 92 106 53 105 116 112 114 127 120 129 38 64 40 58 40 81 88 66 75 60 96 61 98 94 82 89 77 97 104 99 95 105 105 104 Cell g.contains those who graduated and who possessed credit in all high school courses shown. Cell 2.00ntains those who were dismissed and who possessed credit in all high school courses shown. Cell 0 contains those who graduated and who did not possess credit in all high school courses shown. Cell d_contains those who were dismissed and who did not possess credit in all high school courses shown. 107 TABLE IVa OBSERVED CELL FRMUENCIES FOR THE COMPARISCNS SHOWN IN TABLE IV (459 STUDENTS USED IN EACH COMPARISON) Courses 0.11 g" 0.11 g 0.11 g 0.11 g r 199 69 81 90 G 251 94 49 65 A (4%) 159 50 121 109 1 265 126 15 55 P 261 150 19 29 0,9 218 78 62 81 1,2 195 64 85 95 G,T 177 54 105 105 4.0 228 89 52 7o A,P 247 104 55 55 T.P 188 65 92 96 0,0 205 79 75 80 T,C 174 61 106 98 1.0 255 102 47 57 2.3 156 45 144 114 0,3 158 65 122 96 c ,P 251 111 49 48 1,3 184 85 96 76 A.G.P 2T5 75 65 86 4.9.? 175 49 105 110 A.T.P 184 58 96 101 G.T.P 167 49 115 110 4.9.0 205 74 77 85 9.0.? 196 70 84 89 A.T.0 171 56 109 105 A.9.P 220 89 60 70 0.”.O 157 50 125 109 T.0.P 166 57 114 102 A.T.B 155 42 147 117 G030? 151 52 129 107 6.2.8 118 56 162 125 T.9.P 129 45 151 116 4.8.? 175 70 107 89 4.9.8 155 60 125 99 0.3.0 142 55 158 106 TsBso 121 1'02 159 117 4.3.0 165 68 117 91 table continued on next page TABLE IV. (continued) OBSERVED cam. FREQUENCIES FOR THE 00113111150113 snows IN TABLE Iv (459 STUDENTS USED IN EACH comalson) 108 Courses QQQ WHWHQHW?QQHH.C*PC« QQ Q Q Q 0s Q QQ WWI-QPBWFB Omommwomawooaoe we on QQQ Q Q Q Q Q Q Q Q QQQ QQ QQ has:s-a>>>e:cze»>’>_c:>»>->»e- rso~000~0mkmwmmc~urs Q Q Q Q Q Q Q 'U'OO'O’U'U me vs OOWCDOO I. vs 0.11 g’ 0.11 g 0.11 g 0.11 g 165 194 156 165 150 148 116 126 158 112 140 119 156 107 116 149 136 110 106 114 105 1.01.1553;D 102 51 115 86 124 117 150 152 164 154 142 168 140 161 124 175 164 151 '144 170 174 166 177 178 115 94 114 107 115 110 126 119 112 124 109 120 98 124 118 118 115 127 127 121 125 128 Q school courses shown. school courses shown. high school courses shown. Cell 5 contains those who received a good grade in first term algebra and who possessed credit in all high Cell 2 contains those who received a good grade in first term algebra and who did not possess creditin all Cell 3; contains those who did not receive a good grade in first term algebra and who possessed credit in all high Cell ¢_'l_ contains those who did not receive a good grade in first term algebra and who did not possess credit in all high school courses shown. TABLE Va OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TABLE V 109 Courses >0 0 QQ QQ Q>>5J>QQ unnac‘cnrara'o Q Q Q Q Q Q Q Q Q Q Q Q Q QmOFSHWQfiHHOQai-S WO’UUW’UOOO'O'U'DH’U Q Q Q Q Q ?F>FQQ>?Q>Q??O QQ fi'c>>->->’GIGI>'>->’>'cH> iii-38535361980908?) u:u:u:u:>aon?ac>cn19>a<578 cacarovscn'otoroca'o<5~070 Q Q Q Q Q Q QQ QQ QQ Q Q Q Q Q Q Q Q QQQQ QQQ QQQ Q Q Q 251 256 218 177 195 205 221 240 158 167 174 210 196 185 157 169 196 151 118 151 215 142 150 165 150 155 189 155 162 158 112 115 144 125 107 117 94 115 78 54 61 79 79 96 65 49 46 67 7O 55 5O 55 57 52 56 1.0 85 55 52 1.4 6 57 98 69 70 39 22 117 108 86 50 79 79 118 95 64 124 157 151 49 155 110 95 125 105 71 125 100 157 165 145 116 157 168 145 table continued on next page, 55 12 71 64 100 105 102 114 91 77 114 0.11 5‘ 0.11 g 0.11 g 0.11 g Total 424 587 424 424 537 424 582 587 424 424 582 582 424 587 424 587 582 424 424 587 587 424 582 582 424 110 filBLE vs (continued) OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TABLE V Courses 0.11 3" 0.11 g 0.11 g 0.11 g Total A,G,T,C,P 149 58 111 84 582 A.G.T.B.P 110 50 150 92 582 A .G.B.O.P 152 41 128 81 582 G.T.B.G,P 105 54 172 115 424 A.G,’1‘,B.C 105 50 155 92 582 A.T.B.0.P 115 56 149 89 587 A.G.T.B.O.P 102 29 158 95 582 * Cell g_ccntains those who received a good grade in first term algebra and who possessed credit in all high school courses shown. Cell §.contains those who did not receive a good grade in first term algebra and who possessed credit in all high school courses shown. Cell‘g contains those who received a good grade in first term.algebra and who did not possess credit in all high school courses shown. 0911‘9 contains those who did not receive a good grade in first term.algebra and who did not possess credit in all high school courses shown. OBSERVED CELL FREQUENCIES FOR THE COMPARISONS snows 1N TABLE VI (459 S'IUDENTS USED IN EACH COMPARISON) TABLE Vla 111 Courses Cell 5. Cell 3 Cell 3 Cell 3 G -Q Q Q ~62 Q QQQ >Q>G>>>>QQ Q OW'UOGQ- O-i'Uv-a Q - -Q Q ’98 Q WQOWQHFBHOf-JHQHQQ. Q owmwowwomowmwe- Q Q Q Q Q Q Q Q Q>>Q>>Q?.G¢D>>G>>> QQ QQ J Q- ~02 Q Q 'o'u Q 863.6636 0005-3313 Q '0'0'oc3*o. ”5 Q Q Q Q Q Q ’>§)>>’> on Q Q Q Q Q table continued on next page 256 178 222 196 229 250 254 209 255 162 259 176 176 177 167 220 185 158 199 171 119 154 207 154 225 161 145 165 166 166 157 150 197 165 104 55 85 68 92 95 105 88 122 69 119 50 55 54 50 84 62 51 77 6O 56 44 87 57 104 68 59 45 45 49 5O 47 76 56 44 102 58 84 51 50 45 71 25 118 41 104 104 105 115 60 95 122 81 109 161 146 75 126 55 119 155 115 114 114 125 150 85 117 ' 55 102 114 111 110 109 112 85 105 112 ThBLE VIa (continued) OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TKBLE VI (459 STUDENTS USED IN EACH COMPARISON) Courses 0611 3f 0611 g, 0611 g_ 0.11 2 1.0.1.8 118 55 162 124 1.0.8.? 155 56 127 105 1.1.8.? 127 42 155 117 G.T.B.P 112 55 168 124 0.3.0.? 140 52 140 107 0.1.8.0 108 55 172 124 1.1.8.0 119 41 161 118 1.0.8.0 144 58 156 101 1.0.1.0.? 149 46 151 115 1.0.1.8.? 111 54 169 125 1.0.1.8.0 107 54 175 125 1.0.8.0.? 159 51 141 108 0.1.8 0.? 105 54 177 125 1.1.8.0.? 114 40 166 119 1.0.1.8.O.? 102 55 178 126 ' Cell g_contains those who received a good grade in first term algebra and who possessed credit in all high school courses shown. Cell 2 contains those who did not receive a good grade in first term algsbra and who possessed credit in all high school courses shown. Cell gncontains those who received a good grade in first term algebra and who did not possess credit in all high school courses shown. Cell g_contains those who did not receive a good grade in first term algebra and who did not possess credit in all high school courses shown. TABLE VIIa OBSERVED CELL FREQUENCIES FOR 1118 001?1818018 SHOWN 1N TABLE v11 (419 STUDENTS USED IN EACH COMPARISON) 115 Courses 0.11 5" 0.11 g 0611 g 0.11 _c_1_ 1 182 82 79 76 1 246 152 15 26 0 210 106 51 52 1 (4/) 141 64 120 94 1.1 180 75 81 85 0.1 161 65 100 95 0.? 198 90 65 68 1.? 175 74 88 84 1.? 227 115 54 45 1.0 207 101 54 57 1.0 159 75 102 85 0.0 186 91 75 67 1.1.? 171 67 9O 91 1.0.1 159 60 102 98 1.0.? 195 85 66 75 0.1.? 155 58 108 100 0.0.? 179 80 82 78 1.1.0 157 67 104 91 0.1.0 145 60 118 98 1.0.? 155 67 108 91 1.0.0 185 87 78 71 1,0,? 201 100 60 58 1.1.8 122 51 159 107 0.1.8 108 44 155 114 0.8.? 158 62 125 96 table continued on next page 114 TABLE Vlla (continued) OBSERVED CELL FREQUENCIES FOR THE COIPARISONS SHOUN IN TABLE VII (419 STUDHITS USED IN EACH COMPARISCN) Courses 0.11;" 0.11 g 0.11 g 0.11 9_ 1.0.1.? 151 55 110 105 1.0.0.? 176 76 85 82 1.1.0.? 151 61 110 97 0.1.0.? 158 54 125 104 1.0.1.0 141 56 120 102 1.0.1.8 107 40 154 118 1.0.8.? 156 58 125 100 1.1.8.? 116 48 145 110 0.1.8.? 105 42 158 116 1.0.1.0.? 156 50 125 108 1.0.1.8.? 102 58 159 120 1 .G.B .0 .? 124 55 157 105 ' Cell a contains those who received a good grade in second term algebra and who possessed credit in all high school courses shown. Cell 2 contains those who did not receive a good grade in second term algebra and who possessed credit in all high school courses shown. Cell 2 contains those who received a good grade in second term algebra and who did not possess credit in all high school courses shown. ‘ Cell _c_l_ contains those who did not receive a good grade in second term algebra and who did not possess credit in all high school courses shown. 115 TABLE VIIIa OBSERVED CELL FREQUENCIES FOR 1118 COMPARISONS SHOW IN TABLE VIII (415 STUDENTS USED IN EACH CODIPARIBON INVOLVING TRICONOMETB!) (552 STUDENTS USED IN THE COMPARISON WITH ANALYTIC.GEOHETRY) 0811 5‘ 0.11 g 0.11 g 0.11 _c_1_ Courses G 225 90 48 52 T 189 75 84 69 A 255 119 18 25 G,P 210 78 65 64 1.6 221 86 52 56 G.T 169 57 104 85 A.T l85 68 88 74 1.? 178 68 95 74 0.0 197 79 76 65 A .P 254 105 59 59 T.C 166 64 107 78 A.G.P 206 74 67 68 A.G.T 166 55 107 89 0.1.? 159 55 114 89 1.13? 174 65 99 79 0.0.? 188 71 85 71 A.G.C 195 76 80 66 0.1.0 150 55 125 89 A.T.C 162 60 111 82 T.0.P 158 61 115 81 A.G.T.P 156 49 117 95 A.G,C.P 184 68 89 74 .A.G.T.C 147 50 126 92 G,T,C,P 145 50 150 92 A.T,C.P 154 57 119 85 A.G.T.B 110 57 165 105 table continued on next page 116 TABLE VIIIa (continued) OBSERVED CELL FREQUENCIES FOR THE CCHPARISONS SRONN IN TABLE VIII (415 STUDENTS USED IN EACH COMPARISON INVOLVING TRIGONOMETRY) (552 STUDENTS USED IN THE COMPARISON H1111 ANALYTIC GEOMETRY) Courses 0611 g‘ 0.11 g 0.11 g 0.11 g ' Cell a contains those who received a good grade in the college course and who possessed credit in all high school courses shown. Cell‘g contains those who did not receive a good grade in the college course and who possessed credit in all high school courses shown. Cell c contains those who received a good grade in the college course and who did not possess credit in all high school courses shown. Cell.d contains those who did not receive a good grade in the college course and who did not possess credit in all high school courses shown. 117 TABLE IXa OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TABLE IX (579 STUDENTS USED IN EACH COMPARISON) Courses 0.11 5". 0611 g 0611 _<_:_ 0611 g ? 275 68 14 22 C 271 74 18 16 1 262 69 27 21 0 255 59 56 51 1.? 258 55 51 55 0.? 252 55 57 57 0.? 226 44 65 46 1.0 246 56“ 45 54 0.0 214 46 75 44 1.0 250 54 59 56 1.0 177 55 112 55 1.? 186 40 105 50 0.1 175 57 116 55 1.1 190 45 99 47 1.0.? 257 42 52 48 A :60? 225 )9 66 51 0.0.? 209 55 80 55 1.0.0 211 42 78 48 1.0.? 172 29 117 61 1.1.0 174 51 115 59 1.1.? 185 55 106 55 0.1.? 167 50 122 60 1.0.1 171 52 118 58 0.1.0 160 29 129 61 8.0.? 177 58 112 52 1.8.0 172 57 117 55 1.8.0 122 25 167 67 table continued on next page TABLE IXa (continued) OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SHOWN IN TABLE IX (579 STUDENTS USED IN EACH COMPARISON) 118 0.11 3" 0611 g 0611 g 0611 9 Courses 1.0.0.? 206 1.1.0.? 169 1.0.1.? 165 0.1.0.? 156 1.0.1.0 158 1.8.0.? 166 0.8.0.? 146 1.1.8.0 120 1.8.0.? 119 A.G.B.P 151 0.1.8.0 109 1.0.8.0 146 1.0.1.0.? 154 1.0.8.0.? 145 A.T.B.C.P 117 1.9.1.8.0 107 0.1.8.0.? 107 1.0.1.8.0.? 105 51 25 25 25 25 51 26 21 21 51 19 50 19 24 19 17 17 15 65 120 124 155 151 125 145 169 170 158 180 145 155 146 172 182 182 184 59 65 65 67 65 . courses shown. courses shown. Cell ghcontains those who received a good grade in chemistry and who possessed credit in all high school Cell g,contains those who received a good grade in Cell 2_contains those who did not receive a good grade in chemistry and who possessed credit in all high school chemistry and who did not possess credit in all high school courses shown. school courses shown. Cell g'contains those who did not receive a good grade in chemistry and who did not possess credit in all high OBSERVED CELL FREQUENCIES TABLE Xa FOR THE COMPARISONS SRONN IN T1BLE x (547 STUDENTS USED IN EACH COMRARISON INVOLVING FIRST TERM) (516 STUDENTS USED IN EACH COM?1RISON INVOLVING SECOND TERM) (515 STUDENTS USED IN EACH COMPARISON INVOLVING THIRD TERM) 119 First Term. P Physics >QO> QQ Q s czcacaca 70'0'008 ?Q>> Q Q O’U’U'U Q Q Q ’CJOCD as OO’U'O Q Q Q Q to u O'U'UO'U'U weaken Q Q QG>>’Q u c Q Q Q Q Q Q >’>-c1c2>-#~c)>- cooafiomao memoir-311700 ca'ucaca‘0'u<5~c'c Hue Q Q Q Q QQ 252 218 210 172 151 124 1% 95 .91 1W 150 86 76 74 51. 71 6O 65 58 61 54 27 57 24 24 29 16 15 27 27 107 68 109 116 102 145 149 110 110 ontinued on next pagg Courses Cell 3' Cell 2 Cell 2 Cell g 20 ‘ T‘W‘VTF 120 TABLE Xa (continued) OBSERVED CELL FREQUENCIES FOR THE COMPARISONS SROVN IN TABLE x (547 STUDENTS USED IN EACH COMPARISON INVOLVING FIRST TERM) (516 STUDENTS USED IN EACH COMPARISON INVOLVING SECOND TERM) (515 STUDENTS USED IN EACH COMPARISQV INVOLVING THIRD TERM) Courses Cell 5" Cell 3 (Cell 2 Cell _d_ A,G,T,C,P 128 24 112 52 G.T,B.C.P 89 14 151 62 A,G,B,C.P 121 25 119 55 A,G,T,B.P 95 16 147 60 .A.G.T.B.C 89 15 151 61 A.G.T.B.C.P 87 14 155 62 Third ..A.P 214 59 24 18 Term Physics A.T.P 155 56 85 41 1.0.1,p 159 50 99 47 1.1.0.? 159 51 99 ‘6 A,G,T,C,P 127 26 111 51 * Cell g_contains those who received a good grade in physics and who possessed credit in all high school courses shown. Cell 2 contains those who did not receive a good grade in physics and who possessed credit in all high school courses shown. Cell 3 contains those who received a good grade in physics and who did not possess credit in all high school courses shown. Cell d contains those who did not receive a good grade in physics and who did not possess credit in all high School courses shown. ' ' 215-w can... 9*- I“. I‘LL—1A . Demco-293 Date Due WHITMANWWW "WIVES 31293 0106] 5755