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Bowers, Francis Andrew Imaikalani, Jr.
DISCRIMINANT ANALYSIS APPLIED TO PREDICT SUCCESS IN ADVANCED PLACEMENT MATHEMATICS: CALCULUS AB OR CALCULUS BC
The Ohio State University Ph.D. 1984
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University Microfilms International DISCRIMINANT ANALYSIS APPLIED TO PREDICT SUCCESS
IN ADVANCED PLACEMENT MATHEMATICS:
CALCULUS AB OR CALCULUS BC
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Francis Andrew Imaikalani Bowers, Jr., B.A., M.A.
it it it it it
The Ohio State University
1984
Reading Committee: Approved By
Professor F. Joe Crosswhite
Professor Jon L. Higgins
Professor Arthur L. White
Department of Science and Mathematics Education © Copyright by Francis Andrew Imaikalani Bowers, 1984 All Rights Reserved DEDICATION:
To the grandparents and mother
of our six children,
and to the indomitable six "We have met the enemy and he is us."
Pogo
"To be what we are, and to become what we are capable of becoming, is the only end in life."
Robert Louis Stevenson
"What is honored in a country will be cultivated there."
Plato ACKNOWLEDGMENTS
To the many individuals whose counsel, cooperation, and support made this study possible, I express my sincere thanks: mahalo nui loa.
Specifically, recognition is due the following:
The late Professor John W. Riner, Jr., Vice-Chairman of the Department of Mathematics, and the other dedicated members of my Dissertation Committee, Professors F. Joe Crosswhite, Jon L. Higgins, Harold C. Trimble, and Arthur L. White, have my appreciation for their guidance and professional counsel.
Dr. Harlan P. Hanson, Director of the Advanced Placement Program, The College Board, for his prompt responses to my many queries; and to Dr. Carl H. Haag and Mariette Reed at Educational Testing Service for their five-year effort to keep me supplied with national data.
Dr. Roderick F. McPhee, President of Punahou School, for my sab batical and years leave-of-absence to pursue my studies; and to Professor J. Philip Huneke, Vice-Chairman of the Department of Mathe matics, for the repeated offers of encouragement and employment in what I most enjoy— teaching mathematics.
Robert L. Steele, Alan H. Price, and Beverly Bell O'Sullivan at Punahou and Lawrence S . Braden at Iolani for their long distance support in data collection; and to Dr. Winston Healy, Jr., Punahou Academy Principal, and Charles G. Proctor, Acting Headmaster at Iolani School for permission to use the data.
Particularly, I owe more than can be expressed to my mentor and major advisor, Professor F. Joe Crosswhite, whose thoughtful profession
alism and personal interest made the return to studies enjoyable and the
absence of family at least bearable.
And finally, to my wife and children, whose encouragement and
support over five thousand miles never waivered, my love and gratitude;
and to Mom and Dad, whose weekly letters over the past five years were an inspiration, what can I add but me ke aloha pumehana.
iv VITA
20 October 1927 ...... Born - Honolulu, Territory of Hawaii
1950...... B.A. in Economics, University of Hawaii, Honolulu, Territory of Hawaii
1951-53 ...... Mathematics Teacher, Punahou School, Honolulu, Territory of Hawaii
1953-56 Teaching Associate, Department of Mathematics, University of Kansas, Lawrence, Kansas
1956...... M.A. in Mathematics, University of Kansas, Lawrence, Kansas
1956-68 ...... Mathematics Teacher and Department Chairman, Punahou School, Honolulu, Hawaii
1960...... National Science Foundation Summer Institute, Stanford University, Stanford, California
1968-70 U.S. Army, Hawaii and Republic of Vietnam
1970-79 ...... Mathematics Teacher, Punahou School, Honolulu, Hawaii
1971-72 ...... U.S. Army War College, Carlisle Barracks, Carlisle, Pennsylvania
1979-81, Summers 1982-84. Instructor, Department of Mathematics, The Ohio State university, Columbus, Ohio
FIELDS OF STUDlf
Major Field: Mathematics Education
Studies in Mathematics Education. Professor F. Joe Crosswhite
Studies in Educational Research. Professor Arthur L. White
Studies in Mathematics. Professor John W. Riner, Jr.
Studies in Statistics, professor Douglas A. Wolfe TABLE OF CONTENTS
Page EE HE CA T I O N ...... ii
EPIGRAMS...... iii
ACKNOWLEDGMENTS...... iv
VITA ...... V
LIST OF T A B L E S ...... ix
LIST OF...... FIGURES...... xiii
CHAPTER
I . INTRODUCTION TO THE S T U D Y ...... 1
Background of the P r o b l e m ...... 1 Statement of the Problem...... 10 Design and Procedures ...... 12 Delimitations of the Study...... 17 Chapter Outline ...... 22
II . REVIEW OF RELATED RESEARCH AND LITERATURE...... 23
Introduction: Bounds on Review...... 23 General and Differential Predictors of Academic Success ...... 25 S u m m a r y ...... 94
III . THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS: ITS GROWTH NATIONALLY AND ATPUNAHOU SC H O O L ...... 96
Historical Perspective...... 96 The National Growth of Advanced Placement Mathematics as a College Board Program...... 110 The Growth of Advanced Placement Mathematics at Punahou S c h o o l ...... 155
IV. DATA PRESENTATION AND EE SIGN OF THE S T U D * ...... 173
Data Presentation...... 173 Design of the S t u d y ...... 188
vi TABLE OF CONTENTS— Continued
CHAPTER Page
V. DATA REDUCTION, PREDICTION, AND ANALYSIS...... 195
Reducing the Number of Class Variables...... 195 Deriving the Prediction Equations ...... 223 The Discriminant Analysis ...... 227 Internal Validity Application ...... 234 External Validity Application ...... 235
VI . SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS...... 236
S u m m a r y ...... 237 Conclusions...... 242 Recommendations ...... 244
BIBLIOGRAPHY...... 246
APPENDIXES...... 270
A. Calculus AB and Calculus BC: Topical Course Descriptions, 1969-1984 ...... 271
B. Advanced Placement Mathematics: Course Topics, 1954-1968 ...... 275
C. Table 46: Advanced Placement Participation, 1954-1984 ...... 277
D. Table 47: Grade Distributions of Mathematics Candidates, 1954-1984 ...... 279
E . Extracts from the Group Session Report on Mathematics Curriculum Development, Advanced Placement Institute— August, 1965 ...... 283
F. Table 48: AP Examination Participation by Subject for Selected Years...... 289
G. Table 49: Advanced Placement Participation by State in 1984 ...... 291
H. Table 50: Means, Standard Deviations, and Frequencies of Study Population Subgroups ...... 293
vii TABLE OF CONTENTS— Continued
APPENDIXES Page
I. Table 51: Forty Colleges and Universities Receiving the Largest Number of Advanced Placement Examination Candidates— May 1984...... 303
J. Table 52: Punahou AP Calculus Course Frequency by Grade Distribution in ADVM and CALC by Course, 1969-1984 ...... 305
K. Table 53: Punahou AP Calculus Course Frequency by Grade Distribution in SATM and ACHL2 by Course, 1969-1984 ...... 307
viii LIST OF TABLES
Table Page
1. Number of Institutions Reporting Indicated Grade as Minimum Required for Routine Credit(C) or Qualified Credit(Cq) ...... 8
2. Number of Punahou Calculus Students by Year, Course, and Sex...... 13
3. Number of Iolani Calculus Students by Year, Course, and Sex...... 16
4. percentage of Successful Students by Course and PSAT/NMSQT Mathematics Score ...... 34
5. National Number of Calculus Candidates by Year, Course, and Sex...... 46
6. Hawaii's Ethnic Stock; 1853 to 1980...... 53
7. Predictive Validity Comparisons of Selected Procedures...... 57
8. Advanced Standing Programs by Type Institution and Y e a r ...... 66
9. Advanced Placement Participation, 1954-1960...... 114
10. College placement by Grade of Mathematics Candidates in 1959 ...... 121
11. Grade Distributions of Mathematics Candidates, 1954-1960...... 122
12. Advanced Placement Participation, 1961-1968...... 125
13. Grade Distributions of Mathematics Candidates, 1961-1968...... 128
14. Advanced Placement Participation, 1969-1984...... 144
15. Grade Distributions of Mathematics Candidates, 1969-1984...... 147-48
ix LIST OP TABLES— Continued
Table Page
16. Composite Calculus Grade Distributions, 1969-84 ...... 149
17 . Percentage Grade Distributions of College Students and AP Candidates...... 152
18. Comparison of Intellect Descriptions, National Norm Percentages, and College Board ATP Scores...... 161
19. Punahou AP Calculus Grade Distributions, 1959-19 68...... 165
20. Punahou AP Calculus Grade Distributions, 1969-1984...... 168
21. Composite Punahou AP Calculus Grade Distributions, 1969-1984 ...... 169
22. Basic Statistics for Initial Study Population Variables, 1969-1984...... 174
23. Candidate Frequency by COOP Interval and AP Scores on Each Calculus Examination, 1973-1984 ...... 176
24. Number of Punahou AP Students by Year, Course, NOR, and S e x ...... 178
25. Candidate Frequency by COOP Interval, Calculus Course, and Advanced Placement Interest (API), 1973-1984...... 179
26. Candidate Frequency by HNIQ Interval and AP Scores on Each Calculus Examination, 1969-1982 ...... 182
27. Punahou Candidate Frequency on College Board Aptitude and Achievement Tests, by Test Score Interval and Calculus Course ...... 183
28. Candidate Frequency by JGPA Interval and AP Scores on Each Calculus Examination, 1969-1984 ...... 185
29. Grade Distributions of Punahou Candidates in Four Previous Mathematics Courses, Their Average, and the Criterion Calculus by Calculus Course, 1969-1984. . . . 187
x LIST OF TABLES— Continued
Table Page
30. Number of Punahou Students by COOP Test Interval and Calculus Course (CRSE), 197 3-19 84 ...... 197
31. Calculus AB: Intercorrelations between All Criteria and Predictor Variables Used in Analysis...... 200
32. Calculus BC: intercorrelations between All Criteria and Predictor Variables Used in Analysis...... 201
33. Computer Outputs for Separate Regressions: Even and Odd Years, 1973-1984...... 203
34. Computer Outputs for Combined Regressions: All Years 1973 through 1984 ...... 204
35. Computer Outputs for Separate Regressions: Years with and Years without Calculators...... 206
36. Number of Calculus Students by COOP Test interval and Racial Extraction (NOR), 1973-1984...... 210
37. Number of Punahou Calculus Students by AP Grade, Course, and Racial Extraction (NOR), 1969-1984...... 211
38. Number of Punahou Students by Final Calculus Grade, Course, and Racial Extraction (NOR), 1969-1984...... 214
39. Candidate Frequency by AP Examination Score and Interest, by Course, 1969-1984...... 217
40. ANCOVA Computer Output on COOP Test by Four Factors with Five Covariates, 1973-1984 ...... 219
41. Calculus AB Analysis of Covariance for Criteria APEX, COOP, and CALC, 1973-1984 ...... 221
42. Calculus BC Analysis of Covariance for Criteria APEX, COOP, and CALC, 1973-1984 ...... 222
43. SC0AP Equations: Maximal and Optimal Prediction Equations for COOP by API L e v e l ...... 226
xi LIST OF TABLES— Continued
Table Page
44. Two- and Three-Group Classification Function Coefficients...... 232
45. Optimal Classification Function Coefficients...... 235
46. Advanced Placement Participation, 1954-1984 ...... 277
47. Grade Distributions of Mathematics Candidates, 1954-1984 279
48. AP Examination Participation by Subject for Selected Years...... 289
49. Advanced Placement Participation by State in 1984...... 291
50. Means, Standard Deviations, and Frequencies of Study Population Subgroups ...... 293
51. Forty Colleges and Universities Receiving the Largest Number of Advanced Placement Examination Candidates— May 1984...... 303
52. Punahou AP Calculus Course Frequency by Grade Distribution in ADVM and CALC by Course, 1969-1984 ...... 305
53. Punahou AP Calculus Course Frequency by Grade Distribution in SATM and ACHL2 by Course, 1969-1984 307
xii LIST OP FIGURES
Figure Page
1. Case Plot and Classification of Two-Group Discriminant Function ...... 231
2. Case Scatterplot and Classification Results by Three-Group Discriminant Function...... 233
xiii CHAPTER I
INTRODUCTION TO THE STUDY
Background of the Problem
In its 1975 analysis of the fundamental problems involved in evaluating mathematics achievement at the national level, the National
Advisory Committee on Mathematics Education (NACOME) suggested the need for a broader collection of measurement techniques and instruments to evaluate student achievement and program effectiveness. Specifically, the Committee charged that "most of the methods currently used for evaluating and reporting program effectiveness are not sensitive to the specific objectives of the programs and are inefficient in terms of the time and effort required."'*' It was recommended that critical attention be paid to the selection of appropriate evaluation instruments only after program and individual goals had been identified. This study treats one aspect of a program whose specific objectives and individual goals have achieved national recognition over its thirty-year history— the Advanced Placement Program in Mathematics.
National leadership in giving direction for future mathematics programs was exhibited by the Commission on Mathematics of the College
Entrance Examination Board two decades ago. Realizing that the demands
^National Advisory Committee on Mathematics Education of the Conference Board of the Mathematical Sciences, Overview and Analysis of School Mathematics, Grades K-12 (Reston, Va.: National Council of Teachers of Mathematics, 1975), p. 134. of an increasingly complex technological society were creating an urgent
"national need for mathematical manpower," the Commission recommended substantial changes in the secondary mathematics curriculum in the form 2 of a proposed new program for the upper four grades.
With the enunciation of its nine-point program, the Commission gave this nation's students, teachers, college professors, and evalu ators a specific set of goals. Its influence on mathematics educators, textbook authors, curriculum specialists, and those involved in national evaluation was far-reaching. For the first time on such a grand scale, there was an implied acquiescence to accept a major testing service into the area of curriculum influence through the standardized testing of 3 college-capable students. Undue curriculum influence did not become a major issue in the Advanced Placement Program. The Program was initiated as a joint effort of schools and colleges; and when the College Board accepted the responsibility for the program, it wisely retained the school and college representatives on the development committees which made up the course descriptions and the examinations to test mastery of 4 the courses described.
What the Commission had to say about the place of calculus in the proposed curriculum is as pertinent today as it was in 1959:
2 Commission on Mathematics, College Entrance Examination Board [CEEB], Program for College Preparatory Mathematics (New York: College Entrance Examination Board, 1959), p. 7. 3 Marilyn N. Suydam and Alan Osborne, The Status of Pre-College Science, Mathematics, and Social Science Education: 1955-1975, Volume II: Mathematics Education (Columbus: Center for Science and Mathematics Education, The Ohio State University, 1977), p. 78. 4 The origins of the Advanced Placement Program in both The Andover Study and the School and College Study of Admission with Advanced Standing are detailed in chapter III. 3 The Commission takes the position, held generally in the united States at present, that calculus is a college-level subject. A reasonable immediate goal for most high schools is a strong college- preparatory mathematics curriculum that will have students ready to begin calculus when they enter college. Such is the curriculum described in this report. At the same time, however, the Commission recommends that well-staffed schools offer their ablest students a year of college-level calculus and analytic geometry as recommended in the Advanced Placement Program.^ It is essential, though, that such a year be firmly based on a full pre-calculus program, com pleted early by some form of acceleration.
The Program is described on p. 15. An encouraging fact is the rapid increase in the number of schools offering Advanced Placement courses in mathematics. In 1956, 45 schools had 386 candidates for the Advanced Placement Examination in mathematics; in 1959, accord ing to an advance estimate, over 360 schools will have over 3,200 candidates.^
Many of the failures of students participating in the Program could have been avoided if their school decision-makers had realized
just how essential to success were the implications of several key phrases: 'well-staffed schools', 'ablest students', 'a year of college- level calculus and analytic geometry', and 'a full pre-calculus program, completed by some form of acceleration'. Their significance will be discussed in a later chapter as the import bears directly on the success of any secondary school's Advanced Placement Program in mathematics.
It was the opinion of the late Edward G. Begle, well-known mathematician and mathematics educator, that the formation of the Com mission on Mathematics composed as it was of mathematics educators,
Commission on Mathematics, Program, p. 14. The advance esti mate given for 1959 was optimistic. The actual number of candidates who took the 1959 Advanced Placement Examination in mathematics was 1,870, as reported in Daniel T. Finkbeiner, John D. Neff, and S. Irene Williams, "The 1969 Advanced Placement Examinations in Mathematics— Complete and •Unexpurgated'," The Mathematics Teacher 64 (October 1971): 499. How ever, growth has been encouraging. Recent, comparable figures are those for the May 1984 administration: 3,525 schools had 30,583 Calculus AB candidates; and 1,136 schools had 9,379 Calculus BC candidates. Advanced Placement Program of the College Board, "Table of Candidate Grade Distri bution for May 1984 Advanced Placement Examinations," Princeton, 1984. 4 university mathematicians, and classroom teachers was "probably the most important step in the improvement of the mathematics curriculum in the 0 United States."
In the last chapter of his book Critical Variables in Mathematics
Education, written just before his death in March 1978, Professor Begle chose five critical variables for research from among the many his pre vious chapters had indicated would be appropriate for further research.
Two of these five critical variables he insisted should receive the highest priority were predictive tests and acceleration. With respect to acceleration, he wrote:
We can be fairly certain that our brightest students will profit, and no one else will be harmed, if they are separated from the rest of the student body and given a special mathematics program. How ever, we are not yet sure what kind of a program to provide— acceleration or enrichment. My guess is that for the very top layer of students acceleration is preferable but that, perhaps, for the next layer down either procedure is satisfactory. In any case, since our very brightest students do constitute a priceless national asset, this question needs to be answered.7 and on predictive tests:
There will always be sane choice points in the mathematics pro gram of any student. The most prominent of these comes during the junior high school years when a decision has to be made whether the student should or should not attempt a formal course in algebra. Since an erroneous decision, in either direction, is damaging to the student, we need to improve the quality of the information we use in arriving at these decisions. One commonly used piece of information comes from tests which "predict" performance in the algebra course. There are numerous such tests available, but none of them are as
0 Edward G. Begle, "The Reform of Mathematics Education in the United States of America," Mathematics Education in the Americas, ed. Howard F. Fehr (New York: Columbia University Teachers College, 1963), p. 137, quoted in Alan Osborne and F. Joe Crosswhite, "Forces and Issues Related to Curriculum and Instruction, 7 - 12," A History of Mathematics Education in the United States and Canada, ed. Phillip S. Jones (Wash ington, D.C.; National Council Teachers of Mathematics, 1970), p. 266.
Edward G. Begle, Critical Variables in Mathematics Education (Washington, D.C.: Mathematical Association of America and the National Council Teachers of Mathematics, 1979), p. 154. discriminating as we would like. Research aimed at improving these predictive tests could have valuable results for a very large number of our students. In the same way, we need more accurate tests to be used in deciding which students should be separated from the bulk of the student body and given either enrichment materials or acceler ation.®
Participation in the Advanced Placement Program in mathematics
implicitly requires both early identification of the candidates and some
sort of acceleration in order that students complete the full four years
of the normal pre-calculus curriculum by the end of their junior year.
Knowing which critical variables are most predictive of success on, and
best discriminate between each of the two Advanced Placement examinations
in mathematics would at least identify the caliber of junior who should
be accelerated. Some future study, with a goal of deciding which junior
school students should be separated from those taking the normal curric
ulum, might find it profitable to search for predictors highly correlated
with those found to be critical in this study, but ones available earlier
in the student’s academic career.
The Advanced Placement Program (APP) is "a cooperative program of
curriculum enrichment and articulation between high schools that offer
introductory college-level courses to able students and colleges that 9 accept these courses as the practical equivalents of their own.” Since
school year 1955-56, the Program has been the national responsibility of
the College Board which provides secondary schools with descriptions of
college-level courses, administers examinations each May covering the
material outlined in the course descriptions, and supplies the students,
their schools, and colleges with test results on a standardized scale.
8 Begle, Critical Variables, p. 155. 9 College Entrance Examination Board, The College Board Today (New York; College Entrance Examination Board, 1978), p. 6. "An Advanced Placement course in mathematics consists of a full academic year of work in calculus and related topics comparable to xO courses in colleges and universities."x Published course descriptions are revised biennieally by the Committee of Examiners in Mathematics to keep pace with trends in school and college mathematics instruction. A major change in content was announced for school year 1968-69, when the
College Board offered a choice of two calculus course descriptions and two corresponding examinations, entitled Calculus AB and Calculus BC.
Calculus AB is a one-year course in elementary functions and introductory calculus intended for students who have a thorough knowl edge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. "If students are to be adequately prepared for the Calculus AB examination, well over half of that year’s course must be devoted to the topics in differential and 12 integral calculus."
Calculus BC is a one-year course, considerably more intensive and extensive than Calculus AB, in the calculus of functions of a single variable. It presumes a sound knowledge of elementary functions in addition to the prerequisites for Calculus AB, and is comparable in con tent and emphasis to a first course in calculus designed for the mathe matically and scientifically oriented freshman in many colleges and universities. Including topics in differential equations and infinite
10The College Board, Advanced Placement Course Description, Mathematics: Calculus AB, Calculus BC, May 1981 (New York: College Entrance Examination Board, 1980), p. 1.
^ S e e appendix A for topical course descriptions of the two mathematics courses: Calculus AB and Calculus BC, 1969-1984. 12 The College Board, AP Math Course Description, May 1981, p. 1. series in addition to the differential and integral calculus, "the con tent of Calculus BC is designed to qualify the student for placement and 13 credit one semester beyond that granted for Calculus AB."
The Advanced Placement mathematics examinations are each three hours long, and seek to determine how well a candidate has mastered the subject matter detailed in the corresponding course description. Tradi tionally, each examination has been weighted equally on two parts: a ninety-minute objective section and a ninety-minute essay section which requires written solutions. In the objective section were forty-five, multiple-choice questions which tested proficiency in a variety of skills and topics; the essay, or free-response, section had seven problems requiring candidates to demonstrate their ability to carry out non trivial proofs and solve problems involving a more extended chain of reasoning. Approximately 55 percent of the objective and 40 percent of the essay questions have been common to both the Calculus AB and 14 Calculus BC examinations.
The essay sections are graded in June by a representative sample of several hundred school and college teachers familiar with the Program.
Although the essay sections are made public after each examination, the machine-scored, objective sections are held secure to provide stability to the grading scale from year to year. Combining the essay and the
13 The College Board, Advanced Placement Course Description, Mathematics: Calculus AB, Calculus BC, May 1983, May 1984 (New York: College Entrance Examination Board, 1982), p. 2. 14 In May 1983 two other changes were instituted: the use of hand calculators was permitted, but not required, for the first time; and the number of essay questions was reduced to five with a concomitant reduc tion to seventy-five minutes for that section. The allotted time for the objective section was extended to 105 minutes; but its equal weighting with the essay section and the number of objective questions did not change. objective results yields a raw score which is converted to the Pro gram's five-point scale: 5— extremely well-qualified; 4— well-qualified;
3— -qualified; 2— possibly qualified; and 1— no recommendation. These grades are communicated to the student, his secondary school, and his prospective college, at least 1,176 of which have an announced policy of advanced placement or credit in force. The vast majority of colleges and universities grant advanced placement or credit only for grades greater than or equal to three on each of the two mathematics exami nations, as shown in table 1:
TABLE 1
NUMBER OP INSTITUTIONS REPORTING INDICATED GRACE AS MINIMUM REQUIRED FOR ROUTINE CREDIT(C) OR QUALIFIED CREEQ[T(Cq)a
Calculus AB Calculus BC AP Grade Routine(C) Qualified(Cq) Routine(C) Qualified!Cq)
5 21 0 13 0 4 172 41 158 34 3 616 265 582 251 2 5 25 18 29 1 0 0 0 0
Total 814 331 771 314
SOURCE : The College Board, College Placement and Credit by Exami nation (New York: College Entrance Examination Board, 1978), p. 401.
a (C) Credit offered routinely to virtually all students. (Cq) Credit granted with possible qualifications, such as use of other predictors, departmental review, exception in certain programs, etc.
Since table 1 shows that less than three percent of the colleges allow routine credit (less than nine percent allow qualified credit) for grades less than three, it is not surprising that the College Board should report to participating schools the 'percent of current year grades 3 or higher' on their Candidate Grade Roster, or that schools should consider a grade of three to be the minimum acceptable criterion for 'success'. Commenting on the grades of all AP candidates, the Col lege Board said, "Even among this able group approximately 15 to 30 per- 15 cent receive non-qualifying grades of 1 or 2." In AP mathematics, the percentage receiving non-qualifying grades has exceeded forty percent; but in the 1980s has declined to about 30 percent in Calculus AB and 25 percent in Calculus B C . Nationally in 1984, over ten thousand calculus candidates failed to achieve qualifying grades. This intensifies the need to identify and properly counsel able, college-capable students considering participation in AP mathematics; and it puts a premium on proper placement of these students into the correct calculus sequence, if their chances of 'success' on the AP examination is sufficiently great.
With reference to college-capable students and their educational programs, the National Advisory Committee on Mathematics Education ques tioned the effectiveness of 'delayed outcome data' for those students still preparing for college. The Committee granted that some measure of successful performance in college mathematics was a true indicator of secondary school accomplishment, but it felt that the information often came too late to be of real use. "What is really needed is information 16 on that student's likelihood of success prior to entering college."
If this need is taken as an objective, it would seem reasonable to focus one's attention on student input to the Advanced Placement Program where results of success or failure are reported prior to college entrance.
15 Educational Testing Service, "Report to Advanced Placement Teachers," Princeton, N.J., Fall 1982. (Mimeographed.) 16 NACOME, Overview, p. 120. Statement of the Problem
This study addressed the problem of selecting students at the end of their junior, secondary school year for participation in one of the two Advanced Placement Programs in mathematics.
It presumed that such students attend a competently staffed
school capable of providing a curriculum which enable them to complete a four-year, secondary school preparation for the calculus by the end of the junior year, and a faculty trained to teach the subject matter out
lined in the Calculus AB and Calculus BC syllibi. If a qualifying score of at least three on either of the two Advanced Placement mathematics
examinations defines 'success* in that examination, then the specific question addressed by this study was the following:
Which of several selected factors normally available from a given secondary school's record of a student at the end of his junior year are the most reliable predictors of achievement (as measured by scores on an end-of-course Cooperative Calculus test) and success on, and discrimination between, each of the two Advanced Placement examinations in mathematics?
The given study school was Punahou School in Honolulu, Hawaii.
Punahou is a large (3,750 students in grades K-12), independent, college-
preparatory school which has participated in the Advanced Placement Pro
gram since school year 1958-59. Because Punahou has a greater oriental
population than comparable mainland schools, one student characteristic
tested for statistical significance was the nominal variable NOR, an
acronym for 'non-oriental' or 'oriental'. Also tested for their signif
icance so that the subsequent statistical analysis could be controlled
for those found to be significant, were other categorical variables
which partitioned the study population into mutually exclusive and
exhaustive sub-populations: SEX, female or male; YEAR, year in which the 11 annual Advanced Placement mathematics examination was taken; CRSE,
Calculus AB or Calculus BC course and examination taken; and API, an acronym for Advanced Placement Interest, a composite variable which combined both the variable TAP, total Advanced Placement examinations taken, and the variable SCI, an indicator that these examinations included Physics C, other sciences, or no other than a calculus exam.
The utility of the study depended on the predictive strength, reliability, and general availability of the factors finally found to be the most discriminating from among those selected. Fifteen of the original seventeen selected factors were nationally available; eight through the Educational Testing Service, one a nationally known IQ test, and the other six local measures, but ones which every school has in its records— the four previous mathematics course grades, their average, and the junior grade-point-average (JGPA). The eight measures available from the Educational Testing Service included the verbal and Mathematical scores on each of three Scholastic Aptitude Tests: the junior SAT (JSAT) for seventh and eighth graders, the Preliminary SAT (PSAT) for sophomores and juniors, and the SAT for juniors and seniors; and the two Mathematics
Achievement Tests: Level I and Level II. For secondary schools contem plating an Advanced Placement Program in mathematics, a useful purpose of the study was the derivation of classification equations which pre dicted success on either the Calculus AB or Calculus BC examination.
The ultimate purpose of the study was to develop an educationally sound, practical, and efficient means of identifying those able students who would profit and find success in taking the Advanced Placement Program in mathematics. 12 Design and Procedures
The basic methodology and design of this study were those of predictive research. The statistical techniques employed included non- parametric, multinomial tests of hypotheses and analysis of variance to establish the significance or non-significance of certain class vari ables, Pearson product-moment correlation and analysis of covariance to further reduce the number of student characteristic factors, multiple regression to analyze the relationships between criterion variables and predictors and to reduce the set of predictor variables, and discrimi nant analysis to distinguish between critical subgroups in the study population and to generate the classification equations. Data were com puter programmed and machine processed on the Amdahl 470V/6-II at the
Instruction and Research Computer Center of The Ohio State University.
Research in the area of predictability of success in college has centered primarily on the examination of the validity of two traditional predictors: high school scholarship and preadmission test scores. The criterion most often used as a measure of success has been freshman first quarter, first semester, or first year grade-point-average. This study avoided the difficulty of the probable non-equivalence of freshman grade-point-averages at different colleges by using the Advanced Place ment Program's five-point scale as a criterion. The other criteria were final course grade, an internal success indicator, and the achievement criterion, which was an end-of-course objective test score on the
Educational Testing Service's Cooperative Calculus Test (Form A ) .
There were two sets of grades in both the five-point scale (APEX) and the final course grade (CALC), one each in Calculus AB and Calculus
B C . Neither of these criteria were directly comparable between the two 13 courses. Linking the two courses as their only common measure was the third criterion, scores on the Cooperative Calculus Test. It was locally administered to all students in both courses one week before the AP exam.
The data used in this study were collected by the author at his parent institution from the records of Advanced Placement students in mathematics over the past sixteen years. Punahou School granted per mission to use this data in summary form, without reference to individual students, for statistical analysis and possible inclusion as source data in this dissertation. A feasibility study was conducted on the data set for the three years indicated in table 2, which displays the number of students by year course and examination were taken, by course, by sex.
TABLE 2
NUMBER OF PUNAHOU CALCULUS STUDENTS BY YEAR, COURSE, AND SEX
Calculus AB Calculus BC Composite AB & BC Year Female Male Sum Female Male Sum Female Male Sum
1968-69 7 29 36 7 29 36 1969-70 10 31 41 10 31 41 1970-71 11 26 37 11 26 37 1971-72 12 26 38 12 26 38 1972-73 8 14 22 19 29 48 27 43 70 1973-74 12 16 28 11 13 24 23 29 52 1974-75 2 12 14 7 24 31 9 36 45 1975-76 12 6 18 3 20 23 15 26 41 1976-77 8 16 24 4 22 26 12 38 50 1977-78a 20 14 34 8 16 24 28 30 58 1978-793 14 28 42 11 24 35 25 52 77 1979-80a 10 14 24 11 27 38 21 41 62 1980-81 34 27 61 7 21 28 41 48 89 1981-82 29 19 48 10 26 36 39 45 84 1982-83 22 19 41 12 30 42 34 49 83 1983-84 42 32 74 14 31 45 56 63 119
Total 213 217 430 157 395 552 370 612 982
aYears on which the feasibility study was conducted. Of the seventeen, independent variables originally considered for analysis as possible predictors, two (educational level of parents and number of extracurricular activities) were eliminated by the feasibility study, leaving the following fifteen for consideration: final grades in the four previous mathematics courses, the average of these grades, junior grade-point-average, Henman Nelson IQ, JSAT-Verbal, JSAT-Math,
PSAT-Verbal, PSAT-Math, SAT-Verbal, SAT-Math, and Mathematics Achievement
Tests, Level I and Level II. Pearson product-moment correlation, multi ple regression, and analysis of variance and covariance were employed for the purposes of isolating critical variables and data reduction.
It was hypothesized that regardless of year, sex, and race, the best single predictor of achievement and success on each of the Advanced
Placement calculus examinations would be the latest mathematics course grade, but that the Mathematics Achievement test score and the class variable, Advanced Placement interest, would also be significant.
The ultimate problem of distinguishing between students who should be counselled to take Calculus AB, those who should take the more rigor ous Calculus BC, and those who should not take either course was sub
jected to discriminant analysis. Since advanced placement or credit is
normally granted students with qualifying scores of three or better on either of the two Advanced Placement calculus examinations, both two and three groups were formed. The two-group analysis answered the question,
"Which course should a given junior take?" The three-group analysis con sidered the additional possibility of not taking either course. The stepwise method of selecting factors for entry into the analysis was adopted using MINRESID, a method which minimizes the residual variation.
Once the centroids of the discriminating factors were determined, these 15 and the pooled within-groups covariance matrix were used to derive the classification equations, a set of two equations for the two-group anal ysis and a set of three for the three-group analysis. Each classifi cation function was the sum of the classification coefficients multi plied by the raw scores of the final predictors selected for entry, and an additional constant. Given the selected predictors for any student, the two- or three-group classification scores were computed, and the student was classified into the group with the highest score.
An internal validity check on the validity of this discriminant procedure was performed using the input data of the 830 Punahou calculus
students who took the course and examination during the period 1973-84.
By classifying each student and comparing predicted group membership with actual group membership, success in discrimination could be mea
sured by the percentage of students correctly classified. An external
validity check was performed on data solicited from another secondary
school, which had offered both calculus courses for the period 1979-84.
Iolani School is also a private, college-preparatory school in
Honolulu, Hawaii. A former day school for boys, it introduced girls
into the seventh grade in school year 1977-78. By 1981, approximately 17 1,420 boys and 220 girls were enrolled. in 1983 it completed its
transition to a coeducational institution at the secondary level by
graduating its first class with boys and girls. Iolani School granted
permission to use data obtained on their Advanced Placement mathematics
students if no reference was made to individual students. Table 3 dis
plays the number of Iolani students by year, calculus course, and sex.
17 Private Independent Schools, 34th ed. (Wallingford, Conn.: Bunting & Lyon, 1981), p. 212. 16 TABLE 3
NUMBER OF IOLANI CALCULUS STUDENTS BY YEAR, COURSE, AND SEX
Calculus AB Calculus BC Composite AB & BC Year Female Male Sum Female Male Sum Female Male Sum
1978-79 19 19 11 11 30 30 1979-80 23 23 14 14 37 37 1980-81 22 22 21 21 43 43 1981-82 20 20 22 22 42 42 1982-83 9 22 31 7 19 26 16 41 57 1983-84 20 29 49 3 15 18 23 44 67
Total 29 135 164 10 102 112 39 237 276
Punahou School has participated in the Advanced Placement Program since school year 1958-59. It offered the one available calculus course from that year through 1967-68, and then only Calculus BC until 1972-73.
For the last twelve years both courses have been offered and, although the number of candidates in mathematics has varied over the years, at present approximately one-quarter of each year's graduating class of 425 take either the Calculus AB or Calculus BC course and examination.
whether or not a given junior should be counseled to participate in the AP mathematics program, and in which of the two calculus courses he should be enrolled has often been a difficult decision. Until now, this decision has been made by the student's dean and counselor in con sultation with the student and his parents on recommendations made by the mathematics department. Similar decisions are being made each year at some 3,500 other high schools, like Iolani School for instance. The external validity check made on the Iolani data was performed to deter mine if this study had met its designed goal of assisting other schools in making those decisions. 17 Delimitations of the Study
Definitions of Terms
Throughout the study, use is made of certain terms, acronyms, and abbreviations with particular connotations. These are defined below in order to assist in clarifying their usage:
Advanced Placement Program (APP): This is the national, educa tional endeavor of The College Board (College Entrance Examination Board, CEEB) which annually provides schools with descriptions of college-level courses, administers examinations each May covering the material in the course descriptions, and supplies colleges with examination results on it's standardized five-point scale. The Educa tional Testing Service (ETS) provides its operational services.
Advanced Placement (AP): As used in the present study, Advanced Placement refers to the program or policy of any college or univer sity which allows an entering freshman to bypass at least the initial calculus course on the basis of his participation in the APP.
Calculus AB; This is the one-year, college-level course in elementary functions and introductory calculus which is validated by the APP examination having the same name. The College Board's course description is detailed in appendix A.
Calculus BC: This is the more extensive full-year, college-level course in the calculus of functions of a single variable, and also the designation of its validating examination. Appendix A contains a listing of the topics covered in addition to those required for Calculus AB, e.g., sequences and series, elementary differential equations.
APEX: An acronym for Advanced Placement examination used on all computer runs, this ordinal, criterion variable is the grade on the APP five-point scale (from 1— no recommendation to 5— extremely well- qualified) on either the Calculus AB examination (APEXAB) or Calculus BC examination (APEXBC).
CALC: An abbreviation for Calculus grade used on all computer runs, this interval, criterion variable is the numerical equivalent of the final grade in either course, Calculus AB (CALCAB) or Calculus BC (CALCBC) on a twelve-point scale (12 = A+, 11 = A, . . . , 0 = F). Punahou semester course grades are averaged to obtain final grades using this scale; a policy change in school year 1977-78 denied future assignment of the A+ grade.
COOP: An abbreviation for the Educational Testing Service's Cooperative Test in Calculus (Form A), this interval-level, criterion variable is the raw score (0-60) on this sixty-question, objective test administered one week before the APP examinations to all students in both Calculus AB (COOPAB) and Calculus BC (COOPBC). 18 Success on an APP mathematics examination: As used in this study, success indicates attainment of a qualifying grade of three, four, or five on either the Calculus AB or Calculus BC examination.
API; An acronym for Advanced Placement Interest, this composite, nominal-level, class variable combines both the ordinal-level variable TAP, an acronym for total number of APP examinations taken, and the nominal-level, class variable SCI, an indicator that these examinations included Physics C (physics requiring calculus), only other sciences (Biology, Chemistry, Physics B— calculus not required), or no APP examination other than Calculus AB or Calculus BC.
NOR: An acronym for 'non-oriental' or 'oriental', this nominal- level, class variable identifies a student's racial extraction as being at least fifty percent oriental or not.
AVMA: An acronym for average mathematics grade, this interval- level, predictor variable is the average of the final grades in the four, pre-calculus, mathematics courses on Punahou's twelve-point scale. The abbreviations for these four courses are as follows: ALGI = Algebra I; GEOM = Geometry; INTM = Intermediate Mathematics; and ADVM = Advanced Mathematics.
JGPA; An acronym for junior grade-point-average, this interval- level predictor is the grade average of a student's junior-year courses, converted to the four-point grading system.
HNIQ: An acronym for Hinman Nelson Intelligence Quotient, this interval-level, predictor variable is the score on Form B of that instrument. It was administered to all Punahou freshmen until it was discontinued after testing the class of 1982.
ACHLl and ACHL2: Abbreviations for the Level I and Level II Mathematics Achievement tests, these interval-level predictors are the College Board's standardized scores on the test named. The May ACHL2 administration is recommended to all juniors in Advanced Mathe matics being considered for APP mathematics as seniors.
J5ATV and JSATM: Abbreviations for the Verbal and Mathematical scores on the Junior Scholastic Aptitude test, administered to all Punahou seventh-graders until discontinued after testing the Class of 1977, these are interval-level predictors.
PSATV and PSATM: Abbreviations for the Verbal and Mathematical scores on the Preliminary Scholastic Aptitude test, administered in the fall to all Punahou sophomores, this inteirval-level predictor is the College Board's standardized score on the test-section named.
SATV and SATM: Abbreviations for the Verbal and Mathematical scores on the Scholastic Aptitude test required of all Punahou juniors in December, this is an interval-level, predictor, variable on the College Board's standardized scale. 19 Populations Utilized in the Study
The parent population was the set of 358,055 secondary school students who studied an Advanced Placement course in mathematics and took either the Calculus AB or Calculus BC examination during the period
1969 to 1984, inclusive. The study population consisted of 982 former
Punahou seniors over these sixteen years; it was a subset of the parent population. The validating population of 276 Iolani School students also formed a subset of the parent population for the last six years.
Various subgroups of the study population were formed in order to deter mine which student characteristic factors contributed significantly to the prediction of success on, or better discriminate between, the two mathematics examinations. The student characteristics common to such
subgroups included sex, race (NOR), course, year course and examination were taken, Advanced Placement interest (API), previous mathematics grades, and national aptitude, intelligence, and achievement test scores.
Sources of Data
The data for the study, validating, and parent poulations were
drawn from personal records, the official, permanent records of Punahou
and iolani Schools, the transcripts of the Educational Testing Service,
and published reports of The College Board. Specific published sources
of various data will be cited with first use in the study.
Limitations of the study
The most serious limitation of the study was its restriction to
the students in a single school. The need to acquire confidential data
over an extended period of time imposed this obvious limitation. No
direct generalization of the study's results to other secondary schools 20 is necessarily warranted.
The fact that the calculus courses at Punahou have been taught by different teachers over the sixteen years was not considered a serious limitation. The three teachers involved, two at a time, have rotated among the two courses. Any influence exerted by a particular teacher has been neutralized with the consolidation of annual data over time.
There was a time restriction to the years beginning in 1968-69, because this was the first year The College Board offered two course descriptions and corresponding examinations. The problem of proper stu dent placement in one of two courses was a consequence of this decision.
There was also a geographical restriction to students in Hawaii for both the study and validating populations. Except for the possible influence of a larger-than-normal oriental population, which this study addressed, this was not considered a pertinent limitation.
Both schools involved in this study are independent, college- preparatory schools. Attempts were made to obtain public school data for another external validity check; but this limitation remains. Cer tainly many public schools meet the 'competently staffed' and 'faculty trained to teach calculus' assumptions of this study; some have partici pated in the Advanced Placement Program since its inception.
The selection of predictor variables was limited to those most readily available over time; practical considerations played a role in their selection. Although nine of the original seventeen predictor variables (later reduced to fifteen) have national norms, the remaining six were local, student variables. This was not considered to be a serious limitation because any comparable school, which intends to replicate the procedures in the study, has available its own course 21 grades in mathematics, their average, and junior grade-point-average.
The desire to develop a practical procedure for discriminating between those who should, and those who should not, take either Calculus AB or
Calculus BC limited the choice of predictors for success to measures which comparable schools might find readily available.
Implications of the Study
While the restriction to the students in a single school and the use of local predictors implies that the discriminant results of the
study should only be used to classify Punahou students, those results which have their basis in nationally-normed tests have direct practical application to comparable schools which administer the same national tests. The demonstrated, external validity application to data from
Iolani Schools shows how this might be accomplished. Application of the
results for other secondary schools participating in the Advanced Place ment Program in Mathematics will be in direct proportion to the degree to which their school curriculum, students, and faculty are similar to
those at Punahou School, if provision is made for local data. If there
is considerable difference, that portion of the study desired could be
replicated using the same statistical procedures but local, predictor
variable data.
Perhaps the only major value of the study for some schools would
be better identification of the caliber of student likely to find suc
cess in the Advanced Placement Program in Mathematics. Nationally, over ten thousand candidates failed to achieve qualifying scores of three or
better in the 1984 AP calculus administration. If this study's results
serve to reduce the number who fail to qualify, the stud/ will have made
its contribution to mathematics education. Chapter Outline
The present chapter has introduced the study, defined the research problem, and discussed its significance, limitations, design and procedures for solution, and anticipated contribution.
Chapter II contains an exhaustive review of the pertinent literature on general and differential predictors of academic success with emphasis on related research and its present status.
Presented in chapter III is an historical overview of the
Advanced Placement Program in Mathematics both nationally and at Punahou
School. The perceived need for such a program, the problems associated with its initiation, and its growth to present status are covered.
In chapter IV the data on the stud} population of 982 Punahou students who took the Advanced Placement examinations in mathematics over the past sixteen years is presented. Presentation is made through a series of tables which display the tentative predictor and categorical variables. The design of the study in three phases is discussed.
Presented in chapter V are the statistical analyses of the data by non-parametric, multinomial tests of hypotheses, multiple regression, analysis of variance and covariance, and discriminant analysis. The two- and three-group classification functions obtained by discriminant analysis are applied in both internal and external validity checks of the design procedures. The derivation and display of prediction equa tions which might be useful in other schools are also presented.
Chapter VI summarizes the study and its major conclusions. Any recommendations or suggestions for further research will be found in this chapter. CHAPTER II
REVIEW OP RELATED RESEARCH AND LITERATURE
Introduction: Bounds on Review
Since the early 1950's need, knowledge, and technology have combined to yield an outpouring of research literature on the prediction of academic success. Our society's increasing complexity prompted greater demands for talented individuals with extended or specialized training. The high birth rate of the late forties and fifties heightened the competition for college in the sixties and seventies. The desire of admissions officers to make proper selections from a growing number of highly qualified candidates, as well as the demands of a growing techno logical nation to identify and render educational support to students with specialized talent, created the need. New mathematical applica tions, statistical methods, and computer technology provided the means to satisfy this interest in the prediction of academic performance.
In his 1949 review covering the previous twenty-seven years,
Garrett cited nearly two hundred predictive studies."*- Lavin pinpointed 2 some three hundred sources in the eight-year period beginning in 1953.
Fishman and Pasanella located 580 studies made during the fifties and
^"Harley F. Garrett, "A Review and Interpretation of Investiga tions of Factors Related to Scholastic Success in Colleges of Arts and Science and Teachers Colleges," Journal of Experimental Education 18 (December, 1949): 91-138. 2 David E. Lavin, The Prediction of Academic Performance (Hart ford, Conn.: Russell Sage Foundation, 1965), pp. 59-63, 111-21, 150-56. 23 24 suggested that "admission to college and selection of applicants has probably become the most intensively explored topic in educational- 3 psychological research." In the early 1970's, Aiken published three
separate reviews, totalling over two hundred citations, dealing with 4 variables associated with achievement in mathematics.
The local and specific nature of this study has limited the
choice of literature to be reviewed. To gain predictive efficiency in
a field such as mathematics, global predictors indicative of compre
hensive academic excellence yield to variables more highly correlated with the chosen criterion in mathematics. Horst described this technique
as differential predictions
. . . it is important not only that we be able to predict success in college in an over-all or general sort of way for students before they enter the university; it is also important that we be able to predict, as accurately as possible, in what specific areas they have their best or poorest chances for success. It is not enough to be able to say to a student, "You will do well if you go to college." We should also be able to tell him that if he takes courses in mathematics or engineering he will probably do well... . This is what we mean by differential prediction.5
The review includes differential predictors in mathematics, but
not all of these are available to the high school counselor during a
student's junior year when the prediction for his success in the two
3 Joshua A. Fishman and Ann K. Pasanella, "College Admission- Selection Studies," Review of Educational Research 30 (October, I960): 298. 4 Lewis R. Aiken, Jr., "Nonmtellective Variables and Mathematics Achievement: Directions for Research," Journal of School Psychology 8, no. 1 (1970): 28-36; idem, "Intellective Variables and Mathematics Achievement: Directions for Research," Journal of School Psychology 9, no. 2 (1971): 201-12; and idem, "Ability and Creativity in Mathe matics," Review of Educational Research 43 (Fall, 1973): 405-32.
Paul Horst, "The Differential Prediction of Success in Various College Course Areas," College and University 31 (Summer, 1956): 457-58. Advanced Placement (AP) calculus courses must be made. The Advanced
Placement Program's thirty years has allowed sufficient time to produce a substantial body of its own literature: significant historical items are left for Chapter III; but pertinent, predictive studies are reviewed here
General and Differential Predictors of Academic Success
General Intellective Predictors and Criteria
General success criteria
Research in predicting general academic success in college has most often used first-quarter, first-semester, or freshman-year grade- point-average (FGPA) as the criterion for success. This was the case in over 90 percent of the 580 studies reviewed by Fishman and Pasanella.
For the 263 studies which used the high school record as the sole predic- 6 tor of these criteria, they reported a correlation of "roughly .50."
This consolidated statistic obscures the findings of other inves tigators who reported higher correlations between high school and FGPA, 7 than between high school and first-quarter or first-semester GPA's.
Although not as numerous, studies have been made with criteria beyond the first year: both Beatley in 1922 and Kallingal in 1971 used sophomore- g year GPA as criterion; and studies by Anderson and Spencer in 1926,
0 Fishman and Pasanella, pp. 299-301. 7 Cecil B. Read, "Prediction of Scholastic Success in a Municipal University," School and Society (August 1938); 187-88; and Garrett, p. 95. 0 Bancroft Beatley, "The Relative Standing of Students in Second ary School, on Comprehensive Entrance Examinations, and in College," School Review 30 (February 1922): 146; and Anthony Kallingal, "The Predic tion of Grades for Black and White Students at Michigan State University, Journal of Educational Measurement 8 (Winter 1971): 263-65. 26
Hills, Bush, and Klock in 1964, and Siegelman in 1971 used senior or 9 four-year, cumulative GPA as criterion.
Garrett summarized the studies having shorter- and longer-range criteria as follows: "High School scholarship correlates more highly with first-year college grades than with any lesser or greater amount of the entire college record.This conclusion is not unexpected since FGPA data is soon available, the freshman curriculum is the most comparable of any year, and student attrition is less than for longer-range criteria.
High school scholarship as a predictor
To estimate college success, research has centered primarily on the examination of the validity of two traditional predictors: high school scholarship and preadmission test scores. Repeatedly, high school scholarship, as measured by overall grade-point-average (HSGPA) or rank- in-class (HSR), has been found to be the best, single predictor of academic success in college. As early as 1917, Lincoln determined that the high school record of 253 men admitted to Harvard College was a better determinant of both their freshman and sophomore standing than their entrance examination scores.^ Five years later, after the College
Entrance Examination Board had changed to four comprehensive examinations,
9 John E . Anderson and Llewellyn T . Spenser, "The Predictive Value of the Yale Classification Tests," School and Society 24 (September, 1926); 307; John R. Hills, Marilyn L. Bush, and Joseph A. Klock, "Predic ting Grades beyond the Freshman Year," College Board Review, no. 54 (Fall, 1964), p. 25; and Marvin Siegelman, "SAT and High School Average Predictions of Four Year College Achievement," Educational and Psycho logical Measurement 31 (Winter, 1971): 948.
^Garrett, p. 128.
^Edward A. Lincoln, "The Relative Standing of Pupils in High School, in Early College, and on College Entrance Examinations," School and Society 5 (April, 1917); 417-20. 27 Beatley replicated his colleague’s study with 423 men in the Harvard classes of 1920, 1921, and 1922. He reported somewhat lower correlations between entrance examination and freshman standing (r = +.50) versus high school and freshman standing (r = +.56) than did Lincoln; but he was the first to combine the secondary school and examination results to predict 12 freshman standing (r = +.65) and sophomore standing (r = +.62).
The replications have continued. After reviewing 145 studies found in six summaries of the literature between 1934 and 1953, Giusti concluded that the HSGPA was the single best and most stable predictor 13 with median correlations ranging from +.53 to +.66 in the summaries.
Odell in 1927, Weiss in 1970, and Chissom and Lanier in 1975 drew the 14 same conclusion. In the spring of 1967, Gelso and Klock demonstrated that the reliability and stability of the HSGPA or HSR as a predictor was relatively insensitive to the ranking system employed, the year or years 15 included, or the particular courses comprising the average. They qual ified their last statement slightly that summer having found that the total high school average was significantly higher than the average based
12 Beatley, p. 145. 13 Joseph D. Guisti, "High School Average as a Predictor of College Success: A Survey of the Literature," College and University 39 (Winter, 1964): 207. 14 Charles W. Odell, "An Attempt at Predicting Success in the Freshman Year at College," School and Society 25 (June, 1927): 706; Kenneth P. Weiss, "A Multi-Factor Admissions Predictive System," College and University 45 (Winter, 1970): 208; and Brad S. Chissom and Doris Lanier, "Prediction of First Quarter Freshman GPA Using SAT Scores and High School Grades," Educational and Psychological Measurement 35 (Summer, 1975): 463. 15 Charles J. Gelso and Joseph A. Klock, "A Comparison of Basing the Predictive Average Grade on Three Different Methods of Computing High School Averages," College and University 42 (Spring, 1967); 344-45. 28 only on academic courses; however, this significance applied only to applicants with HSGPA's below 2.5 on a four-point scale, and not to 16 entrants with an average of B or better.
Hills, Bush, and Klock used the Gulliksen-Wilks procedure to compare prediction equations over time for seven, successive, entering classes in the University System of Georgia. As a result of their comparisons, they recommended that prediction equations for admissions purposes be used for no longer than three years before being re- 17 evaluated. This is in contrast to the study by Larson and Scontrino which showed that the validity coefficients for equations predicting four-year, cumulative GPA's remained consistently high and remarkably stable when cross-validated over both one- and five-year periods. Based on data from eight years of graduates, this outcome held for correlation coefficients based upon HSGPA as a single predictor, as well as for coef ficients derived from both HSGPA and SAT scores. They also found that including SAT-Verbal and SAT-Math scores increased the predictability 18 slightly for males, but provided no increase for females.
Babbott studied the effect of high school pass/fail grades on college admissions. Although 59 percent of the surveyed admissions
16 Charles J. Gelso and Joseph A. Klock, "The 'Academic' vs. the 'Total' High School Average: A Re-examination," Journal of Educational Measurement 4 (Summer, 1967): 59-61. 17 John R. Hills, Marilyn L. Bush, and Joseph A. Klock, "Keeping College Prediction Equations Current," Journal of Educational Measure ment 3 (Spring, 1966): 33-34. 18 James R. Larson, Jr. and M. Peter Scontrino, "The Consistency of High School Grade Point Average and of the Verbal and Mathematical Portions of the Scholastic Aptitude Test of the College Entrance Exami nation Board, as Predictors of College Performance: An Eight Year Study," Educational and Psychological Measurement 36 (Summer, 1976): 439-43. 29 directors agreed that one academic course taken pass/fail would not make academic qualification difficult, 79 percent said that two such courses would. If an applicant submitted one-half of his courses pass/fail, over
90 percent of the admissions officers said that they would have to put 19 their reliance on other measures.
While high school scholarship may well be the factor with the greatest predictive value for college success, both predictor (HSGPA or
HSR) and criterion (PGPA) measures are based on grades in different
schools and colleges, in courses of varying difficulty, and from teachers whose grading standards differ. Variations in grading standards among
secondary schools and colleges force admissions officers to rely on other
measures in order to increase predictive validity. Generally, these
other measures may be classified as either scaling techniques or tests
of intellectual functioning.
Scaling techniques
Scaling techniques simply relate grades earned in secondary
school to grades earned in college by similar students from the same
school. Two early investigators in prediction scales, Bloom and Peters,
offered three different scaling methods: internal, aptitude, and achieve
ment. Their internal method adjusted school and college grades based on
the performance of prior students from the same secondary school at the
same type of college, a technique recognized by many admissions officers
using experience tables on different secondary schools. The aptitude and
19 Edward F. Babbott, "Effect of High School Pass/Fail Courses in College Admissions," Journal of the National Association of College Admissions Counselors 14 (September, 1969): 25-27. 30 achievement methods converted grades in each school and type college into the mean aptitude or mean achievement score made by similar students in that grade category and school. As an alternative to the aptitude method's mean technique, an aptitude method (regression) was suggested.
This technique determined the scholastic aptitude equivalent of each school grade by computing the regression of scholastic aptitude scores on grades at that school. Substantial increases in the predictive validity of HSGPA were obtained using these scaling methods which adjusted for inter-school variability of grading practices. Assuming an uncorrected grade correlation of +.50 for sake of comparison, the methods employed yielded the following correlations: internal method applied to 4,519 students (r = +.77); aptitude method (regression) applied to 2,959 students (r = +.75); aptitude method (mean) applied to 1,827 students 20 (r = +.72); and achievement method applied to 1,359 students (r = +.65).
Tests of intellectual functioning as predictors
Under the general rubric of tests of intellectual functioning are listed general and special aptitude tests, general and specific achieve ment tests, and general intelligence tests. All correlate positively with success criteria, but not to the same extent as HSGPA. However, when used in combination with HSGPA, the multiple correlations of the two predictors with success criteria usually exceeds that of either alone.
Fishman and Pasanella summarized the use of aptitude tests as intellective predictors with the following:
20 Benjamin s. Bloom and Frank R. peters, The Use of Academic Prediction Scales for Counseling and Selecting College Entrants (New York: Free Press of Glencoe, 1961), pp. 45-68. 31 Among the most commonly used aptitude tests were (in decreasing order of incidence) the Scholastic Aptitude Test (SAT) of the College Entrance Examination Board, the American Council on Education Psycho logical Examination for College Freshman (ACE), and the Ohio State University Psychological Examination (OSPE). Their correlations with comprehensive intellective criteria averaged .47. . . . The usual intellective-predictor combination is an aptitude test plus the high-school record. The multiple correlations of these two predictors with the global college criterion ranged from .31 to .82, with a median of .64 in 24 analyses which did not utilize the College Board SAT. The College Board multiples extended from .34 to .82 (median of .61) for 147 studies predicting freshman average.
Introduced in the same school year that this excerpt was written were two new aptitude tests: the American College Test (ACT) and the 22 Preliminary Scholastic Aptitude Test (PSAT). Within ten years the ACT and PSAT/NMSQT had gained such widespread use and national prominence that they, together with the Scholastic Aptitude Test (SAT), were called 23 "the three major college admissions tests." Numerous studies have com pared the relative predictive efficiency of the scores on these tests.
Lenning and Maxey, in a well-documented, multiple-R comparison among equal-sized, student samples from seventeen colleges, concluded
21 Fishman and Pasanella, p p . 300-1. 22 Founded in 1959, the American College Testing Program first offered the ACT with its battery of four academic tests during school year 1959-60. It now tests approximately 0.9 million students annually. Originally offered by The College Board the same school year, the Preliminary Scholastic Aptitude Test (PSAT) is a shorter version of the SAT and was designed to aid high school counselors in assessing the capabilities of the college-bound and to aid students preparing for the SAT. Jointly sponsored by The College Board and the National Merit Scholarship Corporation since 1971, the PSAT is now properly called the Preliminary Scholastic Aptitude Test/National Merit Scholarship Quali fying Test (PSAT/NMSQT) and has the additional purpose of identifying high-scoring students for scholarships. Statistical data on the numbers taking the PSAT/NMSQT (1.1 million in 1979-80) the first twenty-one years may be found in Educational Research Service, Testing for College Admis sions: Trends and Issues (Arlington: Educational Research Service, 1981), pp . 18-19. 23 Alexander W. Astin, Predicting Academic Performance in College (New York: Free Press, 1971), p. 291. 32
that the ACT battery predicts as well as, and possibly even better than, 24 the SAT. Aleamoni and Oboler found the SAT a more valid predictor than 25 the ACT at a highly selective, midwestern university. Munday deter
mined that the ACT and SAT tests were comparable in predictive validity,
that both were superior to the School and College Ability Test (SCAT),
and that both were more independent of high school grades than either the 2 0 SCAT or the Minnesota Scholastic Aptitude Test (MSAT). Lins, Abell,
and Hutchins demurred, saying that the ACT and SAT were equally limited
in their ability to predict first semester academic performance; and that
because of differences in what the two tests measured and their lack of
test-to-test reliability, "it is impractical if not actually impossible 27 to equate scores on the ACT and SAT." Sassenrath and Pugh disagreed
when they obtained a correlation of +.80 between ACT and SAT composite 28 scores and an identical correlation of +.53 between each test and FGPA.
In an extensive study based on data from sixteen hundred students at
24 Oscar T . Lenmng and E . James Maxey, "ACT versus SAT Prediction for Present-Day Colleges and Students," Educational and Psychological Measurement 33 (Summer, 1973): 403. 25 Lawrence M. Aleamoni and Linda Oboler, "ACT versus SAT in Pre dicting First Semester GPA," Educational and Psychological Measurement 38 (Summer, 1978): 393. 26 Leo A. Munday, "Correlations between ACT and Other Predictors of Academic Success in College," College and University 44 (Fall, 1968): 71-72. 27 L. Joseph Lins, Allan P. Abell, and H. Clifton Hutchins, "Relative Usefulness in Predicting Academic Success of the ACT, the SAT, and Some Other Variables," Journal of Experimental Education 35 (Winter, 1966); 26. 28 Julius M. Sassenrath and Richard Pugh, "Relationships Among CEEB Scholastic Aptitude Test and American College Test Scores and Grade Point Average," Journal of Educational Measurement 2 (December, 1965): 200-1 . 33 fifty-five institutions who had taken at least two of the three major tests while in high school, Astin assessed the comparability of the three tests as predictors of FGPA and generated tables for equating them using equipercentile conversions. In summarizing the results of this portion of research, Astin stated:
The composite scores of the three tests (SAT Verbal and Mathe matical, NMSQT Selection score, and ACT Composite score) are highly interrelated, with an average correlation of about .85. . . . In some instances, of course— with particular types of courses or at particular institutions— one of these tests may be preferable to another, or scores on separate subtests rather than just the compos ite score may be necessary for accurate predictions. (Italics mine.) But our findings do indicate that in general, the SAT, NMSQT, and ACT are interchangeable for the purpose of predicting the student's over all freshman GPA in college; use of scores on the separate subtests of these batteries, instead of the simple composite scores, adds little predictive value.
Carl Haag, the Advanced Placement Program director, suggested that the PSAT/NMSQT mathematical subscores could be used "to help iden tify students who appear likely to profit from an AP course;" but he hastened to add, "the scores, however, should never be used as the sole
criterion for selecting AP students.if a school decides that it has the means and a sufficient number of able students for an AP course, it should also inquire whether each prospective candidate has sufficient general ability, motivation, and an adequate academic preparation.
Table 4 shows the percentage of students within given PSAT/NMSQT
ranges receiving an AP grade of three or better in each of the calculus
29 Astin, pp. 11-12.
"^Carl H. Haag, "Using the PSAT/NMSQT to Help Identify Advanced Placement Students," Princeton, n.d. (Mimeographed.), p. 4. 34 examinations in 1975:
TABLE 4
PERCENTAGE OF SUCCESSFUL STUEENTS BY COURSE AND PSAT/NMSQT MATHEMATICS SCORE3
PSAT/NMSQT Calculus AB Grade Calculus BC Grade Mathematical of Three or above of Three or above
75-79 ___ __ 70-74 86% 89% 65-69 78 73 60-64 65 65 55-59 49 46 50-54 41 18 45-49 17 0 40-44 17 0 20-39 0 0
Percentage at/or above 1975 AP 66% 74% Grade of Three
SOURCE: Carl H. Haag, "Using the PSAT/NMSQT to Help Identify Advanced Placement Students," Princeton, n.d. (Mimeographed), pp. 6-7, tables 7-8.
aTotal number of students taking both PSAT/NMSQT and Calculus AB or Calculus BC in above figures is 2,250 or 993, respectively.
The achievement test is the second test of intellectual function ing considered. Although more often administered as placement or guid ance devices, achievement tests such as the College Board's one-hour tests in fifteen academic subjects, the Cooperative Tests of the Educa tional Testing Service, the Iowa Tests of Educational Development, and various college-constructed tests have a history as selection instruments also. In Garrett's 1949 review of the literature are listed twenty-four
studies between 1922 and 1941 which used general achievement tests to pre
dict scholastic success in college. The median correlation, in a range 35 31 of coefficients from +.23 to +.85, was +.49. These figures have a broader range than those found by Eurich and Cain in 1941, when they drew this conclusion:
General achievement tests provide a close second to high school average as a single basis for predicting college scholarship. Vari ous investigators have found the relationship between the College Entrance Examination Board Tests and general scholarship to range from .39 to .64.
In 1974, while conducting basic validity studies for several liberal arts colleges participating in the College Research Center pro gram, Wilson was able to compare the relative predictive efficiency of rank-in-class, SAT Verbal and Mathematical subscores, and the average of scores presented by a student on the College Board Achievement Tests.
Typically, rank-in-class was the best predictor; but then Wilson said:
The average of scores on all the CEEB achievement tests presented (Av Ach) tends to be the second-best, single predictor of freshman standing.^ . . . for the colleges involved in these studies, it would appear not only that the CEEB Achievement Average is a more valid predictor of performance than the SAT, but also that . . . information provided by the SAT scores adds little of value for predicting college grades after taking into account the high school record and the CEEB Achievement Average.^
31 Garrett, p. 100. 32 Alvin C. Eurich and Leo F. Cain, Prognosis, Encyclopedia of Educational Research (New York: Macmillan Company, 1941), p. 49. 33 Kenneth M. Wilson, "The Validity of a Measure of 'Academic Motivation' for Forecasting Freshman Achievement at Seven Liberal Arts Colleges," Research Bulletin RB-74-29 (Princeton: Educational Testing Service, 1974; Arlington, Va.: ERIC Document Reproduction Service, ED 163 016, 1979), p. 2. 34 Kenneth M. Wilson, "The Contribution of Measures of Aptitude (SAT) and Achievement (CEEB Achievement Average), Respectively, in Fore casting College Grades in Several Liberal Arts Colleges," Research Bulle tin RB-74-36 (Princeton: Educational Testing Service, 1974; Arlington, Va.: ERIC Document Reproduction Service, ED 163 015, 1979), p. 13. 36
In a rare, 1943 study predicting college GPA from the elementary level, Durflinger found the multiple correlation between achievement tests of elementary school subject matter and college grades to be +.533 35 for men and +.542 for women. Surprisingly, he felt that the following conclusion was warranted: "A comprehensive elementary school achievement test is practically as valid in the prediction of college grades as a 36 comparable high school achievement test or high school grades."
Lewis also investigated the relationship between elementary school grades and college GPA. Grade seven was determined to be the earliest year for identification of students with college potential on the basis of a generally increasing correlation coefficient from first grade to high school. Only the correlations between college success and the GPA's of grades seven, eight, and high school exceeded the Henmon- 37 Nelson Test of Mental Ability correlation of +.36.
Intelligence has been a popular factor to study in its relation to college success; but most research reviewers have ranked general intelligence tests or psychological examinations at least third after
HSGPA and achievement tests as a predictor. So said Harris, reviewing investigations made during the 1930's, when he reported correlations 38 between intelligence tests and college GPA ranging from +.33 to +.64.
35 Glenn W. Durflinger, "Scholastic Prediction in a Teachers College," Journal of Experimental Education 11 (June, 1943): 259. 36 Ibid., p. 265. 37 William A. Lewis, "Early Prediction of college GPA Using Pre- College School Grades," Journal of Educational Measurement 3 (Spring, 1966): 35-36. 38 Daniel Harris, "Factors Affecting College Grades: A Review of the Literature, 1930-1937," Psychological Bulletin 37 (March, 1940): 125. 37 In a similar, 1949 summary of ninety-four studies, Garrett found a range of correlations from +.17 to +.67, with a median at +.47. He agreed with Harris that "intelligence tests rank a close third as a means of predicting college success;" and this observation placed intelligence between general achievement (median r = +.49) and general aptitude 39 (median r = +.43), as a predictor variable. Earlier researchers had rated intelligence much higher in the scale of predictors. Hartson ranked it on a par with HSGPA for freshmen entering Oberlin in 1928 and
1929, after he discovered mean correlations of +.473 between intelligence test scores and college scholarship versus +.465 between high school and 40 college scholarship. Later reviewers virtually ignored intelligence as a predictor. For example, in Fishman and Pasanella’s 1960 review, only the following comment on intelligence tests appeared:
Group intelligence tests such as the Otis were less commonly employed, because they have proved generally less satisfactory than tests geared more directly to the measurement of scholastic abilities.
Perhaps the real reason for the declining use of intelligence as a predictor can be explained by the rise of another test. On June 23,
1926, some 8,040 candidates gathered at 318 examination centers to take the College Entrance Examination Board’s newly-designed Scholastic Apti- 42 tude Test, an innovation of experimental psychologist, Carl Brigham.
39 Garrett, pp. 106-12. 40 L. D. Hartson, "The Validation of the Rating Scales Used with Candidates for Admission to Oberlin College," School and Society 36 (September 1932): 415. 41 Fishman and Pasanella, p. 300. 42 Thomas F. Donlon, "Brigham’s Book," The College Board Review, no. 113 (Fall 1979): 29. 38 Brigham believed that studying students' errors could lead to a deeper understanding of the workings of the mind; and he applied this belief in multiple-choice test construction. Fifty-five years later, some 1.5 mil
lion candidates took the modern version of the Scholastic Aptitude Test
(SAT) at over three thousand centers. During the interim, as test con
structors became more skilled in developing aptitude items which measured the ability to reason and draw necessary conclusions from given or known
data, the distinction between aptitude and intelligence tests became
increasingly blurred, at least from a statistical point of view. In
stepwise multiple regression, once HSGPA or other index of achievement,
and aptitude have been applied to predict success, there is evidently
little variance left that an intelligence test can detect. Most modern
investigators agree that practical, predictive efficiency is not really 43 served by employing more than two or three, intellective predictors.
Categorical Background Characteristics
Researchers have found that certain background characteristics
possessed by students affect the accuracy of prediction for success in
college. Generally, these characteristics are nominal-level variables,
43 Astin, p. 279; Joshua A. Fishman, "The Use of Quantitative Tech niques to Predict College Success," Admissions Information (New York: College Entrance Examination Board, 1957), p. 59; R. A. Weitzman, "The Prediction of College Achievement by the Scholastic Aptitude Test and the High School Record," Journal of Educational Measurement 19 (Fall 1982): 190; and Wilson, RB-74-36, p. 13. MacGinitie and Ball made the following commentary on Alexander G. Wesman's article, "Aptitude, Intelligence, and Achievement": "Wesman shows that all tests measure elements of intelligence, aptitude, and achievement, and that content is not entirely predic table from test labels.... Wesman believes that intelligence tests would be better understood if they were called 'scholastic aptitude tests' ." Walter H. MacGinitie and Samuel Ball, eds., Readings in Psychological Foundations of Education (New York: McGraw- Hill, 1968), p. 167. 39 forming mutually exclusive categories into which students can be placed.
A student's sex, race, religion, type of school attended, and the educa tional level of his parents are such categorical variables. A few other characteristics, such as age and family income, are at least ordinal- level variables. The most important of these categorical variables will be considered here.
Sex differences in prediction
A number of general conclusions regarding sex differences in prediction have been found repeatedly in scores of studies. Initially, women tend to be more predictable than men in academic performance.
Seashore found this to be true in both secondary school and in college, despite the fact that women's aptitude test scores and academic grades 44 were less variable than men's. Secondly, "literally hundreds of studies have shown that girls get higher grades in secondary school than 45 boys do;" and the same holds true for freshmen in college. Thirdly however, sex-related differences on tests of mathematical aptitude and achievement favored the men beginning in the mid-teens. This was found to be the case in three national studies (Project Talent, the National
Longitudinal Study of Mathematical Abilities [NLSMA], and the National
44 Harold G. Seashore, "Women Are More Predictable than Men," Journal of Counseling Psychology 9 (Fall 1962): 266. Being more predic table than males, females should have separate prediction equations to increase the accuracy of success prediction. Richard D. W. Bean, "The Predictive Ability of Selected Non-lntellective variables on College Freshmen Grade Point Averages: Towards Improving the Traditional Cog nitive Model," (Ph.D. dissertation, Georgia State University, 1980), DAI 41 (1981), 3540A; Lavin, p. 52; and Julian C. Stanley and Andrew C. Porter, "Correlation of Scholastic Aptitude Test Score with College Grades for Negroes versus Whites," Journal of Educational Measurement 4 (Winter 1967): 200. 45 Astin, p. 4. 40
Assessment of Educational Progress [NAEP]), as exemplified by the follow ing NAEP conclusion:
Male and female mathematical abilities appeared much the same at ages nine and thirteen, but by ages seventeen and adult, males exhibited a definite advantage. This advantage increased in all content areas from age seventeen to adult.
In a ten-country assessment by the International Association for the Evaluation of Educational Achievement (IEA) in 1964 and 1970, similar international results were obtained for both mathematics and sciences
The general pattern of results is one of superior performance by male students in both mathematics and science, but with considerable variation between countries in the extent to which the achievement of boys exceeds that of girls. Furthermore, . . . the sex differences in achievement are in general greater at the pre-university level than they are at an age level where attendance at school is compul-
Although Roberts and McClure in 1970, Keller in 1974, and Guthrie 48 in 1980 failed to find significant sex differences, the majority of
46 National Assessment of Educational Progress, The First National Assessment of Mathematics: An Overview (Washington: NAEP, 1975), p. 45. For Project Talent, see John C. Flanagan, "Changes in School Levels of Achievement: Project Talent Ten and Fifteen Year Retests," Educational Researcher 5 (September 1976): 9-12; and for NLSMA, see James W. Wilson, Patterns of Mathematical Achievement in Grade 11: Z Population, National Longitudinal Study of Mathematical Abilities, no. 17 (Palo Alto: Stanford University Press, 1972). 47 John P. Keeves, "Differences Between the Sexes in Mathematics and Science Courses," International Review of Education 19, no. 1 (1973), p. 57. 48 Fannie M. Roberts, "Relationships in Respect to Attitudes toward Mathematics, Degree of Authoritarianism, Vocational Interests, Sex Differ ences, and Scholastic Achievement of College Juniors," (Ph.D. disserta tion, New York University, 1970), DAI 31 (1970), 2134A; Wesley C. McClure, "A Multivariate Inventory of Attitudes toward Selected Components of Ele mentary School Mathematics," (Ed.D. dissertation, University of Virginia, 1970), DAI 31 (1971), 5941-42A; Claudia M. Keller, "Sex Differentiated Attitudes toward Mathematics and Sex Differentiated Achievement in Mathe matics on the Ninth Grade Level in Eight Schools in New Jersey," (Ed.D. dissertation, Rutgers University, 1974), DAI 35 (1974), 3300A; and Judith B. Guthrie, "A Study of Mathematics Education in the United States: En rollment, Abilities, and Post High School Paths," (Ph.D. dissertation, Claremont Graduate School, 1980), DAI 41 (1981), 3464A. 41 studies reported age-related, sex differences in both achievement and 49 50 attitude in mathematics and researched possible causes (Aiken , Haven , 51 52 53 54 55 Farley , Fennema , Fox , Casserly , and Hilton and Berglund ).
49 Sex differences in mathematical attitude and ability are not biological, but the result of "differing sociocultural •expectations' and reinforcement schedules, complemented by same-sex role modeling." Louis R. Aiken, "Update on Attitudes and Other Affective Variables in Learning Mathematics," Review of Educational Research 46 (Spring 1976): 296. 50 , The two most discriminating functions for able women electing mathematics courses were those "related to the usefulness of high school mathematics to future study or job, and to interest in natural science as opposed to social studies," Elizabeth W. Haven, "Selected Community, School, Teacher, and Personal Factors Associated with Girls Electing to Take Advanced Academic Mathematics Courses in High School," (Ph.D. dissertation, University of Pennsylvania, 1971), DAI 32 (1971), 1747a . 51 Farley found higher achievement for boys at the eleventh, but not at the tenth, grade. "Boys elect further high school mathematics on the basis of previous achievement. Girls do so on the basis of subject preference." Mary de C. Farley, "A Study of the Mathematical Interests, Attitudes, and Achievement of Tenth and Eleventh Grade Students," (Ph.D. dissertation, University of Michigan, 1968), DA 29 (1969), 3039A. 52 "When significant differences did appear they were more apt to be in the boys' favor when higher-level cognitive tasks were being mea sured and in the girls' favor when lower-level cognitive tasks were being measured." Elizabeth H. Fennema, "Mathematics Learning and the Sexes: A Review," Journal for Research in Mathematics Education 5 (May 1974): 137. 53 Girls were more willing to join an all-female accelerated math class than a mixed-sex class; however, post-test achievement was not significantly better. Lynn H. Fox, "Facilitating the Development of Mathematical Talent in Young Women," (Ph.D. dissertation, The Johns Hopkins University, 1974), DAI 35 (1975), 3553B. 54 Casserly found that the supportive role played by counselors, parents, teachers, and peers was more important to girls than to boys in electing advanced calculus and science courses. Patricia L. Casserly, "Helping Able Young Women Take Math and Science Seriously in School," in Colangelo-Zaffrann, New Voices in Counseling the Gifted (Dubuque: Ken dall/Hunt, 1979; reprint ed., New York: The College Board, 1979), p. 6. 55 Investigating sex-typed interest as a possible cause of sex differences in achievement, these authors found parallel differences, emerging after fifth grade, in the percentage perceiving mathematics as interesting and as likely to be an asset in earning a living. Thomas L. Hilton and Gosta W. Berglund, "Sex Differences in Mathematics Achievement — A Longitudinal Study," Journal of Educational Research 67 (January 1974): 231-237. 42
Prediction equations for success, particularly in the physical sciences and engineering, should take into account these sex-related differences 56 m mathematics achievement and attitude.
Jacobs examined forty males and forty females at each of the seventh- and eleventh-grade levels in order to bracket the ages where changes in mathematical achievement and attitude might occur. She found no significant sex differences in either attitude or achievement at the lower grade level, nor in attitude at the eleventh; but, like Parley, she did find that "eleventh grade males had significantly greater achievement 57 in mathematics than eleventh grade females had." In high schools which
require only two years of mathematics for graduation, this statement may
be misleading since many able females fail to elect further mathematics.
In a small, but significant, pilot study at the University of
California at Berkeley in 1973, Sells found that 57 percent of the men,
In predicting college first quarter GPA, sex was found to be a significant main effect as well as having two-factor interaction with HSGPA, ACT Social Studies and ACT Natural Science scores. Lois H. Worth ington and Claude W. Grant, "Factors of Academic Success: A Multivariate Analysis," Journal of Educational Research 65 (September 1971); 7-10. Alan L. Gross, Jane Faggen and Karen McCarthy "substantially reinforced the previously reported finding that females are more predic table than males in academic settings," in a massive study involving 17,745 New York students. "The Differential Predictability of the Col lege Performance of Males and Females," Educational and Psychological Measurement 34 (Summer 1974): 363; S. B. Khan, "Sex Differences in Pre dictability of Academic Achievement," Measurement and Evaluation in Guid ance 6 (July 1973): 88-92; Julian C. Stanley, "The Predictive Value of the SAT for Brilliant Seventh- and Eighth-Graders," College Board Review, no. 106 (Winter 1977), pp. 31-37; and John Paraskevopoulos and L. F. Robinson, "Comparison of Regression Equations Predicting College Perfor mance from High School Record and Admissions Test Scores for Males and Females," College and University 45 (Winter 1970): 211-16. 57 Judith E. Jacobs, "A Comparison of the Relationships between the Level of Acceptance of Sex-Role Stereotyping and Achievement and Attitudes toward Mathematics of Seventh Graders and Eleventh Graders in a Suburban Metropolitan New York Community," (Ph.D. dissertation, New York University, 1974), DAI 34 (1974), 7585a . 43 but only 8 percent of the women, entering Berkeley had the prerequisite four years of secondary mathematics for calculus, chemistry, physics, and statistics. Since fifteen of the twenty majors at Berkeley require either calculus or statistics, 92 percent of the women had, in effect, limited themselves to only five fields of study: the humanities, music, elementary education, social work, and guidance and counseling. Sells concluded that "inadequate preparation in mathematics presents a serious 58 constraint in choice of undergraduate major in college," a condition 59 Tobias described as "a critical vocational filter."
The seriousness of the failure of high school girls to elect more 00 than the required two years of mathematics prompted Perl's study. Her data base included about 13,000 observations of 10th-12th graders from one of the two secondary populations of NLSMA. Ability and achievement proved to be the major discriminator for both sexes between those who elected (electors) and those who failed to elect (non-electors) to con tinue their courses in mathematics. The second significant discriminator in electing, and the first one to show significant sex differences, with higher means for males, was attitude. A confidence and anxiety factor did not appear as a discriminating factor between electors and non electors, but was a significant debilitating factor for females. The
58 Lucy W. Sells, "High School Math as the Critical Filter in the Job Market," Research report, 31 March 1973 (Bethesda, Md.: ERIC Document Reproduction Service, ED 080 351, 1973), p. 1. 59 Sheila Tobias, "Why Janie Can't— or Won't— Do Math," Independent School 37 (May 1978): 42.
^°The National Council of Teachers of Mathematics (NCTM) has recommended that "at least three years of mathematics should be required in grades 9 through 12." An Agenda for Action: Recommendations for School Mathematics of the 1980s (Reston, Va.: National Council of Teachers of Mathematics, 1980), p. 20. 44 fourth factor, a composite spatial ability and measure of preference for typical masculine vocations, was a discriminating factor for females but not for males. Because of the size of the sample, Perl's finding "that the correlation between ability and achievement is higher for girls with female teachers than for any of the remaining three groups studied," is 61 probably noteworthy.
Certain affective variables have been hypothesized to be impor tant factors influencing sex-related differences in mathematics achieve ment or continued election. Fennema and Sherman studied eight of them:
The affective variables selected were attitude toward success in mathematicsj the stereotyping of mathematics as a male domain; the perceived attitudes of mother, father, and teacher toward one as a learner of mathematics; effectance motivation in mathematics; confi- fi? dence in learning mathematics; and usefulness of mathematics.
Of the eight, all but 'attitude toward success' and 'effectance motivation' peaked in significance at the ninth- and tenth-grade levels and, therefore, were considered causal factors for "the increasing differential in the enrollment of the two sexes in advanced mathematics
61 Teri H. Perl, "Discriminating Factors and Sex Differences in Electing Mathematics," (Ph.D. Dissertation, Stanford University, 1979), pp. 145-49. This conclusion contradicts a 1953 study which found the highest correlation for boys with men teachers, and the lowest for girls with female teachers; however, the number of students involved was only 253. Robert S. Carter, "Non-Intellectual Variables Involved in Teachers' Marks," Journal of Educational Research 47 (October 1953): 91. 62 Elizabeth H. Fennema and Julia A. Sherman, "Sex-Related Differ ences in Mathematics Achievement and Related Factors: A Further Study," Journal for Research in Mathematics Education 9 (May 1978): 190-91. Ina A. Cauthen found that "there were definite differences in the relationship between mathematical achievement and personality variables for male and female subjects." Highly correlated factors for females included Reserved, More-Intelligent, Timid, and Self-Sufficient; while those for males were the More-intelligent and Self-Controlled factors. "Selected Demographic and Personality Variables Related to Mathematical Achievement in Men and Women," (Ph.D. dissertation, Texas ASM University, 1979), DAI (1980), 4456A. 45 63 classes." The authors concluded "that when relevant factors are con trolled, sex-related differences in favor of males do not appear often, and when they do they are not large;" but they recommended that further research be conducted on the negative influences which may exist within 64 the schools themselves.
Casserly limited her study to a sampling of those "remarkable high schools" which had at least twice the national proportions of girls participating in Advanced Placement couries in chemistry, physics, and mathematics.65 These schools' success in retaining girls in mathematics depended not at all on especially designed programs promoting mathematics
for girls. Instead, these were schools which had established a favorable
climate "for mathematics, just as there is for English," where mathe matics was presented to all students as important, and where counselors
and teachers encouraged "students to continue the study of mathematics to
keep later career options open, pointing out the increasing range of 66 fields, not just math and science, that mathematics serves."
In the spring of 1979, the Advanced Placement Program Service
Officer, Harlan P. Hanson, offered the following update on the present
63 Fennema and Sherman, p . 202. 64 Ibid., pp. 201-2.
65Casserly gave the following percentages, on a national basis, of the female Advanced Placement candidates in the indicated subject: 17.2 percent in Chemistry; 12.9 percent in Physics B; 6.8 percent in Physics C; and 27.9 and 21 percent in Calculus AB and BC, respectively. "Helping Able Young Women Take Math and Science Seriously in School," p. 5. 66 Patricia L. Casserly, "What Educators Can Do," College Board Review, no. Ill (Spring 1979), p. 24. 46 status of women taking AP examinations:
There aren't significant differences in women's grades and men’s grades on Advanced Placement [AP] Examinations, because candidates are usually subject to a self-selection. Fewer women take the Exami nations, but their percentage of total participation has increased in the last five years. . . . Since 1970, the female representation on the AP calculus, biol ogy, and chemistry Examinations has also increased— from 23.7 to 33.2 percent of total participation in those fields.^
Mariette Reed, the associate program director of Advanced Place ment examinations, furnished the most current distributions by sex for
the two mathematics examinations. Table 5 shows these figures and the
continued increase in percentage of females participating in the last
four years.
TABLE 5
NATIONAL NUMBER OF CALCULUS CANDIDATES BY YEAR, COURSE, AND SEX
Calculus AB Calculus BC Composite AB & BC Year Sex Number % Number % Number %
1980 F 7,174 35.7 2,154 27.7 9,328 33.5 M 12,922 64.3 5,629 72.3 18,551 66.5 1981 F 8,392 37.2 2,333 29.1 10,725 35.1 M 14,145 62.8 5,688 70.9 19,833 64.9 1982 F 9,302 39.0 2,303 28.5 11,605 36.4 M 14,523 61.0 5,790 71.5 20,313 63.6 1983 F 10,444 39.1 2,700 30.7 13,144 37 .0 M 16,262 60.9 6,083 69.3 22,345 63.0
SOURCE: Mariette Reed, Associate Program Director, College Board Programs, Educational Testing Service in personal letter to author, 16 September 1983.
While Hanson’s initial statement is undoubtedly true for total
participation in all twenty-four AP Examinations, it may not hold for
^Harlan P. Hanson et a l ., "The College Board and Women," College Board Review, no. Ill (Spring 1979), p. 25. 47 grade distributions by sex in the two calculus examinations. The preponderance of the literature would seem to indicate a significant, sex-related difference in mathematical achievement for students of this age as measured by their grades.
Racial and Religious Differences in Prediction
In the 1960's, as the pressure for a college education became more intense among minority groups, questions were raised regarding the predictive validity of high school scholarship and national test scores 68 for such students. Stanley and Porter compared the predictive validi ties within three Negro colleges with those in three non-Negro colleges and found the SAT subtests as predictively valid for blacks as for 69 whites. Cleary obtained similar results for blacks and whites in two integrated colleges in the east; but in one college in the southwest, she rejected the hypothesis of equality of regression intercepts and stated that "the Negro students' scores were overpredicted by the use of 70 white or common regression lines." Her regression model criterion of equality of slopes and intercepts among different groups was based on a definition of fairness as 'equal regression lines'; but she failed to
68 Fishman and Pasanella, pp. 301-2; Robert L. Linn and Charles E. Werts, "Consideration for Studies of Test Bias," Journal of Educational Measurement 8 (Spring 1971): 1-4; and Richard B. Darlington, "Another Look at 'Cultural Fairness'," Journal of Educational Measurement 8 (Summer 1971): 71-82. 69 Julian C. Stanley and Andrew C. Porter, "Correlation of Scholastic Aptitude Test Scores with College Grades for Negroes versus Whites," Journal of Educational Measurement 4 (Winter 1967): 199-218. 70 T. Anne Cleary, "Test Bias: Prediction of Grades of Negro and White Students in Integrated Colleges," Journal of Educational Measure ment 5 (Summer 1968): 123. 48 test the equality of the standard errors of estimate.
Others have used the same regression model to investigate test
bias. Three 1971 studies by Temp, Kallingal, and Pfeifer and Selacek all
determined that blacks in integrated colleges were accurately predicted
or overpredicted using combined group regression equations; but only Temp 72 tested for equality in errors of estimate. Bowers rejected the hypoth
esis of a common multiple regression equation predicting freshman GPA for
black and white men and black and white women at the University of
Illinois; however, the blacks were being admitted under a Special Educa
tional Opportunities Program and "significant slope differences among
the four equations were explainable in terms of significant sex differ- 73 ences." Reschly and Sabers used Cleary's definition of test bias to
show that an individual, intelligence test (the Wisconsin-R) was equally
valid for four ethnic groups (Anglo, Chicano, Black, and Native American)
of exceptional children at five levels from first- through ninth-grade,
as a measure of academic aptitude. This they claimed despite the fact
71 The Gulliksen-Wilks testing procedure to determine if the errors of estimate and the regression lines of different groups may be regarded as being the same, is sequential: first, one must test that the standard errors of estimate are equal; second, one tests for the equality of slopes; and finally, one tests for equality of intercepts. Signifi cant results at any stage invalidate the hypothesis that the regression lines are identical. Harold Gulliksen and Samuel S. Wilks, "Regression Tests for Several Samples," Psychometrika 15 (June 1950): 96-100. 72 George Temp, "Validity of the SAT for Blacks and whites in Thirteen Integrated Institutions," Journal of Educational Measurement 8 (Winter 1971): 245-51; Kallingal, pp. 263-65; and C. Michael Pfeifer and William E. Sedlacek, "The Validity of Academic Predictors for Black and White Students at a Predominantly White University," Journal of Educational Measurement 8 (Winter 1971): 253-61. 73 John [E.] Bowers, "The Comparison of GPA Regression Equations for Regularly Admitted and Disadvantaged Freshmen at the University of Illinois," Journal of Educational Measurement 7 (Winter 1970): 224. 49 that "the standard errors of estimate in the analysis of all four groups were significantly different in four of the 10 comparisons 74 (p < .05)
Borup's study of 996 Texas A & I University freshmen, partitioned by sex and ethnic groups (Anglo- and Mexican-American), concluded that high school, quarter rankings were more valid predictors of first semester GPA than the ACT. He based this conclusion on two findings: (1) males had significantly lower mean-quarter ranking and first semester GPA than females; yet they scored significantly higher than the female students on the ACT cumulative and all sub-scores with the exception of English; and (2) neither the mean-quarter rankings nor the first semester GPA's were significant; however, the Anglo-American students scored significantly higher than the Mexican-American students 75 on the ACT cumulative as well as on the four sub-scores.
In the fall of 1980, the five states abutting the Pacific Ocean had the highest percentages of non-Black minorities enrolled in public, elementary and secondary schools: 33 percent of the Hispanic, 56 percent of the Asian or Pacific islander, and 26 percent of the Native
74 Daniel J. Reschly and Darrell L. Sabers, "Analysis of Test Bias in Four Groups with the Regression Definition," Journal of Educational Measurement 16 (Spring 1979): 3. According to Gulliksen and Wilks, if the errors of estimate differ significantly, the hypothesis of a common regression system should be rejected. However, Reschly and Sabers followed Humphreys' suggestion that "the assumption of inequality of errors of estimate can, within limits, be violated." Ibid, citing L. Humphreys, "Implications of Group Differences for Test Interpreta tion," Proceedings of 1972 Invitational Conference on Testing Problems (Princeton, N.J.: Educational Testing Service, 1973). 75 Jerry H. Borup, "The Validity of American College Test for Discerning Potential Academic Achievement Levels— Ethnic and Sex Groups," Journal of Educational Research 65 (September 1971): 3-6. American. In California, a mecca of minorities, Goldman paired with others to perform a number of studies with one or more of these groups.
Goldman and Richards conducted two studies comparing Chicanos and
Anglos; each study used SAT sub-scores to predict GPA. Both studies
showed similar zero-order correlations of SAT-Verbal and GPA but
significant differences, favoring Anglos, for SAT-Math. Substantial
overpredictions of Chicano GPA occurred if Anglo regression equations 7 8 were used. Yet two year later, in a 1976 study involving 5,500 black,
Chicano, Oriental, and white undergraduates at four campuses of the
University of California, Goldman and Hewitt found little evidence of any significant, ethnic differences in regression systems predicting GPA
from HSGPA, SAT-Verbal, and SAT-Math. Correlations for Blacks, Chicanos,
Orientals, and Whites were .33, .38, .42, and .43, respectively. The
7 6 Percentages given are for school enrollment in Alaska, Hawaii, California, Oregon, and Washington, in only four states does the percen tage of the state's enrollment for non-Black minorities exceed 30 percent Texas, 32; California, 33; New Mexico, 55; and Hawaii, 74 percent. Over half of the Asian or Pacific Islander minority are taught in California and Hawaii. W. Vance Grant and Leo J. Elden, Digest of Education Statistics (Washington, D.C.: National Center for Education Statistics, 1982), p. 43, table 34. Similar percentages of enrollment in institu tions of higher education obtain for non-Black minorities in the Pacific states: 28 percent of the Hispanic, 58 percent of the Asian-American, and 26 percent of the Native American. U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States: National Data Book and Guide to Sources, 1982-83 (Washington, D.C.: Government Printing Office, 1982), p. 162, table 266. 77 Roy D. Goldman, "Hidden Opportunities in the Prediction of College Grades for Different Subgroups," Journal of Educational Measure ment 10 (Fall 1973): 205-10; and Roy D. Goldman and Mel H. Widawski, "A Within-Subjects Technique for Comparing College Grading Standards: Implications in the Validity of the Evaluation of College Achievement," Educational and Psychological Measurement 36 (Summer 1976): 381-90. 7 8 Roy D. Goldman and Regina Richards, "The SAT Prediction of Grades for Mexican-American vs Anglo-American Students of the University of California, Riverside," Journal of Educational Measurement 11 (Summer 1974): 129-35. 51 results were controlled for major field of interest, but this did not alter the findings. The authors admitted that the issue of differential validity was less clear, stating that "in general GPA’s of Orientals and
Whites are more validly predicted by the combined use of HSGPA and SAT 79 scores than are the GPA's of Blacks and Chicanos."
Published studies involving Oriental-Americans are rare. In
Breland's examination of the validity of college entrance predictors in
140 studies, only one study included Orientals. Unfortunately, Wilson's study at two institutions simply aggregated Blacks, American Indians,
Latins and Orientals in a single group entitled 'miniority'. Multiple
R's using all predictors (SAT's, high school rank, and average Achieve ment Test scores) and four-year cumulative GPA "were .57 (nonminority) and .42 (minority) at the selective institution and .57 (nonminority) and 80 .59 (minority) at the nonselective institution."
Studies involving Jewish students are also rare. Silverman,
Barton, and Lyon found that regression equations developed for the total sample of black, white, and Jewish students at Roosevelt University, significantly overpredicted black performance while underpredicting 81 Jewish performance for both first- and second-year GPA. To support
79 Roy D. Goldman and Barbara N. Hewitt, "Predicting the Success of Black, Chicano, Oriental and White College Students," Journal of Educational Measurement 13 (Summer 1976): 115. 80 Hunter M. Breland, Population Validity and College Entrance Measures (New York: College Entrance Examination Board, 1979), p. 64, citing Kenneth M. Wilson, Predicting the Long-Term Performance in College of Minority and Nonminority Students: A Comparative Analysis in Two Collegiate Settings, College Board Research Bulletin, RB-78-6 (Princeton: Educational Testing Service, 1978). 81 Bernie I . Silverman, Florence Barton, and Mark Lyon, "Minority Group Status and Bias in College Admissions Criteria," Educational and Psychological Measurement 36 (Summer 1976): 405. 52 the thesis that "with regard to religion, Jewish students outperform non-Jewish students," Lavin cited a study by Strodtbeck who had drawn that conclusion based on a comparison of Italian Catholic and Jewish 82 high school students. Lavin advised further study suggesting that, rather than religion, socio-economic status or differences in their respective value systems might be the real differential: "relative to the Italian Catholic value system, the Jewish culture places greater emphasis on the value of education and confers more prestige upon the 83 scholar." In Hawaii, Lind made a similar statement with respect to
Oriental-Americans:
On the other hand, the Chinese, Japanese, and Koreans, although of peasant origin, also place a high value upon scholarship! the teacher and learned person enjoys a place of dignity and prestige in the community. This fact, together with a growing recognition that education is indispensable to advancement on the economic and social scale in American society, is responsible for the rapid rise in the proportion of older oriental children attending school. A pronounced tendency in oriental families some decades ago to favor boys over the girls for such educational opportunities had by 1960 almost disappeared.®^
With the annexation of Hawaii as a territory in 1898, the United
States acquired not only a strategic military base in the Pacific but also, because of its multiracial, polyglot population, an educational problem of immense magnitude. The decline in the number of native
Hawaiians from a primative population estimated at over 250,000 to a population at annexation of approximately 30,000 and the desire for
82 Lavin, p. 131, citing Fred L. Strodtbeck, "Family Interaction, Values and Achievement," in David C. McClelland, Talent and Society (Princeton, N.J.: D. Van Nostrand, 1958), pp. 135-94. 83 Lavin, p. 131.
Andrew W. Lind, Hawaii1s People (Honolulu: University of Hawaii Press, 1980), p. 97. 53 cheap labor on the plantations prompted the enterprising, European and
American planters in concert with the Hawaiian government to bring nearly 200,000 immigrants to the islands beginning in the 1850's. The majority of these aliens were orientals, the Japanese being predominant.
Table 6 shows the dramatic decline in the numbers of pure Hawaiians and the change in the ethnic stock of Hawaii's population for roughly each thirty-year period from 1853 to 1980.
TABLE 6
HAWAII'S ETHNIC STOCK: 1853 TO 1980
1980 Ethnic stock 1853 1890 1920 1950a Number %
All races . 73,137 89,990 255,912 499,769 831,864 100.0
Hawaiian .... 70,036 34,436 23,729 12,245 9,366 1.1 Part Hawaiian. 983 6,186 18,027 7 3,845 164,086 19.7 Caucasian .... 1,687 18,939 49,140 114,793 178,918 21.5 Chinese ..... 364 16,752 23,507 29,501 46,716 5.6 Japanese .... - 12,610 109,274 180,508 215,930 26.0 Korean ...... - - 4,950 5,111 11,300 1.4 Filipino .... 5 - 21,031 53,382 98,889 11.9 Negro ...... -- 348 1,928 3,304 0.4 Puerto Rican . - - 5,602 6,944 5,719 0.7 Samoan ...... ---- 9,095 1.1 b Mixed ...... -- - 20,336 79,202 9.5 Other ...... 62 1,067 310 1,176 9,339 1.1
SOURCE: Data for 1853-1920 from Lind, p. 28; data for 1950 from U.S. Census of Population: 1950, Bulletin P-C52, pp. viii and 38 cited in Department of Planning and Economic Development, The State of Hawaii Data Book: A Statistical Abstract (Honolulu: State of Hawaii, 1972): 19, table 12; and 1980 from Department of Health and Department of Planning and Economic Development, Population Characteristics of Hawaii, 1980 (Population Report, No. 13, October 1981): 6, table 2. 'Partly estimated for Koreans, Negroes, Puerto Ricans, and others. Mixed category does not include Part Hawaiians. 54 Vaughan MacCaughey, an American educator at the newly-founded
(1907), land-grant College of Hawaii, was appalled at the magnitude of the social, economic, and educational problems of "establishing American ideals among a population that is largely male, alien, homeless, land less, and non-English-speaking."85 In 1910, according to MacCaughey,
57 percent of the population of the Territory of Hawaii could not speak
English, and more than one-quarter of the population over the age of ten were illiterate.88 To the lasting credit of its educators, Hawaii was able to reduce its illiteracy rate from a high of 51 percent in 1890 to
1.9 percent by 1970 and, by 1980 to raise the median years of school completed by persons over twenty-five to 12.7 years with 20.3 percent of 87 these residents completing four or more years of college. How much of this improvement was due to the zeal of the educators and how much to the ethnic thirst-for-knowledge characteristic of the oriental, 40 per cent of the population is unknown. Because of the large percentage of
Oriental-Americans, studies dealing with Hawaii's school subpopulations, which mirror the racial percentages in the parent population, should take into account the possible effect of racial differences.
Examining the correlations between 38,681 freshman GPA’s and corresponding student's race, religion, and socioeconomic background before and after controlling for high school grades, academic aptitude,
85 Vaughan MacCaughey, "The Racial Element in Hawaii's Schools," Education 39 (January 1919): 289.
86 , . _ Ibid. 87 U.S. Department of Commerce, Bureau of the Census, Statistical Abstract of the United States: 1984 (Washington, D.C.: Government Printing Office, 1983): 145, table 224; and Robert C. Schmitt, Historical Statistics of Hawaii (Honolulu: University Press of Hawaii, 1977): 229, table 9.11. 55 and college selectivity, Astin drew the following conclusions:
White students of both sexes obtain higher freshman GPA's than Negro students. These differences are entirely attributable to the better high school grades and higher aptitude test scores of the white students. Orientals obtained the same average freshman GPA as did the non-Oriental students. In short, differences between the races in their average academic performance during the freshman year in college are entirely attributable to differences in ability and past achievement, not to any effects of race per s e . . . . In summary, the student's race, religion, and his parents' educational and economic status have only very small effects on his QQ academic achievement during the freshman year in college.
Student Personal Characteristics
Studies have demonstrated statistically significant increases in prediction beyond high school grades and national test scores for certain personal characteristics of students obtained through biographical data inventories (biodata), recommendations, interviews, and interest or personality measures.
Fishman and Pasanella reviewed twenty-three studies in which biodata were used to predict college grades. They found predictive correlations ranging from .01 to .63, with a median of .13; but most of these studies had not used other available predictors in order to find 39 the incremental value of biodata. In a summary of the incremental validations reported in five studies between 1960 and 1973, Breland
reported predictive correlations from .53 to .70 with increments ranging 90 from .01 to .16 due to use of biodata. After controlling for high
88 Astin, p. 14. 89 Fishman and Pasanella, p. 302. 90 Hunter M. Breland, Assessing Student Characteristics in Admissions to Higher Education (New York: College Entrance Examination Board, 1981): 8, table 3. Examples of biodata factors included indepen dence, socioeconomic status, class or club officer, honor society member, intellectualism, health, class size, scholarship, and high expectations. 56 school grades, aptitude test scores, and college selectivity, Astin examined over one hundred personal characteristics and found that thir teen of these improved the correlation with freshman GPA from .54 to .59 for men, and from .58 to .61 for women. The statistically significant, positive correlates were scholastic honor socity membership, private high school attendance, and high self-ratings on drive to achieve and on academic ability.^
Both recommendations, in the form of ratings and letters of reference, and interviews have been used extensively for employment and academic admissions. However, both the reliability and validity of each of these as predictive measures are questioned because of their subjec tivity. On the other hand, interest measures such as the Kuder
Preference Record, the Strong Vocational Interest Blank, and the newer
Strong-Campbell interest inventory have high reliability and validity, according to Breland, but yield lower correlations, .05 to .40, than 92 personality measures. In Fishman and Pasanella's review of twenty- six studies which used personality measures such as the Rorschach,
Minnesota Multiphasic Personality Inventory, and the Manifest Anxiety
Scale, they reported correlations ranging from .01 to .62 with median 93 correlation of .26 However, Lavin's more extensive review concluded
. . . many of the relationships . . . are tenuous at best, and it is undoubtedly true that the state of knowledge regarding the relation between personality variables and achievement is still so tentative that it cannot be used confidently for practical purposes, such as for college admissions
91 Astin, pp. 12-13. 92 Breland, Assessing Student Characteristics, pp. 24-28. 93 Fishman and Pasanella, p. 303. 94 Lavin, p. 101. 57 Combining predictors such as grades, test scores, and one or more of the procedures used to obtain student personal characteristics yield higher correlations for academic criteria than using any one predictor alone. The most extensive assessment summarizing studies of multiple prediction was done by Breland, who chose to separate academic and non- academic outcomes as much as possible. In twenty-four studies using multiple prediction of academic outcomes, he found correlations ranging from .22 to .81j and in a similar comparison of nonacademic outcomes, 95 the correlations ranged from .08 to .67 in nineteen studies.
TABLE 7
PREDICTIVE VALIDITY COMPARISONS OF SELECTED PROCEDURES
Median Predictive Correlations
Procedures Academic Nonacademic Outcomes Outcomes
Biodata ...... 43 .35 Recommendations .... .20 .18 Interviews ...... 15 .10 Interests measures . . .12 .26 Personality measures .28 .13 Combined procedures3 .60 .29
SOURCE: Hunter M. Breland, Assessing Student Charac teristics in Admissions to Higher Education (New York: College Entrance Examination Board, 1981): 54, table 19.
includes grades and test scores.
Table 7 shows predictive validity comparisons of the various procedures for both academic and nonacademic outcomes. Nonacademic outcomes include "faculty ratings of medical students and resident
95 Breland, Assessing Student Characteristics, pp. 47-52. 58 interns on clinical performance, scientists' income, creative achieve- 96 ment, peer ratings, and extracurricular activities in college." This
table illustrates clearly the superiority of combined procedures which
use statistical techniques of regression in predicting academic outcomes,
over the combination of the same types of data made in the clinical or
subjective sense (as in recommendations and interviews) for nonacademic outcomes. Of the five procedures considered alone, only biodata and interest measures have generally high reliability, validity, and accep tability; and when used in the combined procedure, only biodata promises
any substantial increase in predictive ability.
Differential Prediction in Mathematics
Following Horst's lead, an investigation by Fisher determined
that for the differential prediction of first-year grades in mathematics,
science, English, and social studies, the most effective predictive 97 variables were the parallel content areas of the predictive tests.
At a women's college, Gussett computed the correlations between
final grades in freshman mathematics and SAT-Verbal, SAT-Math, and SAT-
Total. Not surprisingly, he found the correlations to be .48, .62, and 98 .63, respectively. in a second study six years later, he investigated
the predictive validity of the SAT subscores and corresponding College
Board Achievement test scores in predicting the same College Level
96 Ibid., p. 47. 97 Joseph T. Fisher, "The Value of Tests and Records in the Prediction of College Achievement," (Ed.D. dissertation, University of Nebraska Teachers College, 1955), DA (1955), 2097. 98 James C. Gussett, "College Entrance Examination Board Scholas tic Aptitude Test Scores as a Predictor for College Freshman Mathematics Grades," Educational and Psychological Measurement 34 (Winter 1974): 954. 59 Examination Program (CLEP) subject area tests. His subjects were three hundred women who had taken the SAT's and at least one similar College
Board Achievement and CLEP subject area test during the six-year period
1973-1978. The SAT-M and Math-ACH scores yielded substantial corre lations (.79 and .77 respectively) with the CLEP test scores in College
Algebra and Trigonometry; but the correlations in English and American 99 History were much more modest.
Tittle, Weiner, and Phelps were concerned with the validity of the CLEP Mathematics and English Composition test scores in awarding college credit at the City University of New York, Satisfactory validity was obtained for the CLEP Mathematics test which correlated .64 and .62 with raw scores and grades on a faculty-constructed, final examination in the first-year mathematics course. However, the authors found "little relationship between CLEP English Composition scores and present college placement procedures for first-year English.
Omizo examined the predictive validity of the Differential Apti tude Tests (DAT) using GPA and senior grades in engineering, mathematics, and science as criteria, and the differential validity of the eight DAT scales in comparison of high and low achievers at a high school for engineering. Of the eight DAT scales, clerical speed and accuracy (CSA) and abstract reasoning (AR) had the highest correlates (.60 and .47 respectively) with the mathematics grade criterion. In the two-group
99 James C. Gussett, "Achievement Test Scores and Scholastic Aptitude Test Scores as Predictors of College Level Examination Program Scores," Educational and Psychological Measurement 40 (Spring 1980): 213-18.
^^Carol K. Tittle, Max Weiner, and Fred D. Phelps, "Validity of Awarding College Credit by Examination in Mathematics and English," Educational and Psychological Measurement 35 (Summer 1975); 455. 60 discriminant analysis using the eight DAT scales, all but the mechanical reasoning scale were valid discriminators between high and low achievers with a canonical correlation of .88, significant at the .001 level.
The four previous studies have indicated the contribution made by the mathematical components of national aptitude and achievement tests to the prediction of success in first-year college mathematics. The remaining studies, while still predicting success in freshman mathematics
courses, will be grouped according to whether the study population con
sists of ordinary freshmen or of those who sought advanced standing
because of previous college work done at the high school level.
Studies predicting success in mathematics for ordinary college freshmen
The combination of the high school scholastic record, as measured
by HGPA or rank-in-class, and national test scores has proved to be an
effective predictor for general academic success in college, but more
specific measures are required for improving the prediction of success
in college mathematics courses. As early as 1959, Spindt suggested such
specific measures at a conference held in Berkeley, California;
Specific subject grades sometimes predict specific academic successes or failures better than rank-in-class or average grades do. This seems to be true, however, only in the fields where preliminary courses are prerequisite, or at least propaedeutic to college courses (e.g., mathematics for physical science, or mathematics for advanced mathematics courses). It does not seem to be true, at least in the same degree, in the humanities and social sciences.
Michael M. Omizo, "The Differential Aptitude Tests as Predic tors of Success in a High School for Engineering Program," Educational and Psychological Measurement 40 (Spring 1980); 197-201. 102 Herman A. Spindt, "Improving the Prediction of Academic Achievement," Selection and Educational Differentiation, Report of a Con ference at Berkeley, California, May 25-27, 1959, Field Service Center and Center for the Study of Higher Education, University of California (Berkeley; University of California, 1959); 21. 61 Wick (1963)
Some aspe.ct of the high school mathematics record consistently gave the highest correlation. The best multiple regression equations yielded correlations ranging from .301 to .899, with a median value of .642. There was neither a significant nor a consistent difference between the sexes.
These were three conclusions drawn by Wick in a large study of
1,692 freshmen at six Minnesota and Wisconsin colleges. Course grades
and examination scores in eleven, first-semester mathematics courses at
these institutions were used as criteria of success. Based on the high
est correlates with these criteria, five predictor variables were chosen
for each course from among the following: high school average mathematics
grade, number of semesters in mathematics, rank-in-class, scores on a
mathematics placement and aptitude test, and a scaling measure of high
school quality, quantified by using each school's mean score of its stu
dents on the Minnesota Scholastic Aptitude Test. Testing all thirty-one
combinations of the five selected predictor variables, the regression
equation yielding the highest multiple correlation was developed for
predicting success in each course, by sex. The conclusions followed.
Troutman (197 8)
The final grade in a finite mathematics course (FMG) for 123
freshmen was used as criterion by Troutman to determine the predictive
validity of intelligence quotients (Stanford-Binet IQ), high school rank
(HSR), SAT-Math, and high school mathematics grade-point-average (HSMG).
In a stepwise, multiple linear regression analysis, Troutman obtained a
correlation coefficient of .611 for the full set of predictors with the
103 Marshall E . Wick, "A Study of the Factors Associated with Achievement in First-Year College Mathematics," (Ph.D. dissertation, University of Minnesota, 1963), DA 24 (1963), 1891. 62 best order of stepwise entry being SAT-M (.50), SAT-M and HSR (.602),
SAT-M, HSR, and HSMG (.610). Addition of the factor IQ to the three- predictor regression equation contributed little incrementally (.001)j 104 but all multiple correlations were significant at the .01 level.
Spranke1 (1976)
Sprankel evaluated the significance of placement tests and other predictor variables with respect to success as measured by the grades received in the six mathematics courses in which a student could place.
Aside from the mathematics placement test score (MPT), the other pre dictor variables included high school rank (HSR) and mathematics grade- point-average (HSMG), and the ACT English (ACTE), Mathematics (ACTM),
Natural Science (ACTS), and Composite (ACTC) scores. Stepwise regression was used to develop equations for each of the six mathematics courses.
These equations, which included all significant predictor variables for grades in a particular course, had multiple correlations ranging from
.375 to .654. Sprankel found that the best single predictor was the mathematics placement test score (MPT) which appeared in first or second place in five of the six regression equations. MPT was followed in importance by both the high school measures, HSMG and HSR, since each appeared in first or second place three times; and these by ACTM, which occupied third place three times in the six regression equations
104 James G. Troutman, "Cognitive Predictors of Final Grades in Finite Mathematics," Educational and Psychological Measurement 38 (Summer 1978): 401-4. 105 Charlene M. Sprankel, "The Validity of Placement Tests and Other Predictor Variables in the Placement of Students in Beginning Mathematics Courses at Southern Illinois University at Carbondale," (Ph.D. dissertation, Southern Illinois University, 1976), DAI 37 (1976), 3575a . Turner (1968), Farmer (1970), and Knauss (1975)
In separate studies Turner, Farmer, and Knauss used essentially the same predictor variables; but because their study populations and some criteria were disparate, their conclusions regarding the best predictor variables were different. Common predictor variables in these studies were the ACT subscores, previous mathematics grades, and the following high school measures: rank-in-class (HSR) or overall grade- point-average (HSGPA), mathematics grade-point-average (HSMG), and the number of semesters of mathematics (HSMS). Knauss was the only one to use scores on a locally-designed, mathematics placement test (MPT).
Knauss's criteria were final grades in Intermediate Mathematics,
College Algebra, Trigonometry, Analytic Geometry, and Calculus I for all freshmen enrolled in these courses at Northern Michigan in 1971. The MPT score proved to be the best single predictor in the first three courses; but the English ACT score was reported to be best for Analytic Geometry, and the previous (eighth semester) mathematics grade for Calculus I.
Although the number of semesters of high school mathematics (HSMS) was not a significant predictor in any of Knauss’s five multiple regression equations, the mathematics grade-point-average (HSMG) proved significant 106 in College Algebra, Trigonometry, and Calculus I .
Farmer's study population was the set of students who began their college mathematics with College Algebra, Trigonometry, Integrated Algebra and Trigonometry, or Analytical Geometry and Calculus, part I . These four courses began four possible sequences ending with Analytical Geometry and
106 Thomas L. Knauss, "The Mathematics Placement Program for Five Freshman Level Mathematics Courses at Northern Michigan University," (Ph.D. dissertation, University of Michigan, 1975), D M 36 (1976), 6257a . 64 Calculus, part III. Criteria were final grades in the six courses; but subsequent courses in a sequence had previous college course grades as additional predictors. Without a mathematics placement test (MPT) as a variable, Farmer found that the best, single predictor in three of the four beginning courses was the mathematics grade-point-average (HSMG); in Integrated Algebra and Trigonometry, the best was the ACT mathematics subscore; and in any subsequent course in a sequence, the best predictor 107 was the grade in the previous college mathematics course.
Turner's criteria were final grades in Calculus I and Abstract
Algebra, and Calculus I, II, and III GPA and college major GPA for 169 college graduates who had majored in mathematics. The mathematics grade- point-average (HSMG) played an important role in all four of the multiple regression equations in this study; but Turner found the best, single predictor to be HSGPA in three equations and HSR in the fourth. Unlike its absence in Knauss's results, the number of semesters of high school mathematics (HSMS) ranked second in predicting both the Abstract Algebra grade and the Calculus I, II, and III GPA. The ACT Natural Science sub score played a tertiary role, yet still significant, in the multiple 108 regression equations for Calculus I and college major GPA.
Prouse and Turner (1969)
Calculus II was the single criterion in a study by Prouse and
Turner, who considered the same predictor variables as Farmer and Turner,
107 Loyal Farmer, "The Predictive Validities, as Measured by Multiple Correlation, of Certain Mathematics Grades and a Test Battery Using Academic Achievement as Criteria," (Ed.D. dissertation, North Texas State university, 1970), DAI 32 (1971), 1850A. 108 Veras D. Turner, "Prediction of Success as a Mathematics Major at the Minnesota State Colleges," (Ph.D. dissertation, University of Oklahoma, 1968), DA 29 (1969), 2099A. 65 but added the nominal variables which indicated high school class size
and intended major. Significant high school correlates with Calculus II
at the .05 level included rank-in-class (HSR), all four mathematics
courses, and the ACT Mathematics subscore. Higher college correlates
with calculus II were found with the previous mathematics course grades:
Calculus I (.67), College Algebra (.51), and College Trigonometry (.51).
The order of entry for the first six predictor variables in multiple
regression was Calculus I, College Algebra, high school class size,
College Trigonometry, Algebra I, and Plane Geometry. "This combination
yielded a multiple correlation coefficient of .76 as compared to .78 109 using all 15 predictor variables."
These studies have indicated that more specific measures than the
general ones of overall high school achievement and national test scores
are required to improve the prediction of success in college mathematics
courses. Mathematics placement tests (MPT), high school mathematics
grade-point-average (HSMG), and previous mathematics grades have been
used successfully to improve this differential prediction. Where the
goal at a given college is the selection of the proper pre-calculus or
initial calculus course for an incoming freshman, the position of the
mathematics placement test (MPT) in predicting success is preeminent.
However, once the level of entry has been established and an initial
mathematics course grade becomes available, it becomes the best predictor
of success in the next sequential course. Even in the higher level
mathematics courses, a significant contribution to prediction of success
is still made by the high school mathematics grade-point-average (HSMG).
109 Howard Prouse and Veras D. Turner, "Factors Contributing to Success in Calculus II," The Journal of Educational Research 62 (July- August 1969): 439-40. 66 Studies predicting success in mathematics for advanced standing students
Advanced standing refers to college programs which permit an entering freshman, on the basis of his previous record or examination, to bypass certain courses, for which he may or may not receive credit, and to be placed in a more advanced course than is usual. The term is not synonymous with, but does include, the Advanced Placement Program (APP) in Mathematics, which refers specifically to the College Board's dual course program of a full year of calculus taught in the high school, for which most colleges grant credit and advanced placement. The variety of options open to the student of advanced standing and its changing nature can be seen in table 8, where the percentage increase of institutions which award college credit is particularly dramatic between 1965 and 1970.
TABLE 8
ADVANCED STANDING PROGRAMS BY TYPE INSTITUTION AND YEAR
Percentages Universities Public Colleges Private Colleges
Institutions 1960 1965 1970 1960 1965 1970 1960 1965 1970
Having Advanced Standing Programs (including APP) 95% 95% 95% 53% 82% 73% 63% 92% 96% Having Such Programs Which Grant Credit for 1. College Algebra & Trig 40% 30% 34% 34% 31% 99% 29% 20% 95% 2. Analytic Geometry 35% 34% 29% 28% 19% 22% 27% 20% 4% 3. Calculus 46% 44% 95% 12% 19% 62% 20% 25% 68% a 4. Courses above Calculus — 53% —— 16% —— 7%
SOURCE: Conference Board of the Mathematical Sciences, Under graduate Education in the Mathematical Sciences, 1970-71, by John Jewett and C. Russell Phelps with the technical assistance of Clarence B. Lind quist, Report of the Survey Committee, Volume IV (Washington, D.C.: Conference Board of the Mathematical Sciences, 1972), p. 64, table 4.7.
aBlanks indicate that this data was not collected prior to 1970. 67 For the following 1970-75 period, over 75 percent of the insti tutions reported a decline in student training and ability due, they sur mised, to poorer secondary school preparation, lack of student interest in mathematics, and lower admission standards. This decline prompted more colleges to require national test scores for admission and to use local placement examinations for the algebra and trigonometry level.
Between 1970 and 1975, the use of mathematics placement examinations increased at universities from 57 to 74 percent, at public colleges from
68 to 72 percent, and at private colleges from 37 to 53 percent.
Not all of the college respondents to the Conference Board were as pessimistic. A substantial number reported an improvement in student mathematical training, "suggesting a pattern observed elsewhere that 'the best have gotten better, but the balance weaker'The Conference
Board of the Mathematical Sciences concluded,
The exception to this pattern is the advanced placement testing program. Nearly all institutions have programs of advanced standing in mathematics. . . . In the great majority of these schools calculus is the course for which college credit may be entered on the student's record. But a substantial number allow credit for college algebra and/or trigonometry
The studies conducted on advanced standing students reflect both the concern of mathematics educators for the proper placement of the high school senior and college freshman, and the evolution of the Advanced
^^Conference Board of the Mathematical Sciences, Undergraduate Mathematical Sciences in Universities and Four-Year Colleges, and Two-Year Colleges, 1975-76, by James T. Fey, Donald J. Albers, and John Jewett with the technical assistance of Clarence B. Lindjuist, Report of the Survey Committee, Volume V (Washington, D.C.: Conference Board of the Mathema tical Sciences, 1977), pp. 63-64 and table 4.2.
m ibid., p. 63. 112 Ibid., p. 65. Placement Program in Mathematics from its meager, controversial origin by twelve colleges and eighteen participating schools by the time the 120
candidates took the first calculus examination in 1954, to its present accepted status which, in 1983, involved almost 35,500 candidates from 113 about 4,200 high schools seeking entrance to nearly 1,500 colleges.
This increased involvement of high schools and colleges forced more and more mathematics educators to face the problem of proper advanced placement. Several early studies compared the college achievement of those who had had some calculus in high school with those who had not.
Tillotson (1962)
This University of Kansas investigator sought to determine the difference in achievement between freshmen who had studied calculus for two to twelve weeks in high school and those who had not. Criteria of achievement were the first-semester, final examination score and final grade in Calculus and Analytic Geometry I . Regression analysis, using
normalized high school^rank-in-class and mathematics placement test ' •*
scores, yielded a multiple R of .655 in the prediction of the final examination score. Two analyses of covariance were made, one for each
criterion. In each case Tillotson found no significant difference in achievement between those who had been introduced to calculus in high school and those who had not. He concluded that no recommendation could be made either supporting or condemning a short study of calculus in high
113 Donald B. Elwell, "A History of the Advanced Placement Program of the College Entrance Examination Board to 1965," (Ed.D. dissertation, Columbia University, 1967), p. 309; College Entrance Examination Board, "What Happened to Them in College," College Board Review 28 (Winter 1956) 7; and The College Board, Advanced Placement Course Description; Mathe matics, May 1985, May 1986 (New York: College Entrance Examination Board, 1984), p. 29. 69 114 school on the basis of its later effect on college achievement.
Crosswhite (1964)
At Ohio State university, Crosswhite sought to improve the exis ting placement procedures for advanced standing students in mathematics.
He used multiple regression analysis on fifteen variables to predict both an achievement and a success criterion. The sum of the scores on two examinations in the initial analytic geometry and calculus course was the
criterion of achievement; and the final course grade, the criterion of success. The regression population was also examined for the incidence variables sex, college, and whether or not the high school preparation
had included at least one semester of analytic geometry or calculus.
Both maximum and optimum prediction equations were derived for each cri
terion. The optimum prediction equations produced multiple correlations
of .531 with the achievement criterion and .436 with the criterion for
success, using the mathematics placement test score and grade-points in high school mathematics as predictor variables. Only these two predic tors were retained in the recommended placement procedure.
Crosswhite found that first-quarter freshmen admitted with advanced standing to calculus did significantly better (at the .01 level) than continuing students in the same course, whether the criterion was
achievement or success; and that their superiority in grade-point-average
was maintained through the following calculus course.^5
114 Donald B. Tillotson, "The Relationship of an introductory Study of Calculus in High School to Achievement in a University Calculus Course," (Ph.D. dissertation, University of Kansas, 1962), DA 24 (1963), 577. 115 F. Joe Crosswhite, "Procedures for Admission with Advanced Standing in Mathematics at The Ohio State University," (Ph.D. disser tation, The Ohio State University, 1964), pp. 47-173. McKillip (1965)
In a study similar to Tillotson's, McKillip investigated the effects of either one or two semesters of high school calculus on the first-semester grades of students who were repeating calculus in college.
From the parent population of 1963 University of Virginia freshmen, the subgroup which had taken no calculus in high school was used to derive separate regression equations for public and private school graduates.
Their first-semester calculus grade was used as criterion; and their high school rank-in-class (HSR), mathematics grade-point-average (HSMG), and
College Board scores on the SAT-Math, SAT-Verbal, and mathematics achieve ment tests were used as independent variables. These equations were used to predict, for each of the eighty-three subjects who had not been given advanced placement for the one or two semesters of high school calculus, the grade he might have earned had he not taken any calculus previously.
The signed differences between actual and predicted grade for each sub
ject were tested for significance with a Wilcoxon matched-pairs, signed- rank test; and the mean differences, with a t-test.
McKillop concluded that a two-semester calculus course in high
school makes a significant contribution to the improvement of first-
semester calculus grades in college when advanced placement is not given;
but that a one-semester, high school course has no significant effect.
He also found that public and private high schools offer equally effec-
txve courses xn• calculus. i i 116
12.6 William D. MCKillip, "The Effects of High School Calculus on Students' First-Semester Calculus Grades at the University of Virginia," The Mathematics Teacher 59 (May 1966): 470-72; and William D. MCKillip, "The Effects of Secondary School Analytic Geometry and Calculus on Stu dents' First Semester Calculus Grades at the University of Virginia," (Ph.D. dissertation, University of Virginia, 1965), DA 26 (1966), 5920-21. 71 Robi nson (19 6 8)
Robinson repeated McKillip's study design, but restricted his study population to graduates from five, selected Utah high schools over a three-year period. His null hypothesis proposed that a two-semester calculus course in high school would have no significant effect on first- and second-quarter calculus performance at the University of Utah. Final grades in the first two quarters of college calculus were criteria. For each high school, regression equations were calculated based on data from the control group of students who had taken Analytic Geometry, but no calculus, in high school. Predictor variables included ACT English and
Mathematics subscores, and the following high school measures: Analytic
Geometry grade (AG), rank-in-class (HSR), and mathematics grade-point- average (HSMG). These equations were used to determine predicted grades in first- and second-quarter calculus for the study group of graduates from the same high school, who were repeating their calculus in college.
The Wilcoxon matched-pairs, signed-rank test was applied to the signed differences between actual and predicted grades.
Robinson rejected his null hypothesis at the five percent level, and claimed that two semesters of high school calculus has a significant effect on the achievement in first- and second-quarter college calculus 117 of students who must repeat these courses.
Shimizu (1969)
At the University of Hawaii, Shimizu compared the achievement in two freshman courses (pre-calculus and Calculus I) of 124 graduates of
117 William B. Robinson, "The Effects of Two Semesters of Secondary School Calculus on Students' First and Second Quarter Calculus Grades at the University of Utah," (Ed.D. dissertation, University of Utah, 1968), DA 29 (1968), 2990-91. 72 seven, Oahu public schools who had studied four different courses in
advanced mathematics as seniors. She surmised that preparation in calcu
lus would result in higher freshman mathematics achievement than courses
in analytic geometry, probability, or survey of modern mathematics.
National test scores on four Cooperative Mathematics Tests (OOOPI-IV),
Sequential Tests of Educational Progress (STEP), School and College Abil
ity Test (SCAT), and SAT-Math were obtained for the study population of
advanced mathematics students and a randomly selected, control group of
nonadvanced mathematics students from the single high school which had
offered calculus. The summary statistics for the study and control group
of students from the one school revealed only twenty-one members in each
subgroup, but that the advanced mathematics study group had consistently
higher means and smaller standard deviations on all national tests than
the control group. Shimizu observed significant correlations between
grades in college pre-calculus and high school analytic geometry (.71),
and between grades in Calculus I and grades in high school calculus and
survey of modern mathematics (.81 and .68, respectively).
On the basis of her study, Shimizu was unable to conclude that
high school calculus students achieve significantly higher grades in
first-year college mathematics than students with preparation in other
advanced mathematics courses. However, this generalization is suspect;
because her evidence depended on the insignificant differences between
correlations involving only two preparations, a single school, and a
calculus sample of only nine students. This latter fact Shimizu acknowl- 118 edged by recommending replication with a larger sample.
118 Mildred T. Shimizu, "Achievement in Senior Advanced Mathe matics and First-Year College Mathematics," The Mathematics Teacher 62 (April 1969): 311-15. 73 Wight (1969)
Wight conducted a state-wide search for all Utah high schools teaching a full year of advanced placement calculus, and investigated the achievement of their 1965-67 graduates in first-, second-, and third- quarter calculus at the University of Utah. Tested were the hypotheses that students with a full year of high school calculus would not earn grades in any of the three quarters of college calculus significantly different from grades they might have earned had they not taken this course in high school. Multiple regression techniques were employed; grades in the three quarters of college calculus were dependent variables, and the following high school measures were independent variables: back ground in calculus or not, grade-point-average in mathematics (HSMA) and overall (HSGPA), ACT-Math subscore, and a coding variable to designate one of the seven, contributing high schools.
Wight determined that the best predictor for success in first- quarter college calculus was taking high school calculus; HSGPA was also significant. The best predictors for success in both the second- and third-quarters was not taking high school calculus but the school atten- 119 ded and HSGPA, with certain interaction terms also being significant.
Paul (1970)
At Ohio State University, Paul also compared the achievement in a first-quarter calculus course of students grouped according to their high school preparation. 526 freshmen were partitioned into five groups: one year of calculus, one semester of calculus, only a short unit of calculus,
119 Theodore A. Wight, "An Analysis of the Advanced Placement Pro gram in Mathematics in the State of Utah," (Ed.D dissertation, University of Utah, 1969), DAI 30 (1970), 4270B. 74 analytic geometry and no calculus, and neither analytic geometry nor
calculus. Each of these groups was again partitioned by regression analysis into condition groups with similar expectancies of success, where success was measured by the final first-quarter grade. Analysis of variance was used to detect significant differences in achievement among
condition groups. Achievement was measured by the scores on four midterm examinations a :d the final. Paul found a significant difference in mean
achievement, favoring those groups with a high school calculus back
ground; and he concluded:
In brief, students with a year or semester of high school cal culus as background preparation achieved higher than those without that preparation. The differences in achievement, apparent early in the quarter, appeared to diminish as the quarter progressed.
Bingham (1972)
Bingham investigated the contribution made by the College Board's
Level I Mathematics Achievement Test (ACHMI) to the placement of entering
freshmen and their success, as measured by final grades, in the beginning
calculus course at the university of Texas at Austin. Multiple linear
regression analysis was used to test the significance of the difference
between means of three student groups: (1) entering freshmen, in the fall
of 1970, enrolled in beginning calculus as their first college mathematics
course; (2) freshmen from the same class who enrolled in beginning cal
culus after taking a pre-calculus, elementary functions course; and (3)
entering freshmen the following year on whom the prediction equations,
derived from the first two groups, were tested. Besides ACHMI, input data
on each student included scores on SAT-Math, SAT-Verbal, Achievement in
120 Howard W. Paul, "The Relationship of Various High School Mathe matics Programs to Achievement in the First Course in College Calculus," (Ph.D. dissertation, The Ohio State University, 1970), p. 104. 75 English Composition (ACHE), sex, and high school rank-in-class (HSR).
Bingham found a significant difference between the two 1970 groups in the mean course grades in beginning calculus, favoring those who began their calculus immediately upon entry over those who took pre-calculus first.
Because a significant interaction with ACHMI was found to exist between these two groups, an ACHMI cutoff score was established for placement purposes. As an alternative placement procedure, a linear regression model using the other predictors was constructed and tested for effec tiveness using classification tables. A practical test of the relative effectiveness of these procedures was conducted on freshmen the next year.
As a consequence of this test, Bingham determined that students with lower ACHMI scores probably benefited from having enrolled in the pre-calculus, elementary functions course prior to beginning calculus; but that the multivariate regression equation was a more effective 121 predictor of success in beginning calculus than ACHMI cutoff scores.
Sommers (1973)
In a study similar to Crosswhite’s, and with the same purpose of determining a practical placement procedure for deciding which freshmen should take calculus, Sommers designated 133 freshmen, who were com pleting a calculus course as their initial college experience in mathe matics, as his regression population. Comparison groups with previous experience in college mathematics were chosen from the study population, which consisted of all students at Hope College enrolled in either the pre-calculus or beginning calculus course. Defining the initial calculus
121 Ralph L. Bingham, "An Investigation into the Relationship between Advanced Placement in Mathematics and Performance in First Semes ter Calculus at the university of Texas at Austin," (Ph.D. dissertation, University of Texas at Austin, 1972), DAI 33 (1973), 4865A. 76 course grade as the measure of success and the final examination score as an achievement criterion, maximum prediction equations for both success and achievement were derived with regression analysis. A multiple-R of .705 was obtained with the success criteria for the regression popu lation; and multiple correlations of .693, .699, and .678 were associated with the achievement criteria of the three sub-populations.
Sommers determined that the best, single predictor of success was the high school mathematics grade-point-average (HSMA). It accounted for approximately 39 percent of the variance in success explained by his optimal prediction equation. Two other predictors, the score on the HELP
(Hope Entrance Level Placement) examination and the sum of the SAT sub scores, each contributed about 27 percent of the explained variance. The optimal prediction equation with this three-variable combination had 122 corrected multiple correlations close to the maximum obtainable.
* Austin (1975)
At the University of Virginia, Austin studied the achievement in first-semester calculus of twenty-four freshmen who had taken calculus in high school. Achievement was considered in three areas: manipulative skills, problem solving skills, and theoretical concepts. Achievement was measured by the scores on five progress tests; and on each test, the subscores in the three areas were evaluated separately. The sum of these subscores was the overall criterion of achievement. Ranked by their individual SAT-Math scores, the students were then placed into upper and lower ability groups.
122 Dean D. Sommers, "A Study of Selected Factors Predictive of Success in Calculus at Hope College," (Ph.D. dissertation, The Ohio State University, 1973), pp. 105-113. 77 Analysis of variance techniques were used to derive Austin’s three findings: (1) Significant differences, between the mean scores of freshmen with high school calculus and those without this advantage, existed in overall achievement and in each of the three achievement sub areas; (2) There were significant differences in mean scores between upper- and lower-ability levels on all measures of achievement; and
(3) Interaction between ability level and time interval indicated a sig nificant change in the differential effect on theoretical concepts over 123 the period of the semester.
Dykes (1980)
Dykes employed Pearson product-moment correlation and stepwise regression techniques between five predictor variables and a criterion of success, measured by final grade, to formulate a prediction equation for success in college algebra. The stepwise order of entry into the pre diction equation was high school mathematics grade-point-average (HSMA),
Algebra II Cooperative Mathematics Test score, ACT-Math subscore, and overall high school grade-point-average (HSGPA). Although each of the five predictor variables had a significant correlation with the success criterion at the .05 level, the ACT-Composite score made no significant contribution to the prediction equation. A multiple-R of .743 with the college algebra final grade indicated that Dykes' first four, predictor 124 variables explained about 55 percent of the variation.
123 Homer W. Austin, "The Effects of High School Calculus on Achievement during the First Semester in College Calculus for First Year Students," (Ph.D. dissertation, University of Virginia, 1975), DAI 36 (1975), 2728A. 124 Isaac J. Dykes, "Prediction of Success in College Algebra at Copiah-Lincoln Junior College," (Ed.D. dissertation, The University of Mississippi, 1980), DAI 41 (1981), 4630A. O'Neal (1980)
A comparison of predictors to determine a Calculus I final grade criterion was made between one hundred, randomly selected students at the
University of Mississippi and a like number at a nearby junior college .
In both prediction equations, high school mathematics grade-point-average
(HSMA) and ACT-Math subscore were the first two entrants in the stepwise multiple regression. These were the only two predictors in the best ju nior college equation, which had a multiple-R of .43. The best equation for predicting university success was considered significantly different; its multiple-R was .613, and it also included the number of units of high 125 school mathematics (HSMU) and HSGPA as additional predictors.
Williams (1980)
Multiple regression analysis was used to assess the differences in Calculus I, II, and III achievement between junior college transfer and university, non-transfer (native) students. Predictor variables were grades in three pre-calculus courses and ACT subscores; achievement cri teria were grades in the three calculus courses. Williams found that, while ACT-Math and Composite scores were significant, the best predictor for Calculus I achievement was the grade in Analytic Geometry. There was no significant difference in Calculus II between transfer and native stu dents; but in Calculus III, native students did better. No significant 126 difference between male and female achievement existed in Calculus III.
125 Larry D. O'Neal, "A Comparison of the Predictors of Success of University and Junior College Students in the Initial Calculus Course," (Ph.D. dissertation, University of Mississippi, 1980), DAI 41 (1981), 4620A. 126 Raymond Williams, "A Study of Differences in Achievement in Pre-Calculus and Calculus Courses by Junior College Transfer and Non- Transfer Students at the University of Southern Mississippi," (Ph.D. dis sertation, University of Southern Mississippi, 1980), DAI 41 (1980), 1994A. 79 Descriptive studies of mathematics students in the Advanced Placement Program
The Advanced Placement Program (APP) was originated by a group of twelve colleges and twelve schools under the title The School and College
Study of Admission with Advanced Standing (The study). Its avowed pur pose was to improve American education by strengthening secondary schools so that they could provide their ablest students with the equivalent of first-year college work, for which colleges would allow credit and place ment into the next sequential course. It presumed school and college cooperation, able and motivated students, and knowledgeable teachers and administrators at both the high school and college level. Unfortunately for many of the participating students, one or more of these assumptions 127 did not hold; and students failed to receive advanced placement.
Scores of articles, dissertations, and other studies treating various aspects of the Advanced Placement Program have been written in order to better understand or further implement the program. Although
127 Perhaps it is understandable why so few of the first year's candidates in 1954 received academic credit or advanced course placement, since only 69 of those surveyed entered ten of the twelve Study colleges, while the other 337 students enrolled in 61 colleges which had not en tered into The Study's agreement to consider these students for advanced standing. Although all of The Study colleges and a few others receiving a substantial number of candidates awarded higher percentages of credit or advanced placement (e.g. 64 percent in mathematics), "of the 406 can didates, 20 percent received advanced placement and credit, 23 percent placement only, and 3 percent credit only." College Entrance Examination Board, "What Happened to Them in College," p. 7. It is less understandable why the program should be misunderstood five years later. In a follow-up study on the placement of 1,585 of the 1,870 candidates who took the 1959 Advanced Placement examination in mathematics, Vance and Pieters reported the following percentages, by examination grade, of those who had been awarded neither credit nor placement: fives, 4 percent; fours, 7 percent; and threes, 27 percent; yet 25 percent of the twos and 6 percent of the ones did receive credit or advanced placement'. E. p. Vance and R. S. Pieters, "The Advanced Place ment Program in Mathematics," The American Mathematical Monthly 68 (May 1961): 496. 80 not a founding member of the twelve colleges in The Study, Harvard, with its early policy of automatically awarding credit and advanced placement for a grade of three or better on the Advanced Placement examination, provided prestigious college leadership in recognizing the program’s worth and, in turn, became one of the leading college recipients of its 128 participants. In 1960, the director of the program of advanced stand ing at Harvard explained his college’s policies andexperiences with the 129 Advanced Placement Program.
Dissertation studies have covered many aspects of the Advanced
Placement Program at the secondary level: Kerr (1964) developed guide- 130 lines for establishing such a program; Ralston (1961) discussed the 131 identification of capable candidates; Kaloger (1970) suggested factors 132 which might influence student's success; Wilcock (1972) compared the
128 In 1954, The Study's first year, "Harvard University had eighty of the examinees, more than all the participating colleges com bined, and almost 20 percent of the total group in the study." Elwell, p. 95. In 1983, Harvard-Radcliffe received 2,805 examinations from 919 candidates, ranking it fourth behind University of California-Berkeley (4,114), University of Michigan (3,183), and Cornell university (2,999) in examinations received. Advanced Placement Program of the College Board, "Two Hundred Colleges and Universities Receiving the Largest Number of Advanced Placement Examinations from May 1983 Candidates," Princeton, 1983. (Mimeographed.) 129 Edward T. Wilcox, "Advanced Placement at Harvard," College Board Review 41 (Spring 1960): 17-20.
^3°Donald L. Kerr, "Establishing an Advanced Placement Program in the Public High School: Guidelines for School Personnel," (Ed.D. disser tation, Columbia University, 1964), DA 26 (1965), 4788. 131 Nancy C. Ralston, "A Study of the Advanced Placement Program in the Cincinnati Public Schools," (Ph.D. dissertation, Indiana Univer sity, 1961), m (1961), 3074-75. 132 Although Kaloger found IQ test scores to be statistically sig nificant indicators of success, extracurricular activities, chronological age, and sex had no relationship to success. James H. Kaloger, "Charac teristics of the Gross Point High School Students in Advanced Placement Programs," (Ph.D. dissertation, University of Michigan, 1970), DAI (1971), 6440A. 81 personality characteristics of effective and ineffective Advanced Place- 133 ment teachers; McGregor (1962) determined the significance of the 134 program to students, instructors, and counselors; and Felder (1965),
Maclay (1968), Montgomery (1968), and Hepp (1979) analyzed and evaluated the accelerated mathematics programs leading to Advanced Placement parti- 135 cipation. Three studies were concerned with the factors causing attrition in these accelerated, secondary mathematics programs} Morris
(1964) compared the characteristics of persisting and nonpersisting X 36 females; Helton (1964) did a similar study for males at the same 137 institution; and Turner (1980) studied the accelerated mathematics
133 Jack A. Wilcock, "The Relationship between Teacher Character istics and Student Success in the Advanced Placement Program," (Ed.D dis sertation, Utah State University, 1972), DAI 33 (1973), 5457a . 134 Warren M. McGregor, "The Significance of the Present Advanced Placement Program at Massapequa High School, Massapequa, New York— with Recommendations for its Future Development," (Ed.D. dissertation, Columbia University, 1962), DA 23 (1963), 4191-92. 135 Idus D. Felder, Jr., "The Advanced Placement Program in the High Schools of Fulton County," (Ed.D. dissertation, University of Georgia, 1965), DA 27 (1966), 700; Charles W. Maclay, Jr., "The Influence of Two Prerequisite Programs on Achievement in the High School Advanced Placement Calculus Course," (Ed.D. dissertation, University of Virginia, 1968), DA 29 (1969), 3917A; Warren G. Montgomery, "An Analysis and Appraisal of the Sioux City, Iowa, Secondary School Accelerated Mathema tics Program," (Ed.D. dissertation, University of South Dakota, 1968), DA 29 (1969), 2489A; and Donald A. Hepp, "The Effects of a Gifted Mathe matics Program on the Attitudes and Achievement of Secondary School Stu dents Identified as Academically Gifted," (Ph.D. dissertation, University of Pittsburg, 1978), D M 40 (1979), 1326a . X 3 6 Ruby P. Morris, "A Comparative Analysis of Selected Character istics of intellectually Superior Female Students Who Persisted and Those Who Did Not Persist in an Advanced Placement Program," (Ed.D. dissertation, North Texas State University, 1964), DA 25 (1964), 3402-3403. 137 William B. Helton, "A Comparative Analysis of Selected Charac teristics of Intellectually Superior Male Students Who Persist and Those Who Do Not persist in an Advanced Placement Program," (Ed.D. dissertation, North Texas State University, 1964), DA 25 (1964), 3394-95. 82 138 program, searching for means to reduce attrition.
Four studies dealt with administrative aspects of the program:
Clark (1961) analyzed the program's operation in light of a particular 139 school's objectives; Konde (1969) evaluated ninety-seven practices, advocated by educational authorities and used by principals to administer 140 the program in their schools; Linkhart (1968) investigated the methods 141 and procedures used to administer the program in six Arizona schools; and Beougher (1968) determined how various administrative, school, and 142 community factors were related to a successful program.
The remaining group of studies treats the Advanced Placement student in college. The first three are follow-up studies; but the remainder, which concern placement and comparative achievement, will be covered in greater detail.
Lefkowitz (1966) based her follow-up study on 182 responses from
1957-65 graduates of a large public high school, where each respondent
138 Jonathan Turner, "Factors that Affect Attrition in an Accel erated Secondary Mathematics Program," (Ph.D. dissertation, University of Iowa, 1980), DAI 42 (1981), 123A. 139 Donald M. Clark, "Selected Aspects of the Establishment and Operation of the Advanced Placement Program in the Orchard Park Central School: An Historical and Survey Analysis in the Light of School Objec tives, Criteria for the Program, and the Views of Selected Authorities," (Ed.D. dissertation, University of Buffalo, 1961), DA 27 (1961), 2245. 140 Anthony J. Konde, "Selected Practices Used in Administering the Advanced Placement Program in the Secondary Public Schools of the State of New York," Journal of Experimental Education 33 (Spring 1965): 263-76. 141 Bennie R. Linkhart, "Current Patterns in Selected Advanced Placement Mathematics Programs within the State of Arizona," (Ph.D dis sertation, University of Arizona, 1968), pp. 101-115. 142 Elton E. Beougher, "Relationships between Success of Advanced Placement Mathematics Programs and Various Administrative, School and Community Factors," (Ph.D. dissertation, University of Michigan, 1968), DAI 30 (1969), 195A. 83 had been in a mathematics Advanced Placement class. This select group, with mean IQ 144 and mean HSGPA 92.4, entered forty-five different col leges, where 52 percent of them were offered placement, but only 32 per cent received credit. Not until 1964 did all candidates with at least a three on the Advanced Placement examination receive placement or credit.
Although 44 percent of the respondents repeated first-semester calculus,
90 percent said that they would recommend the program to other students.
Lefkowitz thought that "an important shortcoming of the program, perhaps, 143 is the fact that so many received neither placement nor credit."
Brubacher (1967) examined the effects of the Advanced Placement
Program (APP) upon student academic experiences by making comparisons between APP and non-APP students, and among APP students partitioned by whether or not they had received academic credit. APP students rated the quality of their preparation significantly higher than did the non-APP
students. APP-Credit students felt that they had been placed accurately;
but a large percentage of APP-Non-Credit students complained of boredom 144 and duplication as a result of inaccurate placement.
Casserly (1968), a senior research assistant at Educational Test
ing Service, interviewed over fifty freshmen and 350 upperclassmen at twenty colleges to evaluate how Advanced Placement participation had
influenced their high school and college experiences. Hers was a general
study that covered all Advanced Placement offerings; but consensus of the
143 Ruth S. Lefkowitz, "A Study of the Advanced Placement Program in Mathematics at a Large New York City Public High School," (Ed.D. dis sertation, Columbia University, 1966), DA 27 (1967), 4025b . 144 Paul W. Brubacher, "A Study of the Effects of the College Entrance Examination Board's Advanced Placement Program upon Student Academic Experiences at the University of Michigan," (Ph.D. dissertation, University of Michigan, 1967), DA 28 (1967), 2472a . 84 students made specific recommendations possible: increase the number of course offerings available at the secondary level; improve the academic counseling system in college as well as in secondary school; reassess college placement policies and practices through local research on stu dents; and encourage all qualified students to take the appropriate 145 Advanced Placement examinations.
Two studies in the final group exemplify the college research being done on entering Advanced Placement candidates in order to strengthen placement policies with respect to these bright students. The first study is included because it does examine the academic performance in college of Advanced placement (AP) students, even though they may not all be AP mathematics students. However, the basic design of Bergeson's study was used as a model by several other investigators who dealt speci fically with Advanced Placement students in mathematics.
Bergeson (1967)
A study population of 108 matched pairs of 1962-64 freshmen at
Northwestern University was selected. In each pair, one student had been granted credit and advanced standing as a result of successful performance on an Advanced Placement examination; the other member of the pair was a regular progress (RP) student who had not been accelerated, but had taken the prerequisite courses in college, and was now matched with the AP stu dent on the basis of SAT subscores, sex, and participation in the grade criterion course. Tests revealed no significant differences between the
SAT subscore means of the AP and RP students. The Chi-Square Test was
145 Patricia L. Casserly, What College Students Say about Advanced Placement (New York: College Entrance Examination Board, 1968; reprinted from The College Board Review 69 (Pall 1968), 70 (Winter 1968-69)), p. 15. 85 used to determine whether a significant difference in the final grades of common courses existed between the accelerants and their RP mates.
Bergeson found that the difference in the academic performance between AP and RP students in the same course was not significant at the
.05 level. He concluded that "students who had participated in the
Advanced Placement Program while in high school and consequently did not take some preliminary courses in college did as well in subsequent 146 courses, as their regular-progress counterparts."
Wagner, Baker, Wagner, and Baker (1967)
These investigators reviewed the policy of allowing advanced standing credit at the University of California at Davis to students scoring at least three on any Advanced Placement examination (APE), while allowing neither credit nor advanced standing for APE scores below three.
Data from this study would be considered by the Admissions Office as a basis for possible changes in policy. The academic achievement of 174 students who had participated in the Advanced Placement Program from 1958 through 1965 was examined in four subject areas. Only the results in mathematics will be detailed here. Independent variables included APE score, APE GPA, subject area of APE, and sex. APE related first course grade, first course GPA, APE subject related GPA, and first semester GPA were the criteria. To report student achievement, the following descrip tive subgroups were established: (1) the high group received a course grade of A or B, or had a GPA of 2.8 or better (on a 4.0 scale); (2) the average group received a C as course grade, or had a GPA of 2.0 to 2.79;
146 John B. Bergeson, "The Academic performance of College Stu dents Granted Advanced Standing as a Result of Participation in the Advanced Placement Program," The Journal of Educational Research 61 (December 1967): 151-52. 86 and (3) the low group had a course grade of D or F, or a GPA below 2.0.
These investigators referred to an APE score of 1 or 2 as 'failing', and scores of 3, 4, or 5 as 'passing' the original test.
Sixty-three percent of those in the study population who had taken the APE in mathematics failed; this figure was almost twice the national percentage. But of those who failed the APE and had to repeat first-semester calculus, 83 percent received a grade of B or better and their average first-course GPA was 3.23. All of those who passed the APE in mathematics were in the high group of the next sequential course with an average first-course GPA of 3.90. These figures were significantly higher than the 2.19 average GPA of all students taking comparable mathe matics courses. Of the students scoring 1 on the APE, 50 percent were in the high group in first-semester calculus; and 100 percent of the students scoring 2 were in the same group. To the investigators, the last percen tages seemed to justify a policy change to pass those scoring 2 on the
APE in mathematics, but not to pass those scoring 1. Only in mathematics 147 was this recommendation made to the Admissions Office.
Ruch (1968)
As in Bergeson's study, Ruch employed a matched-pair design on students from the same high school to evaluate the freshman achievement and staying power of Advanced Placement (AP) students who attended simi lar colleges. One student in the pair had taken an AP course and the other had not, but was matched on the basis of sex, class rank within eighty-five places, and SAT scores within one standard deviation.
147 S. J. Wagner et al., "Advanced Standing Credit Based on the Advanced Placement Examination at the University of California at Davis," College and University 42 (Spring 1967)s 308-24. 87 The ten male and eleven female pairs attended twelve different colleges, where only six of the AP members were offered credit or placement.
Because the appropriate Chi-square Test was significant at the
.01 level, Ruch was able to claim that AP students were more inclined to
continue with the same subject during their first year in college than
non-AP students. However, when nine non-AP students failed to take the
next AP related course, he was unable to find a significant difference
between AP and non-AP members in the second semester grades earned by the
twelve remaining pairs. No statistical significance was found between
the freshman grade-point-averages of AP and non-AP students. Ruch sug- 148 gested replication of his work with a larger study population.
Pry (1973)
Supported with funds for data collection by the College Entrance
Examination Board (CEEB), Fry's subject data was collected from nine of
the one hundred institutions having the largest number of advanced place- 149 ment candidates in 1969. ' Specifically, his study population consisted
of 256 matched pairs of advanced placement (AP) and regular progress (RP)
students, and 131 matched pairs of College Board (AP-CEEB) and non-
College Board (AP-INST) advanced placement students, selected from all
sections of third-and fourth-term calculus and linear algebra at these
nine institutions. In Fry's study, an advanced placement (AP) student
148 Charles Ruch, "A Study of the Collegiate Records of Advanced Placement and Non-Advanced Placement Students," College and University 43 (Winter 1968): 207-10. 149 The nine institutions were Cornell, Harvard, Michigan State, Princeton, Vanderbilt, Washington University, and the Universities of Pennsylvania, Texas at Austin, and Virginia. Dale E. Fry, "A Comparison of the College Performance in Calculus-Level, Mathematics Courses between Regular-Progress Students and Advanced Placement Students," (Ph.D. dis sertation, Temple University, 1972), p. 97, Appendix E. 88 was a second-term, 1970-71, freshman who had been allowed to bypass one or more prerequisite calculus courses. On the other hand, a regular progress (RP) student was at least a third-term collegian who had pro ceeded through a normal calculus sequence, and who was matched with an AP student on the basis of having the same course and instructor, an SAT-
Math subscore within twenty-five points, and, if possible, the same sex.
Final grades in calculus II, III, and IV, or in linear algebra were used as criteria of achievement for the matched pairs. The effect of indepen dent variables on these grades relied on the use of nonparametric statis tical techniques such as the Wilcoxon signed-rank test, Kendall partial- rank correlation, or the Chi-square test. The study also considered the effect of certain confounding variables: SAT-Verbal subscore, intended major, sex, residence in dormatory, opinion on correct level of place ment, and at least one semester of high school calculus for RP students.
Fry's major objective was to judge the worth of the Advanced Placement
Program in mathematics by comparing the college performance of AP stu dents trained in high school calculus, with the academic performance of equally intelligent non-accelerants.
The study design and subsequent tests allowed Fry to draw seven conclusions: (1) AP students who bypassed calculus I had grades in calcu lus II significantly higher than RP students when results from all insti tutions were combined; but in calculus III or linear algebra, AP and RP students got essentially the same grades; (2) AP students, who bypassed two calculus courses, received significantly higher grades than their RP counterparts in calculus III or linear algebra, and in calculus IV for all institutions combined; (3) AP-CEEB students, who bypassed calculus I on the basis of their success on the national AP examination, received 89 significantly higher grades in their first two calculus courses than their RP mates in all institutions combined, and at least as well in each institution considered separately; (4) AP-CEEB students, who bypassed two calculus courses, did at least as well on each of the next two courses as their RP counterparts in each of the institutions; (5) AP-INST students, who bypassed calculus I on some basis other than the national College
Board examination, had grades in calculus II and calculus III at least as high as their RP mates; but in linear algebra, grades of RP students were significantly higher; (6) AP-INST students, who bypassed calculus I and calculus II, did at least as well as their matched-pair mates in each of the next two calculus courses; and (7) the confounding variables, indi vidually and collectively, were found to have little or no influence on student grades in calculus III, calculus IV, or linear algebra. In Fry's evaluation of his results, he concluded:
The fact that the AP students did at least as well as the RP students in the same classes with the same instructors and having the same scholastic aptitude in mathematics indicates that the high school calculus program has been a success, at least as handled by the nine institutions in this study. . . . Finally, it appears that the fact that the AP student had a high school calculus course and was given recognition for it was the contributing factor for the higher grades for AP students.
Pocock (1974)
A combination of objective data obtained from college records and subjective data obtained by a questionnaire-interview procedure directly from collegeans who had taken the 1968-70 Advanced Placement calculus examinations, was used by Pocock to determine how the Advanced Placement
Program in mathematics influences success in college mathematics. His subjects were sophomores, juniors, and seniors at two, liberal arts
^^Ibid., p. 75. 90 colleges in New York; and he grouped these students by college, class, and success on the AP examinations. For each AP student, a matched-pair control student, who had not begun his calculus until he entered college, but who had the same or similar major, cumulative GPA, and SAT-Math and
SAT-.erbal subscores, was identified. At both colleges, the successful students who received a 3, 4, or 5 on an AP calculus examination were offered two semesters credit and advanced placement, while those with 1 or 2 received no credit and were enrolled in the initial calculus course.
Within the experimental group whose members had taken an AP calculus examination, the subgroups classified by success were designated as
Credit or Non-Credit.
As a result of his study, Pocock reported that AP Non-Credit stu dents viewed their AP and college courses as similar in content and in rigor. Most thought that they had been underplaced and, in comparison to their controls, had experienced excessive duplication and lower intellec tual challenge in the college course. Pocock proposed optional placement of AP Non-Credit students in second-term calculus with first-term credit being conditional on success in the next course. At both colleges, a larger proportion of AP Credit than Non-Credit students became mathematics majors; and they considered the AP calculus teacher and course to be the key factors influencing their decision. In post-calculus, mathematics courses, both AP Credit and Non-Credit students did at least as well as their controls; and at one college, the AP Non-Credit students did sig nificantly better than their counterparts. This prompted Pocock to recommend that "college officials, including mathematics departments, should take an active part in identifying and placing in the freshman mathematics sequence, via personal interview, all AP mathematics students, 91 151 not just those who obtain credit or placement automatically."
Sklar (1980)
Sklar compared three, senior-year mathematics programs for high ability and gifted secondary school students by assessing the differen tial results of these programs on their later decisions to pursue the study of mathematics and to major in mathematics or science in college.
Subjects were 290 students in one of the following programs: the Advanced
Placement (AP) honors sequence, ending in analytic geometry and calculus, at a midwestern school; the Advanced Mathematics (AM) seminar, offered as a fifth-year course at a southern California school, which included top ics in probability and statistics, modern algebra, introduction to calcu lus, and machine programming; and the Standard Mathematics (SM) program at another southern California school, which ended a traditional sequence with college algebra and trigonometry. All students were classified as either high ability, if their I. Q. score was in the range 110-129, or gifted, if their I. Q. score was 130 or above.
Sklar found significant differences among the three programs with respect to the choice of mathematics courses in college. Of those in the
AP and AM programs, 93 percent elected at least one college mathematics course, while only 78 percent of the SM students did so. For the high ability group, first-term calculus was the initial college mathematics course for 57 percent of the AP, 76 percent of the AM, and 43 percent of the SM students. In addition, 29 percent of the AP group started their mathematics with second-term calculus. For the gifted group, there was a
^^Richard C. Pocock, "Advanced Placement Calculus as a Factor in the Study of College Mathematics," (Ed.D. dissertation, Columbia Univer sity, 1974), p. 107. 92 significant differences in the first mathematics course taken between the
AP and AM students: no AP student, but 11 percent of the AM group began at the pre-calculus level; in first-term calculus were 29 percent of the
AP and 72 percent of the AM students; and 54 percent of the AP gifted students received one or two semesters of calculus credit and began their college mathematics with the next sequent course. Sklar concluded that gifted students benefited more from the AP program than did the high ability students, and that the high ability students from the traditional program were not as well prepared for the study of college mathematics as were students in the other two programs.
No conclusion could be drawn as to which program was more effec tive in influencing gifted students to major in mathematics or science, because the percentages of gifted AP and AM students making these choices were too close, 55 and 52 percent, respectively. But significant differ ences were found among the three programs as to undergraduate choice of major for the high ability group: 62 percent of the AP, 56 percent of the
AM, and 30 percent of the SM students majored in mathematics or science.
To better prepare students for college mathematics, Sklar's final recom mendation was that gifted students should be enrolled in the AP program, while the high ability students should take the Advanced Mathematics pro- 152 gram in high school followed by calculus in college.
Frisbie (1980)
This coordinator of placement and proficiency testing at the
University of Illinois at Urbana-Champaign (UIUC) studied the achievement
152 Martha R. Sklar, "A Study of Three Distinct Senior-Year High School Mathematics Programs for High Ability and Gifted Students Relating to the Further Study of Mathematics," (Ph.D. dissertation, Northwestern University, 1980), pp. 45-147. 93 in their first mathematics course of students who were awarded credit for the Calculus AB or Calculus BC Advanced Placement examinations. The 328
Calculus AB and 199 Calculus BC students who submitted their AP exami nation scores in the fall of 1979 were considered separately, and their grades in the first mathematics course were compared to the grades of
non-AP calculus students enrolled in the same course. Credit and place ment into one of two three-course tracts at UIUC is a function of the AP
course and the grade of 1-5 received on the pertinent AP examination.
Calculus AB students with scores 3-5 were allowed eight semester
hours for the first two calculus courses and placement in the third.
Those who accepted credit and third-term calculus placement earned sig
nificantly higher grades than their non-AP peers (mean 3.84 vs 3.46 on a
5 point scale). Calculus AB students with scores of 2 received five
semester hours for the first calculus course and placement in the second.
Those who entered the second course in the first track earned grades which were slightly lower (mean 3.25 vs 3.39) than their non-AP peers; those who
entered the second course in the second track performed significantly
higher (mean 4.95 vs 3.53) than their counterparts. Those Calculus AB
students scoring 1 on the AP exam received no credit and were enrolled in
the first calculus course, where they significantly outperformed their
non-AP mates (mean 4.04 vs 3.74). On the average, Calculus AB students
earned a solid B in their first mathematics course, while their non-AP
counterparts were on the B- and C+ borderline.
Calculus BC students with scores 2-5 were allowed eight semester
hours credit for the first two calculus courses and placement in the third.
The 95 percent who accepted this credit and placement significantly out
performed their non-AP peers by earning average grades in third-term 94 calculus which were at least one full grade-point higher (mean 4.51 vs
3.46). Calculus BC students scoring 1 on the AP examination received no credit and had to repeat the first course in calculus; their grades were exceptional and significantly higher than their peers (mean 4.90 vs 3.74).
The average grade earned by all Calculus BC students was on the B+ and A- borderline. Except for a recommendation of a change in track for AB stu dents scoring 2 on the AP examination, Frisbie concluded that his report supported the existing credit and placement policies at UIUC and gave strong evidence that such policies put no Advanced Placement student at a disadvantage. The performance of the Calculus BC students was so impres sive that he recommended that students with BC scores of 4 or 5 be awarded credit for the third course in the calculus sequence as well as 153 for the first two. This recommendation is consistent with the College
Board's statement that "the content of Calculus BC is designed to qualify the student for placement and credit one semester beyond that granted for 154 Calculus AB.”
Summary
While the high school scholastic record combined with national test scores has proved to be effective in predicting general college success, more specific measures are required to predict success in college mathematics. Studies have indicated that mathematics placement tests, previous mathematics grades or grade-point-average, and the mathe matical component of national aptitude or achievement tests have been used
153 Davxd A. Frisbie, "Comparison of Course Performance of APP and Non-APP Calculus Students," (Research Memorandum No. 207, University of Illinois at Urbana-Champaign, 1980), pp. 1-10. (Mimeographed.) 154 The College Board, Math AP Course Description, May 1984, p . 2. 95 successfully to improve this differential prediction. Prediction equations for success in mathematics have often been refined by con sidering certain categorical background characteristics of the subjects, such as sex, race, or other personal attributes obtained through bio graphical data or interest measures.
A variety of statistical tools have been used in these prediction studies, but the most prominant of these have been Pearson product-moment correlation and stepwise multiple regression analysis. The increased sophistication of computer programs has promoted the use of other tech niques such as analysis of variance and covariance, factor analysis, and discriminant analysis. The studies by Perl at Stanford and Wagner et al. at the University of California at Davis, with their large study popula tions, would not have been possible without the computer. Nonparametric statistical procedures, such as Wilcoxon signed-rank test, Kendall’s correlation coefficient, and the Chi-square Test have also been used effectively, as in the studies by Bergeson, Ruch, and Fry.
Programs of advanced standing have been present in the colleges for years; but at no time have the levels of freshman placement changed so dramatically as in the early 1970s. The steady increase in the num bers of students participating in the Advanced Placement Program in math ematics and the change from one to two AP mathematics courses in 1969 contributed to the increase in the percentage of institutions offering different and higher levels of advanced standing. CHAPTER III
THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS:
ITS GROWTH NATIONALLY AND AT PUNAHOU SCHOOL
Historical Perspective
The increasing complexity of our growing technological society has placed extraordinary demands on American education to fulfill the need for leadership in mathematics and science. The "Rockefeller Report" on Education described both the reasons for the demand and the need:
The heart of the matter is that we are moving with headlong speed into a new phase in man's long struggle to control his environment, a phase beside which the industrial revolution may appear a modest alteration of human affairs. Nuclear energy, exploration of outer space, revolutionary studies of brain functioning, important new work on the living cell— all point to changes in our lives . . . The immediate implications for education may be briefly stated. We need an ample supply of high calibre scientists, mathematicians and engineers. Quantitative arguments over the shortages in these fields are beside the point. We need quality and we need it in considerable quantity'.
To fulfill this need, the educators of the 1950s focused their attention on the pool of able high school students and the problem of their transition to college. Eliminating unnecessary duplication of work in college, particularly on the part of superior high school students,
and providing a continuous program of study in the transition from school to college curricula were not new educational problems. They were rec
ognized by such groups as the 1893 Committee of Ten, the 1899 Committee on College Entrance Requirements, and the 1934 Eight Year Study. For at
Rockefeller Brothers Fund, The Pursuit of Excellence: Education and the Future of America, Panel Report V of the Special Studies Project, America at Mid-Century Series (Garden City: Doubleday a Co., 1958), p. 28. 96 97 least twenty years the University of Buffalo reduced duplication by offering college credit for work done in high school beyond graduation requirements, if the work was compatible with their own. Credit was granted for successful performance on college credit examinations, which were designed and graded by the separate college departments in subject areas corresponding to their own oourses. From 1932 to 1946 a total of
1,496 students took 2,730 examinations, and 81 percent of the written examinations resulted in credit. This experimental program at a single university had a number of elements in common with today's Advanced 2 Placement Program.
Renewed interest in improving the articulation between school and college was stimulated by the Fund for the Advancement of Education established by the Ford Foundation in 1951. Two projects, supported by the Fund from 1952 to 1955, will be described in detail because they were the progenitors of the Advanced Placement Program. Support was given to these projects, according to the Fund's vice president, because "we con
sider the area of articulation between the parts of the school system as one of the most critical in American education. Much of the effective
ness of the entire system is lost because our program of education is not
continuous."^ That both The School and College Study of General Education
(The Andover Study) and the School and College Study of Admission with
Advanced Standing (The Kenyon Plan) were solidly based on close cooper ation between middle and secondary school, and between school and college
2 Harold A. Anderson, "Educational News and Editorial Comments The University of Buffalo Experiment," The School Review 61 (October 1953): 385-86. 3 Alvin C. Eurich, "The Able Student— A Challenge to School and College," The College Board Review 18 (November 1952): 299. 98 will be clear in their description. The evolution of the Advanced
Placement Program (APP) from experimental projects into a national pro gram under the College Entrance Examination Board is considered before examining its introduction into the curriculum and growth at a particular secondary school in Honolulu, Hawaii, the site of the study population.
The Andover Study
The Andover Study, first proposed in May 1951 by John M. Kemper,
Headmaster at Phillips Academy of Andover, Massachusetts, was jointly sponsored by three, private secondary schools— Andover, Exeter, and
Lawrenceville— and three, ivy League universities— Harvard, Princeton, and Yale. Officially titled 'The School and College Study of General
Education,' its purpose was to plan and test a continuous curriculum over 4 the last two years of high school and the first two years of college.
Alan R. Blackmer, a spokesman of the six committee members, one from each institution, believed that in order to effectively raise the standards of college preparatory education and reduce the wasteful duplication in cer tain curricula, these four years in which a student received most of his general education should be viewed and planned as a wholes
Often the college must spend the greater part of a year in picking up the pieces of a student's fragmented and impoverished secondary education. Conversely, many good schools, both public and private, must carry their ablest boys and girls into 'college work', if they are to offer them any real stimulus. Too frequently the result is repetition in college of work well done in school. For well-prepared students this means boredom, loss of intellectual momentum, and serious waste of time in moving towards intellectual and professional objectives.5
4 Elwell, pp. 36-44j and Marian P. Franklin, "Advanced Placements Past, Present, and Future," The Educational Forum 29 (March 1965): 349. 5 Alan R. Blackmer, "The Three School, Three College Plan," The College Board Review 18 (November 1952): 300. 99 To obtain evidence of duplication and gaps in training, the joint
committee made a careful examination of the academic records from grade eleven through college of the 344 graduates of the three schools in the
class of 1951 at the three colleges. Detailed surveys of the material
taught in ten subject areas over grades 11-14 were conducted to determine
any duplication or discontinuities in study programs. Recommendations
based on these investigations, a series of panel discussions, and a
twenty-page questionnaire to a sample of 1952 graduates were published
in a book entitled General Education in School and College. Herein the
committee members recommended that able students should be provided
. . . with the opportunity of moving ahead at their own pace in a field where they are particularly strong. We shall call this progression in strength to distinguish it from what is usually called "acceleration," that is, moving ahead by a term or year in all subjects.6
Proposed also were a set of advanced placement tests, to be con
structed under the direction of the College Board and offered to all qualified students on a national basis, in the major subjects taught in
high school, "which would enable the colleges supporting these examina
tions to give an entering student advanced placement in a subject."7 The
appendix to the report made specific proposals for the implementation of
such a program to include the role of the College Board and the Educa
tional Testing Service. The similarity of these proposals to those
actually adopted by the Advanced Placement Program attests to the depth
and influence of the study.
0 Alan R . Blackmer, et a l ., General Education in School and College; A Committee Report by Members of the Faculties of Andover, Exeter, Lawrenceville, Harvard, Princeton, and Yale (Cambridge: Harvard University Press, 1953), cited by Jack N. Arbolino, "Progression in Strength," The College Board Review 115 (Spring 1980): 16.
?Ibid. 100 The Kenyon Plan
The School and College Study of Admission with Advanced Standing, which became widely known as The Kenyon Plan, originated in discussions
President Chalmers of Kenyon College had with his faculty in 1950-51 regarding the possible revision of their bachelor's degree requirements
"to encourage able students in strong secondary schools to pursue a liberal arts education at a pace appropriate to their ability and their 0 teachers' interests and skills." The exigencies of the Korean conflict and "the increase of professions depending on graduate work . . . put - g increasing emphasis on efficient use of the years available for study."
Chalmers admitted that he had at least three prejudices:
1. For the bright student who is well taught, the American system wastes time. 2. The best place for a schoolboy is in school. 3. The best teachers of 17-year-olds are as likely to be found in schools as in universities.'*’®
Chalmers firmly believed that the advancement of American edu cation required the strengthening of the secondary schools in the academic disciplines, particularly for the able and ambitious student.
Specific subject matter goals in the disciplines would have to be pre scribed. Curricular continuity in school and college would be possible if colleges could "reach agreement upon a common statement of standards
0 William H. Cornog, "Initiating an Educational Program for the Able Students in the Secondary School," The School Review 65 (Spring 1957): 49. 9 Gordon K. Chalmers, "Advanced Credit for the School Student," The College Board Review 18 (November 1952): 309.
^Ibid. Chalmers's second point was made "to stress the differ ence between the Study's methods and objectives and those of another Ford Fund for the Advancement of Education program, Early Admission, to which President Chalmers, among others, referred as 'the rob-the-cradle plan'." William H. Cornog, "The Advanced Placement Program: Reflections on Its Origins," The College Board Review 115 (Spring 1980): 15. 101 and achievement in the central subjects of a college freshman year with a view to granting advanced credit at entrance to a limited number who qualify
To accomplish this, Chalmers adopted a pragmatic approach, a
basic characteristic of The Kenyon Plan. By the fall of 1951, he had persuaded twelve college deans or presidents to form a Committee on
Admission with Advanced Standing charged with the task of determining the
advanced credit requirements which would be acceptable to the twelve par- 12 ticipants and broadly applicable to similar institutions nationwide.
By May of 1952, twelve principals or headmasters of strong secondary
schools had joined the twelve college heads in the Central Committee.13
Dr. William Cornog, President of Central High School in Philadelphia, was
named executive director of the study. The Central Committee enlisted
11Chalmers, p. 309. 12 The twelve colleges of the study were Bowdoin, Brown, Carleton, Haverford, Kenyon, Massachusetts Institute of Technology, Middlebury, Oberlin, Swarthmore, Wabash, Wesleyan and Williams. William H. Cornog, "The High Schools Can Educate the Exceptionally Able Student," National Association of Secondary-School Principals Bulletin 39 (April 1955): 381. 13 The exceptionally strong, secondary schools, both independent and public, whose headmasters, principals, or superintendents were invi ted into the central committee of the study were Bronx High School of Science, New York; Central High School, Philadelphia; Evanston Township High School, Evanston, Illinois; Germantown Friends School, Philadelphia; Horace Mann School, New York; Newton High School, Newtonville, Massachu setts; St. Louis Country Day School, St. Louis; High School, Brookline, Massachusetts; Brooks School, North Andover, Massachusetts; Lower Merion Senior High School, Ardmore, Pennsylvania; western Reserve Academy, Hud son, Ohio; Department of Public Schools, Providence, Rhode Island; and Oak Park and River Forest High, Oak Park, Illinois. Chalmers, p. 309. The first seven schools named were the pilot schools of the study during 1953-54. Able students in these schools took the revised courses of advanced college-level work provided by the eleven, subject-matter committees, in order to be prepared for the 'advanced standing exami nations' to be given for the first time in May 1954. College Admission with Advanced Standing: Announcement and Bulletin of Information ([Philadelphia]: The School and College Study of Admission with Advanced Standing, 1954), p. 4. 102 the aid of eighty-one school and college teachers to form the following eleven subject matter committees: English composition, literature, Latin,
French, German, Spanish, mathematics, biology, chemistry, physics, and history. Each of the committees, typically composed of four college and three high school teachers, met throughout the school year 1952-53 to produce a description and definition of freshman work in their field and set standards to be met by examination. Their final reports, submitted to the Central Committee in June, were published in August and circulated to the twelve college faculties for their vote in the early fall of 1953.
In an amazing show of unanimity and support for the fledgling program, the twelve, sovereign college faculties,
. . . with no college dissenting, voted approval of the experimental plan to consider for admission with advanced credit able students who had received instruction at the level defined by the committees and had met the standard set by the advanced credit examinations to be administered in the spring of 1954.^
A strong foundation for the Advanced Placement Program in mathe matics was laid by the Committee on Mathematics of the School and College 15 Study in their comprehensive report. The Committee made extensive suggestions in regard to appropriate goals, texts, source materials, and
14 Cornog, "The High Schools Can Educate the Exceptionally Able Student," p. 383. 15 The Study's Committee on Mathematics had the following members: Julius Hlavaty, Bronx High School of Science, New York; Elsie P. Johnson, Oak Park and River Forest High School, Illinois; Charles Mergendahl, Newton High School, Massachusetts; Heinrich W. Brinkmann, Swarthmore College, Pennsylvania; George B. Thomas, Jr., M. I. T., Massachusetts; Elbridge P. Vance, Oberlin College, Ohio; and Volney H. Wells, Williams College, Massachusetts. Mergendahl, Brinkmann, Thomas, and Wells also served on the Study's Committee of Examiners in Mathematics. College Admission with Advanced Standing; Bulletin of Information, pp. 86-90. To provide test reliability in the program transfer to the Col lege Board, Brinkmann was the first Chief Reader of the 1956 Advanced Placement Examination in Mathematics; Vance also served. Thomas was named to the Board's Commission on Mathematics. College Entrance Examin ation Board, 55th Report of the Director— 1956 (New York: College Entrance Examination Board, 1957), pp. xii and 120. 103
sample examination questions, and presented "a detailed program in mathematics for the last three years of secondary school which culminates 16 in a course that will be acceptable for advanced standing in college."
With this detailed program, the Committee presaged the report of the
Commission on Mathematics, five years later, as suggested by Elwells
Perhaps of all the committees the one on mathematics recommended the most comprehensive curriculum reform in order properly to equip secondary school students to do the college-level work in their senior year.-*-
School and College Study of Admission with Advanced Standing: Reports of Committees, 1952-1953 (Philadelphia: Central Committee of the Study, August 1953), p. 106.-^
In the fall of 1953, approximately 550 students in eighteen high
schools, including the three schools in The Andover Study, enrolled in 18 the courses which had been defined by the subcommittees. Among these were seven pilot schools, in which pilot study directors had been desig
nated to examine the problems of student and faculty selection, state
requirements governing curricula, and the sequence of subjects in their
current curriculum. Although the pilot schools differed considerably,
they generally selected students for the college program on the basis of
the following criteria: past scholastic achievement; recommendation by
previous instructors and departmental chairmen; objective data, such as
junior College Board and I. Q. scores; anecdotal data of guidance offi
cers; and parental approval. Teachers of the advanced classes were
departmental chairmen, those who had taught college classes, or teachers
16 College Admission with Advanced Standing: Bulletin of Infor mation, p. 37.
17Elwell, pp. 76-77. 18 Cornog, "The High Schools Can Educate the Exceptionally Able Student," p. 383. 104 19 with "wide and varied experience."
In each subject a Committee of Examiners, usually three college professors and two school teachers assisted by a test consultant from the
Educational Testing Service (E . T . S .), devised a three-hour examination to be administered in May and read in June by school and college teachers 20 under the direction of a chief reader from the Committee. The Study contracted with E. T. S. to administer the first examinations in the eighteen experimental schools and, for validation purposes, to a selected sample of freshmen in the twelve colleges. In May 1954, 532 school can didates took 959 examinations, including 120 in mathematics, and entered
94 colleges. In the May 1955 examinations, the last under supervision of the Study, the corresponding figures were 38 schools, 925 students, 1,522 21 examinations (including 265 in mathematics), and 134 colleges.
No validating college examinations were used in the second-year administration. The comparative results from the 1954 examination, reported separately to the twelve colleges by E . T . S ., had convinced these college faculties of the strength of the school candidates and the program, according to David Dudley:
The fact that the high school students did very well indeed in comparison with the college students . . . was one of the strongest factors in convincing many college faculties in the early part of the program of the ability of the high school students.
19 College Admission with Advanced Standing: Bulletin of Infor mation, pp. 78-80. 20 David A. Dudley, "The Advanced Placement Program," National Association of Secondary-School Principals Bulletin 42 (December 1958); 2. 21 The College Board, Advanced Placement Program (New York; College Entrance Examination Board, 1956), p. 7; and Elwell, appendix F, p. 309. 22 David A. Dudley, "The Beginnings of the Advanced Placement Program," (Chicago; Illinois Institute of Technology, 1963), p. 15 (mimeographed), quoted in Elwell, p. 93. 105 A survey of the 532 school candidates who took the first examin ation in 1954 was conducted by Marjorie Olsen of E. T. S. Seventy-one of the ninety-four colleges responded with data on 406 (76 percent) of the examinees. The percentage of these who were awarded advanced placement or credit was disappointingly lows 20 percent received advanced placement and credit, 23 percent placement only, and 3 percent credit only. About
54 percent of the sixty-nine candidates, who matriculated at one of the ten Study colleges familiar with the program, received advanced placement or credit. Surprisingly, the three colleges of The Andover Study got a
disproportionate share of the candidates:
Harvard University had eighty of the examinees, more than all the participating colleges combined, and almost 20 percent of the total group in the study. Princeton University received thirty-nine of the students and Yale University, thirty-eight. On the other hand, twenty-nine of the colleges had only one candidate each."*"
^College Admission with Advanced Standing: Final Report and Summary of the June 1955 Evaluating Conferences of the School and College Study ([Philadelphia): School and College Study of Admission with Advanced Standing, 1956), pp. 73-76.^
The results of the survey indicated that the award of advanced placement or credit varied not only between colleges, but also from one subject to another. For example, the percentages of awards by the Study
colleges differed noticeably in the following subjects: German, 100 per- 24 cent; mathematics, 64 percent; and physics, 26 percent.
Although first-year candidates' proper placement or credit was disappointing, the academic performance of these students as college freshmen was not. Rank-in-class information obtained for 204 candidates showed "32 percent ranked in the top sixth of their freshman class, 65
23Elwell, p. 95. 24 The College Board, "What Happened to Them in College," p. 7. 106 percent in the middle two-thirds of the class, and only 3 percent in the 25 bottom sixth of the class." Had this been a typical group of freshmen,
only 17 percent would have been in the top or bottom sixth, respectively.
The apparent success of its first-year operation and the intense
interest generated in schools and colleges by The Kenyon Plan, coupled
with the impressive arguments of the Andover Report, General Education in
School and College, led to a joint effort on the part of the study's
executive committee and the officers of the College Board "to explore the 26 possibility of extending the experiment." In October 1954, at the
behest of its Committee on Examinations and a $50,000 grant from the Ford
Fund, the College Entrance Examination Board agreed to assume the respon
sibility, beginning school year 1955-56, for the continuation and expan
sion into a national program of the work done by the two studies. The
examinations scheduled for 1956 were announced:
Advanced placement tests for admitted candidates will be offered by the College Board for the first time in May 1956. The new program will maintain continuity with the experimental work conducted for sev eral years by the School and College Study of Admission with Advanced Standing and the School and College Study of General Education. The purpose of the program is to encourage superior preparation of college candidates in the secondary schools by providing a method whereby able students with superior training can demonstrate their proficiency and qualify for advanced placement. The college will decide whether or not advanced credit will also be granted. Tests to be offered in the first administration will be in the fields of American History, Biology, Chemistry, Composition, European History, French, German, Latin, Literature, Mathematics, Physics, and Spanish. They will be based on course descriptions prepared by examining committees for the information of candidates, schools and colleges. Each description will attempt to indicate the scope and content of a typical course which the colleges would consider ade quate preparation for advanced placement.
25Ibid. 26 Cornog, "Initiating a Program for Able Students," p. 51. 27 College Entrance Examination Board, "News of the College Board," The College Board Review 24 (Fall 1954): 2. 107 Two changes are evident in this announcements under the College
Board, the tests would be called 'advanced placement' examinations, and their number would be increased by one "with the introduction of exami nations in American and European History. (Previously, the SCSAAS has had a non-examination offering in American History and nothing in Euro- 28 pean History.)" To facilitate the transition of the program to Board control, the College Board appointed a Subcommittee on Advanced Exami- 29 nations which included the directors of both major studies.
The impact of the comprehensive 1954 report of Brinkmann’s mathe matics subcommittee was discussed at the Evaluation Conference, held simultaneously June 22-25 with the grading of the Study's last 'advanced standing' examinations in 1955. The conferees unanimously agreed that,
. . . in view of the relatively small number of students who have thus far benefited from the program carried on under the report; in view of the long-range planning required by most schools before a change in curriculum and a full test of the program can be made; in view of the considerable number of schools here represented who wish to undertake to implement the subcommittee's report in the coming year; and, in view of the potential enrichment of the mathematical training at all levels which may result, . . . We therefore respectfully recommend that the present subcommittee be continued . . . in order that this experienced group mays (1) carry out a vigorous continuation of the program, (2) conduct systematic evaluations of the program's results, (3) publicize the facts regarding the program among schools and colleges, [and] (4) work with the College Entrance Examination Board in ways which will help the program fulfill its objectives.3®
28 Harlan P. Hanson, personal letter written 9 September 1982. 29 Chaired by Archibald Macintosh of Haverford College, the Sub committee on Advanced Examinations included Alan R. Blackmer, Phillips Academy, Andover; William H. Cornog, President, Central High School, Philadelphia; and Marion Tait, Dean of vassar College. College Entrance Examination Board, 54th Report of the Director— 1955 (New Yorks College Entrance Examination Board, 1956), p. viii.
^College Admission with Advanced Standings Final Report and Summary of the June 1955 Evaluating Conferences of the School and College Study, quoted in Vance and Pieters, "The APP in Mathematics," p. 495. 108 Influenced by the reports of the June 1955 Evaluating Conferences and their own Committee of Examiners, which had observers with each sub committee of the study, the College Board appointed new Advanced Place ment Examination Committees from a larger group of schools and colleges than originally represented, but retained sufficient members of the original committees to provide continuity. Fortuitously, the Committee of Examiners in Mathematics raised the issue of the dichotomous alter natives in their test construction: either to test the mathematics actually taught in the majority of secondary schools, "although in adopting this alternative they would not test readiness for college mathematics, particularly in engineering schools and in programs for majors in mathematics and science;" or to test in terms of existing col- 31 lege standards, which would be unfair to the majority of students.
Acting on these reports in July 1955, the College Board appointed 32 a Commission on Mathematics "to review the existing secondary school mathematics curriculum, and to make recommendations for its modernization, 33 modification, and improvement." It began its work in August of that
Frank H. Bowles, Admission to College: A Perspective for the 1960's; 57th Report of the President, College Entrance Examination Board (New York: College Entrance Examination Board, 1960), p. 26. 32 The Commission on Mathematics had the following appointees: Albert W. Tucker, Princeton University, Chairman; Carl B. Allendoerfer, University of Washington; Edwin C. Douglas, Taft School, Connecticut, and Chairman of the Mathematics Examiners Committee (ex officio); Howard F. Fehr, Columbia University; Martha Hildebrandt, Proviso Township High School, Illinois; Albert E. Meder, Jr., Rutgers University, Chairman of the Committee on Examinations (ex officio); Frederick Mosteller, Harvard University; Eugene P. Northrop, University of Chicago; Ernest R. Ranucci, Weequahic High School, New Jersey; Robert E. K. Rourke, Kent School, Connecticut; George B. Thomas, Jr., M. I. T.; Henry Van Engen, Iowa State Teachers College; and Samuel s. Wilks, Princeton university. College Entrance Examination Board, 55th Report of the Director— 1956, p. xii.
Commission on Mathematics, Program, p. xi. 109 year, just as the College Board accepted the transfer of the Study's experimental program as an integral part of its own activities with respect to administrative and financial support, but continued as a distinctive, coooperative program between school and college.
Operational control of the program passed smoothly to the College
Board under the guidance of its Director, Frank H. Bowles, and his two
Associate Directors, William C. Fels and S. A. Kendrick. Named as the first Director of the Advanced Placement Program was Charles R. Keller, rebel leader of the historians’ non-examination revolt under the Study and, during 1955-56, on sabbatical leave from his post as chairman of the 34 history department at Williams College. To assist the Director in fos tering close cooperation between school and college was the Commission on 35 Advanced Placement with Bayes M. Norton of Kenyon College as Chairman.
It is an ironic twist of Advanced Placement's own history that, during his tenure as Director from 1955 to 1957, Professor Keller "became the program's most eloquent and indefatigable public defender, promoter, 36 advocate, and ambassador." The College Board's new Advanced Placement
Program needed such early leadership in order to survive and grow.
34 College Entrance Examination Board, 55th Report of the Director — 1956, p p . vii and 5.
^~*The first Commission on AP had the following members: Norton; Gordon K. Chalmers, President, Kenyon College; William H. Cornog, Super intendent of Schools, New Trier Township, Illinois; Robert N. Cunningham, Dean, Phillips Exeter Academy, New Hampshire; Harold Gores, Superinten dent of Schools, Newtonville, Massachusetts; Harlan P. Hanson, Director, Program of Advanced Standing, Harvard University; Harold Howe, Principal, Walnut Hills High School, Ohio; Frank R. Kille, Dean, Carleton College; Otto F. Kraushaar, President, Goucher College; Morris Meister, Principal, Bronx High School of Science, New York; Robert H. Pitt, Dean of Admis sions, University of Pennsylvania; Edward M. Read, Headmaster, Saint Paul Academy, Minnesota; and Albert E. Meder, Jr., Chairman, College Board Committee on Examinations (ex officio). Ibid., p. ix. 36 Cornog, "APP: Reflections on Its Origins," p. 17. 110 The National Growth of Advanced Placement Mathematics as a College Board Program
The Early Period: 1955 to 1960
In the summer of 1955 after the grading of the May examinations, the College Entrance Examination Board assumed the responsibility for the continuation and expansion of the Advanced Placement Program into a
national program by providing schools with practical descriptions of
college-level courses, administering examinations each May covering the material outlined in the syllabi, and supplying colleges with examination
results on a standardized scale. For the first College Board exams in
May 1956, Professor Brinkmann, an experienced holdover from the School
and College Study, was appointed chairman of the Advanced Placement 37 Examiners in Mathematics. By 1960 it was generally acknowledged that
the College Board had met its mandate to retain the fundamental features
of the School and College Study while successfully transforming the
experimental program, catering mainly to schools and colleges in the
northeastern United States, into a program of national dimensions.
By its promise to keep the basic features of the Study's experi
mental program, the College Board committed itself to essay-type exami
nations, the designation of joint school and college committees of
Examiners and Readers to make and grade the examinations, and the annual
Advanced Placement Conferences to which school and college teachers and
37 The normal composition of the early committees of examiners was four college professors and two school teachers, one private and one pub lic. The first committee of examiners in mathematics had the following members: Heinrich W. Brinkmann, chairman, Swarthmore College; Edward G. Begle, Yale University; Garrett Birkhoff, Harvard University; Richard S. Pieters, Phillips Academy (Andover, Mass.); Harry D. Ruderman, Bronx High School of Science (Bronx, N.Y.); and George B. Thomas, Jr., Massachusetts Institute of Technology. The College Board, Advanced Placement Program [1956], p. 134. Professor Brinkmann, as chairman, was also designated Chief Reader for the 1956 Advanced Placement examination in mathematics. Ill administrators were invited
. . . to discuss the examinations, their grading, the syllabi, teaching methods, course organization and content, policies with respect to the granting of advanced placement and credit, and other topics related to the program. Curiously enough, these conferences, at the time they were inaugurated . . . were . . . the only program of regularly organized meetings between school and college teachers to discuss their joint instructional concerns, a fact which is a sufficient commentary on the progress of articulation between school and college.®®
Their desire to improve this imperfectly articulated system had led the College Board to cooperate with three of the four experiments
sponsored by the Fund for the Advancement of Education. Officers of the
Board saw in each of these studies, especially in The Kenyon Plan, the means to strengthen the college preparatory programs in the secondary
schools, avoid repetition at different levels of the system, and shorten the span of education for the ablest students who were the ones most 39 likely to continue on to professional or graduate work. They encour
aged Chalmers in his efforts to bring schools and colleges together; in
fact, Director Bowles had been the one to recommend Dr. Cornog as an
ideal director of the Study because of his experience as both an English 40 professor at Northwestern and a secondary school principal.
Others had wanted the College Board to become involved in the
testing of students who had completed work of college caliber in their
secondary schools much earlier than the Board actually did. As early as
1951, Frank Kille, Dean of Carleton College and a member of the Executive
38 Bowles, Admission to College, p. 29. 39 William C. Fels, "How Tests May Be Used to Obtain Better Articulation of the Total Educational System," Speech to the National Association of Secondary-School Principals, February 22, 1954, reproduced in Frank [H .] Bowles, The Refounding of the College Board, 1948-1963 (New York: College Entrance Examination Board, 1967), pp. 125-26. 40 Elwell, p. 64. 112 Committee of the College Board, had urged that college credit tests in
English, mathematics, and foreign languages, "satisfactory to educators in both secondary schools and colleges," be developed "by experts . . . assisted by the best thought of instructors in both secondary school and 41 college and by adequate funds for research." He called attention to the increased use of Board tests in granting exemptions from certain college requirements, the national requirement for military service, and concluded that "the Board could perform no greater service than to make available tests of such quality that colleges would use them to grant 42 some college credit to the unusually able and ambitious student."
As Director, later President, of the College Board, Prank Bowles had resisted the Board’s adoption of the program because the commitment to essay-type examinations involved a return to the costly written exami nation techniques and grading procedures which characterized the Board 43 program prior to 1942. Acceptance of the program by the College Board meant that arrangements would have to be made to finance an expanding program without endangering the values developed during the experimental period when the expenses were being underwritten by the Ford Fund. After careful study of various fee scales and considerable controversy, the decision was made to establish the May 1956 examination fee "at $10 per 44 candidate so that the fee should not be an obstacle to any candidate."
41 Frank R. Kille, "A Dean Looks at Admissions," The College Board Review 14 (May 1951): 201.
43 Bowles, Admission to College, p. 15.
44 College Entrance Examination Board, Report of the President— 1957; Fifty-Sixth Annual Report (New York: College Entrance Examination Board, 1958), p. 46. These receipts, together with the generous subvention of $30,000 for
1955-56 from the Ford Fund, would cover only a fraction of the expenses; therefore, "in taking this action, the Board accepted the fact that the 45 program would not be self-sustaining in the near future." The Execu tive Committee made the unselfish decision that "the program should be supported to the fullest extent necessary to insure its wide under standing and acceptance, even if, in order to do so, it proves necessary 46 to draw upon the Board's reserve funds."
During this entire early period, the Advanced Placement Program operated at a deficit, which was made up from the Board's operating reserves: more than $100,000 was allocated in each of the first two years and, even "a new scale of candidate fees which establishes a registration fee of $5 and a charge of $8 for each examination" could not prevent the average annual deficit from reaching approximately $150,000 by the end of 47 the period. The astonishing growth of the Advanced Placement Program, 48 "proportionally by far the largest recorded by any Board program," forced the Board to invest almost "one million dollars in subsidizing the 49 program's operating deficit" by the end of 1961. Table 9 shows this growth in terms of the numbers of students, schools, colleges, examina tions in all subjects, and examinations in mathematics for this period,
46 College Entrance Examination Board, 54th Report of the Director — 1955, p. 13 47 Bowles, Admission to College, p. 71. 48 Ibid., p. 70. 49 Frank H. Bowles and Richard Pearson, Admission to College: A Program for the 1960's; 58th Report of the President, College Entrance Examination Board (New York: College Entrance Examination Board, 1962), p. 29. 114 as compared to the corresponding figures for the experimental period 50 under the School and College Study.
TABLE 9
ADVANCED PLACEMENT PARTICIPATION, 1954-1960
Number Number Number Number of Exams of Exams Number Year of of in All in Mathe of Schools Candidates Subjects matics Colleges
1954a 18 532 959 120 94 1955a 38 925 1,522 265 134 1956 104 1,229 2,199 386 130 1957 212 2,068 3,772 724 201 1958 355 3,715 6,800 1,177 279 1959 560 5,862 8,265 1,870 391 1960 890 10,531 14,158 2,908 567
SOURCES: Harlan P. Hanson, personal letter written 22 August 1984; [Educational Testing Service], "Distribution of Candidate Grades Advanced Placement Examinations," Princeton, 1956-60. (Mimeographed single sheets, by year); The College Board, Advanced Placement Program [1956], p. 7; Col lege Entrance Examination Board, Report of the President— 1957, p. 42; Bowles, Admission to College, p. 70; The College Board, Advanced Placement Program: 1966-68 Course Descriptions (New York: College Entrance Exami nation Board, 1966), p. 15.
aConducted as the School and College study of Admission with Advanced Standing.
Comparative growth between categories represented in table 9 can be appreciated by using the statistic AAPI, the average annual percentage increase. For the first five years under the College Board, the AAPI for the number of mathematics candidates was 66 percent, which compares favorably with the AAPI for the number in each of the following cate gories: schools, 72 percent; all candidates, 71 percent; examinations in all subjects, 61 percent; and colleges, 45 percent.
See appendix C: Advanced Placement Participation, 1954-1984 for entire data set. 115 During its first full year of operation as a Board program, the Advanced Placement Program underwent a thorough review and partial
reorganization which included a restatement of the program's purposes, redesign of the examinations scheduled for the week of 7-11 May 1956, and
revision of the syllabi of the twelve college-level courses for inclusion
in the new College Board publication, Advanced Placement Program. ^
The Advanced Placement mathematics seniors studying calculus and
related analytic geometry that year were fortunate that the Study's
Mathematics Committee had done such a comprehensive job the previous year
because this descriptive booklet was not published until May 1956. The
twelfth-year course outline and consequently the scope of the Advanced
Placement examination in mathematics required no revision and remained 52 the same as it had been under the Study. The primary objective of the
course remained unchanged: "to give a substantial training in differential
and integral calculus, with sufficient applications to bring out the 53 meaning and importance of the subject
The Advanced Placement examination in mathematics has always been
three hours long and divided into two parts: Part I, a multiple-choice,
objective section testing proficiency in a wide variety of topics; and
Part II, an essay or problem section requiring detailed, written solutions
51The College Board, Advanced Placement Program (New York: Col lege Entrance Examination Board, 1956) was the first in the annual series of 'Acorn Books'. It "not only received a certificate of special merit at the annual exhibition of the New York Employing Printers Association, but curiously enough had its cover selected for reproduction in Modern Publicity, an annual of international advertising art published in London." College Entrance Examination Board, Report of the President— 1957, pp. 22-23. 52 See appendix B: Advanced Placement Mathematics Course Topics, 1956-68. 53 The College Board, Advanced Placement Program [1956], p. 118. 116 which involve a more extended chain of reasoning. During the entire early period, Part I had thirty, multiple-choice questions to be covered in one hourj and Part II had ten longer problems for which the examinees had two hours. The first part has always been machine-graded, while the second is graded in June by a representative sample of college and secondary school teachers under the direction of the Chief Reader.
Like the Chief Readers in Mathematics who succeeded him, Heinrich
Brinkmann was a mathematics professor charged with developing the exami nation, the fairness and reliability of the reading, and the initial grading standards. The fairness and reliability are insured by taking such precautions as "the assignment of each examination book to several readers, the deliberate obscuring of previous readers' grades as well as of the students’ names and schools, and the continuous monitoring of 54 readers' work by table leaders." until the mid 1970s, the student’s
Part II paper, his grade on the standardized one-to-five scale, an out line of his secondary school's course, and his teacher's specific recom mendation were sent to the student's college. The college could make an informed decision on credit or advanced placement based on this infor mation. Lately, as more colleges gained confidence in the grading scale, only the student's Part II paper and standardized grade have been sent.
Two of the early difficulties faced by the Advanced Placement
Program concerned the tendency of the readers to recognize only superla tive performance and the interpretation of the grades by the colleges:
It was found, not entirely to the surprise of the administrators, that the school and college teachers assembled to read and grade the advanced placement papers were so determined to allow no softening of standards that they actually set standards unrealistically high
54 The College Board, College Placement and Credit by Examination (New York: College Entrance Examination Board, 1978), p. x. 117 and, in fact, higher than they would ordinarily have held them in their own institutions. . . . Moreover, coupled with this tendency to rigorous grading on the part of the readers was the tendency of the colleges to accept only honor grades, i.e., the equivalent of their own A ’s and B's, for credit in college. If this tendency, like the tendency to over severity in marking, had not been detected and guarded against from the very outset, it is probable that a serious, if not damaging blow, would have been dealt to the entire program.^
In order to protect both the candidates and the colleges against unrealistic grading standards, the College Board arranged for their first two examinations to be administered to quasi-control groups of college students. These examinations were administered at thirty-nine institu tions in 1956 and twenty-nine in 1957. As a result of a thorough study by the Educational Testing Service of the difficulty level, depth, scope, and quality of performance demanded on the 1956 examinations and with consideration of both the readers' comments and the subject-matter con ference reports, significant improvements were made in the 1957 exami nations. In the reading of this second set of examinations, a serious attempt was made to give "the grading what might be called a normal college-intensity, that is, to approximate the same level between lenien cy and severity that exists in the average college of substantial
strength." The report concluded:
To help determine how well the achievement of the secondary school students qualified as college-level achievement, and how well the grading intensity met the standard sought, the examinations were administered to comparison groups of students from 29 different col leges. The fact that the school students did better than most of the college comparison groups confirms the reasonableness of accepting the school students' achievement for credit and placement on a basis directly comparable to that used by a college for its own students taking a comparable course in the college.
55 College Entrance Examination Board, 55th Report of the Director — 1956, pp. 8-9.
56College Entrance Examination Board, Report of the President— 1957, p. 44. 118 The tangible evidence of a student's success in an Advanced
Placement course is not only a qualifying grade of at least a '3* on the
May examination, but also college recognition of his achievement by the award of credit or advanced placement. of the 130 colleges receiving the
1955-56 Advanced Placement candidates, 76 considered students for both credit and advanced placement, 26 others gave advanced placement only,
8 reported that their existing policy prevented any type of award, and 57 the policies of the remaining 20 colleges were undetermined.
During 1956-57, numerous additional colleges adopted an advanced placement policy; and a College Board study of the reports from over 50 percent of the colleges attended by the 1957 examinees claimed that
"almost all colleges receiving advanced placement students recognized the program through appropriate placement (and the majority of these colleges 58 gave placement and credit)." Generally, however, while most colleges recognized the 'honor' grades of '5' and '4', a goodly portion chose to ignore the middle grades of '3* and '2'. The College Board felt impelled to make the following addition to the Advanced Placement grade interpre tation leaflets and quote it in the revised syllabus of 1958:
The grades 2 to 5 are considered to be passing college grades by the committee of examination readers. It would be highly undesi rable, and impossible anyway, for any committee to try to set up "national" standards. The Director of the Advanced Placement Program, however, has asked each reading committee to state through its use of defined grades its own concepts of failing, creditable, and honor marks at the college level. . . . The committees have therefore set . their "pass" and "creditable" grades not according to secondary school standards, but according to college standards.^
^Ibid. 5 8 The College Board, Advanced Placement Program (New York: College Entrance Examination Board, 1958), pp. 14-15. 59 Ibid., p. 15. 119 In the Advanced Placement Program [1958], the student was advised that, because the program was new, he should not expect all colleges to
. . . follow the same, or even similar, advanced standing proce dures . It is hoped that greater uniformity in the administration of advanced standing policies, such as that achieved by the cooperating colleges in the School and College Study, may unfold as schools and colleges gain experience with the p r o g r a m . ^
To achieve this end, the indefatigable program director, Charles
Keller, traveled extensively during his two-year tenure to visit some 61 113 schools and 70 colleges and to give 43 talks at 52 meetings.
Unfortunately, the unanimity found in the cooperating colleges of the
School and College Study still has not been achieved nationally.
While no one questions the right of each college to establish its own placement and credit policies, it seems incongruous that many selec tive colleges should grant placement and credit routinely for scores of three or higher, while other institutions question fours. In a recent
letter to the principals of participating secondary schools, the program
director urged them to question a particular college’s Advanced Placement policy if it appeared to be inappropriate:
It is disheartening to hear that departments in some colleges either do not have an AP policy or have one that recognizes only grades of 4 or 5 (and/or may award only a third or a half year of placement and credit to successful AP candidates. . . . It is quite proper for you to question a college's AP policy if it appears out of date or inappropriate, in a number of instances, misunderstandings have been cleared up bv such inquiries from secondary school teachers and officials. ^
60 Ibid., p. 12. 61 College Entrance Examination Board, 55th Report of the Director — 1956, p . 27 and Report of the President— 1957, p p . 44-45. Charles R. Keller, Director of the Advanced Placement Program from 1955 to 1957, was succeeded by David A. Dudley (1957-58), Jack N. Arbolino (1958-65), and finally by Harlan p. Hanson (1965 to present). 62 Carl H. Haag, Letter to principals, July 1982. 120 A strong proponent of the Advanced Placement program from the very beginning, Harvard College has always had a policy of routinely accepting AP scores of three or higher without departmental review or further ratification. Succeeding Harlan P. Hanson as its director of the program of advanced standing was Edward T. Wilcox, who argued forcefully for such a policy:
If students get a qualifying grade [AP score '3' or higher], they can expect advanced placement at Harvard as a matter of course. Regardless of subtle, parochial arguments in its favor, there is something sticky and confusing about a college policy that encourages a student to spend a hard year in an advanced placement course, challenges him to get a qualifying grade on the appropriate Advanced Placement Examination, and then points out that he still must face a kind of placement by ambush, a mysterious re-examination by the college.
Wilcox believed that the ultimate justification, besides the practical one that Harvard garnered a disproportionate share of the early 64 Advanced Placement candidates, was the performance of these students as freshmen taking sequent advanced courses. Of the 806 students given cre dit and advanced placement in the five-year period 1954-58, exactly half were in mathematics. The percentages of these 403 students in each grade category of the sequent mathematics course were as follows: A— 36 percent;
B— 31 percent; C— 26 percent; D--5 percent; and E— 2 percent. Wilcox concluded that
. . . the performance of these students . . . makes it clear, at this point at least, that our policy has been a sound one. I don't feel we have dangerously overplaced students as a result of this rather naked acceptance of the College Board grades.^
63 Edward T. Wilcox, "Advanced Placement at Harvard," The College Board Review 41 (Spring 1960): 19. 64 In the first seven years, Harvard received 8 percent of all AP candidates. See table 9; and Edward T . Wilcox, "Seven Years of Advanced Placement," The College Board Review 48 (Pall 1962): 31. 65 , Wilcox, "Advanced Placement at Harvard," p. 20. 121 For every college, like Harvard, which displayed early leadership
in their acceptance of the Program's basic premise that "college work
completed in school deserves college credit," there were colleges which
did not understand the Advanced Placement Program and either denied
credit or placement even for 'honor' AP grades or allowed credit or 66 placement for 'failing' grades. Concerned about the extent of this
lack of knowledge, the College Board conducted follow-up studies on the
placement of candidates each year. Table 10 shows the disposition made
by colleges of 1,585 reporting candidates out of the total of 1,870 stu
dents who took the Advanced Placement examination in mathematics in 1959.
TABLE 10
COLLEGE PLACEMENT BY GRADE OF MATHEMATICS CANDIDATES IN 1959
Total Number of Number Awarded Percent AP Grade Number of Candidates Placement of Number Candidates Reporting and/or Credit Reporting
5 (highest honors) 225 196 188 96 4 (honors) 283 254 236 93 3 (creditable) 586 510 371 73 2 (pass) 329 276 70 25 1 (fail) 447 349 22 6
Totals 1,870 1,585 887 56
SOURCE: Elbridge P. Vance and Richard S. Pieters, "The Advanced Placement Program in Mathematics," The American Mathematical Monthly 68 (May 1961): 496.
Within any discipline, the course description is the key to an
understanding of the preparation expected of a recommended candidate for
it delimits both the scope of the course and the annual examination. The
66 jack N. Arbolino, "What's Wrong with the Advanced Placement Program," National Association of Secondary-School Principals Bulletin 45 (February 1961): 30. 122 outlines are kept as general as possible within the restraints imposed by the disciplines. The science and mathematics syllabi are more prescrip tive than most. Secondary schools which failed to distinguish between a program of the honors variety, involving a broader and deeper study of normal high school work, and the college-level Advanced Placement Program in mathematics found that even superior students were not properly pre pared for the May examination. The need to make a distinction between the two types of program and to adopt a curriculum leading up to the cal culus, such as that recommended by the Commission on Mathematics, was made by different schools at varied rates. Table 11 shows the annual grade distributions in mathematics during this early period of change.
TABLE 11
GRADE DISTRIBUTIONS OF MATHEMATICS CANDIDATES, 1954-1960
Number of Candidates (N) and Percentage Below Grade (%B) Each Year Grade 1954a 1955 1956 1957 1958 1959 1960 N %B N %B N %B N %B N %B N %B N %B
5 5 96 12 95 44 89 80 89 115 90 225 88 176 94 4 8 89 23 87 109 60 169 66 235 70 283 73 404 80 3 35 60 70 60 87 38 176 41 334 42 586 42 606 59 2 34 32 74 32 100 12 117 25 217 23 329 24 711 35 1 38 0 86 0 46 0 182 0 276 0 447 0 1011 0
Total 120 265 386 724 1177 1870 2908
SOURCES: John R. Valley, "Annual SCSAAS Report [of E.T.S. Program Director]," Table B-7, Princeton, 1954. (Mimeographed); and Paul H. Haz- lett, Jr., "Advanced Placement Evaluation Report," ([New York: College Entrance Examination Board, 1961]), First Draft, Appendix B, Table 1, p. 7. (Mimeographed.) Citation is nonstandard because the First Draft, like the annual report, was never published but distributed privately.
aThe five-point scale was not adopted until 1955. The 1954 mathe matics distribution is only available by composite scores and percentiles. In order to be uniform, figures in this column were derived by the author from the given distribution and made to coincide, as closely as possible, with the percentile distribution in 1955, the other year under the SCSAAS. 123 Table 11 shows that, in the first five-year period under the
College Board, approximately three-fourths of the mathematics candidates had passing scores of 2 or better. Even though the average annual per centage increase (AAPI) in mathematics candidates was 66 percent, the
number who made qualifying scores of 3 or more remained fairly stable at about 60 percent, except for 1960. The 1960 deflation of grades troubled the Board because, if it continued, it might have a serious effect on the
Program's further acceptance by both schools and colleges.
Three theories for the grade deflation were advanced: possibly, because of the rapidity of the Program’s growth in new schools, students were not as well prepared; the 1960 examination was more difficult; or the examination readers were too stringent in their grading. Malcolm's in depth analysis, which examined the performance of students with simi lar SAT scores in old, intermediate, and new schools, determined that
"The decline was simply not due to an influx of poorly prepared candi- 67 dates from schools new to the program." No further evidence was available to indicate whether it might have been the difficulty of the examination or the severity in grading, but the matter was left for the
Committee of Examiners to take into consideration for the following year.
The Program's growth, with its attendant implications of the increased ability of schools to produce able candidates and of the growing willingness of colleges to accept them, convinced many educators that the
Program was exerting a powerful influence on American education. As this
^Donald J. Malcolm, An Analysis of Scores Received by Candidates on Some of the Advanced Placement Examinations in Relation to the Length of Time Their Schools Had Participated in the Program (Princeton: Educa tional Testing Service, 1961), quoted in Paul H.- Hazlett, Jr., "Advanced Placement Evaluation Report," ([New York: College Entrance Examination Board, 1961]), First Draft, pp. 29-30. (Mimeographed.) 124 early period ended, James B. Conant wrote, "The success of the Advanced
Placement Program in the last few years is one of the most encouraging 68 signs of real improvement in our educational system." Lloyd Michael, a superintendent-principal in Evanston, Illinois praised another aspect:
One of the most promising developments in curriculum articulation is the Advanced Placement Program. . . . One of the most successful and worth-while [sic] features of the Program has been the series of summer conferences in the various subject fields attended by school and college teachers to discuss courses, teaching methods, and the examinations.^
The impact of the Advanced Placement conferences as a means of improving curricular articulation can be appreciated by knowing that the nine subject area conferences in 1960 were attended by some 1,500 school 70 and college teachers. Considering this early period, Copley concluded that "the program has passed its probationary stage and deserves to be 71 regarded as an established institution."
The Middle Period: 1961 to 1968
The beginning of this eight-year period was marked by a change in the mathematics examination; and the end, by a change to two mathematics courses with corresponding examinations. During the interim, the Program grew approximately four-fold but the rate of growth slowed considerably during the last half of the period. Table 12 shows this growth.
68 James B. Conant, Slums and Suburbs (New York: McGraw-Hill Book Co., 1961), p. 89. 69 Lloyd S. Michael, "Articulation Problems with Lower Schools and Higher Education," The Bulletin of the National Association of Secondary- Principals 43 (February 1959): 54. 70 Jack N. Arbolino, "Proper Placement— Key to Articulation," The College Board Review 44 (Spring 1961); 12. 71 Frank 0. Copley, The American High School and the Talented Student (Ann Arbor: The University of Michigan Press, 1961), p. 33. 125 TABLE 12
ADVANCED PLACEMENT PARTICIPATION, 1961-1968
Number Number Number Number of Exams of Exams Number Year of of in All in Mathe of Schools Candidates Subjects matics Colleges
1960-61 1,126 13,283 17,603 3,609 617 1961-62 1,358 16,255 21,451 4,190 683 1962-63 1,681 21,769 28,762 5,848 765 1963-64 2,086 28,874 37,829 7,710 888 1964-65 2,369 34,278 45,110 9,021 994 1965-66 2,518 38,178 50,104 9,630 1,076 1966-67 2,746 42,383 54,812 10,675 1,133 1967-68 2,863 46,917 60,674 11,623 1,193
SOURCES: [Advanced Placement Program of the College Board], "Annual Advanced Placement Program Participation," Princeton, [1984]. (Mimeographed); and "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1961-68. (Mimeographed single sheets, by year.)
Having considered the analysis of the previous year's test results, the Committee of Examiners decided to change the format of the
May 1961 mathematics examination; the change remained in effect for the next twenty years. The time allotted for the multiple-choice, objective
Section I was increased from one hour to ninety minutes with a propor tional increase in the total number of questions. Increasing the number of short-answer questions from thirty to forty-five permitted wider cov erage of the syllabus. The number of longer essay problems in Section II was reduced from ten to seven to compensate for the reduction in time from two hours to ninety minutes. To determine results on the standar dized five-point scale, each section would be equally weighted. No changes were made in the syllabus for the examination or in the types of questions in each section. The average time per question in Section I remained two minutes, and only increased from twelve to about thirteen 126 minutes per question in Section I I . With the projected increase in the number of mathematics candidates, the change promised not only better course coverage,, but also greater cost effectiveness by reducing the num- 72 ber of trained readers to correct the longer Section II problems.
The rate of growth during the middle period, as shown in table 12 and measured by the average annual percentage increase (AAPI), was con siderably less than during the previous early period. The AAPI for the number of mathematics candidates was down from 66 to 19 percent, and a similar condition existed for the other categories: schools, 16 percent; all candidates, 21 percent; examinations in all subjects, 20 percent; and colleges, 10 percent. A phenomenon common to all categories was the dis parity between the AAPI's in the first and second halves of the period.
For example, the AAPI for mathematics candidates in the first half was 28 percent; while in the second half, it dropped to 11 percent.
A growing interest in the Program was exhibited by state univer sities and departments of education during this period. For example, in
June 1962 only one state university hosted an annual subject-matter con- 73 ference— the University of Michigan in English. In June 1963 the annual conferences were held on eight campuses, three of which were state univer sities: Michigan State, biology; University of North Carolina, history; 74 and University of California at Davis, mathematics. By June 1966 there
72 ,, . College Entrance Examination Board, "Advanced Placement Exami nations— May 1961: Grade Reports," New York, 1961. (Mimeographed), p. 1. 73 Jack N. Arbolino, "Advanced Placement Program," Annual Report of the College Board 1961-62 (New York: College Entrance Examination Board, 1963), p. 19. 74 Jack N. Arbolino, "Advanced Placement Program," Annual Report of the College Board 1962-63 (New York: College Entrance Examination Board, 1964), pp. 32-33. 127 were state universities sponsoring five of the eight annual conferencesj and the previous fall, "the state universities of Michigan, Illinois, and New York (their several campuses combined) were among the first six institutions in order of candidates received, with the University of 75 Michigan receiving most of all." The Board supported statewide efforts to educate the schools about the Program in California, Hawaii, New York, and Utah through a series of workshops and institutes. For example, an
Advanced Placement Institute was held 16-20 August 1965 at the University of Hawaii for school and college teachers and was supported, in person, by College Board representatives Robert Cameron, Eugene Ferguson, and 76 the new Advanced Placement Director, Harlan P. Hanson.
While the number of mathematics candidates increased four-fold during the period (from 2,908 in 1960 to 11,623 in 1968), the grade dis tributions of these candidates remained relatively stable. Approximately
80 percent continued to score passing marks of 2 or better, and slightly 77 more than half received qualifying scores of 3 or more. During 1965-66 the College Board’s Committee of Advanced Placement changed the verbal interpretations of the five-point scale of grades "to relate more sharply 78 to the examiners' requirements." The change is illustrated in table 13.
75 Harlan P. Hanson, "Advanced Placement Program," Annual Report of the College Board, 1965-66 (New York: College Entrance Examination Board, 1966), p. 46. 7 6 See appendix E for extracts from W. Eugene Ferguson, "Report of the Group Session— Mathematics Curriculum Development, Advanced Placement Institute, University of Hawaii, 16-20 August 1965." (Mimeographed.) 77 See appendix D: Grade Distributions of Mathematics Candidates, 19 54-1984. 7 8 Hanson, "Advanced Placement Program," Annual Report of the College Board, 1965-66, p. 48. 128 TABLE 13
GRADE DISTRIBUTIONS OF MATHEMATICS CANDIDATES, 1961-1968
Number of Candidates (N) and Percentage Below Grade (%B) Each Year AP Exam Grade 1961 1962 1963 1964 N %B N %B N %B N %B
5 (Highest Honors) ...... 296 92 443 89 359 94 480 94 4 (Honors) ...... 627 74 749 72 809 80 1051 80 3 (creditable) ...... 1162 42 1269 41 1941 47 2313 50 2 (Pass) ...... 875 18 1012 17 1550 20 2033 24 1 (Fail) ...... 649 0 717 0 1189 0 1833 0
Total ...... 3609 4190 5848 7710 Mean Grade...... 2.7 2.8 2.6 2.5 Standard Deviation. . . . 1.2 1.2 1.1 1.2
1965 1966 1967 1968 AP Exam Grade N %B N %B N %B N %B
5 (Extremely Well Qualified) 661 93 763 92 660 94 965 92 4 (Well Qualified) ...... 1161 80 1148 80 1595 79 1667 77 3 (Qualified)...... 2532 52 2828 51 2680 54 3259 49 2 (Possibly Qualified) . . . 2846 20 2848 21 3241 23 2894 24 1 (No Recommendation). . . . 1821 0 2043 0 2499 0 2838 0
Total ...... 9021 9630 10675 11623 Mean Grade...... 2.6 2.6 2.5 2.6 Standard Deviation. . . . 1.2 1.2 1.2 1.2
SOURCE: [Advanced Placement Program of the College Board], "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1961-68. (Mimeographed single sheets, by year.)
The first attempt to collect and publish college statements of their Advanced Placement policies was made by the College Board in 1963. 79 Only thirty-seven institutions were included. The effectiveness of the
Program depended on college policies which honored appropriate Advanced
Placement work done by entering students, yet there was no central source
79 College Policies on Advanced Placement at Thirty-Seven insti tutions (New York: College Entrance Examination Board, 1963). 129 of such information. In the spring of 1967, the Board asked all four- year colleges and some forty two-year colleges to indicate the lowest examination grade they considered acceptable for granting advanced stand ing or credit in each of the Program's twelve courses. The survey's usable returns on 869 institutions were consolidated and published that 80 fall. In general, the survey indicated an increased willingness among colleges to honor Advanced Placement work: 60 percent of the reporting colleges honored grades in at least eight of the twelve subjects; 43 per cent granted some form of recognition (most commonly for qualifying marks of 3 or better) in all twelve subjects; and 14 percent said that "they do not normally guarantee to recognize any Advanced Placement grade in any 81 subject." The Board planned to update this booklet periodically.
Table 13 shows that the mean grades and standard deviations in mathematics varied little from 2.6 and 1.2, respectively, over these eight years. In fact, none of the other twelve subjects had means lower than the means in mathematics and physics, which remained close together at the bottom within one-quarter of a standard deviation of one another. From
1959 through 1967, the grade reports sent to the student, his school, and his college gave his scaled scores on the objective and essay portions of the mathematics test separately, together with his overall, composite score. Over this period, the mean grade on the mathematics essay portion was consistently one-sixth to one-quarter of a standard deviation below the corresponding mean objective grade. This consistent disparity con cerned many at E.T.S. and the College Board, among others. In 1965, the
80 College Advanced Placement Policies, 1967-68 (New York: College Entrance Examination Board, 1967). 81 "News of the College Board," The College Board Review 65 (Fall 1967); 34. 130 retiring Advanced Placement Director had been one of the early advocates of a second Advanced Placement mathematics course and examination:
Although the APP has thrived and earned widespread recognition, there have been few changes in its scope since the beginning. Today, 10 years later, there is a need for broader coverage in some subjects at the college level, particularly mathematics, and there may also be a need for new subjects, such as Russian.
In her study of the responses of 182 AP math graduates, 1957-65,
Lefkowitz found that the students themselves realized their deficiency:
The most frequent reaction of the students concerned their insufficient background in theory in the Advanced Placement course. They found that, when placed in classes with other students who had studied elementary calculus in college, the college-prepared students had a stronger theoretical background. However, the teacher of the Advanced Placement class at the high school felt that the students in the class did not have the mathematical maturity for an emphasis on rigor in the course. . . . Recognition of this problem is probably one of the reasons for the adoption of two Advanced Placement g^ogram syllabi in mathematics beginning with the school year 1968-69.
Lefkowitz's argument referred to the following distinction made between the two courses in the College Board's new 1968 syllabus:
Aside from differences of content, the major distinction between Calculus AB and Calculus BC is that Calculus AB is primarily con cerned with an intuitive understanding of the concepts of calculus and experience with its methods and applications whereas Calculus BC requires deeper knowledge of the theoretical tools of calculus. Use of the word "intuitive" is not meant to suggest a reduction of either clarity of concept or precision of expression. Rather it attempts to distinguish between a calculus course that emphasizes precise proofs of all theorems— rigor in the formal sense— and a calculus course that states definitions and theorems correctly but that frankly defers some proofs until a later course.
82 Jack N. Arbolino, A Report to the Trustees of the CEEB: The Council on College-Level Examinations (New York; College Entrance Exami nation Board, 1965), p. 17. 83 Ruth S. Lefkowitz, "The First Nine Years— A Study of the Advanced Placement Program in Mathematics," Journal for Research in Mathematics Education 2 (January 1971): 34. 84 The College Board, 1968-69 Advanced Placement Mathematics (New York: College Entrance Examination Board, 1968), p. 149. 131 Lefkowitz's study school erred in the matter of student selection and the teacher, in the decision to limit the theoretical aspects, for if a student did not have the mathematical maturity required for the calcu lus course prescribed by the syllabus at the time, the student should not have been in the Advanced Placement class. The argument is fallacious
(post hoc, ergo propter hoc), because the decision to adopt two mathe matics courses was based on a much broader consideration of national need than the recognition of a problem of insufficient background in theory.
When the Commission on Mathematics presented its college prepara tory program for grades 9-12, it was quite specific with respect to its
Elementary and Intermediate Mathematics courses for grades 9, 10, and 11.
The Advanced Mathematics course for grade 12 was more flexible; the core was at least a one semester course, Elementary Functions, which stressed the study of polynomials, the exponential and logarithmic function, and the circular functions. If this course was not extended to a full year, then the Commission recommended either a probability course with statis- 85 tical applications or a semester of modern algebra. The Commission's college preparatory program ideally filled four years, but could be cur tailed to three-and-a-half if Elementary Functions was limited to a semes ter. With respect to the place of calculus, the Commission was specific:
The Commission takes the position, held generally in the united States at present, that calculus is a college-level subject. . . . At the same time, however, the Commission recommends that well-staffed schools offer their ablest students a year of college-level calculus and analytic geometry as recommended in the Advanced Placement Pro gram. It is essential, though, that such a year be firmly based on a full pre-calculus program, completed early by some form of accel eration"! [Italics mine.]8
85 Commission on Mathematics, Program, p. 30. 86 Ibid., p. 14. 132 A year of college-level calculus and analytic geometry for any high school senior presumes that he has met the prerequisites for the
Advanced Placement course by the end of his junior year. Assuming that a given 'well-staffed school' has a sufficient number of 'able students' to offer an AP mathematics course, it must still organize a pre-calculus curriculum which allows three and one-half to four years of work to be taught in three years, or allows the study of algebra to begin one year earlier in the eighth grade. Every school contemplating an AP mathema tics course must resolve this curricular problem, and not have it resolved administratively at the expense of time taken from the mathematics class
room. Large high schools with a greater span of control over seventh, eighth, and ninth grade curricula have an easier task designing such a
pre-calculus program. A joint publication by the College Board and E.T.S.
lists five methods used successfully by schoolsj but each school must 87 determine the method which best fits its students and circumstances.
To determine what the nation's high schools were doing to imple
ment the recommendations of the Commission on Mathematics, questionnaires
were sent to approximately 2,700 seniors who took either the Level I or
Level II Mathematics Achievement tests of the College Board in December
1965 or January 1966. Seventy percent responded; and of these, 1,186 had
taken the Level I test and 724, the Level II. Relevant to a pre-calculus
program was the fact that at least 17 percent stated that they had taken
Intermediate Mathematics in grade 10. Also, 17 percent said that they
"were enrolled in an advanced placement course in mathematics for which
87 Advanced Placement Program of the College Entrance Examination Board, Beginning an Advanced Placement Mathematics Course, comp, and ed. S. Irene Williams, Chancey Jones, and James Braswell (New York: College Entrance Examination Board and Educational Testing Service, 1975), p. 8. 133 they hoped to gain advanced standing and/or credit in college, with 5 percent taking a College Board advanced placement course and 12 percent taking an advanced placement course other than that sponsored by the 88 College Entrance Examination Board." The latter group was asked to indicate which selected topics would receive substantial attention in their non-College-Board course by the end of their first semester in grade 12. Of those not in the College Board program, 96 percent said that elementary functions would be covered in that period, but only about half of that percentage could say the same for differential (57 percent) 89 and integral (47 percent) calculus topics.
Here then was the crux of the problem: a substantial number of
Advanced Placement aspirants had not been able to complete the Elementary
Functions prerequisite course by the end of their junior year. What was needed in mathematics was a second advanced placement course which would combine elementary functions and calculus in a single year, but in a pro portion which would devote well over half of that year course to the cal culus . This roughly describes the new Calculus AB course and gives the rationale for the adoption of two Advanced Placement courses in 1968-69 .
A school contemplating an AP mathematics program must be 'well- staffed' . The growth shown in the middle period meant the introduction of the Program to many new schools and their teachers. perhaps it was inevitable that during this period there would be uncomplementary charges made against schools which failed to realize what strengths were required in an Advanced Placement mathematics teacher:
88 S. Irene Williams, "A Progress Report on the implementation of the Recommendations of the Commission on Mathematics,” The Mathematics Teacher 63 (October 1970): 467. 89 Ibid., p. 467, table 7. 134 The first problem a school must solve when planning for its superior students is to get superior teachers
Not every high school teacher is capable of teaching a course at the collegiate level. Competence in teaching at the high school level is one thing: the capacity to deal with college-level materials is quite another .^1
I know that some of you have had some sad experiences with students who have had calculus in high school, but most of those failures, I am sure, came from ill-conceived calculus courses, taught by ill- prepared teachers, taken by ill-prepared students
The liberal objections are that the teaching of mathematics is too dogmatic; too much devoted to rule-of-thumb and memorization, too little devoted to intuition, argument, reason, and deduction. The technical objections are primarily that the student is not prepared for an early entrance into the sciences, . .
A crash program to teach calculus in response to various pres sures might result in a course's being taught by inadequately pre pared teachers to less than superior pupils. The course could become a meaningless mechanical manipulation of symbols that results in a less than adequate preparation in an area of mathematics so important to the understanding of more advanced work
As for the teacher, calculus is a hard subject to teach well, and should be handled by only the best trained and thoroughly experienced persons. I deplore the use of the average high school teacher in a calculus course, and equally deplore the use of inexperienced grad uate students in such courses in college
These are six, truthful indictments of the many expressing con
cern for the vital role of the Advanced Placement mathematic teacher. In
90 George Grossman, "Advanced Placement Mathematics— for Whom?" The Mathematics Teacher 55 (November 1962): 564. 91 j. Quentin Jones, "Advanced Placement— Taking a Hard Look," NASSP Bulletin 59 (October 1975); 69. 92 [W.] Eugene Ferguson, "Calculus in the High School," The Mathematics Teacher 53 (October I960): 451. 93 Albert A. Blank, "Remarks on the Teaching of Calculus in the Secondary School," The Mathematics Teacher 53 (November 1960): 537 94 Albert Beninati, "It's Time to Take a Closer Look at High School Calculus," The Mathematics Teacher 59 (January 1966): 29. 95 Carl B. Allendoerfer, "The Case against Calculus," The Mathematics Teacher 56 (November 1963): 484. 135 general, the characteristics sought in the teacher of an AP calculus course are the same ones to be found in any excellent teacher, but per haps to a higher degree: mastery of his subject field; infectious enthu siasm and ability to communicate this knowledge and its relevancy effec tively to his students; insistence on high standards, not only for his students, but for himself; and the honest desire to teach, and not feel threatened by, students whose academic potential may far exceed his own.
In the Commission's 'well-staffed school', the likelihood of finding such a paragon, not only as a teacher of the Advanced Placement calculus, but also as teachers of the prerequisite courses leading to the calculus, is high. Such teachers continually increase their own knowledge and skills, while demanding from their students slightly more than the students may believe is possible. This skill in challenging the forefront of student ability in order to stimulate bright youngsters into exercising and developing their talents is another mark of the experienced AP teacher.
When the middle period began, the forty-fourth annual convention of the National Association of secondary School Principals [NASSP] was devoted to the theme, "Quality Education: Today's Priority." Summarizing the presentations made in the session on the benefits of Advanced Place ment and its implications for the curriculum, Kenneth W. Lund concluded:
It is not enough to group separately the talented seniors and assign a capable teacher, adopt a college textbook, and label a course as advanced placement. . . . On the contrary, an advanced placement class is the last step and a natural one that follows from some decisions that are made and some action taken in the curriculum and instructional programs of the school and affecting the student as he enters high school. These decisions include the following concepts and their implementation: 1. The reality of individual differences
96 , The College Board, 1968-69 Advanced Placement Mathematics, pp. 146-47. 136 2. The acceptance of ability grouping and the concomitant curricular and instructional adjustments 3. The need for acceleration and its value for certain individual students 4. The establishment of a sound guidance program and a policy of individual planning for each pupil In short, the entire faculty, student body, and their parents will feel the effects of this program^
In the same session, Jerry Gerich, a principal in Grosse Pointe,
Michigan, summarized the effects of the Program in other areas:
Advanced Placement programs have affected teaching, course plan ning, high-school-college communication, and the educational goals of secondary schools and institutions of higher education in the United States. While much progress has been made during the fifties, it appears the sixties should be even more productive.^
The middle period was more productive with 1,973 new schools pro viding AP courses for their able students. Dr. Conant used the expanding
Program as a measure of any high school's quality. In his view, two of the minimum five criteria which should be met by any widely comprehensive high school were that the school "provide one or more advanced placement 99 courses" and "provide instruction in calculus." For budgetary reasons, not all would agree with Conant's revised statement that "every high school ought to strive to provide the opportunity for advanced placement in at least one subject, no matter how few candidates there may be."'*'^
97 Kenneth W. Lund and Jerry J. Gerich, "How Can Advanced Place ment Programs Benefit Qualified Students? What Are the Implications for the Secondary-School Curriculum?" The Bulletin of National Secondary- School Principals 44 (April 1960): 217. 98 Ibid., p. 216. 99 James B. Conant, The Comprehensive High School (New York: McGraw-Hill Book Co., 1967), pp. 16-17. The widely comprehensive high school referred to any school of medium size (enrollment between 750 and 2,000) and type, "as measured by the percentage of graduates in 1965 who pursued a formal education (25 per cent to 75 per cent)." Ibid., p. 11.
"^^Conant, Slums and Suburbs, p. 93. The Recent Period, 1969 to 1984
After fifteen years, the Advanced Placement Program offered des criptions of two calculus courses and two corresponding AP examinations, entitled Calculus AB and Calculus BC, for the school year 1968-69.'^'*'
Prior to that year only one course description and validating examination had been given. Its present counterpart is Calculus BC, which "is an intensive full-year course in calculus that places due emphasis on theo retical aspects of the calculus of functions of a single variable and 102 that includes topics in infinite series and differential equations."
The new Calculus AB course combines elementary functions and the calculus in a single year, but the emphasis for well over half the year must stress the differential and integral calculus. The calculus content of the AB course is somewhat less than that in previous course descrip tions, but the content of Calculus BC is considerably more extensive with the addition of the topics in infinite series and elementary differential equations. Depending on their size, curriculum, availability of experi enced teachers, and extent of student talent in mathematics, schools could decide to offer both courses, only one of them, or neither course.
Each AP candidate, based on the extent and degree of his course and prep aration, would take the most appropriate examination in May.
That both courses described in the 1968-69 syllabus represented college-level mathematics courses, for which colleges could feel confi dent in granting credit and advanced standing, was clear. Each course followed recommendations made in the 1965 report to the Mathematical
^^See appendix A: Calculus AB and Calculus BC: Topical Course Descriptions. 102 The College Board, 1968-69 Advanced Placement Mathematics, p p . 146-47. 138 Association of America by its Committee on the Undergraduate Program in
Mathematics (CUPM), which was charged with making recommendations for the
improvement of college and university mathematics curricula. This 1965 report, entitled "A General Curriculum in Mathematics for Colleges"
(GCMC), proposed a general college curriculum in mathematics consisting of a descriptive list of semester-course offerings, "from which a multi- 103 plicity of individual student programs can be constructed." Among the
lower division courses were the following semester offerings: Mathematics
0, Elementary Functions and Coordinate Geometry; Mathematics 1, Introduc- 104 tory Calculus; and Mathematics 2, Mathematical Analysis. In composing the two Advanced Placement mathematics course descriptions, the College
Board accepted the CUPM recommendations: "Advanced Placement Calculus AB
corresponds approximately to GCMC Mathematics 0 and 1, whereas Calculus
BC is similar to GCMC Mathematics 1 and 2."^^ This choice of recom
mended semester courses would make it easy for most colleges, which offer
a calculus sequence of several semester courses, to place an AP candidate
within their sequence according to his results on either Advanced Place
ment examination. It also explains the reason for the College Board's
Committee on the Undergraduate Program in Mathematics, "A Gen eral Curriculum in Mathematics for Colleges," report to the Mathematical Association of America, Berkeley, California, January 1965, p. 3. 104 Ibid., 9-11. Mathematics 1: Introductory Calculus (second version) is a three semester-hour course of 39 lessons in the differential and integral calculus of the elementary functions with associated analytic geometry. Ibid., pp. 33-34. Mathematics 2: Mathematical Analysis (first version, which com pletes single variable calculus before multivariable.) This is a 3 or 4 semester-hour course of 39 lessons in the techniques of one-variable cal culus, limits, and series. Differential equations, 15 lessons, is a topic taken from the next course, Mathematics 4. Ibid., pp. 37-38.
^ ^The College Board, 1968-69 Advanced Placement Mathematics, p. 147. unequivocal statement that "the content of Calculus BC is designed to qualify the student for placement one semester beyond that granted for
Calculus AB
The Advanced Placement mathematics examinations are each three hours long, and they are meant to determine how well a candidate has mas tered the concepts and techniques of the subject matter described in the corresponding course. Because Calculus AB and Calculus BC have the com mon semester of material from the recommended GCMC Mathematics 1 course, it is not surprising that, between 1969 and 1982, approximately one-third of the forty-five objective questions and three of the seven essay ques tions on the two examinations were identical. With the reduction to five essay questions in the last two years, only two questions have been the same. Both examinations are demanding. This means that any attempt to offer the Calculus BC students less than a full 'year of college-level calculus' or the Calculus AB students less than a full semester, by giv ing superficial coverage to some topics included in the course descrip tions, is futile. At entry in September, each prospective candidate must be fully prepared to spend eight to ten hours per week until mid-May acquiring the calculus.
The sequential nature of mathematics requires that a student com plete at least three and one-half years (through elementary functions) of the Commission's four-year program before undertaking the Calculus BC course. Many school mathematics departments argue that students will be better prepared for this demanding course if, in addition to the semester of elementary functions in their fourth year, they also have a semester of analytic geometry, enriched with such topics as polar coordinates, 140 sequences and series, and limits of a function. Schools have various names for this two-semester sequence: advanced mathematics, mathematical analysis, pre-calculus mathematics, or simply Math 4. Whatever the name, it is a fitting fourth-year course for those who intend to take their calculus as college freshmen; but it must be completed in the 11th grade by those destined for Calculus BC as seniors. For the latter, some form of curriculum revision is necessary to accelerate them by one year.
Schools have found varying degrees of success with one of the following methods of curricular revision; (1) short compression— identify participants during grade 8, and compress the three and one-half to four years of the Commission's program into grades 9, 10, and 11; (2) summer school compression— identification and compression as in (1), except that students take one of che four prerequisite courses during summer school in classes which usually meet four hours per day, five days per week, for 107 six to eight weeks; (3) long compression— identify participants by the end of grade 6, teach accelerated arithmetic in grade 7, and teach the 108 Commission's program in the four grades 8 through 11.
In 1960, Principal O. Meredith Parry of William Penn Senior High School in York, Penn, reported that "about thirty per cent [of a summer school enrollment over 700] accelerate their regular program by this method so they may study Advanced Placement or College Level courses in the eleventh and twelfth grades; i.e., advanced physics, mathematics, or chemistry, etc." John J. Condon et al., "How Can Summer Schools Enrich or Accelerate the Educational Program of Capable Students?" Bulle tin of the NASSP 44 (April I960); 126. See also Paul W. Harnly and Harry D. Lovelass, "How Have Summer Schools Been Used to Enrich the Educational Program for the Academically Talented?" Bulletin of the NASSP 43 (April 1959): 182-186. 108 APP of the College Board, Beginning an Advanced Placement Mathematics Course, p. 8. Excellent examples of three junior high school programs which have accelerated arithmetic in grade 7, algebra in grade 8, and geometry in grade 9 are described in Clarence A. Brock et al., "How Can the junior High School Provide Quality Education for the Academically Talented Student?" Bulletin of the NASSP 44 (April 1960): 139-145. 141 Both schools involved in the present study use long compression.
Selected participants are grouped in an enriched, honors sequence begin ning with accelerated arithmetic in grade 7. By completing algebra in grade 8, they have been accelerated in the Commission’s recommended four- year sequence by one year over their college-capable classmates. Later entry into the honors sequence, by newcomers to the school or by ’’late bloomers" when identified by previous schools, teachers, or counselors, 109 is possible through use of summer school compression.
Those concerned with improving the quality of education for the academically talented or gifted eventually must decide whether their pro posed program should include provisions for acceleration, enrichment, or grouping. Ability grouping, whether in multitracks or by subject matter, is fairly well understood. But, both in the literature and in application by schools, there is little consensus on a clear definition of either acceleration or enrichment. Some use the terms distinctly, speaking of acceleration, grouping and enrichment as three separate "administrative devices" or "educational techniques" employed in providing specially for superior students; others speak of enrichment as a catchall, which 110 includes acceleration and grouping as subordinate techniques.
109 A wide variety of different special programs and school prac tices for the academically talented in twenty-eight cities may be found in National Education Association and National Association of Secondary- School Principals, Administration; Procedures and School Practices for the Academically Talented Student in the Secondary School (Washington, D.C.: National Education Association, 1960), pp. 138-172.
^^Ib i d . , pp. 83-94. See also National Education Association and National Council of Teachers of Mathematics, Mathematics for the Academically Talented Student in the Secondary School, ed. Julius H. Hlavaty (Washington, D.C.; National Education Association, 1959), pp. 29-38; and Milton J. Gold, Education of the Intellectually Gifted (Columbus, Ohio: Charles E. Merrill Books, 1965), p. 158. 142 The Southern Regional Project for Education of the Gifted, in their justification of special education for gifted students, made the initial assumption that "gifted children as a group differ from others in learning ability; they learn faster and remember more, and they tend to think more deeply with and about what they learn.Copley suggested that enrichment and acceleration should be thought of as twin responses to the fact that talented students are able to learn more, on the one hand; and learn at a faster rate, on the other;
And to prevent any overlap— for obviously the student who learns fas ter may also learn more, and vice versa— let us think of any subject as having a central core consisting of those fundamental parts with out a mastery of which no one can be said to know it at all. . . . Acceleration means that the student is set to mastering this core of essentials at a rate of efficiency and speed higher than the ordinary. Besides this core, the subject also has ramifications and amplifi cations, relations with other subjects, and so on— some more, some less significant, but all contributing to its totality. Enrichment means that the study of a judicious selection of these is added to mastery of the core.'*'8
18 This definition is my own. it is based on observation of what the schools are actually doing rather than on educational theory. I fully realize that, by my definition, acceleration and enrichment could take place concurrently 2
Copley's definitions of enrichment and acceleration are adopted for use in this study with the slight modification that acceleration is the process by which selected students are allowed to advance more quickly in "mastering the core of essentials" than their chronological peers.
While this may be accomplished in kindergarten or elementary school by early admissions or grade skipping, it more normally is accomplished in junior and senior high school by compressed curricula, e.g. accelerated
Southern Regional Project for the Education of the Gifted, The Gifted Student; A Manual for Program Development (Atlanta; Southern Regional Education Board, 1962), p. 22. 112 Copley, American High School, pp. 19-20. 143 arithmetic, or by taking more courses per year, e.g. enrolling in summer school. For example, if the extra course taken during summer school is part of the Commission's core curriculum, this is considered acceleration but not enrichment. Enrichment does characterize courses in the honors sequence in which the very talented have been grouped, because invariably additional material is added to the core.
While acceleration is necessary to enable Calculus BC students to complete the Commission's core curriculum by the end of their junior year, the Calculus AB course permits entry for students who have not completed elementary functions. Many schools which had not been able to organize a curriculum to accelerate their students by at least one semester decided that they did have the capability of offering the new Calculus AB course. Some schools already participating in the Program decided that the two additional chapters required for Calculus BC (infinite series and elementary differential equations) could not be added immediately to their existing calculus curriculum; these schools counseled their stu dents to take Calculus AB also. The outcome was not surprising.
The sixteen-year, recent period began in 1968-69 with a percen tage annual increase in schools almost double that of the previous year; in mathematics candidates, it was more than doubled. From a total of
11,623 Advanced Placement mathematics candidates in 1967-68, the number of candidates increased to 13,954 the following year with 10,280 taking the new Calculus AB examination and 3,674 the more demanding Calculus BC examination. Table 14 shows the continued growth in Advanced Placement participation, 1969-1984, in all categories with one curious exception.
In 1972-73 for some inexplicable reason, but possibly economic, there was a decrease in every category for the first, and only time in thirty years. 144 TABLE 14
ADVANCED PLACEMENT PARTICIPATION, 1969-1984
Number Number Number Number of Exams of Exams Number Year of of in All in Mathe of Schools Candidates Subjects matics Colleges
1968-69 3,095 53,363 69,418 13,954a 1,288 1969-70 3,186 55,442 71,495 14,379 1,368 1970-71 3,342 57,850 74,409 14,673 1,382 1971-72 3,397 58,828 75,199 15,186 1,483 1972-73 3,240 54,778 70,651 14,310 1,437 1973-74 3,357 60,863 79,036 16,038 1,507 1974-75 3,498 65,635 85,786 17,090 1,517 197 5-76 3,937 75,651 98,898 19,065 1,580 1976-77 4,079 82,728 108,870° 20,317 1,672 1977-78 4,323 93,313 122,561 22,510 1,735 1978-79 4,585 106,052 139,544 24,727 1,795 1979-80 4,950 119,918 160,214 27,879 1,868 1980-81 5,253 133,702 178,159 30,558 1,955 1981-82 5,525 141,626 188,933 31,918 1,976 1982-83 5,827 157,973 211,160° 35,489 2,130 1983-84 6,273 177,406 239,666 39,962 2,153
SOURCES: [Advanced Placement Program of the College Board], "Annual Advanced Placement Program Participation," Princeton, [1984]. (Mimeographed); and "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1969-84. (Mimeographed single sheets, by year.)
aColumn figures include both Calculus AB and Calculus BC exams.
This 1978-79 figure includes the 1 millionth candidate.
C1976-77 figure includes 1 millionth exam; 1982-83, 2 millionth.
To date, a total of 427,811 candidates have taken an Advanced
Placement mathematics examination, 69,756 in the first fifteen years and
358,055 during the recent sixteen-year period. Of the latter, 258,821 or
72 percent chose the Calculus AB examination and 99,234 took Calculus B C .
During this period, the number of schools doubled and the number of can didates, total exams, and exams in mathematics increased at least three fold; but the average annual percentage increase (AAPI) was less than it 145 had been during the previous middle period. To better show the compara tive growth between categories for both periods, AAPls for the eight-year middle period in each category are presented first, followed by the AAPls for the first and second halves (each eight years) of the present periods number of schools— 16, 3, and 6 percent; candidates— 21, 6, and 11 per cent; examinations in all subjects— 20, 7, and 12 percent; examinations in mathematics— 19, 7, and 10 percent; and colleges— 10, 4, and 4 percent.
Note the common pattern within categories for the three, consecutive, eight-year terms: the average annual percentage increase (AAPI) was high in 1961-68, dropped to a low during 1969-76, and then partially recovered in 1977-84. For the first and second halves of the present period, the
AAPls in the two calculus courses were dissimilar: Calculus AB— 4 and 11 percent; Calculus BC— 7 and 6 percent. This shows the steady growth of
Calculus BC, slightly diminishing in its rate during the second half, versus the volatile growth of the new Calculus AB, beginning slowly and then almost tripling during the second half as more schools and candi dates realized that it was within their capabilities.
One of the reasons for the increase in total number of candidates during 1969-84 was that the College Board doubled the number of course offerings from twelve to twenty-four. In addition to the second mathe matics examination in 1969, an additional one was offered in physics:
Physics C requires the calculus, while Physics B does not. In 1971,
French Literature made a reentry; and the following year music and two offerings in art (Art History and Studio Art) were added. In 1973, the physics with calculus was separated into two semester courses and exami- minations: Physics C— Mechanics and Physcis C— Electricity and Magnetism.
Spanish and music added an extra course in 1977 and 197 8, respectively, 146 as did art and English in 1980. Computer Science was the latest addition in 1984, so that the Program currently provides course descriptions and 113 examinations on twenty-four college courses in twelve fields.
One aspect of the growth of Advanced Placement during its first ten years was its geographical imbalance. Led by New York state, which
"consistently provided about one-fourth of all candidates," five of the eight states presenting the greatest number of candidates were in the 114 Northeast. But by 1984, although New York still led with 16.2 percent of the candidates followed closely by California with 15.6 percent, the
AP center had moved westward. The Midwest and Far West now have greater representation, particularly in percentages of schools participating and 115 AP examinations per 100 thousand population. Reviewing the Program's first twenty-five years, AP Director Harlan P. Hanson observed:
So, too, the program has commended itself to educational authorities in entire states. New York and Utah, then California and Connecticut, and more recently, Pennsylvania and the Carolinas, have all made APP an integral part of their educational plans. Per capita participation is highest in Utah and Alaska, with New York, Delaware, and Colorado, followed by Massachusetts, Connecticut, and Hawaii, close behind. But these same states are not only highest in their rates of use. They— and others like Oregon, Virginia, and North Carolina— grow most in this respect each year.
Table 15 reflects the growth in the two calculus courses and the improvement in school proficiency teaching each course, as evinced by the fifteen percentage point increase in qualifying students over the period.
113 See appendix F: AP Examination Participation by Subject for Selected Years. 114 Elwell, p. 116.
^^See appendix G: Advanced Placement Participation by State in 1984. This includes top ten ranking in schools, candidates, and exams.
^^Harlan P. Hanson, "Twenty-Five Years of the Advanced Placement Program: Encouraging Able Students," The College Board Review 115 (Spring 1980): 9. TABLE 15
GRADE DISTRIBUTIONS OP MATHEMATICS CANDIDATES, 1969-84
AP Grade Number of Candidates (N) and Percentage Below Grade (%B), Calculus AB and Calculus BC, By Year
1969 1970 1971 1972 1973 1974 1975 1976 LaiCUlUS A.D N %B N %B N %B N %B N %B N %B N %B N %B
5 931 91 935 91 1,241 88 1,114 90 1,289 87 1,045 91 1,431 88 1,648 87 4 1,738 74 1,848 73 1,926 70 1,838 73 1,734 69 1,974 73 2,593 66 2,828 66 3 2,726 48 2,565 48 2,345 47 3,355 41 2,814 41 3,727 40 3,783 34 4,241 33 2 2,974 19 2,924 20 3,281 16 2,533 17 2,798 13 2,273 20 2,558 12 2,604 13 1 1,911 0 2,001 0 1,675 0 1,771 0 1,236 0 2,194 0 1,439 0 1,755 0
Subtotal 10,280 10,273 10,468 10,611 9,871 11,213 11,804 13,076 Mean Grade 2.7 2 .7 2 .8 2 .8 2.9 2.8 3.0 3.0 Stnd. Dev. 1.2 1 .2 1 .3 1 .2 1.2 1 .2 1.2 1.2
1969 1970 1971 1972 1973 1974 1975 1976 N %B N %B N %B N %B N %B N %B N %B N %B
5 554 85 603 85 900 79 765 83 1,232 72 814 83 1,306 75 1,345 78 4 770 64 822 65 899 58 946 63 910 52 1,062 61 1,255 52 1,327 55 3 943 38 965 42 1,109 31 1,085 39 1,001 29 1,209 36 1,329 26 1,539 30 2 989 11 1,191 13 747 13 816 21 903 9 694 22 888 10 1,085 12 1 418 0 525 0 550 0 963 0 393 0 1,046 0 508 0 693 0
Subtotal 3,674 4,106 4,205 4,575 4,439 4,825 5,286 5,989 Mean Grade 3.0 2 .9 3 .2 2 .9 3.4 3 .0 3.4 3.3 Stnd. Dev. 1.2 1 .3 1 .3 1 .4 1.3 1 .4 1.3 1.3
Total 13,954 14,379 14,673 15,186 14,310 16,038 17,090 19,065
'■‘J TABLE 15— Continued
AP Grade Number of Candidates (N) and Percentage Below Grade (%B), Calculus AB and Calculus BC, By Year
1977 1978 1979 1980 1981 1982 1983 1984 U1 Us) rxD N %B N %B N %B N %B N %B N %B N %B N %B
5 1,939 86 2,168 86 2,329 87 2,759 86 3,173 86 3,581 85 3,905 85 4,639 85 4 2,935 65 3,181 66 3,740 66 4,079 66 4,762 65 4,996 64 5,879 63 7,071 62 3 4,336 35 5,281 33 5,949 32 6,970 31 7,759 30 8,011 30 8,962 30 10,255 28 2 3,099 13 3,233 12 3,286 13 3,423 14 3,952 13 4,288 12 4,685 12 4,843 12 1 1,780 0 1,911 0 2,311 0 2,865 0 2,891 0 2,949 0 3,275 0 3,775 0
Subtotal 14,089 15,774 17,615 20,096 22,537 23,825 26,706 30,583 Mean Grade 3 .0 3 .0 3 .03 3 .02 3.06 3.08 3 .09 3 .13 Stnd. Dev. 1 .2 1 .2 1 .20 1 .22 1.21 1.22 1 .21 1 .21
1977 1978 1979 1980 1981 1982 1983 1984 N %B N %B N %B N %B N %B N %B N %B N %B
5 1,560 75 1,696 75 1,705 76 1,700 IQ 1,840 77 1,978 76 2,362 73 2,525 73 4 1,558 50 1,484 53 1,570 54 1,608 57 1,937 53 1,764 54 1,853 52 2,013 52 3 1,654 23 1,825 26 1,724 30 2,064 31 2,147 26 2,267 26 2,376 25 2,580 24 2 753 11 901 12 1,236 12 1,265 15 1,105 12 1,111 ‘12 1,198 11 1,039 13 1 703 0 830 0 877 0 1,146 0 992 0 •973 0 994 0 1,222 0
Subtotal 6,228 6,736 7,112 7,783 8,021 8,093 8,783 9,379 Mean Grade 3 .4 3 .3 3 .28 3 .19 3.32 3.33 3.39 3 .38 Stnd. Dev. 1 .3 1 .3 1 .33 1 .34 1.30 1.31 1.31 1 .33
30,558 31,918 39,962 Total 20,317 •22,510 * 24,727 27,879 35,489
SOURCE: [Advanced Placement Program of the College Board], "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1969-84. (Mimeographed single sheets, by year.) (-■ 149 Exceeded only by the number of candidates in English, 427,811 candidates have taken an Advanced Placement mathematics examination in the past thirty-one years. Of these, 84 percent took either Calculus AB or Calculus BC in the last sixteen years. More schools offer the AB course and this number has more than doubled in the last twelve years— from 1,653 in 1973 to 3,525 in 1984. The corresponding figures for Cal culus BC are 706 in 1973 to 1,136 in 1984. Many schools offer both courses, which automatically poses the problem of student selection for each course. Table 15 shows the improvement of the mathematics candi dates in both calculus courses even as their number more than tripled: the mean grades in each course improved by about one-third of a standard deviation and the number of qualifying candidates increased in Calculus
AB from 50 to 70 percent, and in Calculus BC from 60 to 75 percent.
Table 16 shows a consolidation of the sixteen distributions for each course into a single composite distribution. The greater strength of the Calculus BC candidates is apparent in this comparison.
TABLE 16
COMPOSITE CALCULUS GRACE DISTRIBUTIONS, 1969-84
Number of Candidates (N) and Percentage Below Grade (%B) AP Grade Calculus AB Calculus BC N %B N %B
5 34,127 87 22,885 77 4 53,122 66 21,778 55 3 83,079 34 25,817 29 2 52,754 14 15,921 13 1 35,739 0 12,833 0
Total 258,821 99,234 Mean Grade 2.99 3.26 Stnd. Dev. 1.22 1.32 150 Some might argue that it is improper to consolidate distributions from different tests over time. In the present case, it is not the raw scores that are being consolidated, but the numbers of candidates at each level of the grading scale. The stability of the grading scale over time is a fundamental tenet of the Program; colleges depend on it for the granting of credit and advanced standing. It is achieved by holding the objective, Section I portion of each year's test secure so that scxne questions may be reused the following year for equating purposes. Each syllabus has commented on the stability of the grading scale;
This reuse of some of the multiple-choice questions provides essential information about the relative abilities of populations taking the examination in different years and about the level of difficulty of various examination forms. The statistical information obtained guides the chief reader in determining the range of scores that will yield reported grades on a 1 to 5 scale that are comparable from year to year. The equating process is at present the best method available for providing students, colleges and universities, and secondary schools with a grading scale that remains stable for many years. The stability of the grading scale is basic to the wide acceptance by colleges and universities of the Advanced Placement Program in mathematics
In addition to maintaining the stability of the grading scale
from one year to the next, the College Board periodically obtains distri
butions of their objective tests taken by a sample of college students in
order to have a comparative check on AP grading standards. Between 1974
and 1980, the Board conducted a national study to determine the validity of the Advanced Placement assessment procedures. College faculty members were asked to administer the objective section of the AP examination in their field to non-AP students at the end of the appropriate course.
Representative groups, from among the 150 institutions receiving over
117 The College Board, Advanced Placement Course Description, Mathematics; Calculus AB, Calculus BC (New York; College Entrance Examination Board, 1980), pp. 11-12. 70 percent of the AP candidates annually, responded to provide data on 118 ten Advanced Placement examinations. In all ten subject areas tested,
•'Advanced Placement students achieved a higher mean score than did col lege students, indicating at least slightly superior performance to that 119 of college students." For example, of the maximum 45 points possible on each calculus examination, the mean scores for AP candidates and non-
AP collegeans in Calculus AB were 24.5 and 16.5, and in Calculus BC were 120 23.5 and 15.4, respectively.
The Calculus BC validity study was conducted in 1974, and the
Calculus AB test in 1977 . The percentage distributions in table 17 pro vide a comparison of the course grades earned by non-AP college students with the grades AP candidates received on their AP examinations, and the
AP grades (estimated on the basis of the objective section scores) that the non-AP college students would have received on the AP examination, in both calculus courses, table 17 shows that a greater percentage of college students received high grades than did the superior AP candidates on their qualifying examinations. For example, 63 percent of the college freshmen in Calculus BC received A ’s or B ’s, while only 48 percent of the
AP candidates were granted honor grades of 5 and 4. Yet, only four per cent of the college students had comparable 5 and 4 estimated AP grades.
118 Institutions participating in the 1974-80 validity studies and among those receiving the greatest number of candidates are indicated in appendix is Forty Colleges and Universities Receiving the Largest Number of Advanced Placement Examination Candidates, May 1984. 119 Carl H. Haag, "Comparing the performance of College Students and Advanced Placement Candidates on AP Examinations," Advanced Placement Program of the College Board, Grading the Advanced Placement Examinations in Mathematics, A report by Patricia Henry. (New York: College Entrance Examination Board, 1981), p. 33. 152 TABLE 17
PERCENTAGE GRADE DISTRIBUTIONS OF COLLEGE STUDENTS AND AP CANDIDATES
Percent of Students in Each Grade Level Advanced Placement Examination Scale A B C D E (5) (4) (3) (2) (1)
Calculus AB Course Grades Earned by College Students (A-E) 24 31 30 13 2 College Students' Estimated AP Grades (5-1) 3 9 23 31 34 AP Grades Earned by AP Candidates (5-1) 14 21 31 22 12
Calculus BC Course Grades Earned by College Students (A-E) 29 34 27 8 2 College Students' Estimated AP Grades (5-1) 1 3 15 36 46 AP Grades Earned by AP Candidates (5-1) 28 20 23 20 9
SOURCE: Carl H. Haag, "Comparing the Performance of College Stu dents and Advanced Placement Candidates on AP Examinations," Advanced Placement Program of the College Board, Grading the Advanced Placement Examinations in Mathematics, A Report by Patricia Henry. (New York; College Entrance Examination Board, 1981), p. 33.
A comparison of the course grades below C received by the college students with the AP grades below 3 awarded to the AP candidates would suggest that it is easier for college freshmen to receive C's or higher than for a superior group of AP candidates to receive qualifying scores.
The validity studies, particularly the two in mathematics, provide evi dence that an assumption of equivalency between the Program's five-point grading scale and the five, school and college, letter grades is false.
Experienced AP teachers know that it is likely that their C students will probably receive a 2 on the AP examination and fail to qualify. Carl H.
Haag, Director of the College Board Placement Test Program, agreed:
The data from the validity studies confirm that high standards of grading have been maintained in the Advanced Placement Program. For 153 example, AP candidates with grades of 3 are more comparable to col lege students receiving B, while an AP grade of 4 most often is simi lar to a college A. It appears that in many subjects an AP grade of 2 most often corresponds to a grade of C at college; [italics mine] this fact suggests that AP candidates with grades of 2 should be given special consideration to make certain they are appropriately placed and credited for their accomplishments .^1
Recognition, in terms of appropriate credit and placement, of college-level work done in secondary school continued to be a concern of the College Board. Its College Advanced Placement Policies, 1968 was a boon to school counselors and AP teachers who had to advise their AP stu dents. For each participating college, the booklet listed the lowest AP grade in each examination that the institution would normally accept as qualifying for credit or advanced standing. Revised editions, updating college policies and adding new colleges, were published in 1971, 1973,
1975, and 1978. The 1973 edition contained the AP policies of 1,450 institutions, including 93 colleges and universities which declared pro- 122 visions for granting immediate Sophomore Standing.
In 1970 the Trustees of the College Board approved the use of the
Advanced Placement Program as a criterion for membership eligibility, the college requirement being the award of credit and advanced standing for 123 grades of 3 or better in 70 percent or more of the examinations. A number of colleges which had more stringent requirements reviewed and revised their AP policies. The 1975 edition was considerably larger for it contained college policies for 19 examinations in the Advanced
Ibid. 122 The College Board, College Advanced Placement Policies, 1973 (New York: College Entrance Examination Board, 1973), pp. v-vi. 123 "News of the College Board," The College Board Review 78 (Winter 1970-71): 13. 154 Placement Program and five general and forty-one subject examinations in 124 the new College-Level Examination Program (CLEP). The 1978 edition was even more extensive. It contained policy information from 1,541 col leges on twenty-one AP examinations, five general and fifty CLEP subject examinations, the Admissions Testing Program, Sophomore Standing, and 125 maximum credit by examination. Appendix 1 of this edition was the source for table 1 which shows how many institutions report each AP grade in Calculus AB and Calculus BC as the minimum for routine or qualified credit. That less than 9 percent of the colleges grant even qualified credit for a grade of 2 on the calculus examinations is still a matter of concern to AP teachers, counselors, and the College Board.
For the 1983 and 1984 AP mathematics examinations, two changes were made: the use of calculators was permitted, although not required; and the number of essay questions was reduced from seven to five with a
concomitant reduction from 90 to 75 minutes in time allowed for this sec tion. The use of calculators will not be permitted on the calculus examinations beginning May 1985. The Committee on Mathematics believes that calculators can be used effectively in a calculus course, but was
concerned that the use of calculators with different technical capabili ties would cause inequities among students and the fact that students who 126 made substantial use of them in 1983 did not do as well as others.
124 The College Board, College Placement and Credit by Examination, 1975: Advanced Placement Program, College-Level Examination Program (New York: College Entrance Examination Board, 1975), p. v. 125 The College Board, College Placement and Credit by Examination (New York: College Entrance Examination Board, 1978), pp. xiii-xv. 126 The College Board, Advanced Placement Course Description, Mathematics: Calculus AB, Calculus BC— May 1985, May 1986 (New York: College Entrance Examination Board, 1984), pp. 12-13. The Growth of Advanced placement Mathematics at Punahou School
Punahou School in Honolulu, Hawaii is an independent, college- preparatory, coeducational day school with a student body of about 3,700 from kindergarten to grade 12. Founded in 1841 by Congregational mis sionaries, it is the oldest college preparatory school west of the Rocky
Mountains, and the second oldest school west of the Mississippi River.
In 1853 Punahou was chartered as a non-profit, non-sectarian school by the Hawaiian Monarchy. Maintained by income from tuitions, endowments, and gifts, it has a 76 acre campus with plant valued at $50 million and 127 general endowment approximately $15 million. "Although small by pub lic school standards, Punahou . . . is America's largest non-parochial 128 independent school." College admissions authorities rate Punahou among the top ten U.S. secondary schools.
Punahou has two major divisions, each with its own principal.
The Junior School, kindergarten through grade 8, has about 55 percent of the student body; and the Academy, grades 9-12, has the remainder— about
1,700 students. The Academy's primary goal is the rigorous intellectual training in the liberal arts disciplines required for the scholastic
preparation and self-discipline that students need to be successful in
college. The student body mirrors Hawaii’s cosmopolitan character with members from every racial, religious, social, and economic group.
Punahou has had only two presidents since 1944; Dr. John F. Fox,
1944-1967, and Dr. Roderick F. McPhee, 1968 to present. Under Fox, the
size of the student body was tripled, from approximately 1,150 to 3,500.
127 Punahou School, "Punahou Handbook," Honolulu, 1983. (Mimeo graphed.), pp. 18-22. 128 Punahou School, "Aims," Honolulu, 1967. (Mimeographed.), p. 2. 156 The increased tuition revenue was used primarily to raise faculty sala ries; but teacher pay increments were based on strong classroom perfor mance and continued college course work in one's teaching area at an average rate of three semester hours per year. A teacher employed with out a master's degree was expected to acquire one in his teaching field.
By the end of his term as president, Fox had a professional staff and faculty of about 235, three-quarters of whom had at least a master's degree. Major construction of plant facilities accompanied the student body expansion under Fox and have continued to completion under McPhee.
Dr. Thomas H. Hamilton, former President of the University of Hawaii, once said in an address to the PTA, "Punahou isn't just a school. It's 129 a way of life."
Punahou is the site of Hawaii's first College Board test center.
Word of the Advanced Placement Program came to Hawaii in 1956 with the visit of Frank Bowles, President of the College Board. At the time, the mathematics department in the Academy was teaching the standard program: algebra in grade 9, demonstrative geometry in grade 10, intermediate algebra and trigonometry in grade 11, and advanced algebra and solid geometry in grade 12. Plans were made to alter the program for at least two tracks— regular and honors. The regular college preparatory program would have elementary algebra in grade 9, plane and solid geometry in grade 10, a continuation of algebra, trigonometry, and analytic geometry in grade 11, and a pre-calculus course stressing elementary functions and analytic geometry in grade 12. The honors track would take the entire program advanced by one year, beginning with an accelerated arith metic in grade 7 so as to start the elementary algebra in grade 8.
129 Ibid., p. 1. 157 Student Selection: Gifted, Academically Talented, or College Bound
As a college preparatory school, Punahou is determined to give its Academy students as strong a background in the fields of literature, mathematics, science, history, languages, the arts, music, and physical development as it is possible for students to acquire in four years. To the mathematics department, this means that the college bound student in the regular track who completes four years of college preparatory mathe matics should be able to take calculus immediately upon college entry.
The brightest of these students who enjoy and do especially well in mathematics should be grouped in honors sections, given an accelerated arithmetic in grade 7, allowed to take the four-year sequence a year earlier than their college bound classmates and, if their interest and ability remains high, should be given the college-level Advanced Place ment calculus course as seniors.
The task of selecting these students was clearly a responsibility of the junior school, but guidelines were discussed with the Academy mathematics department. 1958-59 was set as the target date for algebra in grade 8. Because this class would not graduate until 1963, an elabo rate acceleration plan for the interim classes 1959-62 which made use of summer school and extra classes was developed; and the entire package was presented, first to the Academy principal, then the school president, and finally the Board of Trustees by the chairman of the mathematics depart ment for approval. Permission to initiate the preparation phase in 1957-
58 and the full program in 1958-59 was granted to include special dispen sation which allowed AP classes to be less than the desired minimum of twenty students. Only the very brightest students would be selected during this interim period. Guidance was found in many sources, but 158 particularly The "Rockefeller Report1' on Education, Dr. Conant's studies, and the report of the Commission on Mathematics.
The Commission recommended substantial changes in the secondary mathematics curriculum in the form of a proposed new program for the four upper grades. It also advised that the college-capable be grouped by ability and study mathematics for at least three, preferably four, years:
Moreover, it should be emphasized that the recommendation of three years of mathematics for the college-capable is minimal. . . . [The Commission] believes that school counselors, teachers, and parents have a duty to see that as many as possible of the college- capable should study mathematics in high school for four years. The most talented [italics mine] should attempt the Advanced P^gcement Program (described on p. 15) if it can be made available.
The "Rockefeller Report” on Education indicates the unanimity among educational leaders on the priority which should be given four years of mathematics in the curriculum for the academically talented:
In addition to the general education prescribed for all— four years of English, three to four years of social studies, one year of mathematics and one year of science— the academically talented [italics mine] student should have two to three additional years of science, three additional years of mathematics, and at least three years of a foreign language .^3^
As used by Dr. James B. Conant, the expression 'academically talented' referred to those students who "were able to study effectively and rewardingly a wide program of advanced mathematics, science, and 132 foreign languages." While this definition clearly suggests superior ability in the traditional disciplines intended for the college-capable,
^3^Commission on Mathematics, Program, p. 12. 131 The original source is explained by this parenthetical remark: "(These recommendations are based partly upon a recent study by Dr. James B. Conant and partly upon the findings of a conference sponsored by the National Education Association of 'Education of the Academically Tal ented.')" Rockefeller Brothers Fund, The Pursuit of Excellence, p. 27. 132 James B. Conant, The American High School Today (New York: McGraw-Hill Book Co., 1959), p. 20. 159 administrators and guidance counselors might prefer a working definition such as the one suggested by Copleys
IQ of about 120 or better, SAT scores in the 600's, grades consistently A's and B's— leaven this statistical lump with the opinions of wise and experienced teachers and advisers, and we shall have something resembling a working criterion of "academic talent." Probably the one mistake that must be avoided at all costs is rigidity, for although the academically talented have some charac teristics in common, they are also subject to wide variation.6 0 See NEA and NASSP, Administration; Procedures and [School] Practices for the Academically Talented Student (Washington, D.C.; [National Education Association], 1960), pp. 34-44.
Thus, identification of the academically talented stems from two major sources; the judgement of teachers, to provide information on traits of character and academic achievement; and national test scores, to provide comparison measures of ability in, and aptitude for, academic work. In Slums and Suburbs, Conant acknowledged the contribution of these two sources; and, having examined the national test scores of thou sands of students who were achieving success although carrying heavy academic loads, suggested his own operational criterion;
On the basis of such evidence, I suggest that if a student in grade 9 has a consistent score on a scholastic aptitude or intelli gence test which places him in the upper 15 or 20 per cent of the high school population on a national basis, then the presumption is that he is academically talented. In my judgment, this student ought to think in terms of a professional career and ought to elect a heavy academic program beyond the minimum required of all students.
This operational definition of the academically talented has gained general support; it was adopted in early conferences instituted by the National Education Association, where "it was recommended that the
Academically Talented Project focus on the upper 15 to 20 percent in
133 Copley, The American High School and the Talented Student, p. 5. 134 Conant, Slums and Suburbs, p. 89. 160 academic achievement, or more precisely, on the population one standard 135 deviation above the mean.”
Caution should be exercised because often the terms 'gifted' and either 'academically talented' or 'talented' are used interchangeably.
Historically, the term 'gifted' was used to characterize either those who were mentally superior or those who had some exceptionally well-developed
talent. At one time the United States Office of Education only consid
ered the top half of one percent of the intelligence distribution as
gifted. Present usage of the term includes the top 2 to 3, or even 5,
percent of the distribution, and usually includes successful achievement
as well as high intelligence. For purposes of Federal education pro
grams, the following definition of 'gifted and talented' was established
by an advisory panel to the United States Commissioner of Educations
Gifted and talented children are those identified by profes sionally qualified persons who by virtue of outstanding abilities, are capable of high performance. . . . It can be assumed that utilization of these criteria [general intellectual ability, specific academic aptitude, creative or productive thinking, leadership ability, visual and performing arts, or psychomotor ability] for identification of the gifted and tal ented will encompass a minimum of 3 to 5 percent [italics mine] of the school population
The Cooperative Committee on the Teaching of Science and Mathe
matics of the American Association for the Advancement of Science and the
U.S. Office of Education offered an expanded interpretation of those
considered to be talented in mathematics and science: "The talented and
135 Kenneth E. Anderson, ed., Research on the Academically Tal ented Student (Washington, D.C.: National Education Association, 1961), p. 13. 3.36 U.S., Department of Health, Education and Welfare, Education of the Gifted and Talented, by S. P. Marland, Jr., Vol. Is Report to the Congress of the United States by the U.S. Commissioner of Education (Washington, D.C.: Government Printing Office, 1971), p. ix. 161 rapid learners in mathematics and science will be interpreted to mean the pupils who are among the upper 20 percent of the students in general 137 intelligence and who seem to be apt xn science and mathematics."
Experienced admission officers, curriculum planners, and guidance counselors are aware of the variation in usage of these terms, particu larly when translated into comparable national test scores. Table 18 summarizes these descriptions of intelligence or achievement, and com pares them on a percentage basis with equivalent scores on College Board 138 tests administered in the Admissions Testing Program (ATP).
TABLE 18
COMPARISON OF INTELLECT EESCRIPTIONS, NATIONAL NORM PERCENTAGES, AND COLLEGE BOARD ATP SCORES
Upper Percentage Comparable ATP Scores Description of National Norm Exceeded by Percentage
Early 'gifted' ...... 0.5 758 Present 'gifted' .... 2 - 3 688 - 705 'Gifted and talented'. . 3 - 5 664 - 688 'Academically talented'. 15 - 20 584 - 604
aData based on an assumed mean of 500 and standard deviation 100.
137 U.S., Department of Health, Education, and Welfare and Office of Education, Education for the Talented in Mathematics and Science, by Kenneth E. Brown and Philip G. Johnson, Bulletin 1952, No. 15 (Washing ton, D.C.s Government Printing Office, 1952; reprint ed., 1953), p. 2. 138 ATP offers the Scholastic Aptitude Test (SAT), the Test of Standard Written English (TSWE), and sixteen Achievement Tests (ACH) . The verbal and mathematical scores on the SAT and each score on the ACH, including Mathematics Level I and Level II (ACHMLl and ACHML2) are reported on a scale 200-800. Test data for SAT-Math, ACHMLl, and ACHML2 are, respectively: 467, 535, and 651 mean; 114, 94, and 95 standard deviation; 0.91, 0.87, and 0.87 reliability; and 35, 32, and 33 standard error of measurement. Educational Testing Service, The Mathematics Examinations of the College Board (New York: College Entrance Examination Board, 1980), p. 8. 162 A test score, like any measurement, is an approximate indicator rather than an exact measure of a student's intelligence, achievement, or aptitude. However, guidance counselors and curriculum designers who realize the nature and limitations of test scores, also know that the admission test scores are good, though not infallible, predictors of per formance in college. Used as an additional, nationally uniform measure of academic ability to supplement the student's school record and teacher recommendations, these national scores can differentiate between the average students, clustered around the mean, and the academically tal ented or gifted who range at least one or two standard deviations, respectively, above the mean.
This latter distinction of a full standard deviation between the academically talented and gifted lowest cut-off points was thought to be sufficiently important by Conant to merit a separate recommendation;
For the highly gifted pupils some type of special arrangement should be made. These pupils of high ability, who constitute on a national basis about 3 per cent of the student population, may well be too few in number in some schools to warrant giving them instruc tion in a special class. In this case, a special guidance officer should be assigned. , . . The identification of the highly gifted might well start in the seventh or eighth grade or earlier. If enough students are available to provide a special class, these students should take in the twelfth grade one or more courses which are part of the Advanced Placement Program. . . . This program should be adopted not only because of the benefits which accrue to the students involved, but because it may well have a good influence on students of somewhat less ability by raising the tone of the whole academic program.
Both the Commission on Mathematics and the "Rockefeller Report"
supported Conant's views on the Advanced Placement Program for the most talented and gifted. The challenge facing educators in general, and
Punahou in particular, in each of the major subject areas was clears to
139 Conant, The American High School Today, pp. 62-63. 163 develop programs broad enough to account for individual differences in all students; to modernize the content and improve the quality of courses offered to the academically talented; to provide the most talented and gifted, the upper 3 to 5 percent of the student population, the opportu nity for challenging study in the Advanced Placement Program; and to strengthen the pre-service and in-service education of teachers to enable them to effectively implement these broader and more demanding programs.
In mathematics, the challenge was answered by the Commission on
Mathematics with its presentation of a nine-point program for the college bound. This program revamped, extended, and modernized the existing mathematics curriculum in the secondary school. It did more than present a modernized four-year sequence of new mathematics courses: algebra in grade 9; plane, solid, and coordinate geometry in grade 10; algebra and trigonometry in grade 11; and a semester of elementary functions, fol lowed by a choice of semester courses in either modern algebra, or ana lytic geometry, or probability with statistical applications for grade 12.
It also addressed the need for curricular revision, the detailed course changes to meet this need, the increased responsibility of mathematics teachers to seek the knowledge required to implement the new program, and the necessity for greater articulation between school and college, and between college mathematicians and mathematics educators in order to improve the courses designed for teachers of mathematics.140
At Punahou in 1957-58, The junior School began the accelerated arithmetic course for the talented and gifted upper 3 to 5 percent in the class of 1963, the first class destined to take their algebra in grade 8, complete four years of school preparation in mathematics in grade 11, and
140 Commission on Mathematics, Program, pp. 17-34. 164 take the Advanced Placement calculus course as seniors. In the Academy, only the gifted in the interim classes 1959-1962 were accelerated: those with strong recommendations from previous teachers, ranked near the top of their respective classes, at least an A- average in previous mathe matics courses, SAT-Math scores above 675, ACH-Math Level I or Level II above 700, and Otis IQ scores over 128. The graduating class of 1959 had
298 students, but only 15 were in the first AP calculus class. They were quality, but the first few years brought some disappointing results.
Advanced Placement Calculus at Punahou, 1959-1968
The mathematics curriculum at Punahou in the late 1950s and early
1960s was in a constant state of change as the department sought to adopt the recommendations of the Commission on Mathematics. This had the pri ority because it affected all of the college bound students. For the upper echelon of these students in the honors program it was a period of challenge and excitement, but the entire Academy became infected with their eagerness. Although only two years of mathematics were required for graduation, the percentage of students taking four years continued to rise until it topped 90 percent. Teachers were not immune— several sought and received National Science Foundation grants for summer study, while others enrolled in higher level mathematics courses at the University of
Hawaii. For the second AP class in 1959-60 the mathematics department finally was consolidated in the newly constructed, twelve-classroom
Bingham Hall. Being together and participating in the Program had an uplifting effect upon all mathematics courses both in the Academy and in the Junior School. Increased contact between the two school divisions for curriculum planning was a decided advantage. Joint testing of the 165 algebra students in grades 8 and 9 was instituted bringing departmental members in the two school divisions into closer contact. Articulation on
such matters as departmental standards, performance objectives, teaching
aids, and successful instructional techniques improved.
Punahou's mathematics department adopted certain policies with
respect to student participation in AP mathematics. One policy, which
has held for twenty-six years, is that seniors who take the AP calculus
course must take the Advanced Placement examination in May. It is con
sidered an inherent part of the AP mathematics course, and no credit is
given without it. The science department has the same policy, if a stu
dent is taking a college-level course, it is presumed that he wants
college-level credit. The experience for the student in preparing for
the three-hour examination and the benefit to the AP teacher and the
school in having an external evaluation of the effectiveness of their AP
course outweigh other considerations, including cost. $38 is a bargain
price for the possibility of two semesters of college credit. Table 19
shows the AP grade distributions for Punahou's first ten calculus classes.
TABLE 19
PUNAHOU AP CALCULUS GRACE DISTRIBUTIONS, 1959-1968
AP Grade 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968
5 1 1 4 3 3 2 5 8 6 8 4 5 5 5 4 5 4 3 8 15 12 3 5 6 8 6 9 10 4 10 11 9 2 3 4 0 2 1 2 4 1 6 3 1 1 2 0 0 1 2 1 0 0 3
Total 15 18 17 15 19 20 17 27 38 35 Mean Grade 3.1 2.9 3.8 3.5 3.4 3.1 3.4 3.9 3.6 3.5 Stnd. Dev. 1.1 1.1 0.8 1.0 1.0 1.1 1.3 0.9 0.9 1.2 166 In these first ten years, 221 Punahou seniors took the AP course and examination. Of these, 96 percent received passing grades of 2 or better, and 84 percent had qualifying grades of 3 or higher. For the same period, the comparable national figures were 78 percent passing and
50 percent qualifying. Punahou's mean grade for the ten-year period was
3.46 with 1.06 standard deviation. The department chairman was the AP teacher for all years except 1961-62 when he was on sabbatical.
The text used in the regular track, Advanced Math course for seniors in 1959-62 was Kline, Oesterle, and Willson's Foundations of 141 Advanced Mathematics. in the Advanced Math Honors course, which was offered to those in the honors track during their junior year, the pre calculus test used during this period was Brixey and Andree's Fundamen- 142 tals of College Mathematics. The selection of Longley, Smith, and Wil
son's Analytic Geometry and Calculus as the text in the AP program for 143 the first two years was an error in judgement. Although strong in the mechanics of differentiation and integration, it was weak in theory and
applications. Abetted by a teacher new to the AP program and unfamiliar
with the degree of emphasis placed on theoretical aspects, the perfor
mance of the first two classes, particularly on the essay section of the
AP examination was disappointing. For all but the last year in this
period, the College Board reported three scores for each candidate: his
objective section score, his essay score, and a composite score. While
141 William E. Kline, Robert Oesterle, and Leroy M. Willson, Foundations of Advanced Mathematics (New York: American Book Co., 1959) . 142 John C. Brixey and Richard V. Andree, Fundamentals of College Mathematics (New York: Henry Holt & Co., 1954; rev. ed., New York: Holt, Reinhart, & Winston, 1961). 143 William R. Longley, Percey F .. Smith, and Wallace A. Wilson* Analytic Geometry and Calculus (New York: Ginn & Co., 1952). 167 the majority of candidates did well on the objective part of the 1959 or
1960 examination, their essay scores were generally lower. When the
nature of the problem was realized, the text was changed to Thomas's
Calculus and Analytic Geometry, a later edition of which is still being 144 used. An alternative fifth-year course in probability and statistics,
using a text by Mosteller, Rourke, and Thomas, was offered to honors stu
dents who had switched back to the regular track or any student for whom 145 an additional year of pre-calculus mathematics was appropriate.
This ten-year period ended with the announcement by the College
Board that descriptions of two calculus courses would be offered for use
in school year 1968-69. At Punahou, the decision was made to offer the
students only Calculus BC, at least for the first few years.
Advanced Placement Calculus at Punahou, 1969-1984
The 982 Punahou students who took the Calculus AB or Calculus BC
courses during the last sixteen years are the subjects on whom data was
collected for this study. The growth from 36 students taking the single
Calculus BC course in 1969 to the 119 students in Calculus AB and Cal
culus BC in 1984 is displayed in table 2 of chapter I . The latter figure
represents 28 percent of the graduating class. Three AP teachers have
been directly involved with these classes, each having an opportunity to
teach both courses but not during the same year. That they have done a
professional job teaching the AP course is clear from table 20, which
shows Punahou's AP calculus grade distributions for these sixteen years.
144 George B. Thomas, Jr., Calculus and Analytic Geometry, 2d ed. (Reading, Massachusetts; Addison-Wesley Publishing Co., 1960). 145 Frederick Mosteller, Robert E. Rourke, and George B. Thomas, Jr., Probability with Statistical Applications (Reading, Massachusetts: Addison-Wesley Publishing Co., 1970). TABLE 20
PUNAHOU AP CALCULUS GRADE DISTRIBUTIONS, 1969-1984
AP Grade Number of Punahou Candidates Participating in Calculus AB or Calculus BC , by Year
Calculus AB 197 3a 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984
5 1 1 1 5 3 15 7 6 11 22 5 20 4 6 9 6 7 10 12 21 9 14 11 16 31 3 7 16 6 6 7 7 10 8 27 13 18 16 2 8 1 1 0 4 0 2 1 6 1 2 4 1 0 1 0 0 0 0 2 0 3 1 0 3
Subtotal 22 28 14 18 24 34 42 24 61 48 41 74 Mean Grade 3.00 3.29 3.50 3.94 3.50 4.24 3.69 3.83 3.39 4.08 3.59 3 .82 Stnd. Dev. 0.93 0.76 0.76 0.80 0.93 0.78 0.98 0.87 1.05 1.01 0.77 1.03
Calculus BC 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984
5 13 12 10 15 29 8 17 12 17 15 20 18 7 18 15 28 4 15 16 11 11 10 9 10 9 5 5 9 10 9 11 14 14 3 6 9 12 8 7 3 2 2 3 2 6 10 8 7 8 2 2 1 4 4 2 2 4 2 0 1 2 0 0 4 0 4 1 1 1 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0
Subtotal 36 41 37 38 48 24 31 23 26 24 35 38 28 36 42 45 Mean Grade 4.06 3.88 3.73 3.92 4.38 3.88 4.35 4.43 4.46 4.38 4.40 4.21 3.68 4.31 3.90 4.53 Stnd. Dev. 0.95 0.95 0.99 1.15 0.89 1.08 0.88 0.66 0.86 0.97 0.77 0.84 1.02 0.79 1.08 0.69
Total 36 41 37 38 70 52 45 41 50 58 87 62 89 84 83 119
aCalculus AB was not offered at Punahou School until 1973. O' oo 169
As was the case nationally, the increase in number of students in
Calculus AB from the time it was first offered in 1973 was more rapid than in Calculus B C . Table 21 provides a consolidation of the sixteen distributions into a single composite distribution for each calculus course. This table may be compared with table 16 which shows similar composite distributions on the national scale for the same period.
TABLE 21
COMPOSITE PUNAHOU AP CALCULUS GRADE DISTRIBUTIONS, 1969-1984
Number of Candidates (N) and Percentage Below Grade (%B) AP Grade Calculus AB Calculus BC N %B N %B
5 97 77 254 54 4 152 42 168 24 3 141 9 95 6 2 30 2 31 1 1 10 0 4 0
Total 430 552 Mean Grade 3.69 4.15 Stnd. Dev. 0.97 0.95
Over the dozen years of participation in Calculus AB, table 21 shows that 98 percent of the 430 Punahou candidates received passing grades of 2 or better, while 91 percent had qualifying grades of 3 or higher. The comparable national figures for Calculus AB are 86 percent passing and two-thirds qualifying from table 16. in Calculus BC, 99 per cent of the 552 Punahou candidates had passing grades and 94 percent had qualifying grades; and these compare favorably to the corresponding national percentages of 98 and 91, respectively, for the same period.
The mean grades in each Punahou course distribution also compare 170 favorably to the corresponding, national mean grades, exceeding them by slightly more than one-half of a standard deviation.
Punahou uses a variable schedule based on a six-day cycle with a modular unit of fifteen minutes. Both calculus and pre-calculus courses meet five days per cycle for four mods (one hour less five minutes for passing time) . Since the AP mathematics examination is usually adminis tered in mid-May, each course has a total of about 130 class meetings prior to the examination. Average class size is twenty-five students.
During this period, the AP teachers chose to use texts by Thomas in both courses: Elements of Calculus and Analytic Geometry was used in 146 Calculus ABj and Calculus and Analytic Geometry, in Calculus BC. In the fourth-year, pre-calculus courses, both the regular track, Advanced
Math students and the honors track, Advanced Math (H) students used Brown and Robbins, Advanced Mathematics: An Introductory Course for a number of years; but for the last two years, the honors group has been using a text 147 by Yunker, Vannatta, and Crosswhite.
Will the Advanced Placement Program Continue to Grow?
Despite the national recognition accorded the Advanced Placement
Program and its steady growth and acceptance by greater numbers of high schools and colleges, "AP still remains shamefully under-utilized in this
146 George B. Thomas, Jr., Elements of Calculus and Analytic Geometry, 2d ed. (Reading, Mass.: Addison-Wesley Publishing Co., 1972); and George B. Thomas, jr. and Ross L . Finney, Calculus and Analytic Geometry, 5th ed. (Reading, Mass.: Addison-Wesley Publishing Co., 1979). 147 Richard Brown and David Robbins, Advanced Mathematics: An Introductory Course (Boston: Houghton Mifflin, 1981); and Lee E. Yunker, Glen D. Vannatta, and F. Joe Crosswhite, Advanced Mathematical Concepts (Columbus, Ohio: Charles E. Merrill Publishing Co., 1981). 171 148 country." Each year about 3 million students graduate from our nation’s 20,000 high schools; and each fall approximately 2.4 million of these and an increasing number of older students present themselves for the first time at some 2,600 colleges, universities and other insti- 149 tutions of higher learning. If one accepts conservative estimates that three and fifteen percent of the 3 million high school graduates are gifted and academically talented, respectively, this would suggest that
some 90,000 gifted should be participating in Advanced Placement, and that 450,000 academically talented could benefit from this program each year. Yet in the May 1984 annual administration, only about 177,000
candidates from 6,300 secondary schools were tested. Of these, only
40,000 candidates from no more than 4,500 high schools, seeking credit
or advanced placement at about 1,500 colleges, had participated in the
Program in mathematics by electing one of the two validating exams.
These last figures represent about 40 percent of the estimated
number of gifted students, less than ten percent of the academically talented, twenty percent of the high schools, and about 60 percent of the
colleges. The disturbing inference is that, while the represented twenty
percent of the high schools may have 40 percent of the gifted graduates,
it is doubtful; and this means that somewhat more than 60 percent of the
gifted, for whom the Advanced Placement Program was designed, are not
being reached by the program in mathematics. The total figures indicate
that at least one-fifth of the academically talented who are not gifted
148 S. P. Marland, Jr., "Advanced Placement; An Above-Average Opportunity," NASSP Bulletin 59 (May 1975); 37. 149 The American Association of Collegiate Registrars and Admissions Officers and The College Board, Undergraduate Admissions; The Realities of Institutional Policies, Practices, and Procedures (New York; College Entrance Examination Board, .1980), p. 1. 172 must be participating. One wonders how many of these students, number
ing perhaps as many as 15,000 in mathematics, were among the 11,000 who
failed to get qualifying scores on the calculus examinations.
The inferences from these figures also indicate the wisdom of the
Commission on Mathematics twenty-five years ago, when it commented on any school's consideration of curricular change in order to include calculus:
A few schools now teach calculus effectively in their regular programs. But schools that can do this are unusual at the present time, in regarding calculus as a college-level subject, the Com mission does not want to discourage those with exceptional facilities from attempting exceptional tasks. Rather, it warns against pre mature general acceptance of a curricular responsibility for which all too few schools are now adequately prepared."^®
While the "few schools" have become thousands over the twenty-
five years, the Commission's caution to schools contemplating the AP Pro
gram in mathematics is still apropos. The Program is not possible at a
secondary school which does not have a strong, underlying program for its
academically talented. The fact that the intended areas of study of
approximately sixty percent of the college-bound seniors require calculus 15 is argument enough to support a full four-year sequence in mathematics.
With such a program in being, schools should feel obliged to follow the
Commission's recommendation on Advanced Placement if the number of gifted
and talented students in the school warrants it. Thousands of schools
have made this transition successfully; more could, and should.
Commission on Mathematics, Program, pp. 14-15. 151 The actual percentages given for the intended areas of study of college-bound seniors are quoted as follows: "Physical sciences and rela ted areas, 15.7%; Biological sciences and related areas, 23.8%; Business, commerce, and communications, 19.5%; Social sciences and related areas, 22.0%; Arts and humanities, 12.5%; and other areas, 6.7%." W. Vance Grant and Leo J. Eiden, Digest of Education Statistics (Washington, D.C.: National Center for Education Statistics, 1980), p. 67, citing The College Board, National Report on College-Bound Seniors, 1978 (New York: College Entrance Examination Board, 1978), [pp. 18-19.] CHAPTER IV
DATA PRESENTATION AND DESIGN OF THE STUDY
Because this study addresses the problem of selecting students at the end of their junior, secondary school year for participation in one of the two calculus courses in the Advanced Placement Program, this chap ter initially presents the pertinent data collected from a given school's records at the end of the junior year. The given school is Punahou. The study population is the set of 982 Punahou seniors who have taken the
Calculus AB or Calculus BC course and examination during the sixteen years both examinations have been offered by the College Board. With
'success' defined as a qualifying score of at least three on either AP examination in mathematics, the design of the study to determine the most reliable predictors of success on, and discrimination between, each of the two AP calculus examinations will be presented and discussed.
Data Presentation
The data available in student records at the secondary level are similar at different schools. The specific data collected on the 982
Punahou students will be displayed by category in a series of tables.
These tentative predictor variables, or their equivalents, are available at most secondary schools with college preparatory programs. They are displayed and discussed in groups thought to be meaningful to the study.
Table 22 lists, and gives the basic statistics for, all initial variables used in the study except for four categorical variables which partition the study population into subgroups for further study. 173 174 TABLE 22
BASIC STATISTICS FOR INITIAL STUDY POPULATION VARIABLES, 1969-1984
NUMBER SAMPLE STANDARD MINIMUM MAXIMUM STD ERROR COEF OF VARIABLE3 N MEAN DEVIATION VALUE VALUE OF MEANb VARIATION0
APEX 982 3.95 0.99 1.00 5.00 0.03 24.99 APEXAB 430 3.69 0.97 1.00 5.00 0.05 26.38 APEXBC 552 4.15 0.95 1.00 5.00 0.04 22.88 CALC 982 9.02 1.49 2.50 11.50 0.05 16.56 CALCAB 430 8.94 1.46 4.00 11.00 0.07 16.35 CALCBC 552 9.09 1.52 2.50 11.50 0.06 16.70 COOP 830 46.75 7 .85 15.00 60.00 0.27 16.79 COOPAB 430 43.17 7.46 15.00 59.00 0.36 17.28 COOPBC 400 50.59 6.28 28.00 60.00 0.31 12.42 AVMA 982 9.73 0.92 5.60 11.50 0.03 9.44 JGPA 982 3.50 0.30 1.82 4.00 0.01 8.58 HNIQ 762 134.15 11.48 101.00 164.00 0.42 8.56 JSATV 249 58.97 7.21 40.00 73.00 0.46 12.23 JSATM 249 61.75 5.68 43.00 71.00 0.36 9 .19 PSATV 969 54.32 8.64 29.00 77 .00 0.28 15.90 PSATM 969 65.04 6.40 40.00 80.00 0.21 9.85 SATV 982 57 .39 8.69 30.00 80.00 0.28 15.90 SATM 982 68.46 6.17 46.00 80.00 0.20 9.02 DSATS 982 64.78 5.95 42.00 79.00 0.19 9.18 ACHL1 200 72.05 6.73 53.00 80.00 0.48 9.35 ACHL2 837 73.80 5.58 56.00 80.00 0.19 7 .56 ALG1 982 9.88 1.26 4.50 12.00 0.04 12.75 GEOM 982 9.73 1.15 5.00 12.00 0.04 11.80 INTM 982 9.71 1.18 4.50 11.50 0.04 12.18 ADVM 982 9.60 1.12 5.00 11.00 0.04 11.62 SCI 982 0.87 0.84 0.00 2.00 0.03 97.16 API 982 2.61 1.16 1.00 4.00 0.04 44.60 TAP 982 2.55 1.37 1.00 9 .00 0.04 53.66
^ o u r categorical variables are not included in this list because basic statistics do not apply. Three are qualitative, nominal-level variables: SEX, female or male; NOR, oriental or non-oriental; and CRSE, Calculus AB or Calculus BC; and one is a quantitative, interval-level variable called YR, the year a candidate took the course and examination.
Technically this is the standard deviation of the distribution of sample means, U/^T ; but figures in this column are estimates, sA^T . £ A logarithmic transformation to achieve uniform variance is appropriate if the coefficient of variation, normally 0/(1, is constant across the populations of interest. Figures in this column obtained by a SAS (Statistical Analysis System) program are the estimates lOOs/x .
^This is a constructed variable: DSATS = (SATV + 2-SATM)/3. 175 The Criterion Variables
The first three variables in table 22 serve as criteria in this study and are defined below in order to clarify their source and usage:
APEX: An acronym for Advanced Placement examination used on com puter runs, this interval-level, criterion variable is the grade on the APP five-point scale (from 1— no recommendation to 5— extremely well-qualified) on either the Calculus AB examination (APEXAB) or the Calculus BC examination (APEXBC).
CALC: An abbreviation for Calculus grade used on all computer runs, this interval-level, criterion variable is the numerical equi valent of the final grade in Calculus AB (CALCAB) or Calculus BC (CALCBC) on a twelve-point scale (12 = A+, 11 = A, . . . , 0 = P). For year courses at Punahou, semester grades are averaged to obtain final grades using this scale; a policy change during school year 1977-78 denied future assignment of the A+ grade.
COOP: An abbreviation for the Educational Testing Service's Cooperative Test in Calculus (Form A), this interval-level criterion is the raw score (from 0 to 60) on this sixty-question, objective test administered in two forty-minute parts one week before the APP examinations to all students in both Calculus AB (COOPAB) and Calculus BC (COOPBC)
By far the most important of these criteria is the APEX score because it is the basis on which colleges rely to award advanced standing or credit. For this reason, a qualifying APEX score was chosen as the measure of success. Although the CALC grade and COOP score are both measures of achievement in calculus, the latter has greater significance in this report because it is the only criterion common to all students regardless of calculus course (CRSE). Therefore, the search for the most reliable predictors of achievement will use the COOP score as the primary measure of achievement. Punahou has offered Calculus BC throughout the sixteen years; but it did not offer Calculus AB, nor administer the Coop erative Test, until 1972-73. Table 23 shows the candidate frequency in each cell of a COOP-interval by APEXAB or APEXBC score matrix.
■'’Educational Testing Service, Cooperative Mathematics Tests: Calculus, Form A (Princeton: Educational Testing Service, 1963). 176 TABLE 23
CANDIDATE FREQUENCY BY COOP INTERVAL AND AP SCORES ON EACH CALCULUS EXAMINATION, 1973-1984
Number of Punahou APP Candidates
Calculus AB Exam Calculus BC Exam COOP Total interval 1 2 3 4 5 £ AB 1 2 3 4 5 £ BC
56-60 12 12 6 96 102 114 51-55 9 41 50 4 43 71 118 168 46-50 1 15 73 33 122 2 20 50 29 101 223 41-45 2 40 58 10 110 10 23 13 7 53 163 36-40 6 65 11 82 3 9 2 1 15 97 31-35 1 4 19 1 1 26 4 5 9 35 26-30 2 16 1 19 1 1 2 21 21-25 3 1 4 4 16-20 3 1 4 4 11-15 1 1 1
Total 10 30 141 152 97 430 1 20 61 114 204 400 830
Table 23 indicates that the correlations between the Cooperative
Test scores and the scores on either the Calculus AB or Calculus BC examinations are high. A student who scores above 40 on the Cooperative
Test will most likely have a qualifying score on his AP examination no matter which course he is taking. On the other hand, a student with a
Cooperative test score of 30 or lower has little chance of qualifying.
The Cooperative test administered one week before the AP examination can
be a strong predictor of success, but it comes far too late to assist in making a student selection for either course.
The Categorical Variables
There are four categorical variables which partition the study
population into mutually exclusive and exhaustive subsets, which can be
used for further study. These are YR, CRSE, SEX, and NOR. In addition, 177 there are three other discrete, quantitative variables which might be used as predictors; but because SCI, API, and TAP partition the study population and embody a measure of student interest and industry, they will be used as categorical variables. The seven are defined as follows:
YR; This discrete, quantitative variable represents the year any senior took either calculus course and AP examination. The range is sixteen years; the values, 69 to 84.
CRSE: An abbreviation for course, this nominal-level variable represents either of the two calculus courses— Calculus AB or Calculus BC.
SEX: This nominal-level variable represents the candidate's gen der, female (F) or male (M).
NOR: An acronym for 'non-oriental' (N) or 'oriental' (0), this nominal-level, class variable identifies a student’s racial extraction as being at least fifty percent oriental or not.
SCI: This nominal-level, class variable indicates whether or not a calculus candidate also has such an interest in science that he took at least one AP science course and examination. A value of 2 was assigned if the AP course and examination included Physics C, a physics which requires calculus; a value of 1 was given those who took at least one of the other AP science offerings, namely Biology, Chemistry, or Physics B— calculus not required. A value of 0 was assigned to those who took an AP calculus, but no AP science, course.
TAP: This is an ordinal-level variable which is an acronym for the total number of AP examinations taken at Punahou. Table 22 indi cates that the TAP values in this study range from 1 to 9.
API: An acronym for 'Advanced Placement Interest', this nominal- level, class variable combines both SCI and TAP in a single measure more precise than SCI alone. The value 1 was assigned if the candi date only took a calculus course and its AP examination; a value 2 was given those who also took an AP course in English, a foreign lan guage, history, art, or music but no science; a value of 3 was assigned to students who took any AP science course except Physics C (physics requiring calculus); and a value of 4 was reserved for those who took Physics C in addition to one of the two calculus courses.
Table 2 in chapter 1 displayed three of the categorical variables by giving the number of Punahou students by year, calculus course, and sex. Table 24 adds the variable NOR to these three class variables in order to show the 7 to 4 ratio of orientals to non-orientals. 178 TABLE 24
NUMBER OF PUNAHOU AP STUCENTS BY YEAR, COURSE, NOR, AND SEX
Calculus AB Calculus BC Composite
Year Non- Non- Oriental Oriental NOR SEX oriental Oriental F M E F M E FM E F M E 0 NF M
1969 2 14 16 5 15 20 16 20 7 29 1970 7 17 24 3 14 17 24 17 10 31 1971 6 17 23 5 9 14 23 14 11 26 1972 11 15 26 1 11 12 26 12 12 26 1973 6 10 16 2 4 6 11 18 29 8 11 19 45 25 27 43 1974 8 10 18 4 6 10 8 7 15 3 6 9 33 19 23 29 1975 1 7 8 1 5 6 6 14 20 1 10 11 28 17 9 36 1976 9 4 13 3 2 5 2 16 18 1 4 5 31 10 15 26 1977 6 5 11 2 11 13 2 14 16 2 8 10 27 23 12 38 1978 13 6 19 7 8 15 4 13 17 4 3 7 36 32 28 30 1979 9 20 29 5 8 13 8 13 21 3 11 14 50 27 25 52 1980 8 10 18 2 4 6 7 19 26 4 8 12 44 18 21 41 1981 26 17 43 8 10 18 3 9 12 4 12 16 55 34 41 48 1982 19 12 31 10 7 17 9 16 25 1 10 11 56 28 39 45 1983 15 10 25 7 9 16 9 16 25 3 14 17 50 33 34 49 1984 29 19 48 13 13 26 11 22 33 2 10 12 81 38 55 64
Total 149 130 279 64 87 151 106 240 346 50 156 206 625 357 369 613
It is interesting to note that the ratio of females to males is almost the same as the ratio of non-orientals to orientals; in fact, of the 982 candidates, 37.6 percent are female and 36.4 percent are non oriental . However, the distribution by SEX and NOR between the two cal culus courses is quite different. In Calculus BC, the ratio of males to females exceeds 2.5 to 1; but in Calculus AB, the same ratio is almost
1 to 1, and the number of oriental females actually exceeds the number of oriental males. Table 6 indicates that about 55 percent of Hawaii's 1980 population was oriental. Appendix G shows that 7 5 percent of the AP can didates in 1983 were non-white. Figures in table 24 support the inference that most of Hawaii’s non-whites taking the AP examinations are oriental. 179 TABLE 25
CANDIDATE FREQUENCY BY COOP INTERVAL, CALCULUS COURSE, AND ADVANCED PLACEMENT INTEREST (API), 1973-1984
Number of Punahou Students by Course in Each API Category9 COOP Calculus AB Calculus BC Composite Interval Total 1 2 3 4 1 2 3 4 1 2 3 4
56-60 4 3 4 1 5 7 16 74 9 10 20 75 114 51-55 15 7 21 7 16 11 28 63 31 18 49 70 168 46-50 35 32 49 6 12 12 47 30 47 44 96 36 223 41-45 40 22 40 8 15 7 19 12 55 29 59 20 163 36-40 41 17 19 5 4 5 3 3 45 22 22 8 97 31-35 10 7 9 3 2 3 1 13 9 12 1 35 26-30 10 8 1 1 1 10 9 2 21 21-25 2 1 1 2 1 1 4 16-20 2 1 1 2 1 1 4 11-15 1 1 1
Total 160 88 152 30 55 44 117 184 215 132 269 214 830 Percent 19 11 18 4 7 5 14 22 26 16 32 26 100
^ v e r y Advanced Placement Interest (API) category includes an AP Calculus course and perhaps more, as follows: 1— only AP Calculus; 2— AP English, History, Languages, or the Arts; 3— AP Biology, Chemistry, or Physics B (calculus not required); 4— Physics C (calculus required).
The SCI variable is incorporated into API and will not be tabled separately. Table 25 shows that a relationship does exist between API and achievement in the calculus, as measured by the Cooperative Test score, both within and between courses.
API was originally chosen as a class variable for a number of reasons. It seemed reasonable that a student who had the interest and industry to take college-level courses in both calculus and Physics C, which requires calculus, would be doing far more homework in calculus, acquiring greater skill in mechanics, and doing more analytic problem solving using the calculus, than another student not taking Physics C.
With such reinforcement the Calculus-Physics C student should do better. It seemed, also, that a student with an interest in science and sufficient ambition to take a college-level science course, even one which did not require calculus, would have some reinforcement in terms of work habits and thought processes: an ability to organize and inter pret data, discover patterns and similarities, an attitude of curiosity and exploration, experience in analytic, step-by-step problem solving, self-discipline, and attention to detail. By transferring such skills, this individual would probably do better in calculus than someone more interested in the arts, languages, or social sciences. But even those students whose interests are in these latter fields, by the mere fact that they are willing to do the additional work of a college-level course in history or languages, might do better than a student whose only college-level course was the one in calculus. Table 25 visibly supports the general notion that the API level increases from 1 to 4 as the achievement level in calculus, as measured by the criterion variable
COOP, increases. The generalization that the greater the interest in science, the greater the achievement in calculus also appears to have some validity. Clearly, the majority of Physics C students not only chose the more demanding Calculus BC course, but they did well in it.
The National Test Variables
The use of national test scores by postsecondary institutions to assess learning that has already occurred and to predict subsequent suc cess in college-level courses has been discussed in chapter I I . Readily available from the College Board’s Admissions Testing Program or the
PSAT/NMSQT are the nationally-normed scores on the JSATV, JSATM, PSATV,
PSATM, SATV, SATM, ACHLl, and ACHL2. Most of these scores appear in the records of Punahou juniors by the end of that year. In addition, the 181 Henmon-Nelson Test of Mental Ability was available on Punahou students who graduated prior to 1983. The nationally available test variables used in this study are defined as follows:
HNIQ: An acronym for Henmon-Nelson Intelligence Quotient, this interval-level, predictor variable is the score on Form B of the Henmon-Nelson Test of Mental Ability. This test was administered to all Punahou freshmen until it was discontinued after the Class of 1982 was tested.
ACHLl and ACHL2: Abbreviations for the Level I and Level II Achievement Tests in Mathematics of the College Board's Admissions Testing Program, these interval-level predictors are the standard ized test scores on either test named. The May or June ACHL2 test administration is recommended for all juniors completing their fourth-year course in Advanced Mathematics, particularly those who are being considered for Advanced Placement mathematics as seniors.
JSATV and JSATM: Abbreviations for the Verbal and Mathematical scores on the Junior Scholastic Aptitude test administered to all Punahou seventh-graders until discontinued after the Class of 1977 was tested, these are interval-level, predictor variables.
PSATV and PSATM: Abbreviations for the Verbal and Mathematical scores on the PSAT/NMSQT (Preliminary Scholastic Aptitude Test/ National Merit Scholarship Qualifying Test), administered in October to all Punahou sophomores, this interval-level, predictor variable is the College Board's standardized score on the test-section named. Somewhat shorter and easier than the SATM, the PSATM is based on a fifty-minute test containing fifty multiple-choice questions.
SATV and SATM: Abbreviations for the Verbal and Mathematical scores on the Scholastic Aptitude Test, which is taken by all Punahou juniors in the December administration, this is an interval-level variable on the College Board's standardized scale of the particular test-section named. The SATM score is based on a sixty-minute test containing sixty multiple-choice questions.
DSATS: An abbreviation for 'double Scholastic Aptitude Tests’, this interval-level, predictor variable combines both verbal and mathematical aptitude components, but double-weights the latter. It is linearly dependent on SATV and SATM from which it is computed using the formula: DSATS = [SATV + 2(SATM)]/3 .
Table 26 shows the frequency distribution of Punahou candidates on each of the AP calculus examinations by Henmon-Nelson IQ interval and
APEX score. It would appear from the table that a stronger statistical
dependence exists between APEX and HNIQ in Calculus BC than in Calculus AB. 182 TABLE 26
CANDIDATE FREQUENCY BY HNIQ INTERVAL AND AP SCORES ON EACH CALCULUS EXAMINATION, 1969-1982
Number of Punahou APP Candidates HNIQ Calculus AB Exam Calculus BC Exam Interval Total 1 2 3 4 5 E AB 1 2 3 4 5 E BC
161-165 2 2 4 4 156-160 1 1 1 2 3 8 14 15 151-155 3 3 3 4 9 23 39 42 146-150 1 4 6 4 15 2 13 23 33 71 86 141-145 2 6 5 13 2 11 18 24 55 68 136-140 13 16 12 41 1 18 17 35 71 112 131-135 1 4 19 15 14 53 1 5 14 22 33 75 128 126-130 1 6 16 17 13 53 1 3 10 22 22 58 111 121-125 1 6 25 22 13 67 5 7 12 22 46 113 116-120 2 3 15 10 6 36 2 7 5 14 50 111-115 1 2 5 6 2 16 3 2 2 1 8 24 106-110 1 2 1 4 2 1 3 7 101-105 1 1 2 2
Total 6 24 103 101 70 304 3 26 83 138 208 458 762 Mean HNIQ 128.7 137 .4 133.9 Stnd. Dev. 9 .7 11.1 11.4 Stnd. Err. 0.6 0.5 0.4
Comparing the results displayed in table 26 with table 23, which shows the strong statistical dependence between APEX and COOP, it is obvious that the relationship between APEX and HNIQ is much weaker and may not even be statistically significant. It is not surprising that the mean HNIQ for Calculus BC candidates is higher, and the standard devi ation greater, than candidates in Calculus AB given the difference in amount of material covered in each course and examination.
The frequency distribution of candidates on the College Board tests by their standardized score intervals shown in table 27 must be viewed with caution. The tests were not administered at the same time, but over a period of years in grades 7, 10, and 11. The numbers of TABLE 27
PUNAHOU CANDIDATE FREQUENCY ON COLLEGE BOARD APTITUDE AND ACHIEVEMENT TESTS, BY TEST SCORE INTERVAL AND CALCULUS COURSE
Number of Verbal (V), Mathematical (M), and Achievement Test Candidates
Score Calculus AB Candidates Calculus BC Candidates Interval JSAT PSAT SAT JSAT PSAT SAT DSATS*ACHL1 ACHL2 DSATS*ACHLl ACHL2 V M V M V M V M V M V M
Max: 80 2 15 3 1 13 30 197 75 - 79 2 3 2 13 15 62 3 46 16 148 45 41 142 70 - 74 1 7 43 9 75 19 9 127 12 9 29 178 65 215 158 43 110 65 - 69 8 8 16 106 30 145 90 16 99 43 74 71 164 100 128 210 10 31 60 - 64 5 15 39 128 59 126 179 17 40 55 57 115 106 114 38 114 6 2 55 - 59 14 14 82 91 97 49 100 10 11 44 44 99 36 115 9 22 1 50 - 54 9 7 102 45 107 17 36 1 28 13 112 10 83 2 45 - 49 9 2 94 7 82 5 5 13 2 67 1 40 1 1 40 - 44 2 56 2 31 1 7 3 37 11 35 - 39 22 10 8 4 30 - 34 5 3 2 3 25 - 29 1
Total 47 47 425 425 430 430 430 70 354 202 202 544 544 552 552 552 130 483 Mean Score 55.7 59.4 51.5 61.9 53.9 64.7 61.3 67.3 70.2 59 .5 62.1 56.6 67 .6 60.2 71.4 67.7 74.6 76.3 Stnd. Dev. 7.3 5.7 8.2 6.3 7.9 5.9 5.2 7.3 5.4 7.3 5.8 8.7 5.8 8.6 5.0 5.1 4.8 4.1 Stnd. Err. 1.1 0.8 0.4 0.3 0.4 0.3 0.2 0.9 0.3 0.5 0.4 0.4 0.2 0.4 0.2 0.2 0.4 0.2
*DSATS = [SATV + 2(SATM)]/3
oo CO 184 candidates taking these tests are not the same; but one can make valid comparisons of the same test results between courses, or between the ver bal and mathematical section scores on the same test, with respect to the latter, it is noted that the mean score in mathematics aptitude on both the PSAT and SAT is at least one standard deviation above the mean verbal aptitude score.
The number of candidates who took the JSAT in seventh grade and the ACHLl as juniors is limited; each test was taken by about 25 percent of the study population. However, the JSAT scores, verbal and mathema tical, were included because they are the earliest national scores obtainable on a candidate, if a sufficiently strong correlation should be found between JSATM and APEX, it might prompt a return to the adminis tration of the JSAT in seventh grade for guidance purposes. ACHLl was included not only because it was available and contributed some infor mation, but also because students in the earlier years often had to make a choice between ACHLl and ACHL2 to satisfy a college they wished to attend. ACHL2 scores are available on over 85 percent of the study popu lation; of the remaining 15 percent, about 13 percent have ACHLl scores.
Scores on the other variables are available for most, if not all, of the study population: PSAT, 98,7 percent; SAT and DSATS, 100 percent.
The Local Predictor Variables
in any junior's file is kept a record of courses taken, grades in those courses, and even summary statistics such as grade-point-averages.
Six of these entries might be termed ’local variables' because they are available only at the particular school until reported to some college.
For the purpose of this study, these six variables represent the grades in the four previous mathematics courses, their average, and the junior 185 grade-point-averagej and they are identified as follows:
JGPA: An acronym for junior grade-point-average, this ratio-level predictor is the grade average of a Punahou student's junior-year academic courses, converted to the usual four-point grading system preferred by colleges.
AVMA: An acronym for 'average mathematics' grade, this ratio- level, predictor variable is the average of the final grades in the four mathematics courses taken in a college preparatory school as prerequisites to the calculus. The computation uses Punahou's twelve-point scale. The abbreviations for the four courses being averaged are as follows: ALGl = Algebra I; GEOM = Geometry; INTM = Intermediate Mathematics; and ADVM = Advanced Mathematics. These four are also used as variables individually.
While JGPA is not cumulative, it is the latest across-subjects measure available on candidates and is indicative of ability to master college preparatory courses. Table 28 displays the number of candidates in each calculus course by JGPA intervals and APEX scores.
TABLE 28
CANDIDATE FREQUENCY BY JGPA INTERVAL AND AP SCORES ON EACH CALCULUS EXAMINATION, 1969-1984
Number of Punahou APP Candidates JGPA Calculus AB Exam Calculus BC Exam Interval Total 1 2 3 4 5 E AB 1 2 3 4 5 E BC
A 3.84-4.00 6 4 4 14 6 23 72 101 115 A- 3.51-3.83 5 35 58 45 143 1 11 46 86 141 285 428 B+ 3.17-3.50 3 15 60 67 40 185 2 13 31 48 35 129 314 B 2.84-3.16 4 4 27 23 7 65 1 6 12 9 4 32 97 B- 2.51-2.83 2 6 10 1 19 1 1 2 4 23 C+ 2.17-2.50 1 1 2 1 1 3 C 1.84-2.16 1 1 1 c- 1.51-1.83 1 1 1
Total 10 30 141 152 97 430 4 31 96 167 254 552 982 Mean j g p a 3.37 3.60 3.50 Std. Dev. 0.32 0.28 0.32 Std. Err. 0.02 0.01 0.01 186 The review of the literature indicated that more specific mea sures than intelligence, aptitude, or overall high school achievement are required to improve the prediction of success in college mathematics courses. Mathematics placement tests, mathematics grade-point-average, and previous mathematics course grades improved this differential pre diction. For this reason, the grades in the four mathematics courses leading up to the calculus were collected for each member of the study population, and the average of these four grades was computed. For the regular student in a college preparatory program, these would be the four secondary school courses recommended by the Commission on Mathematicsj for students at Punahou who took accelerated arithmetic in grade seven and began the four-year sequence with algebra in grade 8, the four grades in ALGl, GEOM, INTM, and ADVM would be available at the end of the junior year.
No distinction was made in this study to indicate that previous mathematics grades were taken in regular or honors sections. Punahou's mathematics department is a firm believer in departmental testing among
different teachers who are teaching a common course, no matter what the grade level of the students within section. Sufficient departmental tests during the semester and the majority of questions on the semester examinations are given all students taking a given course to insure con
sistency in grading among teachers. The grade distributions of the
honors sections are invariably higher than the regular, as they should
be. While most Advanced Math (H) students are recommended for Calculus
BC, not all are. The best students in regular classes who might have
accelerated by taking a mathematics course during summer school may also
be recommended. Table 29 shows the distribution of mathematics grades. TABLE 29
GRADE DISTRIBUTIONS OF PUNAHOU CANDIDATES IN FOUR PREVIOUS MATHEMATICS COURSES, THEIR AVERAGE, AND THE CRITERION CALCULUS BY CALCULUS COURSE, 1969-1984
Grade Number of Calculus AB and Calculus BC Candidates in Each Mathematics Course and a ALGl GEOM INTM ADVM AVMA CALCb Grade interval AB BC E AB BC E AB BC E AB BC E AB BC E AB BC
A 10.6-12.0 102 269 371 40 203 243 37 201 238 21 163 184 12 176 188 41 81 A- 9.6-10.5 111 164 275 98 216 314 120 194 314 81 228 309 130 273 403 105 144 B+ 8.6- 9.5 104 77 181 143 96 239 118 121 239 158 117 275 197 95 292 116 121 B 7.6- 8.5 70 36 106 104 31 135 103 30 133 118 39 157 74 7 81 87 93 B- 6.6- 7.5 30 6 36 32 4 36 34 5 39 40 5 45 15 1 16 45 80 C+ 5.6- 6.5 6 6 12 1 13 11 1 12 10 10 2 2 22 18 C 4.6- 5.5 6 6 1 1 2 6 6 2 2 9 12 c- 3.6- 4.5 1 1 1 1 5 D+- 2.6- 3.5 2 D 1.6- 2.5 1
Total 430 552 982 430 552 982 430 552 982 430 552 982 430 552 982 430 552 Mean Grade 9 .39 10.3 9.88 8.93 10.0 9.56 8.91 10.0 9.52 8.74 9 .92 9 .40 9.10 10.1 9.67 8.73 8.88 Stnd. Dev. 1.38 1.00 1.26 1.19 0.95 1.20 1.30 0.96 1.24 1.12 0.93 1.17 0.87 0.74 0.94 1.50 1.56 Stnd. Err. 0.07 0.04 0.04 0.06 0.04 0.04 0.06 0.04 0.04 0.05 0.04 0.04 0.04 0.03 0.03 0.07 0.07
aAVMA = (ALGl + GEOM + INTM + ADVM)/4
Unlike the five other variables displayed, CALC is a criterion variable. Calculus AB and Calculus BC are different courses with no common examinations except the COOP test. The grade distributions in these two courses are independent; therefore, the distribution of their sum is irrelevant. 188 Design of the Study
The investigation to determine which of the selected variables were the most reliable predictors of achievement (as measured by COOP) and success on, and discrimination between, Calculus AB and Calculus BC
(as measured by APEX) was conducted in three phases: data reduction, discriminant analysis, and validity application. The results of these three phases are reported in chapter V. Data reduction is considered for both categorical and predictor variables; validity application is made both internally and externally. A difficulty inherent in the study is the fact that the major criterion, APEX, when partitioned to measure success or failure, is no more than a class variable. As such, the num ber of statistical techniques which can be employed are limited. Dis criminant analysis becomes the dominant statistical technique in this report because it is one of the few capable of handling interval-level predictors and a categorical variable as criterion. The study design may seem tortuous, but it was conceived to circumvent this difficulty.
Phase I : Data Reduction
Phase I has two goals: the elimination of those categorical or class variables found not to be statistically significant, and the iden tification of a reduced set of predictor variables for use in discrimi nant analysis. With respect to the class variables, YR, CRSE, SEX, NOR, and API were tested for significance; SCI and TAP were not tested because of their inclusion in API. The CRSE variable cannot be tested using either the APEX or CALC criterion because neither criteria is directly comparable between courses. The APEX measure of Calculus AB describes proficiency in that course only; and, although the two AP examinations have some common questions, they are different examinations 189 designed to measure mastery of different courses. Likewise, grades in
Calculus AB and Calculus BC are assigned independently because these are
different courses, having no tests in common except COOP. A chi-square test on the equality of two multinomial distributions of COOP grades will be reported showing CRSE should be retained as a significant variable.
The stability of the APEX grading scale so widely accepted by
colleges and universities from year to year is based on a statistical equating process which involves the reuse of multiple-choice questions.
This is a strong argument supporting the elimination of YR as a variable
of consequence; APEX is designed to be statistically independent of YR.
However, in order to show that YR is not a significant variable
with respect to APEX for this study population, a general linear test
approach was adopted for testing the equality of two regression lines.
Because this procedure had to be applied separately for each course, the
full APEX data set was first partitioned by course into APEXAB and APEXBC.
To determine that the variable YR was not significant in the Calculus BC
course, for example, the APEXBC scores were partitioned further into the
sets of scores for even and odd years, named APEXBCEV and APEXBCOD,
respectively. Analysis of variance procedures were used to obtain two
linear regression models for APEXBCEV and APEXBCOD as functions of their
respective COOP scores. ANOVA was also applied to the combined scores,
and an F-test was performed under the hypothesis that the even and odd
year regression lines were the same, i.e. that both the slope and inter
cept terms were the same. The test indicated that it was not possible to
reject this hypothesis; so the lines were considered to be the same
regardless of even or odd year composition. This procedure was repeated
for APEXAB; and YR was eliminated as a categorical variable . 190 With respect to each criterion, similar procedures were used to test for the statistical significance of each of the three categorical variables SEX, NOR, and API. When APEX was used as criterion in non- parametric tests, an adjustment was required because of the paucity of scores at level 1— no recommendation. Table 21 shows that only 10 of the
430 Calculus AB students and 4 of the 552 Calculus BC students received this grade. Because the Chi-square tests required at least five entries per cell, APEX scores below three were combined. Since these are the two scores for which the majority of colleges do not give credit or advanced standing, their consolidation makes practical sense.
Using APEX as criterion and, for example, SEX as the categorical variable to be tested for statistical significance, three Chi-square tests on the equality of two independent, multinomial distributions were conducted: the first was on all Punahou males and females taking a cal culus examination; the second, on males and females with APEXAB scores; and the third, on those with APEXBC scores. A similar procedure was used to test separately for the significance of NOR and API, except that API was tested for equality of four independent, multinomial distributions.
Pearson product-moment correlations were computed between all
interval-level dependent and independent variables. Those independent
variables showing the highest correlation and greatest linear relation
ship with CALC and COOP were selected to serve as covariates with the
factors CRSE, SEX, NOR, and API in an analysis of covariance. From the
results of this analysis, only two categorical variables were found to be statistically significant— CRSE and API.
Because COOP was so highly correlated with both APEX and CALC within each course (CRSE), multiple regression analysis was used to 191 obtain four regression equations, one for each level of Advanced Place ment interest (API). Each of the four prediction equations represents that linear combination of independent variables which best correlates with COOP for the given API level. To differentiate between actual Coop erative Test scores (COOP) and predicted scores, which are API-dependent, the predicted COOP scores were given the labels SC0AP1-SC0AP4. SCOAP is an acronym for SCI-COOP-AP, which succinctly describes the new variable as one which includes the SCI component of API, predicts the COOP score, and will be used to predict success on the AP examination. The composite column of table 25 gives the apportionment of the 830 candidates who took the Cooperative Test to each API level: 26 percent took only AP calculus;
16 percent also enrolled in AP English, History, Language, or the Arts;
32 percent had AP Biology, Chemistry, or Physics B; and 26 percent took
AP Physics C, which required the calculus. The number in each API cate gory was presumed to be sufficient to provide the prediction equation with the essence of that API level. These were the prediction equations
desired for Phase I I .
Phase II: The Discriminant Analysis
As the major statistical technique utilized in this report, dis
criminant analysis was used to distinguish statistically between either two or three subgroups of APEX. A four-subgroup analysis was also cal
culated but discarded as being too unwieldly. For each choice of sub groups two discriminant analyses were run, one without the SCOAP pre
dictor variable and one including that predictor, in order to determine whether use of SCOAP increased the proficiency of the classification
functions. Because the number of predictor variables was large and
included both nationally and locally available predictors, the stepwise 192 procedure was chosen so that the most useful set of predictors to achieve satisfactory discrimination between groups would result. Obtained were the coefficients for the classification function for each of the sub groups, the number depending on whether a two-group or three-group sub division had been made. The highest score for a given junior would place that student in the classification category to be recommended.
The variable CRSE as dependent variable in a two-group, discrim inant analysis would be used to answer the question, "Which AP calculus course should a given junior take?" The discriminant analysis calculated the effects that the interval-level, predictor variables had on the nomi nal-level, dependent variable CRSE.
The major discriminant analysis, which partitioned APEX into three subgroups, required a new categorical variable TRGP (abbreviation for tri-group). TRGP had the value 1 if either APEXAB or APEXBC had a score below a qualifying score of 3; it had the value 2 if APEXAB was
3 or greater; and TRGP = 3 if the APEXBC score was greater than or equal to three. This discriminant analysis with TRGP as dependent variable would determine the most reliable predictors of success on, and discrimi nation between, each of the two AP calculus examinations. It would pro vide three classification equations, one for each subgroup, and enable the mathematics teacher or guidance counselor to answer the question,
"Should this particular junior be advised to take Calculus AB, Calculus
BC, or not take either course and improve his pre-calculus abilities during his senior year?"
The stepwise selection method was thought to be a wise choice since knowledge of the sequential entry into the analysis of the next best discriminator would allow a relative ranking of the importance of 193 each independent variable to the analysis. Of particular interest was the relative ranking of the new SCOAP variable. In the SPSS package for discriminant analysis there are five stepwise selection criteria avail able . Of these only two seemed to be appropriate for this study: MAHAL, which maximizes the Mahalonobis distance between the two closest groupsj and MINRESID, which is defined by the value R as follows:
R = Z , - ,i ttt » where D. . is the Mahalonobis distance between 1 + (Di j/4) 13 groups i and j. Each term of this summation is an estimate of 1.0 minus the square of the multiple correlation between the set of dis criminating variables being considered and a dummy variable which identifies the corresponding pair of groups, i.e., each term is the proportion of variation in the dummy variable not explained by the discriminating variables under consideration. The objective here is to minimize R, the residual variation.2
MINRESID does separate groups that are close together; but the rationale for selecting this method was based on the nature of the groups being separated. The decision to allow the industrious, straight A, top- ranked student to take Calculus BC is automatic; as is the decision to recommend another year of pre-calculus mathematics to the low C student who has a poor knowledge of fundamentals. The problem lies with the bor derline student; and, rather than maximizing the distance between the two calculus exam centroids, it seemed more appropriate to minimize the proportion of variation not being explained. That gray area of uncer tainty due to student variability will always exist in education. If a small portion of that variability as measured by the discriminating variables can be reduced, so much the better. MINRESID was chosen.
Discriminant analysis is a powerful classification technique which identifies likely group membership in a given case when all that is
2 William R. Klecka, "Discriminant Analysis," in Norman H. Nie et al., Statistical Package for the Social Sciences, 2d ed. (New York: McGraw-Hill Book Co., 1975), pp. 447-48. 194 known are the values on the discriminating variables. The classifi
cation equations are derived from the pooled, within-groups covariance matrix and the centroids for the discriminating variables, one equation for each group. A given case is classified as a member of the group with the highest classification score. The validity of the discriminant
analysis, as indicated by its usefulness and efficiency, is discussed in the third and final phase.
Phase ill: Validity Application
Both internal and external validity checks on the adequacy of the discriminant analysis were conducted. Internal validity was deter mined using Punahou-based data from the study population; the external
validity application was made on a six-year data set collected at Iolani
School. Validity describes a measure's usefulness; and the usefulness
of a procedure can be demonstrated by showing it to be effective. The
relative efficiency of the derived set of classification functions in
both the internal and external validity applications was measured by the
percentage of correctly classified scores. Specifically for the inter
nal validity check, it was found by selecting the discriminant program
option which computed the discriminant score and classification infor
mation for each case. Because the actual group membership is known for
the cases used in the analysis, the percentage of correctly classified
cases can be determined. This is the internal measure of the effective
ness of the discriminant analysis. The external check was similar.
This is the design of the study. Statistical techniques employed
included Pearson product-moment correlation, non-parametric multinomial
chi-square testing, multiple regression analyses, analyses of variance
and covariance, and discriminant analysis. Their application follows. CHAPTER V
DATA REDUCTION, PREDICTION, AND ANALYSIS
As detailed in the design of the study, investigation of the
categorical variables to determine which are significant and which should
be eliminated must precede the identification of a reduced set of pre
dictor variables for use in the discriminant analysis. Therefore, the
order of section presentation in this chapter begins with the data reduc
tion of the class variables by the various methods described, and then
determines the best set of predictor variables by multiple regression for
the remaining class variable categories. Only then are the prediction equations applied in the discriminant analysis. Internal validation with
Punahou-based data is presented in the final section.
Reducing the Number of Class Variables
Course, year, racial extraction, sex, and Advanced Placement
interest (CRSE, YR, NOR, SEX, and API) are the categorical variables con
sidered. Referral will be made to appendix H which contains the computer printout of the sample means and standard deviations of the three depen
dent variables, twelve independent variables, and TAP under all possible
breakdowns of the study population with respect to four of the categorical variables. Year is excluded. Each line (referred to by observation num ber) represents one of the 135 (3*3-3*5 = 135), possible subsets of the study population; and the observation identifies the cell and cell fre quency before giving the sixteen means and standard deviations. For example, if the variable of interest is the total number of AP exams 195 196 taken (TAP), the first line (OBS 1) shows that the mean number of AP examinations (MTAP) taken by the 982 students in the study population was about 2.6 with a standard deviation (SDTAP) of 1.1 approximately. OBS 46 and 47 indicate that, on the average, Calculus BC students took more AP exams than those in Calculus AB, 2.9 to 2.1 respectively. The statement holds unless the comparison is being made between category 4 Advanced
Placement interest students, in which case the mean number of exams taken by students in either course is 3.7 (OBS 51 and 55). There are only two categories with less than five candidates: Calculus AB females, oriental or non-oriental, who also took Physics C (calculus required) .
The class variables are examined in the following order: course, year, oriental or non-oriental, sex, and Advanced Placement interest.
Retention of the variable CRSE
With respect to the success criterion AP examination grade (APEX) and the achievement criterion grade in calculus (CALC), the course (CRSE) taken by the candidate is a necessary categorical variable. It distin guishes between Calculus AB and Calculus BC final grades which are assigned independently in the two separate courses. The two national
AP examinations for these different courses share some common questions; but their frequency distributions are not the same (see appendix D) and
APEXAB and APEXBC grades certainly receive different treatment in most colleges because the Calculus BC content is designed to qualify a candi date for credit and placement one semester beyond that granted for AB.
However, with respect to the other achievement criterion COOP, which serves as the common link between APEX and CALC, the variable CRSE may not be significant. Students in both courses take this Cooperative
Test on the same two days approximately one week before the mid-May AP 197 examinations. The present section answers the question whether or not there is a significant difference in COOP test scores due to the candi date's course. Recall that both the Calculus AB course and the COOP test were introduced at Punahou in school year 1972-73. The COOP test scores for the last dozen years have been grouped in convenient five-score intervals, and the two multinomial distributions are displayed in table
30. Comparison of this table with table 25 shows that the '30 & below' category containing the tails of the two distributions was used because
Calculus BC has no scores below 26.
TABLE 30
NUMBER OP PUNAHOU STUDENTS BY COOP TEST INTERVAL AND CALCULUS COURSE (CRSE), 1973-1984
COOP Calculus Calculus Total Interval AB BC
56 - 60 12 102 114 51 - 55 50 118 168 46 - 50 122 101 223 41 - 45 110 53 163 36 - 40 82 15 97 31 - 35 26 9 35 30 & below 28 2 30
Total 430 400 830 Mean Grade 43.17 50.59 46.75 Stnd. Dev. 7.46 6.29 7 .85
Consider the total of all Calculus AB and Calculus BC students taking the COOP test; their test scores constitute two independent, mul tinomial distributions with parameters n^ and P^j» i = 2, ..., 7 (the seven COOP intervals), j = 1 (CRSE 1 is Calculus AB), 2 (CRSE 2 is Cal culus BC), respectively. Let x^j represent the corresponding score fre quencies for the i-th COOP interval and j-th CRSE. To test the null 198 hypothesis that there is no significant difference in COOP test scores
due to differences in CRSE membership, for i = 1, 2, . . .,7, we let
Ho ! pil = pi2 = pi vs Hl ! pii ^ pi2 for some il
At the a = *05 significance level, we reject H0 in favor of
if and only if the test statistic defined by 2 nj(xil + xi2 > 2 7 x, . - 0 , = E E \ 13 nl + n2 (1) j=l i=l ------nj(xil + xi2)/(nl + n2 )
is such that > 12.59 [chi-square with 2k - 2 - (k - 1) = k - 1, and with k = 7, the degrees of freedom, df = 6 .] Here n^ = 430 and n^ = 400.
Since the test statistic Q &(C0OP & CRSE) = 196.73 > 12.59, we reject the
null hypothesis (a = 0.0); and we conclude that there is a significant
difference in COOP test scores due to differences in CRSE at the (approx
imate) .05 level of significance. The large value of chi-square implies
that some significant relationship exists between CRSE and COOP; there
fore, CRSE is retained as a categorical variable.
As a test of statistical significance, the chi-square test
assumes that both variables are only nominal level but the test statis
tic may also be applied to higher level variables. That is the case
here. With COOP grouped the way it is, it is no longer interval-level
but ordinal. Dichotomies (like CRSE) may be treated as ordinal or
interval-level even though there is no inherent order between categories
because they only have two categories. Considering COOP as the depen
dent and CRSE as the independent variable, there are two appropriate
measures of ordinal association: Kendall’s Tau C and Somer's D (Asym
metric) . In this case, both have the same value (0.54). This measure
of association supports retention of CRSE as a class variable. The Correlation Matrixes
The decision to retain CRSE as a class variable means that the courses Calculus AB and Calculus BC should be treated separately in the analysis. As an initial step toward the development of prediction equa tions for success on the AP examination (APEX) and for achievement in both the common Cooperative Test (COOP) and separate course grades in cal culus (CALC), product-moment correlations were computed for each pair of variables involved in the study. The resulting correlation matrixes, a separate one for each course, are reported in tables 31 and 32.
Care must be exercised in reading these tables because only above the main diagonal, where normally ones would appear, is each a true matrix. The main diagonal carries the number of observations used to calculate the coefficient for the entry-variable. Each x^j entry above the main diagonal gives the Pearson product-moment correlation coeffi cient between the i-th and j-th variable, and the corresponding signifi cance probability for that correlation appears in the x.^ entry below the diagonal. Decimal points are omitted but all entries without parentheses are accurate to the nearest .0005. Significantly high are the intercor relations among the criteria and between criteria and average previous mathematics grades (AVMA), achievement test level II (ACHL2), and the
Advanced Mathematics grade (ADVM), particularly in Calculus BC (table 32).
The variables JSATV, JSATM, and ACHLl have been omitted from these tables because of the small number of observations.
Elimination of the Variable YR
The variable year (YR) which designates the candidates' class and the year the AP examination was taken is a natural to be considered for early elimination. With respect to the criterion COOP, the year that the TABLE 31
CALCULUS AB: INTERCORRELATIONS BETWEEN ALL CRITERIA AND PREDICTOR VARIABLES USED IN ANALYSES
Number of Observations Are in Parentheses, Correlations above, and belowa Variables Probabilities Defined yl y2 y3 X1 X2 X3 X4 X5 X6 X7 X8 X9 xio X11 X12 X13
= APEX (430) 818 701 315 347 213 196 155 200 150 211 406 121 172 225 339 yl = COOP 000 (430) 712 338 345 202 178 147 204 180 237 445 149 176 235 378 Y2 = CALC 000 000 (430) 410 418 120 096 116 139 118 154 360 235 230 288 y3 389 = AVMA 000 000 000 513 115 066 145 X1 (430) 024 124 150 309 741 666 729 610 = JGPA 000 000 000 000 (430) 146 188 054 255 026 147 317 239 337 556 X2 312 = HNIQ 000 000 037 046 011 (304) 612 239 644 314 556 279 015 164 007 113 X3 = PSATV 000 000 049 623 000 000 (425) 334 791 243 575 214 022 078 029 X4 032 = PSATM 001 002 017 010 267 000 000 (425) 279 564 556 348 132 183 060 051 X5 = SATV 000 000 004 175 000 000 000 000 X6 (430) 260 696 284 002 114 012 086 = SATM 002 000 014 002 592 000 000 000 000 (430) 873 367 139 153 060 131 *1 = DSATS 000 000 001 002 000 000 000 000 000 X8 002 (430) 422 106 174 041 138 = ACHL2 000 000 000 000 000 000 000 000 000 000 000 (354) 226 204 185 325 *9 = ALGl 012 002 000 000 000 799 654 006 967 004 028 000 (430) 357 387 296 X10 = GEOM 000 000 000 000 000 004 111 000 018 002 000 000 000 (430) 367 231 X11 = INTM 000 000 000 000 000 900 552 217 217 398 000 X12 799 000 000 (430) 355 — ADVM 000 000 000 000 048 512 293 007 X13 000 074 004 000 000 000 000 (430)
Entries without parentheses are accurate to nearest .0005; those underlined are negative; and o probability > |r| under H0 : rho = 0 . TABLE 32
CALCULUS BC: INTERCORRELATIONS BETWEEN ALL CRITERIA AND PREDICTOR VARIABLES USED IN ANALYSES
. , , Number of Observations Are in Parentheses, Correlations above, and Probabilities below3 Variables Defined Y1 y2 y3 X1 x2 X3 X4 X5 X6 X7 X8 X9 xio X11 X12 X13
= APEX (552) 739 741 413 372 118 149 211 161 202 227 372 233 240 309 481 yl = COOP 000 (400) 807 476 430 287 192 227 242 313 353 529 235 327 368 501 y2 = CALC 000 000 (552) 545 505 146 131 225 194 280 296 475 286 297 434 606 y3 = AVMA 000 000 000 (552) 595 174 176 295 205 321 331 355 714 699 741 637 X1 = JGPA 000 000 000 000 (552) 272 313 181 391 234 381 289 327 365 429 569 X2 = HNIQ 011 000 002 000 000 (458) 643 346 684 303 587 286 124 220 102 044 X3 = PSATV 001 000 002 000 000 000 (544) 342 824 251 641 257 116 220 099 077 X4 = PSATM 000 000 000 000 000 000 000 (544) 279 550 524 355 199 260 214 139 X5 = SATV 000 000 000 000 000 000 000 000 (552) 303 111 331 117 230 115 134 X6 = SATM 000 000 000 000 000 000 000 000 000 (552) 833 502 217 219 219 190 X7 = DSATS 000 000 000 000 000 000 000 000 000 000 (552) 525 209 279 211 207 X8 = ACHL2 000 000 000 000 000 000 000 000 000 000 000 (483) 179 275 292 330 X9 = ALGl 000 000 000 000 000 008 007 000 006 000 000 440 324 X10 000 (552) 411 = GEOM 000 000 000 000 000 000 000 000 000 000 000 000 000 (552) 453 310 X11 = INTM 000 000 000 000 000 029 021 000 007 000 000 000 000 000 (552) 432 X12 ADVM 000 000 000 000 000 343 073 001 002 000 000 000 000 000 000 (552] X13
Entries without parentheses are accurate to nearest .0005j probability > |r| under Hq : rho = 0 . o 202 identical Form A of the test was administered makes no difference. It might be argued that YR is significant with respect to the calculus grades designated by CALC because of changing standards, except that only three instructors taught the two courses over the sixteen years and they gave equivalent tests for the specific purpose of ensuring consistency in grading. YR might be significant with respect to the major criterion
APEX in the parent population, but this is highly unlikely due to the stability of the Advanced Placement grading scale so widely accepted by colleges. However, in order to show that YR is not a significant vari able with respect to APEX for this study population, a general linear test approach was adopted for testing the equality of two regression lines. The testing procedure was applied twice: first, to determine if
YR is significant with respect to APEXj and second, to determine whether the use of calculators made any significant difference. Although dis cussed in detail only for Calculus BC, evidence is presented to show that similar results pertain in Calculus AB.
To show that YR is not significant with respect to AP grades in
Calculus BC, the APEXBC scores were partitioned into the set of scores for even and odd years and named APEXBCEV and APEXBCOD, respectively. To test whether the use of calculators made any difference in scores, APEXBC was subdivided differently— into scores for the years 1973 through 1982
(APEXBC82) and scores for the calculator years, 1983 and 1984 (APEXBC34).
In the first application, two linear regression models were obtained by fitting separate lines for APEXBCOD and APEXBCEV as functions of their respective COOP scores. COOP was selected because of its high correlation with APEXBC {.818) even though it is a criterion variable elsewhere in the study. Temporarily, it acts as an independent variable. 203 To construct the full model, denote the observations from
APEXBCOD aiK* those from APEXBCEV (x^2» Y±2^ ‘ T^e m°del
is expressed as follows:
Y. . = Pn . + p. . + for i = 1, . . . , n.j j = 1, 2. (2) lj Oj lj ij7 7 j 9 It is assumed that the error terms, are independent N(0, 0 )
and have constant variance. But fitting the full model (2) is equivalent
to fitting two separate regression lines, one for APEXBCOD and one for
APEXBCEV. The relevant results from each computer run are presented in table 33. APEXBCOD has n^ = 210 cases and APEXBCEV has n^ = 190 cases.
TABLE 33
COMPUTER OUTPUTS FOR SEPARATE REGRESSIONS: EVEN AND ODD YEARS, 1973-1984
APEXBCOD APEXBCEV
Regression coefficients and 95% C.I. Regression coefficients and 95% C.I. P x = -1.31227 [-1.9104, -0.7141] PQ2 = -1.03186 [-1.7978, -0.2660] P = 0.11021 [ 0.0984, 0.1221] P12 = 0.10422 [ 0.0894, 0.1191]
Analysis of variance: Analysis of variance: Source Sum of squares df Source Sum of squares df
Regression SSR. = 119.15923 1 Regression SSR2 = 67.99755 1 Residual SSE1 = 73.83600 208 Residual SSE2 = 66.68140 188
Total SST01 = 192.99523 209 Total SST02 = 134.67895 189 F = 335.67798 Signif F = 0.0000 F = 191.71072 Signif F = 0.0000
Denote the error sum of squares for the regression of APEXBCOD
SSE^, and that for APEXBCEV as SSE^. It follows that the error sum of
squares for fitting the full model (2) to the entire set of data is:
SSE(F) = SSE1 + SSE2 (3)
= 7 3.83600 + 66.68140 = 140.51740 204 The Gulliksen-Wilks testing procedure for equality of regression lines requires that the standard errors of estimate be tested for equal ity before the tests for equality of slopes and intercepts. The equality of error variances for APEXBCOD and APEXBCEV can be tested by the usual
F-test, using the estimates of the error variances from table 33:
SSEl - 73.83600 _ „ , SSE2 _ 66.68140 _ „ ------_ 0.35498 and--- - = z-r— — = 0.35469 nx - 2 208 . ' n2 - 2 188 so that: F* = = 1.0008 0.35469
Specifying a = .05 as the level of significance, we find the two action limits to be F ( .025; 208, 188) = 0.79 and F ( .975; 208, 188) = 1.34.
Since F* is contained between these two limits, APEXBCOD and APEXBCEV have equal error variances at the .05 level of significance.
To test whether the two regression lines are the same, we
formulate the following null hypothesis and state the alternative:
Ho ! P01 = P02 and P11 = P12 VS Hl: P01 ¥ P02 °r P11 * P12 (4)
Under the null hypothesis, the two regression lines have the
same slopes and intercepts; therefore, the full model (2) is reduced:
= 60 + + ■ 151 where [3^ and are the common parameters. If we fit this reduced model to the data, it is equivalent to fitting a single regression line to the
combined set of data APEXBC, where n^ + n^ = 210 + 190 = 400. A linear
regression has been fitted to APEXBC, and the results arepresented in table 34 from the computer printout. Accordingly, the error sum of
squares from this table is given by the following:
SSE(Reduced model) = SSE(R) = 140.71204.
Since APEXBC has n^ + nobservations, the degrees of freedom associated 205 TABLE 34
COMPUTER OUTPUT FOR COMBINED REGRESSIONS: ALL YEARS 1973 THROUGH 1984
APEXBC
Regression coefficients and 95% C.I.:
3q = -1.19727 [-1.66498, -0.72955] 3X = 0.10769 [ 0.09851, 0.11686]
Analysis of variance: Source Sum of Squares df
Regression SSR(R) = 188.28796 1 Residual SSE(R) = 140.71204 398
Total SSTO(R) = 329.00000 399 F = 532.56711 Signif F = 0.0000
with SSE(R) are n^ + n^ - 2. On the other hand, since n^ - 2 degrees of freedom are associated with the error sum of squares SSE^ for APEXBCOD, and n2 - 2 with SSE2 for APEXBCEV, the number of degrees of freedom asso ciated with SSE(F) = SSE1 + SSE2 are - 2) + {n2 - 2) = ^ + n - 4.
In order to test the equality of the two regression lines, we compute the test statistic F* = ----- SSE(R) - SSB(F) + SSE|F) (n-^ + ^ — 2) - (n^ + ^ - 4) n-^ + ng - 4
_ SSE(R) - SSE(F) nl + n 2 “ 4 F* ------(6) SSE(F) 2
F* = 140.71204 - 140.51740 ^ 210 +190-4 140.51740 * 2
F* = 0.27
Specifying the level of significance at a = .05, we find that in order to reject Hq , F* > F( 1 - a; 2, nx + n2 - 4) = F{0.95; 2, 396) > 2.9957. But
F* = 0.27 ¥ 2.9957j therefore, there is insufficient evidence to reject 206 the null hypothesis, and we conclude that the regression lines for
APEXBCOD and APEXBCEV are the same line. This conclusion implies that the variable YR is not significant with respect to APEXBC and should be eliminated from consideration.
To test whether the use of calculators, which were permitted only in the years 1983 and 1984, made any significant difference in APEXBC scores, the entire procedure just completed was repeated. APEXBC was partitioned into the set of scores for 1973 through 1982 (APEXBC82) and the set of scores for the calculator years (APEXBC34). Table 35 presents the relevant results from the two computer runs which fitted the separate regression lines for APEXBC82 and APEXBC34. The number of observations for APEXBC82 and APEXBC34 are n^ = 313 and n2 = 87 » respectively.
TABLE 35
COMPUTER OUTPUTS FOR SEPARATE REGRESSIONS: YEARS WITH AND YEARS WITHOUT CALCULATORS
APEXBC82 APEXBC34
Regression coefficients and 95% C.I. Regression coefficients and 95% C.I. P = -1.49000 [-2.0462, -0.9338] PQ2 = -0.71944 [-1.6004, 0.1615] Pn = 0.11267 [ 0.1018, 0.1235] P12 = 0.10075 [ 0.0830, 0.1185]
Analysis of variance: Analysis of variance: Source Sum of squares df Source Sum of squares df
Regression SSR^ = 143.56520 1 Regression SSR2 = 47.22904 1 Residual SSE^ = 106.47314 311 Residual SSE2 = 31.62153 85
Total SST01 = 250.03834 312 Total SSTC>2 = 78.85057 86 F = 419.34309 Signif F = 0.0000 F = 126.95365 Signif F = 0.0000
To obtain the error sum of squares for fitting this second full model to both sets of data, we add the error sum of squares for each 207 separate regression given in table 35:
SSE(2F) = SSE1 + SSE2 = 106.473 + 31.622 = 138.095
It is noted that combining the data based on the ten, pre-calculator years (APEXBC82) with the two years using calculator (APEXBC34) yields the same reduced model (APEXBC) for the combined regressions in table 34.
To test whether the regression lines for APEXBC82 and APEXBC34
are the same, again we assume the null hypothesis given by (4). Because the procedure is being repeated, only the essentials are reported. First, we test for the equality of error variances for the two regressions.
From table 35 the estimates of the error variances are obtained.
106.47314 . 31.62153 = 3 U “ 85 °.920
If we wish to control a at the .05 level, we require F( .025} 311, 85)
and F ( .975; 311, 85) which have values approximately 0.75 and 1.41,
respectively.^ Since F* falls between these limits, we conclude that the
two regression lines have equal error variances.
To test the equality of the two regression lines, we compute the
F* test statistic from (6):
_ 140.712 - 138.095 t 313 +87-4 . 138.095 ' 2
Specifying the level of significance at a = .05, we can determine that
2.9957 < F ( .95; 2, 396) < 3.0718. Since F* > 3.0718, we reject the null
hypothesis and conclude that the linear regressions for APEXBC82 and
APEXBC34 are not the same. The next step in the analysis is to determine
whether the two regression lines differ with respect to their slopes or
The limitation of F-distribution tables to entry values other than those shown here requires extrapolation. The results shown are con servative. Egon S. Pearson and H. 0. Hartley, Biometrika Tables for Statisticians, Volume II (London: Biometrika Trust, 197 6 ), p. 179, table 5. 208 their intercepts. First, we should test for the equality of slopes because, if the slopes are equal, the regression lines are parallel and only differ in height. If so, the input data for the two regression lines may still be pooled for purposes of estimating the common slope.
We test for the equality of slopes by obtaining a confidence interval for the difference in slopes, (3^ - 3^2 • Tlie linb-b5 of the 1 - a confidence interval are given by the following:
s2‘bll ' b12> = n / f n2 '. 4 I------? I ‘8> flz(xu ^- v ------m i2 - x2) J s (0.11267 - 0.10075) = 313 +87-4 [ 11307.00 + 4653.52 ]
s2 (0.01192) = 0.000105779
s(0.01192) = 0.0103
To construct 95 percent confidence limits for - 3^2» ^ find that t ( .975; 396) = 1.97 approximately so that the limits are;
0.01192 - (1.97 ) (0 .0103) < g1]L - 312 < 0.01192 + (1.97) (0 .0103)
- 0.00837 < pn - 0 < 0.0322
Since 0 ^ - 0 ^ = 0 is within the 95 percent confidence limits, we conclude that the slopes are the same. This implies that the differ ence between the regression lines for APEXBC82 and APEXBC34 is their
heights because the lines are parallel. A rapid check in table 35 shows that the intercept 0 ^ for APEXBC34 is not only greater than 3q^> but
is actually above the 95 percent confidence interval for 3q^> the inter
cept for APEXBC82. This implies that the Punahou students did slightly 209 better on the Calculus BC AP examination during the two years calculators were allowed than during the non-calculator years. A similar result was found with the Calculus AB students, but this will not be presented since the decision has been made not to allow calculators beginning with the
1985 administration. For our purposes here, the important result is that with respect to the AP examination scores, the categorical variable YR is not significant. It may be eliminated and the data over the years com bined or pooled.
Elimination of the Racial Extraction Variable, NOR
In order to eliminate a categorical variable, it must be shown that it is not statistically significant with respect to the criterion in question. Knowing already that CRSE is a significant class variable with respect to all three criteria, APEX, COOP, and CALC, means that the variable NOR (non-oriental or oriental) should be tested against all three criteria, but separately by CRSE at least for APEX and CALC. The
latter two variables represent grades in independent examinations or
courses, APEXAB-APEXBC or CALCAB-CALCBC; but COOPAB and COOPBC represent
scores on the same examination. The present section answers the question
whether or not there is a significant difference in COOP test scores due
to the candidate’s racial extraction. The same multinomial chi-square
test and test statistic (1) is used as was used to test the significance
of CRSE. The two multinomial distributions by COOP interval and racial
extraction are shown in table 36.
Consider the total of all oriental and non-oriental students
taking the COOP test; their test scores constitute two independent, mul
tinomial distributions with parameters and P^j> i = 1, ...» 7 (the 210 TABLE 36
NUMBER OF CALCULUS STUEENTS BY COOP TEST INTERVAL AND RACIAL EXTRACTION (NOR), 1973-1984
COOP Non- Oriental Total Interval Oriental
56 - 60 72 42 114 51 - 55 113 55 168 46 - 50 152 71 223 41 - 45 104 59 163 36 - 40 57 40 97 31 - 35 18 17 35 30 & below 20 10 30
Total 536 294 830 Mean Grade 46.99 46.29 46.75 Stnd. Dev. 7 .59 8.28 7 .85
seven COOP intervals), j = 1 (NOR 1 is Oriental), 2 (NOR 2 is Non-
Oriental), respectively. Let represent the corresponding score fre quency for the i-th COOP interval and j-th NOR. To test the null hypoth
esis that there is no significant difference in COOP test scores due to
being oriental or non-oriental, for i = 1, . . . , 7, we let
V - Pil = Pi2 = Pi VS Hl: pil * Pi2 for SOme i -
At the a = .05 significance level, we reject Hq in favor of H^
if and only if the test statistic Q & defined at (1 ) is greater than or
equal to 12.59 (chi-square with df = 6 ) . Here n^ = 536 and n^ = 294.
Since the test statistic Q^fCOOP & NOR) = 6.06 > 1 2 .5 9 , there is insuf
ficient evidence to reject the null hypothesis (a = 0.4164); and we con
clude that there is no significant difference in COOP test scores due to
differences in racial extraction at the (approximate) .05 level of sig
nificance. With respect to COOP, NOR has no statistical significance
and can be eliminated as a categorical variable. 211 TABLE 37
NUMBER OF PUNAHOU CALCULUS STUDENTS BY AP GRADE, COURSE, AND RACIAL EXTRACTION (NOR), 1969-1984
Number of Orientals (OR) and Non-Orientals (NOR), by Course AP Grade Interval Calculus AB Calculus BC Composite OR NOR E OR NOR E OR NOR E
5 64 33 97 158 96 254 222 129 351 4 102 50 152 103 64 167 205 114 319 3 85 56 141 65 31 96 150 87 237 1 & 2 28 12 40 20 15 35 48 27 75
Total 279 151 430 346 206 552 625 357 982 Mean Grade 3.71 3.66 3.69 4.15 4.16 4.15 3.95 3.95 3.95 Stnd. Dev. 0.97 0 .98 0.97 0.9 4 0.97 0.95 0.98 1.00 0.99
With respect to the AP examination criterion, APEX, three chi- square tests on the equality of two independent multinomial distributions were conducted: the first on the composite of all orientals and non orientals who took an AP examination at Punahou during 1969-84} the
second and third on orientals and non-orientals who took the Calculus AB or Calculus BC examination. Because of the paucity of scores at the 1- and 2-levels, these scores have been combined to form a single cell.
These are the two scores for which credit and advanced standing are not
awarded, so the consolidation makes practical sense. The composite dis tributions mix APEXAB and APEXBC grades, which is a questionable consol
idation because the examinations are not the same. However, it was done
at the a = .05 significance level only for comparative purposes with the
results of the tests on APEXAB and APEXBC separately, each at the
CL - .025 level. The elimination of NOR as a class variable of conse quence depends on the two separate tests, not the composite. The three
chi-square tests follow the same procedure as that used with COOP. 212 Consider the total of all orientals and non-orientals who took an AP examination at Punahou between 1969 and 1984 (table 37): their dis tribution of examination scores form two independent, multinomial distri butions with parameters n^, P^j» P2j» p3j» anc^ p4j» 3 = 1 (oriental),
2 (non-oriental), respectively. Let X „ , i = 1, 2, 3, 4; j = 1, 2 repre sent the corresponding frequencies of scores at the 1-2 level combined, and the 3, 4, and 5 levels separately. Here n^ = 625 and n = 357. To test the null hypothesis that there is no significant difference in cal
culus AP examination scores due to these racial differences, we let
Ho : pil = pi2 = Pi * 1 = 1»2»3»4 VS Hl ! pil ^ p±2 for some i -
At the a = .05 significance level, we reject Hq in favor of if
and only if the test statistic defined by (1) is such that > 7.81
(chi-square with k - 1 for k = 4, i.e. 3 df).
Since Q 3 (AB & BC) = 0.094 7.81, there is insufficient evidence
to reject the null hypothesis (a = 0.993); and we conclude that there is
no significant difference in composite AP examination scores due to a
candidate being oriental or non-oriental at the (approximate) .05 level
of significance.
The same chi-square test was repeated separately for the candi
dates who took the Calculus AB examination or the Calculus BC examination.
To compare these subsets to the total number taking the AP examinations,
a significance level of a = .025 was chosen for each of the APEXAB and
APEXBC multinomial distribution tests. Taken together, the following two
tests can be compared to the previous one at the a = .05 level.
For the Calculus AB students and for the same parameters defined
above, except that n^ = 279 and n^ = 151, we test the same null hypoth
esis: Hq : p±1 = pi2 = p± , i = 1,2,3,4 VS HlS pi;L f pi2 , for some i. 213
At the a - .025 significance level, we reject Hq in favor of if and only if Q 3(AB) > 9.35 (chi-square with 3 degrees of freedom).
Since Q^(AB) = 2.149 5^ 9.35, the null hypothesis cannot be rejected. We conclude that there is no significant difference in APEXAB scores due to oriental racial differences at the (approximate) .025 level of signifi cance. Here a = 0.542, and the two appropriate measures of ordinal asso ciation, Kendall's Tau C and Somer's D (Asymmetric, APEXAB as dependent variable) have the values - 0.026 and - 0 . 028, respectively.
At a = 0.025, the same test was repeated on the two multinomial populations of orientals and non-orientals who took the Calculus BC exami nation. It was found that = 1-593 ^ 9.35, and again the null hypothesis could not be rejected at the (approximate) 0.025 level. The a = 0.661 was slightly larger than that for APEXAB, and Kendall's Tau C and Somer's D (Asymmetric, APEXBC as dependent variable) were even closer to zero, being 0.013 and 0.014, respectively.
The evidence of the last two tests clearly indicates that the categorical variable NOR is not significant with respect to the AP exami nation grades and can be eliminated as far as APEX is concerned. It is permissible to add the two (approximate) 0.025 levels of significance, and claim that there is no significant difference in APEX scores due to oriental racial differences at the (approximate) 0.05 level.
With respect to grades achieved in the two calculus courses at
Punahou, CAIX3AB and CALCEC, chi-square tests were conducted to determine the equality of the two independent, multinomial distributions shown in table 38. Again the handful of students at the lower end of the dis tributions required that they all be concentrated in the single cell entitled 'C and below'. Also the composite distributions mixed grades 214 TABLE 38
NUMBER OF PUNAHOU STUDENTS BY FINAL CALCULUS GRADE, COURSE, AND RACIAL EXTRACTION (NOR), 1969-1984
Grade Number of Orientals (OR) & Non-Orientals (NOR) by Course and Calculus AB Calculus BC Composite Grade interval OR NOR £ OR NOR £ ORNOR
A 10.6-12.0 34 7 41 51 30 81 85 37 A- 9.6-10.5 73 32 105 95 49 144 168 81 B+ 8 .6- 9.5 72 44 116 76 45 121 148 89 B 7 .6— 8 .5 53 34 87 61 32 93 114 66 B- 6 .6- 7.5 29 16 45 45 35 80 74 51 C+ 5 .6- 6 .5 11 11 22 9 9 18 20 20 C 1.6- 5.5 7 7 14 9 6 15 16 13
Total 279 151 430 346 206 552 625 357 Mean Grade 9 .08 8.68 8.94 9.15 8.98 9.09 9.12 8.85 Stnd. Dev.3 1.45 1.45 1.46 1.49 1.56 1.52 1.52 1.60
For means and standard deviations, see appendix H, OBS 46,47, and 56-59. Composite means and standard deviations were computed directly. from two independent courses; the sums would be of interest to principals or department chairmen to check grading practices, but the distribution of the sum is irrelevant to this discussion.
Consider separately all orientals and non-orientals who received a final grade in either Calculus AB (CALCAB) or Calculus BC (CALCBC) between 1969 and 1984 (table 38): their distributions of grouped, final grade averages form two independent, multinomial distributions with parameters n^. and p „ for i = 1, . . . , 7 (grade intervals) and j = 1
(oriental), 2 (non-oriental), respectively. Let represent the cor responding frequencies of scores at the i-th grade level and j-th racial extraction class. Initially, to test the null hypothesis in Calculus AB that there is no significant difference in final grades between orientals and non-orientals (n^ = 279 and n^ = 151), for i = 1, . . . , 7 we let: 215
V Pil = Pi2 = Pi VS Hl ! Pil ¥ Pi2 ’ f°r S°me i>
At the a = 0.025 significance level, we reject Hq in favor of if and only if Q &( AB) > 14.45 (chi-square with 6 degrees of freedom).
But Q &(AB) = 11.36 5^ 14.45, so that rejection of Hq is not possible at the a = .025 level. Therefore, there is no significant difference in
CALCAB grades between orientals and non-orientals at this level. We
find that a = 0.078 with the measures of ordinal association as follows:
Kendall's Tau C = -0.147 and Somer's D (Asymmetric) = - 0,161.
We replicate this procedure at the CC = 0.025 level on the distri
butions of CALCBC grades and find that O.(BC) = 3.71 ^ 14.45. Here too, D ~- the null hypothesis cannot be rejected, and we conclude that there is no
significant difference in CALCBC grades between non-orientals and orien tals at the (approximate) .025 level. The smallest significance level
at which we could reject Hq is a = 0.717; and the measures of associ
ation in this case are Kendall’s Tau C = - 0.054 and Somers's D (Asym
metric, grade interval as dependent variable) = - 0.057. The results of
these two tests taken together imply that, with respect to final grades
in either Punahou calculus course, there is no significant difference in
achievement between orientals and non-orientals at the 0.05 level.
The chi-square, multinomial tests on grouped COOP, APEX, and CALC
scores or grades have all reached the same conclusion: there is no sig
nificant difference in any of the three criteria between non-orientals
and orientals. Therefore, NOR can be eliminated as a class variable.
Consideration of the Variables SEX and API
Originally, it had been intended to treat the elimination or
retention of all class variables, including sex and Advanced Placement 216 interest (API), using the same procedure just demonstrated with NOR.
However, it was found that the seven-by-four matrices for grouped COOP or
CALC scores versus API often did not have the minimum cell frequency required, particularly when partitioned by course. For example, grouped
COOP scores and API for the Calculus AB course had 25 percent of its cells below minimum frequency; Calculus BC was 39 percent below. Because of its significance to this report, the chi-square multinomial test to answer the question whether or not there is a significant difference in
AP examination scores due to Advanced Placement interest is presented before application of analysis of covariance to the problem.
Consider by Advanced Placement interest (API) all Punahou candi dates who took the Calculus AB examination between 1973 and 1984. Their distribution of exam scores constitute four independent, multinomial dis tributions with parameters n^,, P^j» ^or i anc^ j = • • • » 4, where i represents an AP examination level (grades 1 and 2 combined) and j desig nates a level of Advanced Placement interest. Let represent the cor responding frequencies of scores displayed in table 39. Here n^ = 160, n2 = 88, n3 = 152, and = 30. To test the null hypothesis that there is no significant difference in Calculus AB examination scores due to
Advanced Placement interest, we formulate the following null hypothesis:
Ho : Pil = Pi2 = pi3 = Pi4 = Pi VS Hl: Pij * Pik » for SOmS i = 1, . . . , 4, j = 1, . . . , 4, k = 1, . . . , 4, and j ^ k. At the a = 0.025 level of significance, we reject Hq in favor of H^ if and only if the test statistic given in (1) is such that Qn (AB) > 19.02 (chi- y — square with 9 degrees of freedom). Since Qg (AB) = 25.78 > 19.02, we reject the null hypothesis and claim that there is a significant differ ence in APEXAB scores due to a difference in Advanced Placement interest. 217 TABLE 39
CANDIDATE FREQUENCY BY AP EXAMINATION SCORE AND INTEREST, BY COURSE, 1969-1984
Number of Punahou Students by Course in Each API Category3
AP Grade Calculus AB Calculus BC Composite
1 2 3 4 E 1 2 3 4 E 1 2 3 4
5 21 22 45 9 97 27 26 46 155 254 48 48 91 164 4 53 31 59 9 152 34 29 46 58 167 87 60 105 67 3 65 32 36 8 141 23 17 21 35 96 88 49 57 43 2 & 1 21 3 12 4 40 11 7 4 13 35 32 10 16 17
Total 160 88 152 30 430 95 79 117 261 552 255 167 269 291 Mean Grade 3.4 3.8 3.9 3.7 3.7 3.8 3.9 4.1 4.4 4.2 3.6 3.9 4.0 4.3
ain addition to an AP Calculus course, candidates’s Advanced Placement interest levels are defined as follows: 1— only AP Calculus; 2— no science, but AP English, History, Languages, or Arts; 3— AP Biol ogy, Chemistry, or Physics B (no calculus); 4— Physics C (with calculus).
The same chi-square test conducted on the four multinomial popu lations by AP interest for candidates taking the Calculus BC course resulted in Qg (BC) = 44.37 > 19.02. We reject the null hypothesis and infer that there is a significant difference in APEXBC scores due to
Advanced Placement interest at the .025 level. Here a = 0.0000 (compared to a = 0.0022 for APEXAB) and the values for the measures of association are Kendall's Tau C = 0.197 and Somers's D (Asymmetric) = 0.216. These compare favorably with the corresponding values of 0.157 and 0.171 for
APEXAB. The results of both tests taken together imply that there is a significant difference in AP examination scores due to Advanced Placement interest at the (approximate) 0.05 level of significance.
Because the effects (or lack of effects) of the categorical vari ables SEX and API are of primary concern, an analysis of covariance 218 design was employed. Added to reduce regression error variance were five, independent predictor variables (covariates) . Predictor selection of the following was based on the highest correlates with all three cri teria as determined in tables 31 and 32: Advanced Math grade (a d v m )j
Mathematics Achievement Test, Level n (ACHL2); average of four, pre vious mathematics courses (AVMA); Junior grade-point-average (JGPA); and
SAT Mathematics score (SATM).
Three computer runs were required: the first on the entire data set for 1973-84; the second, only on the data for students in Calculus
AB; and the third for Calculus BC student data. The first run employed the categorical, independent variables (called factors) CRSE, API, NOR, and SEX with the covariates already mentioned in a regression for the dependent variable COOP. The second and third computer runs did not include the factor CRSE because the regression populations were restric ted to the students taking one of the two courses. The SPSS subprogram,
ANOVA, allows the user to run multiple analyses of covariance on more than one dependent variable at a time. The output prints the decompo sition of explained variance by main effects, covariates, interactions, and residual for each dependent variable separately. Table 40 shows the complete decomposition of explained variance and residual for the cri terion variable COOP on the entire regression population. COOP is the one criterion measuring scores on the common Cooperative Test taken by both Calculus AB and Calculus BC students. In the second and third com puter runs for the two different courses, all three criteria were used.
The ungrouped, raw scores were the measures used for the dependent vari ables, and the outputs allow comparison of main effects and covariates between dependent variables by course. 219 TABLE 40
ANCOVA COMPUTER OUTPUT ON COOP TEST BY FOUR FACTORS WITH FIVE COVARIATES, 1973-1984
COOP BY CRSE, API, NOR, AND SEX WITH ADVM, ACHL2, AVMA, JGPA, AND SATM
SUM OF MEAN SIGNIF SOURCE OF VARIATION SQUARES DF SQUARE F OF F MAIN EFFECTS 14,026.887 6 2,337.814 68.480 0.000 CRSE 11,581.203 1 11,581.203 339.242 0.000 API 2,346.254 3 782.084 22.909 0.000 NOR 96.438 1 96.438 2.825 0.093 SEX 2.992 1 2.992 0.088 0.767 COVARIATES 10,611.430 5 2,122.286 62.167 0.000 ADVM 6,328.977 1 6,328.977 185.392 0.000 ACHL2 3,061.098 1 3,061.098 89.667 0.000 AVMA 721.102 1 721.102 21.123 0.000 JGPA 424.992 1 424.992 12.449 0.000 SATM 75.262 1 75.262 2.205 0.138 2-WAY INTERACTIONS 384.953 12 32.079 0.940 0.506 CRSE API 265.269 3 88.423 2.590 0.052 CRSE NOR 38.049 1 38.049 1.115 0.291 CRSE SEX 55.287 1 55.287 1.619 0.204 API NOR 60.686 3 20.229 0.593 0.620 API SEX 7 .095 3 2.365 0.069 0.976 NOR SEX 0.614 1 0.614 0.018 0.893 3-WAY INTERACTIONS 249.371 10 24.937 0.730 0.696 CRSE API NOR 90.107 3 30.036 0.880 0.451 CRSE API SEX 96.314 3 32.105 0.940 0.421 CRSE NOR SEX 55.789 1 55.7 89 1.634 0.201 API NOR SEX 63.594 3 21.198 0.621 0.602 4-WAY INTERACTIONS 92.270 3 30.7 57 0.901 0.440 CRSE API NOR 92.268 3 30.756 0.901 0.440 SEX EXPLAINED 25,364.910 36 704.581 20.639 0.000 RESIDUAL 27,071.770 793 34.138 TOTAL 52,436.680 829 63.253
There are several important results visible in the output of the covariance analysis in table 40 for the dependent variable COOP. At the a = .05 level of significance, none of the 2-, 3-, or 4-way interactions is significant, although the 2-way interaction of CRSE and API is border line. CRSE and API are both significant at the .05 level; SEX and NOR 220 are not significant, and the smallest significance levels at which null hypotheses that there were no significant differences in COOP scores due to sex or racial extraction would be rejected are a = 0.767 and a = .093 respectively. CRSE and API are both significant at a = 0.000. As far as the criterion COOP is concerned, the categorical variables CRSE and
API should be retained and the variables SEX and NOR eliminated.
For the second and third computer runs on the separate courses, the CRSE categorical variable was withdrawn and the other class variables retained. Tables 41 and 42 portray the results of the two analyses of covariance with the restricted regression populations by course. All three criteria are presented with the sources of variation side-by-side for comparison purposes. Again, consider a = 0.05 level of significance.
In no case, for either course, are the 2- or 3-way interactions signifi cant. Almost all covariates are significant at the .05 level except
SATM. API, as a categorical variable, is significant at a = 0.000 in every case. SEX is not significant in any case, with a's ranging from
0.193 to 0.799 in the separate courses and criteria. NOR is not sig nificant for any of the three criteria in Calculus BC, but shows to be significant for grades in Calculus AB. Recall that in the corresponding test on NOR for grouped CALCAB and CAICBC data, we found 0C = 0.078 and a = 0.717. Sufficient evidence exists at the .05 level of significance that Advanced Placement interest (API) is a significant class variable with respect to all three criteria and should be retained. On the other hand, SEX is not a significant variable with respect to the three cri teria and it, together with the class variable NOR, should be eliminated.
The worth of API as a critical variable should now be apparent.
Course cannot be used in the prediction for success; API becomes the key. TABLE 41
CALCULUS AB ANALYSIS OF COVARIANCE FOR CRITERIA APEX, COOP, AND CALC, 1973-1984
ANALYSIS OF COVARIANCE; APEX, COOP, AND CALC BY API, NOR, AND SEX WITH ADVM, ACHL2, AVMA, JGPA, AND SATM
DEPENDENT VARIABLE; APEX DEPENDENT VARIABLE; COOP . DEPENDENT VARIABLE; CALC
SUM OF SIGNIF SUM OF SIGNIF SUM OF SIGNIF SOURCE OF VARIATION SQUARES DF F OF F SQUARES DF F OF F SQUARES DF F OF F
MAIN EFFECTS 20.26 5 5.41 0.000 836.67 5 3.78 0 .002 49.36 5 6.15 0.000 API 19.59 3 8.72 0.000 697.59 3 5.25 0.001 30.53 3 6.34 0.000 NOR 0.18 1 0.23 0.629 136.20 1 3.08 0.080 16.10 1 10.03 0.002 SEX 0.49 1 0.66 0.418 2.88 1 0.06 0.799 2.74 1 1.70 0.193 COVARIATES 71.76 5 19.16 0.000 5,311.05 5 24.00 0.000 202.75 5 25.26 0.000 ADVM 44.73 1 59.71 0.000 3,225.51 1 72.86 0.000 129.66 1 80.77 0.000 ACHL2 19 .88 1 26.53 0.000 1,527.45 1 34.50 0.000 28.03 1 17 .46 0.000 AVMA 3.61 1 4.81 0.029 302.95 1 6.84 0.009 28.04 1 17 .46 0.000 JGPA 3.30 1 4.41 0.036 193.04 1 4.36 0.037 16.74 1 10.43 0.001 SATM 0.24 1 0.33 0.568 62.10 1 1.40 0.237 0.28 1 0.17 0.677 2-WAY INTERACTIONS 4.55 7 0.87 0.532 202.20 7 0.65 0.712 6.84 7 0.61 0.749 API NOR 3.93 3 1.75 0.157 128.95 3 0.97 0.406 4.54 3 0.94 0.420 API SEX 0.30 3 0.14 0.939 71.96 3 0.54 0.654 0.63 3 0.13 0.941 NOR SEX 0.07 1 0.10 0.756 15.21 1 0.34 0.558 1.58 1 0.98 0.321 3-WAY INTERACTIONS 3.26 3 1.45 0.227 163.91 3 1.23 0.297 1.06 3 0.22 0.882 API NOR SEX 3.26 3 1.45 0.227 163 .91 3 1.23 0.297 1.06 3 0.22 0.882 EXPLAINED 99.83 20 6.66 0.000 6,513.83 20 7.36 0.000 260.01 20 8.10 0.000 RESIDUAL 306,39 409 18,105.41 409 656.57 409 TOTAL 406.22 429 24,619.24 429 916.58 429 to to TABLE 42
CALCULUS BC ANALYSIS OF COVARIANCE FOR CRITERIA APEX, COOP, AND CALC, 1973-1984
ANALYSIS OF COVARIANCE: APEX, COOP, AND CALC BY API, NOR, AND SEX WITH ADVM, ACHL2, AVMA, JGPA, AND SATM
DEPENLENT VARIABLE: APEX EEPENEENT VARIABLE: COOP DEPENLENT VARIABLE: CALC
SUM OF SIGNIF SUM OF SIGNIF SUM OF SIGNIF SOURCE OF VARIATION SQUARES DF F OF F SQUARES DF F OF F SQUARES DF F OF F
MAIN EFFECTS 22.70 5 8.10 0.000 2,224.13 5 19.36 0.000 94.62 5 18.94 0.000 API 22.46 3 13.35 0.000 2,206.54 3 32.01 0.000 91.49 3 30.53 0.000 NOR 0.08 1 0.14 0.709 8.32 1 0.36 0.548 2.07 1 2.07 0.151 SEX 0.16 1 0.28 0.598 9.26 1 0.40 0.526 1.05 1 1.06 0.305 COVARIATES 90.19 5 32.17 0.000 5,214.48 5 45.39 0.000 397.73 5 79.63 0.000 ADVM 59.48 1 106.08 0.000 3,028.82 1 131.81 0.000 286.35 1 286.65 0.000 ACHL2 19.54 1 34.86 0.000 1,623.98 1 70.67 0.000 58.50 1 58.56 0.000 AVMA 10.27 1 18.31 0.000 385.36 1 16.77 0.000 38.93 1 38.97 0.000 JGPA 0.87 1 1.56 0.213 171.06 1 7.44 0.007 13.02 1 13.03 0.000 SATM 0.02 1 0.04 0.837 5.28 1 0.23 0.632 0.92 1 0.92 0.338 2-WAY INTERACTIONS 2.79 7 0.71 0.664 83.00 7 0.52 0.823 2.85 7 0.41 0.897 API NOR 0.00 3 0.00 1.000 32.16 3 0.47 0.706 1.44 3 0.48 0.695 API SEX 2.27 3 1.35 0.258 39.32 3 0.57 0.635 1.47 3 0.49 0.689 NOR SEX 0.60 1 1.06 0.303 29.95 1 1.30 0.254 0.24 1 0.24 0.626 3-WAY INTERSECTIONS 0.80 3 0.48 0.700 6.18 3 0.09 0.966 5.67 3 1.89 0.131 API NOR SEX 0.80 3 0.48 0.700 6.18 3 0.09 0.966 5.67 3 1.89 0.131 EXPLAINED 116.48 20 10.39 0.000 7,527.79 20 16.38 0.000 500.86 20 25.07 0.000 RESIDUAL 212.51 379 8,708.77 379 378.61 379 TOTAL 328.99 399 16,236.56 399 879.47 399
fO to to 223 Deriving the Prediction Equations
In order to derive the prediction equations, backward planning from the goals of the discriminant analysis was necessary. The major goal of the discriminant analysis is to classify an end-of-year junior and possible candidate for Advanced Placement calculus into one of three categories: as a student recommended for the Calculus AB course; as one who, because of his greater potential, should be recommended for Calculus
BC; or as a student who should not take calculus in high school. For the last category of student there should be a "remedial", pre-calculus course to improve his knowledge and skills so that he can take calculus as a college freshman. Because the course to be taken is a criterion for the discriminant analysis, neither the class variable CRSE nor the
APEX criterion itself, partitioned by course, can be used in deriving the prediction equations. As a significant categorical variable, API should be used. An accurate estimate of its value is available because the prospective candidate is in the process of enrolling in courses for his senior year. But how can API be used? A carrier variable is needed.
The answer lies in the criterion COOP which, although not avail able at the end of the junior year, can be used as a dependent variable to be predicted. Tables 31 and 32 show that this variable is highly correlated with grades on the AP examinations. While COOP itself cannot be used as a predictor variable, its predicted value can be.
Advanced Placement interest has been shown to be a significant class variable with respect to both APEX and COOP. Using multiple regression analysis, four regression equations, one for each level of
Advanced Placement interest (API), were derived with COOP as the depen dent variable. Each of the four, prediction equations represents that 224 linear combination of independent variables which best correlates with
COOP for a given API level. The general model of the regression equa tions has the following forms
Y . = g. . + 13, .X, + .X„ + . . . + P. .X. + . . . + P .X , (9) j Oj I3 1 2j 2 rxj 1 nj n* where x^ represent the n = 12 predictor variables, Y ^ are the dependent
COOP scores for the j = 1, . . . , 4 level of API, and P „ (regression coefficients) are the constants to be estimated. In order to differen tiate between the actual Cooperative Test scores (COOP = Y ^) and the
A predicted scores, Y^, the predicted COOP scores were given the label A SCOAP, i.e. Y_. = SCOAP(j), for j = 1, . . . , 4. SCOAP is an acronym for SCI-COOP-AP, which succinctly describes the new variable as a pre dicted COOP score, but one which is API dependent and therefore includes a strong SCI component, and as a predictor variable in its own right when used to predict success in the discriminant analysis.
The SPSS "New Regression" procedure, through a series of sub commands, allows the user to specify not only what independent variables are to be used, but what subset of cases of the study population should 2 be used for the regression population. Two different sets of indepen dent variables were selected: a maximal set of twelve— six national test scores (PSATM, PSATV, SATM, SATV, DSATS, and ACHL2) and six local mea sures (ALGl, GEOM, INTM, ADVM, JGPA, and AVMA); and an optimal set of seven (PSATM, PSATV, SATM, SATV, DSATS, ACHL2, and ADVM). The ACHLl variable was not considered because it was available for only 200 of the
982 total cases; HNIQ was considered, although limited to 762 cases, and eliminated when it was found to be the last variable to enter in a
2 C. Hadlai Hull and Norman H. Nie, "New Regression," SPSS Update 7-9 (New York: McGraw-Hill Book Company, 1981), pp. 94-121. 225 step-wise, multiple regression and increased the multiple-R by less than 0.01. The optimal set of seven independent variables was chosen for its practical application in other schools. It includes the six remaining, national test variables but only one local variable, the grade in the previous mathematics course, ADVM. Restricting the number of local variables was a practical measure; the Iolani data set, on which the external check was made, did not include JGPA or the grades in the mathematics courses prior to ADVM.
Specifying independent variables for use in multiple regression does not guarantee their inclusion in the prediction equation:
To avoid numerical difficulties, all variables are tested for tolerance prior to entry into an equation. A variable’s tolerance is the proportion of variance remaining after the effects of the independent variables already in the equation have been partitioned out. That is, it is one minus the squared multiple correlation of that independent variable with the independent variables already in the equation. Minimum tolerance is defined as the minimum of the recomputed tolerances of the variables in the equation when a variable is entered at the next step. A variable must pass both tolerance and minimum tolerance tests in order to enter a regression equation.^
Thus, for each of the four levels of API, maximum and optimum prediction equations for the dependent variable COOP were derived. The
maximal set of four equations was then used in the discriminant analysis,
while the optimum set of four was reserved for later use in the external
analysis check on Iolani data. The two sets of regression equations are
given in table 43 by API level, order of entry of the independent vari
ables (and multiple-R at each stage), corresponding regression coeffi
cients, beta weights, t-test values, and significance levels of t. The
corresponding maximal and optimal SCOAP are fairly close in terms of
"^Ibid., p. 106. 226 TABLE 43
SCOAP EQUATIONS: MAXIMAL AND OPTIMAL PREDICTION EQUATIONS FOR COOP BY API LEVEL
API Variable Order Regression Coefficients Level of Entry and Beta T Sig. and N Multiple-R B0 B1 B2 B3 B4 Wt p Value of T
Maximal 1 ACHL2 0.498 0.556 .356 5.567 .000 (215) ADVM 0.567 2.161 .305 4.771 .000 SCOAPl (Constant) -16.573 -2.599 .010 2 AVMA 0.545 2.178 .294 2 .994 .003 (132) DSATS 0.604 0.322 .269 3.573 .000 ADVM 0.623 1.220 .212 2.214 .029 SCOAP2 (Constant) - 7.034 -1.153 .251 3 ADVM 0.441 1.456 .216 3.216 .002 (269) ACHL2 0.519 0.360 .273 4.812 .000 JGPA 0.541 5.231 .194 2.943 .004 SCOAP3 (Constant) -12.016 -1.990 .048 4 AVMA 0.645 3.775 .461 5.340 .000 (214) ACHL2 0.729 0.384 .384 7 .001 .000 GEOM 0.739 -1.360 r- . 17 6 -2.513 .013 JGPA 0.747 3.769 .159 2.469 .014 SCOAP4 (Constant) -36.286 -5.793 .000 Optimal 1 ACHL2 0.498 0.556 .356 5.567 .000 (215) ADVM 0.567 2.161 .305 4.771 .000 SCOAPl' (Constant) -16.573 -2.599 .010 2 ADVM 0.502 2.283 .396 5.253 .000 (132) DSATS 0.588 0.387 .324 4.295 .000 SCOAP2' (Constant) - 0.437 -0.075 .941 3 ADVM 0.441 2.192 .326 5.723 .000 (269) ACHL2 0.519 0.391 .296 5.206 .000 SCOAP3' (Constant) - 2.952 -0.560 .576 4 ACHL2 0.632 0.664 .390 6.612 .0 00 (214) ADVM 0.711 2.600 .384 6.633 .000 SATV 0.718 0.095 .105 2.106 .036 SCOAP4' (Constant) -31.207 -4.912 .000
2 percentage of variance explained (multiple R ), with the maximal equation
R being slightly higher except for the identical SCOAPl and SC0AP1'.
The maximal equations are the ones needed for the discriminant analysis. 227 The Discriminant Analysis
In finding the maximal and optimal sets of prediction equations
for COOP* the first major goal of the study has been reached. The inves tigation has determined which of the selected variables are the most
reliable predictors of achievement as measured by the Cooperative Test
scores. To increase the predictive accuracy of the regression equations, partitioning of the study population by the candidates's interest in
Advanced Placement was found to be necessary. API is a significant,
categorical variable. The strongest of the selected variables proved to
be ADVM and ACHL2, the grade in Advanced Mathematics and the score on the
national Mathematics, Achievement Level II test. At least one of these
two variables appeared as an independent variable entrant at every API
level. The beta weight column in table 43 shows the standardized coeffi
cients and allows comparisons of the relative effect of each independent
variable on the dependent SCOAP. A summary of the maximal set of pre
diction equations for COOP is given as follows in descriptive terms:
SCOAPl = - 16.573 + 0.556(ACHL2) + 2.161{ADVM) (10) SCOAP2 = - 7.034 + 2.178(AVMA) + 0.322( DSATS) + 1.220(ADVM) SCOAP3 = - 12.016 + 1,456(ADVM) + 0.360(ACHL2) + 5.231(JGPA) SCOAP4 = - 36.286 + 3.775{AVMA) + 0 „384(ACHL2) - 1.360{GEOM) + 3.769(JGPA)
The major goal of this investigation was to determine which of
the selected variables were the most reliable predictors of success on,
and discrimination between, Calculus AB and Calculus B C . This determi
nation was made by discriminant analysis in answer to one of two ques
tions: "Which of two calculus courses should a given junior take?" or
"Should a given junior take Calculus AB, Calculus BC, or neither course?"
Recommendations for juniors must be made in that year for the courses to 228 be taken as seniors. What the student has achieved, how strong his work habits, how high his motivation, and what he is probably capable of achieving are inherently contained in the vital recommendation of the student's previous teacher. However, in many cases the teacher simply does not know which course a student should take. The likelihood of success in each course becomes an important factor, and it is at this point that discriminant analysis can play a role. Here it is used to distinguish statistically between either two or three subgroups of APEX, the number depending on which question is to be answered.
In order to answer the question "Which course should a junior take?" a two-group discriminant analysis was conducted. Initially, the same twelve variables used in the regression analysis and the derived variable SCOAP were considered as discriminating variables. This was soon reduced to a more manageable number. By discriminant analysis one or more linear combinations of these discriminating variables were formed in such a way as to maximize the separation of the two groups. Opera tionally, the results take the following formj
D. = d. .Z, + d_ ,Z„ + . . . + d. .Z. + . . . + d .Z , (11) g lg 1 2g 2 i] l n] n’ where j = 1, 2 (Calculus AB = 1, Calculus BC = 2), is the score on the discriminant function for the j-th group, the d.,'s are weighted coefficients, and the z.'s are the standardized values of the n dis- l criminating variables used in the analysis. The available options in the SPSS procedure include the classification function coefficients
(Fisher's linear discriminant functions), which have the following form:
°1 = + bj xi + + • • • - O i/i + • • • * K f y ll2) where the x.'s are the values of the independent, discriminating vari ables used in the ana h\si s? and the p....' s are the coefficients to be 229 used in classification. Because the number of discriminating variables was large and included both national and locally available variables with considerable overlap in ability to discriminate, the stepwise procedure was used to find the most useful set of predictors to achieve satisfac tory discrimination. The stepwise selection method allowed a relative ranking of the contribution made by each discriminating variable being entered. In every case, the first stepwise entrant was the SCOAP vari able. Selection of MINRESID, which minimizes the proportion of variation not being explained, appeared to be the most appropriate method among the five, stepwise selection criteria for reasons already discussed.
To answer the question, "Should a given junior be recommended for Calculus AB, Calculus BC, or neither course?" a three-group discrim inant analysis was required. A new class variable TRGP (abbreviation for tri-group) was introduced and designated the dependent variable. TRGP had the value 1 for those who failed to qualify in either course; it was assigned the value 2 for those qualifying in Calculus AB; and it had the
value 3 if APEXBC was greater than two. The same set of discriminating
variables was used in the final analysis for both two-group and three-
group discrimination. The input discriminators in both cases were SCOAP,
ADVM, ACHL2, PSATV, PSATM, SATV, SATM, DSATS, SCI, and TAP (the total
number of AP examinations taken). The classification variables accepted
by the discriminant function would determine the most reliable predictors
of success on, and discrimination between, Calculus AB and Calculus BC,
as measured by scores on the Advanced Placement examinations.
The number of discriminant functions to be derived in any analy
sis is limited to the smaller of either one less than the number of
groups (g - 1) or the number of discriminating variables. One function 230 was derived in the two-group case; and two, in the three-group. Wilks'
lainda and the relative percentage of the eigenvalue associated with the
function were the statistical measures used to judge the relative impor tance of succeeding functions. The total variance in the discriminating variables is represented by the sum of the eigenvalues.
The results of the two-group discriminant analysis accounted for an eigenvalue sum of 1.03. The canonical correlation was 0.7125 which
indicates that slightly over 50 percent of the variance in the discrimi
nant function was explained by the groups. Wilks' lambda, which is an
inverse measure of discriminating power, i.e. that proportion in the
original variables which was not removed, agreed with a value of .4924.
Lambda's equivalent as a chi-square statistic had the value 584.51, which with 6 degrees of freedom has level of significance ot = 0.0. The
order of entry with Wilks' lambda at each stage was: SCOAP (.527), SATM
(.504), TAP (.497), ADVM (.495), PSATM (.493) and ACHL2 (.492). Since
there was only one discriminant function, the visual plot of cases is
simply a histogram of the cases' distribution along the function. This
is depicted in separate plots for Calculus AB = 1 and Calculus BC = 2
and both groups together in figure 1. The classification function
coefficients O^j) appear in table 44 together with the corresponding
classification function coefficients for the three-group case.
The TRGP canonical discriminant functions had eigenvalues for
the first and second functions of 1.137 and 0.031, respectively. The
first function contained 97.3 percent of the variance with a canonical
correlation of 0.7294. Application of the function was chi-squared
valued at 651.54 with significance level a = 0.0. Application of the
second function with canonical correlation 0.1742 left a Wilks' lambda 2M FIGURE 1
CASE PLOT AND CLASSIFICATION OF TWO--CROUP DISCRIMINANT FUNCTION
iiinidcham for group i ah -- CANONICAL DISCRIMINANT FUNCTION I -- «o ♦
F R 60 ♦ E U U 1 F 40 ♦ 1 1 N 1 1 1 1 C I 1 1Mill Y II IIlllll 20 ♦ 1 1 1 1 I1 11 1 1 1 1 1 1 1 1 11 1 1 1 I 1 1 1 1 1 1 1 1 1 11 11 1 1 1 1 (11 1 1 1 1 1 1 11 11 1 1 M 1 1 OUT. -6 -4 -2 9 2 CLASSIFICATION 1 I 1 I 111 I 11111111 I 11 1 1111 111 111 1 II 1 1 1 I 111122222222222222 GROUP CENTROIDS 1 HISTOGRAM FOR CROUP 2 BC — CANONICAL DISCRIMINANT FUNCTION 49 ♦
F 2 R 30 ♦ 2222 E 2222 Q 2 2222 2 2 U 2222222 2 2 E 29 ♦ 22222222 2 2 N 2222222222 222 C 2 2222222222 222 Y 2 22222222222222 19 ♦ 22222222222222222 2222222222222222222 2222222222222222222222 22 222222222222222222222222 OUT...... ♦...... ♦...... ♦...... ♦...... ♦...... ♦...... ♦...... OUT —6 —4 -2 0 2 A 6 CLASSIFICATION I I I 11 11 I I I I I I 11 I I I I I 11 111 1111 I I I 1111 111 I I ' CROUP CENTROIDS 2 ALL-CROUPS STACKED H19T0CRAM — CANONICAL DISCRIMINANT FUNCTION I — 09 + ♦
F R 69 C Q 22 2 0 2 12 2 E 49 21 12 222 ♦ N 1121122222 2 C I 1111111222222 y 11211111112222222 2 2 29 21111111111222222222 2 2 ♦ 1111111111111122222222222 11 I I I 11 I I 1111112222222222222 111111111111111111122222222222 OUT...... ♦ ...... ♦...... ♦ ...... f...... ♦ ...... ♦ ...... ♦ ...... OUT -6 -4 -2 9 2 4 6 CLASSIFICATION II I I I I 11 I I I I I I I I I I I II I I 11 I 11 I I I I I I I I I I 1112222222222222222222222222222222222222222 CROUP CENTROIDS I 2 CLASSIFICATION RESULTS -
NO. OF PREDICTED CROUP MEMBERSHIP ACTUAL CROUP CASES I 2
CROUP I 430 377 S3 AB 07.7* 12.3* CROUP 2 400 70 330 DC 17.3* 02.S*
PERCENT OF “GROUPED- CASES CORRECTLY CLASSIFIED! 09.10*
CLASSIFICATION PROCESSING SUMMARY 030 CASES WERE PROCESSED. O CASES HAD AT t.KAST ONE HISS I NO DISCRIMINATING VARIABLE. <1110 CASKS VO. I IF. USED FOH 1*11 INTO) OUTl’UT. 232 TABLE 44 TWO- AND THREE-GROUP CLASSIFICATION FUNCTION COEFFICIENTS
INDEPENDENT TWO-GROUPs COURSE THREE-GROUP: TRGP VARIABLES CALC AB CALC BC NON-QUAL AB-QUAL BC-QUAL
SCOAP - 3.5176 - 3.1488 3.4009 - 3.2632 - 2.8511 ADVM 13.3431 13.5829 13.0034 13.1698 13.3733 ACHL2 3.7138 3.6683 3.7344 3.7849 3.7450 PSATM 0.7389 0.7748 0.4041 0.4515 0.4531 SATM 1.2867 1.3986 1.5935 1.5125 1.6274 TAP 0.9696 1.2442 0.5008 0.1956 0.4830 (CONSTANT) -181.0854 -208.0455 - 178.9352 -184.3491 -211.5326
of 0.9697, which is equivalent to chi-square of 25.403 having level of significance OC = 0.0001. The order of entry of the discriminating vari ables differed little from the two-group case: SCOAP, SATM, TAP, PSATV,
ACHL2, and ADVM. The territorial map depicting a scatterplot based on the two discriminant functions in which the group symbols 1, 2, and 3 for the non-qualified, Calculus AB qualified, and Calculus BC qualified,
respectively, occupy every printing position, appears in figure 2 .
Wilks' lambda is revealed.
The two discriminant analyses performed to obtain classification
function coefficients used the maximal set of regression equations for the variable SCOAP. The entire procedure was repeated using the optimal
set of regression equations with very similar results but slightly less
discriminating power. The two-group order of entry and Wilks' lambda were: SCOAP (.576), SATM (.543), SCI (.533), ADVM (.522), and PSATM (.518).
The three-group results were similar: SCOAP (.539), SCI (.523), SATM
(.498), ADVM (.489), PSATV (.486), DSATS (.484), and PSATM (.482). The
classification coefficients for both the two- and three-group discrimi nant functions are presented when used for the external validity check. FIGURE 2
CASE SCATTERPLOT AND CLASSIFICATION RESULTS BY THREE-GROUP DISCRIMINANT FUNCTIONS
CAJVORICAL DISCRIMINANT FUNCTION I -0 -6 -4 -2 0 2 4 6 0 0 1133 113 133 1133 1133 1133 6 1133 ♦ ♦ ♦ ♦ ♦ 113 C 133 A 1133 N 1133 0 1133 n 4 ♦ 1133 ♦ ♦ ♦ ♦ ♦ 1 113 c 133 A 1133 L 1133 112333 D 2 4 1222233 ♦ ♦ 4 ♦ 1 112 22333 S 122 222333 c 112 22233 n 122 22333 * i 112 22233 Pf e 4 122 ♦ 22333 ♦ ♦ 4 i 112 * 22233 N 122 * 22333 A 112 22233 H 122 22333 T 112 22233 -2 4122 ♦ 4 22333 ♦ 4 K 112 22233 U 122 22333 N 112 222333 C 122 22233 T 112 22333 1 - 4 122 4 4 + 22233 4 0 112 22333 N 122 22233 I 12 22333 2 122 22233 112 22333 -6 122 ♦ ♦ ♦ ♦ ♦ 22233 112 22333 122 22233 112 22333 122 222333 I 12 22233 “0 122 22333 -0 -6 -4 -2 0 2 4 6 0
CLASSIFICATION RESULTS -
NO. OF PREDICTED CROUP HENBF.RSH1P ACTUAL CROUP CASES 1 2 3
CROUP 1 61 3 47 9 UNQUALIF 0.2* 77.0* 14.0* GROUP 2 390 4 346 40 ADQUAL1F I. 0* 00. 7* 10.3* CROUP 3 379 0 63 316 BCQUAJ.IF 0. OK 16.6X 03. 4*
PERCENT OF •GROUPED* CASES CORRECTLY CLASSIFIED: 60.36*
CLASSIFICATION PROCESSING SUMMARY nao c a s e s w e r e p r o c e s s e d . 0 CASES HAD AT LEAST ONE MISS1NC DISCRIMINATING VARIABLE. 030 CASKS WERE USED FOR PRINTED OUTPUT. 234 Internal Validity Application
The validity of a procedure generally refers to its usefulness in predicting some outcome of importance. If success on the Advanced
Placement examination in mathematics is considered to be sufficiently important, one ought to demonstrate the utility of a proposed procedure by applying it and determining how well it does. This is predictive validity. In the present case, it is certainly appropriate to present evidence that the discriminant analysis is proficient in categorizing students into either Calculus AB or Calculus BC in the two-group case, or into the additional category of a probable 1 or 2 on the AP exami nation in the three-group case.
The presentation is almost anticlimactic since the SPSS sub program Discriminant has an option selection which allows the user to obtain computed discriminant scores and a comparison of the classifi cation results with the actual group membership of the cases used in the analysis. For the two- and three-group discriminant analyses which used the maximal set of SCOAP regression equations, the computer-generated classification results are displayed in figures 1 and 2, respectively.
In the two-group case, 85 percent of the 830 calculus candidates at
Punahou were correctly classified; in the three-group case, only 80 per cent were grouped correctly.
The two- and three-group discriminant analyses were repeated using the optimal set of SCOAP regression equations. The corresponding results for comparison purposes were 82.77 percent correct classifi cation in the two-group case and 76.87 percent in the three-group case.
It is not surprising that these figures using the optimal set of pre dictors are smaller since the multiple-R's in table 43 were smaller. 235 External Validity Application
The two- and three-group discriminant classification functions were applied to an external (to Punahou) data source. Iolani School is another of Hawaii's thirty-four Advanced Placement schools and, although considerably smaller than punahou, enjoys an excellent reputation for scholarship. Permission to use data collected on Iolani's 276 calculus candidates over the six-year period 1979-1984, was generously given.
The classification functions, which used the optimal SCOAP regression equations, were applied to classify all 276 candidates. Table 45 shows the optimal classification function coefficients for both groups.
TABLE 45
OPTIMAL CLASSIFICATION FUNCTION COEFFICIENTS
INDEPENDENT TWO-GROUP: COURSE THREE-GROUP: TRGP VARIABLES CALC AB CALC BC NON-QUAL AB-QUAL BC-QUAL
SCOAP 0.4997 0.6561 0.1202 0.4203 0.5898 ADVM 6.8694 7.6060 7.8436 7.6002 8.3542 PSATM 0.9121 0.9645 0.7906 0.7641 0.8037 SATM 1.5632 1.6894 1.2452 1.2870 1.3882 SCI - 3.1445 - 2.3403 - 2.699 4 - 3.5629 - 2.7774 DSATS 0.4869 0.3277 0.3534 PSATV 0.2148 0.2851 0.2913 (CONSTANT) -120.2333 -147.3330 -123.0659 -126.2270 -153.6379
A program was written which utilized the PRINT option available in the Statistical Analysis System (SAS). Both the two- and three-group classification scores were computed for the 276 candidates. The output of the program printed the highest classification category in each group.
The percentages of the 276 candidates correctly classified were 80.80 in the two-group classification and 75.72 in the three-group classification.
Punahou's corresponding figures were 82.77 and 76.87, respectively. CHAPTER VI
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
This research study addressed the problem of selecting students, who have completed their four years of college preparatory mathematics
by the end of their junior year, for participation in one of the Advanced
Placement Program's two courses in mathematics. Both courses, Calculus
AB and Calculus BC, represent college-level mathematics for which most
colleges grant credit or advanced placement on the basis of qualifying
scores of three or better on the three-hour, comprehensive examinations given each May. Defining a qualifying score of at least three on either
of the two Advanced Placement calculus examinations as 'success', this
investigation sought to answer the following specific question:
Which of several selected factors normally available from a given secondary school's record of a student at the end of his junior year are the most reliable predictors of achievement (as measured by scores on an end-of-course Cooperative Calculus test) and success on, and discrimination between, each of the two Advanced Placement examinations in mathematics?
The need for such a study is apparent if one knows that during the period that the College Board has offered two examinations, 1969-84,
34 percent of the 258,821 Calculus AB candidates and 29 percent of the
99,234 Calculus BC candidates have not achieved success, as defined, on these nationally given examinations. Supposedly, these highly motivated
students, who elected to take college-level work as seniors, were among
the (approximately) 90,000 gifted or 450,000 academically talented
students who graduate each year. Of the many possible reasons which 236 237 might be responsible for their failure to qualify, being in the wrong course because available information on the student wasn't used should not be one of them. Knowledge of reliable predictors of success on each of the Advanced Placement examinations in mathematics might serve the mathematics teacher or counselor well in advising a border line case to take one course or the other, or even neither course.
The study population consisted of the 982 calculus students at
Punahou School who took either the Calculus AB or Calculus BC course and examination during the last sixteen years, 1969-84. Data available in their records at the end of their junior year included nine nationally-available test scores, of which six were finally used, and six locally-available predictors. To the four categorical variables representing the year course and examination were taken, course taken, sex of candidate, and racial extraction were added three other discrete, quantitative variables of science interest, advanced placement interest, and total number of Advanced Placement examinations taken. Advanced
Placement interest, because it embodies a certain qualitative measure of student interest and industry turned out to play an important role in the investigation. Data were computer programmed and machine processed on the Amdahl 470V/6-II at the Instruction and Research Computer Center of The Ohio State University.
Summary
The design of the study called for execution in three phases: data reduction, discriminate analysis, and both internal and external validity application. Because of the large amount of data available, data reduction was a major effort even though it did not have the 238 significant impact of the last two phases. Data reduction had two goals: the elimination of those categorical variables which were not statistically significant, and the identification of a reduced set of predictor variables to be used in the study’s dominant statistical tech nique, discriminant analysis.
A general linear test approach was used to eliminate the class variable year as a first consideration. This would allow pooling of the data over the twelve-year period, 1973-1984, during which time Punahou offered both courses. Analysis of variance procedures were used to obtain linear regression models, one over the six even-numbered years, the other over the odd-numbered years. An F-test performed under the hypothesis that the even and odd year regression lines were the same, i.e. had the same slopes and intercepts, indicated that rejection was not possible. Regressing APEX scores on the highly-correlated COOP test score yielded identical lines over even and odd years. YR, as a categorical variable, was eliminated. An interesting side issue con cerning the 1983 and 1984 test administrations when calculators were allowed was presented. The hypothesis that the regression line over the non-calculator years was the same as the regression line over the calculator years was rejected at the a = .05 significance level'. It was demonstrated that the slopes were statistically equal (which implied support for the reliability of the College Board's five-point grading scale), but that the intercept of the regression line over the calcu lator years was significantly higher than the line over the years with out calculators. The result was not entirely unanticipated.
The variable CRSE was shown to be significant with respect to the criterion variable COOP by means of a non-parametric, multinomial 239 hypothesis test which utilized chi-square. Course was retained as a class variable; however, the racial extraction variable NOR was elimi nated with respect to both COOP and APEX (grouped) using the same three chi-square tests on the equality of two independent multinomial distri butions. The chi-square multinomial tests on grouped COOP, APEX, and
CALC scores or grades all reached the same conclusion; there is no significant difference in any of the three criteria due to differences in racial extraction (oriental and non-oriental) .
One last chi-square test was used to infer that there was a significant difference (a = 0.0000 in Calculus BC, ot = 0.0002 in Calcu lus AB) in AP examination grades due to the class variable API.
An analysis of covariance design which used CRSE, API, NOR, and
SEX as factors with the covariates ADVM, ACHL2, AVMA, JGPA, and SATM first on the criterion COOP showed that CRSE and API were both signifi cant at the .05 level and that both SEX and NOR were not significant.
Making separate computer runs without the CRSE variable factor on the subpopulations of Calculus AB and Calculus BC students had similar results. Course and API were retained as categorical variables, and both SEX and NOR were eliminated. The importance of API as a critical class variable was noted.
The second goal after reducing the number of categorical vari ables was to derive the prediction equations. Multiple regression analysis was used to derive four prediction equations on the twelve predictor variables for the criterion COOP by the four levels of API interest. Actually, two sets of four equations were derived: a maximal set, in which freer use of local variables was made; and an optimal set, in which the only locally-available variable used was the course grade 240 in the immediately preceding course, ADVM. For each level of API, the predicted COOP score was named SC0AP(j), where j = 1, . . . , 4 was the
API level. The two sets of SCOAP equations [maximum SC0AP(j) and opti mum SCOAP(j)'] were the equations desired for the discriminant analysis.
For each set of SCOAP equations, the discriminant analysis pro vided classification function coefficients for use in either two- or three-group analysis. The maximal set of SCOAP equations would be used on Punahou-based data, and the optimal set on Iolani-based data. The classification functions in the two-group case answered the question,
"Which calculus course should a given student take?" The classification function in the three-group case predicted group membership in Calculus
AB, Calculus BC, or neither course. Juniors in the last-named category would be advised that their likelihood of success on either calculus AP examination was too low for them to be considered, and that the recom mended course for them should be at the pre-calculus level where a stronger foundation for calculus in college might be found.
In the internal validity check of the two-group discriminant analysis, the derived functions classified 85.15 percent of the Punahou students correctly; in the three-group analysis, 80.36 percent of the grouped cases were correctly classified. In both cases, the input of the SCOAP variable used the maximum prediction equations for SCOAP.
Discriminant analysis using the optimal set of SCOAP' regression equa tions resulted in 82.77 percent correct classification in the two-group case, and 76.87 percent in the three-group case. When applied in the external validity check on the iolani data to classify their 276 candi dates, correct classification was made on 80.80 percent in the two- group case and 75.72 percent in the three-group case. 241 To answer the original question regarding the most reliable predictors for achievement (as measured by COOP) and success {as mea sured by APEX), both categorical and independent variable predictors must be considered. In the covariance analysis N = 830) for COOP, which eliminated SEX and NOR as variables, the factors had the following eta values: CRSE, 0.47; API, 0.40; NOR, 0.04; and SEX, 0.13. Since 2 (eta) indicates the proportion of variation in COOP exjplained by each factor, it can be determined that CRSE explained approximately 22 per cent; API, about 16 percent; and SEX and NOR, a negligible amount less than one percent. Adjustments for factors and covariates gave an over all multiple R = 0.689, implying that about 47 percent of the total variation in COOP was explained by the model. CRSE and API should be considered as reliable predictors for achievement; but, of these, only
API is known at the time recommendations for the senior year are due for juniors. API becomes a critical variable.
The independent variables allowed entry by multiple regression into the SCOAP prediction equations included the variables ACHL2, ADVM,
AVMA, DSATS, JGPA, and GEOM for the four maximal equations, but only the first two appeared in at least three equations. Beta weights also con firmed the preeminence of ACHL2 and ADVM as the most reliable indepen dent variables for predicting COOP in the optimal SCOAP equations:
SC0AP1' = - 16.573 + 0.556(ACHL2) + 2.161{ADVM) (13) SCOAP2* = - 0.437 + 2.283(ADVM) + 0.387(DSATS) SCOAP3' = - 2.952 + 2.192(ADVM) + 0.391(ACHL2) SCOAP4' = - 31.207 + 0 .664{ACHL2) + 2.600(ADVM) + 0.095(SATV) SCOAP*' = - 22.186 + 0.493(ACHL2) + 1.635(ADVM) + 1.778(AVMA)
Either ACHL2 or ADVM were first entry predictors in all optimal
SCOAP regression equations. The respective multiplex's for the five equations were 0.567, 0.588, 0.519, 0.718, and 0.659. SCOAP*' is a 242 dependent variable which is not API dependent, i.e. the regression popu lation was the entire subpopulation of 830 students without partitioning for Advanced Placement interest. The equation for SCOAP*' should not be used (and wasn't) unless, for some reason, the API category is unknown.
To determine the most reliable predictors for success on, and discrimination between, each of the two Advanced Placement mathematics examinations, the intial entrants (with respective Wilks' lambda) into the discriminant classification functions are considered. In both the two- and three-group analyses, using either maximum or optimum SCOAP equations, produced the SCOAP variable as first entrant. For example, in the two-group case, the order of entry (with respective Wilks' lambda) was: SCOAP (.527), SATM (.504), TAP (.497), ADVM (.495), PSATM (.494), and ACHL2 (.492). Recall that Wilks' lambda is an inverse measure of the discriminating power in the original variables which has not yet been removed by the discriminant function. After the SCOAP variable entered, the discriminating power was only improved by all five other variables 0.035. This quantity is so small in comparison to SCOAP that all variables with the exception of those contained in SCOAP can be ignored. ACHL2 and ADVM are the most reliable predictors.
Conclusions
The findings of this investigation support certain conclusions which may be of interest to other Advanced Placement mathematics teach ers, The College Board, or other investigators. These conclusions are statistical in nature because that was the nature of the study. They are not meant to overemphasize the role of statistics or prediction when dealing with human beings. But if additional information from 243 their use serves to assist juniors in making the proper decision on which calculus course to take, or whether to take calculus in high school at all, the study will have served its purpose.
There is ample evidence to support a conclusion that certain categorical variables are statistically significant to achievement in calculus and success on the Advanced Placement examinations in mathe matics, while others are not. Specifically, the course one takes and the level of interest in Advanced Placement, as defined in terms of the kinds of other college-level courses taken in high school, are statis tically significant at the a = .05 level, while other class variables such . sex and the fact that a person is oriental or not are not sig nificant . The large percentage of orientals in Hawaii encouraged the investigation of the latter question. At the a = .05 level, there was no significant difference in Advanced Placement calculus grades due to differences in either sex or racial extraction.
Because API was a significant class variable, the conclusion that interest in science is significant with respect to both success and achievement in mathematics is supported. The evidence lies in the fact that three of the four API levels were defined by the SCI variable; no other AP science course; an AP course in Biology, Chemistry, or Physics
B (calculus not required); or Physics C (calculus required).
Evidence supported the College Board's claim that their grades reported on a 1 to 5 scale to the candidates and their colleges remain stable over years. The consistency in grading from year to year was not altered by the conclusion that Punahou students did fractionally better during the two calculator years. The slopes of the regression lines over calculator and non-calculator years were the same. 244 Evidence was presented which supports a conclusion that the two most significant predictor variables from among the nine national test scores and six local variables which were considered, were the scores on the Admission Testing Program's Achievement Test in Mathematics,
Level II and the grade in the pre-calculus Advanced Mathematics course.
Each of these variables was significant at the a = .05 level with respect to all three criteria: AP examination grade in calculus, final grade in the calculus course, and Cooperative Test score in calculus.
It can also be concluded that discriminant analysis is a signif
icant statistical procedure for classifying students either into the two separate courses, Calculus AB and Calculus BC, or into three sepa
rate categories, a partitioning which adds the third category of
"neither course" to the two previously mentioned. However, for the dis
criminant analysis to be significant input must include not only the two predictor variables ACHL2 and ADVM, but Advanced Placement interest.
Finally, it can be concluded that correct classification both internally to a school which provides the data and externally to other
schools whose curriculum, faculty, and students are similar to the
first, is possible for at least 76 percent of the cases to about 85 per
cent of the cases, by using the discriminant analysis described.
Recommendations
The following recommendations are made based on the objectives
of this investigation and the conclusions just derived.
1. Selection procedures for admitting students into Calculus AB or Calculus BC should include consideration of other Advanced Placement
course interest, their Mathematics Achievement Level II test scores, and 245 their pre-calculus mathematics course grades. If necessary to make a determination for a difficult border line case, the optimal classifi cation function provided in table 45 should be used.
2. Colleges which do not give at least qualified credit or advanced standing to students with grades of 2 on either Advanced Place ment examination, should do so. The results of the College Board study
(table 17) which concluded that an AP grade of 2 was comparable to a college grade of C is supported in this study by table 52. College mathematics departments should reconsider and make provisions for the students with grades at this "possibly qualified" level.
3. The College Board and the Educational Testing Service should publish tables (similar to table 53) of candidate frequency by Achieve ment Level I and Achievement Level II interval and AP examination score for each calculus examination to assist teachers and counselors in making proper placement decisions, in addition, ETS should replicate the present study procedure as far as derivation of the SCOAP prediction equations and discriminant analysis using these equations, using the raw scores on the AP examination instead of the COOP scores.
4. Educational Testing Service should publish an updated version of the Cooperative Test in Calculus in two forms: one each for Calculus
AB and Calculus BC in two 40-minute parts. ETS should publish norms for these tests, which might even include previously used objective ques tions from actual AP examinations which are no longer being used.
5. Recommendations for further study: the procedures utilized in the present study design with respect to SCOAP regression equations and discriminant analysis should be repeated in other AP examination fields,
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. Advanced Placement Course Description, Mathematic s : Calculus AB, Calculus BC. New York: College Entrance Examination Board, 1980.
______. Advanced Placement Course Description: Mathematics, Calculus AB', Calculus BC, May 1983, May 1984. New York: College Entrance Examination Board, 1982.
______. Advanced Placement Course Description: Mathematics, May 1985, May 1986. New York: College Entrance Examination Board, 1984.
______. College Advanced Placement Policies, 1973. New York: College Entrance Examination Board, 1973.
______. College Policies on Advanced Placement at Thirty-Seven Insti- tu't'ions'. New York: College Entrance Examination Board, 1963.
______. College Placement and Credit by Examination, 1975: Advanced Placement Program, College-Level Examination program. New York: College Entrance Examination Board, 1975.
______. College Placement and Credit by Examination. New York: College Entrance Examination Board, 1978.
______. "News of the College Board." The College Board Review 78 Winter 1970-71.
______. 54th Report of the Pirector--1955. New York: College Entrance Examination Board, 1956.
______. 55th Report of the Director—1956. New York: College Entrance Examination Board, 1957.
______. Report of the President—1957; Fifty-Sixth Annual Report. New York: College Entrance Examination Board,' 1958.
______. "What Happened to Them in College." The College Board Review 28 (Winter 1956): 7.
Commission on Mathematics. College Entrance Examination Board, 55th Report of the Pirector--1956. New York: College Entrance Examination Board, 1957.
Educational Testing Service. Cooperative Mathematics Tests: Calculus, Form A. Princeton: Educational Testing Service, I§63.
______. The Mathematics Examinations of the College Board. New York: College Entrance Examination Board, 1980.
______. "Report to Advanced Placement Teachers." Princeton, N.J., Fall 1982. (Mimeographed.) 262
Fels, William C. "How Tests May Be Used to Obtain Better Articulation of the Total Educational System." Speech to the National Association of Secondary-School Principals, February 22, 1954; In Frank H. Bowles. The Refounding of the College Board, 1948-1963. New York: College Entrance Examination Board, 1967.
Ferguson, W. Eugene. "Report of the Group Session—Mathematics Cur riculum Development, Advanced Placement In stitu te , University of Hawaii, 16-20 August 1965." (Mimeographed.)
Fey, James T.; Albers, Donald J.; and Jewett, John. Undergraduate Education in the Mathematical Sciences 1975-731 Report of the Survey Committee,' vol. V. Washington, D.C.: Conference Board of the Mathematical Sciences, 1977.
Fishman, Joshua A. "The Use of Quantitative Techniques to Predict College Success." In Admissions Information. New York: College Entrance Examination Board, 195/.
Frisbie, David A. "Comparison of Course Performance of APP and Non- App Calculus Student." Research Memorandum No. 207, Univer sity of Illin o is at Urbana-Champaign, 1980. (Mimeographed.)
Haag, Carl H. "Using the PSAT/NMSQT to Help Identify Advanced Place ment Students." Princeton, n.d. (Mimeographed.)
Hanson, Harlan P. "Advanced Placement Program." Annual Report of the College Board, 1965-66. New York: College Entrance Examina- tion Baord, 1966.
Jewett, John and Phelps, C. Russell. Undergraduate Education in the Mathematical Sciences, 1970-71, Report of the Survey Committee, vol.' IV. Washington, D.C.: Conference Board of the Mathematical Sciences, 1972.
Malcolm, Donald J . , An Analysis of Scores Received by Candidates on Some of the Advanced Placement Examinations in Relation to the Length of time Their Schools Had Participated in the Program TPnnceton: Educational'Testing Service, 1961), In Hazlett, Paul H., "Advanced Placement Evaluation Report," ([New York: College Entrance Examination Board, 1961]). (Mimeographed.)
National Council of Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va.: National Council' of Teachers of Mathematics, 1980.
National Council of Teachers of Mathematics. P rio rities in School Mathematics, Executive Summary of The PRISfo Project. Reston, Va.: National Council of Teachers of Mathematics, 1981. 263
National Education Association and National Association of Secondary- School Principals. Administration: Procedures and School Practices for the Academically Talented Student in the Secondary School. Washington, D.C.: Nationa1 Educ ation Association, 1960.
National Education Association and National Council of Teachers of Mathematics. Mathematics for the Academically Talented Student in the Secondary- School, ed. Julius H. Hlavaty. Washington, D.C.: National Education Asssociation, 1959.
Punahou School. "Aims.11 Honolulu, 1967. (Mimeographed).
______. "Punahou Handbook." Honolulu, 1983. (Mimeographed).
Rockefeller Brothers Fund. The Pursuit of Excellence: Education and the Future of America. Panel Report V of the Special Studies Project, America at Mid-Century Series. Garden City: Doubleday & Co., 1958.
Southern Regional Project for the Education of the Gifted. The Gifted Student: A Manual for Program Development. Atlanta: Southern Regional Education Board, 1962.
Spindt, Herman A. "Improving the Prediction of Academic Achievement." In Selection and Educational Differentiation, pp. 15-29. Report of a Conference at Berkeley, University of California, May 25-27, 1959, Field Service Center and Center for the Study of Higher Education, University of California (Berkeley: University of California, 1959.
Wilson, Kenneth M. "The Contribution of Measures of Aptitude (SAT) and Achievement (CEEB Achievement Average), in Forecasting College Grades in Several Liveral Arts Colleges." Research Bulletin RB-74-36. Princeton: Educational Testing Service, 1974; Arlington, Va.: ERIC Document Reproduction Service, ED 163 U15, 1979.
______. "Predicting the Long-Term Performance in College of Minority and Nonminority Students: A Comparative Analysis in Two Collegiate Settings." College Board Research Bulletin, RB-78-6 (Princeton: Educational Testing Service, 1978; Bethesda, Md.: ERIC Document Reproduction Service, ED 210 325, 1978.
______. "The Validity of a Measure of 'Academic Motivation' for Forecasting Freshman Achievement at Seven Liberal Arts College." Research Bulletin RB-74-29. Princeton: Educational Testing Service, 1974: Arlington, Va.: ERIC Document Reproduction Service, ED 163 016, 1979. 2 64
Theses and Dissertations
Austin, Homer W. "The Effects of High School Calculus on Achievement during the First Semester in College Calculus for First Year Students." Ph.D. d issertatio n , University of Virginia, 1975. DM 36(1975): 2728A.
Bean, Richard D. W. "The Predictive Ability of Selected Non- Intel lective Variables on College Freshmen Grade Point Averages: Towards Improving the Traditional Cognitive Model." Ph.D. d issertatio n , Georgia State University, 1980. DM 41(1981): 3540A
Beougher, Elton E. "Relationships between Success of Advanced Place ment Mathematics Programs and Various Administrative, School and Community Factors." Ph.D. d issertation, University of Michigan, 1969. DM 30(1969): 195A.
Bingham, Ralph L. "An Investigation into the Relationship between Advanced Placement in Mathematics and Performance in First Semester Calculus at the University of Texas at Austin." Ph.D. d issertatio n , University of Texas at Austin, 1972. DAI 33(1973): 4865A.
Brubacher, Paul W. "A Study of the Effects of the College Entrance Examination Board's Advanced Placement Program upon Student Academic Experiences at the University of Michigan." Ph.D. dissertation, University of Michigan, 1967. DA 28(1967): 2472A.
Cauthen, Ina A. "Selected Demographic and Personality Variables Related to Mathematical Achievement in Men and Women." Ph.D. d issertatio n , Texas A&M University, 1979. DAI 40(1980): 4456A.
Clark, Donald M. "Selected Aspects of the Establishment and Operation of the Advanced Placement Placement Program in the Orchard Park Central School: An Historical and Survey Analysis in the Light of School Objectives, C riteria for the Program, and the Views of Selected Authorities." Ed.D. dissertation, Univer sity of Buffalo, 1961. DA 27(1961): 2245.
Crosswhite, F. Joe. "Procedures for Admission with Advanced Standing in Mathematics at The Ohio State University." Ph.D. disser tatio n , The Ohio State University, 1964.
Dykes, Isaac J. "Prediction of Success in College Algebra at Copiah- Lincoln Junior College." Ed.D. dissertation, The University of Mississippi, 1980. DM 41(1981): 4630A. 265
El well, Donald B. "A History of the Advanced Placement Program of the College Entrance Examination Board to 196b." Ed.D. disser tatio n , Columbia University, 1967.
Farley, Mary de C. " A Study of the Mathematical Interests, A ttitudes, and Achievement of Tenth and Eleventh Grade Students." Ph.D. dissertation, University of Michigan, 1968. DA 29 (1969): 3039A.
Farmer, Loyal. "The Predictive Validities, as Measured by Multiple Correlation, of Certain Mathematics Grades and a Test Battery Using Academic Achievement as C riteria." Ed.D. dissertation, North Texas State University, 1970. DAI 32(1971): 1850A.
Felder J r ., Idus D. "The Advanced Placement Program in the High Schools of Fulton County." Ed.D. dissertation, University of Georgia, 1965. DA 27(1966): 700.
Fisher, Joseph T. "the Value of Tests and Records in the Prediction of College Achievement." Ed.D. d issertatio n , University of Nebraska Teachers College, 1955. DA 15(1955): 2097.
Fox, Lynn H. "F acilitating the Development of Mathematical Talent in Young Women." Ph.D. d issertatio n , The Johns Hopkins University, 1974. DM 35(1975): 3553B.
Fry, Dale E. "A Comparison of the College Performance in Calculus- Level, Mathematics Courses between Regular-Progress Students and Advanced Placement Student." Ph.D. d issertation, Temple University, 1972.
Guthrie, Judith G. "A Study of Mathematics Education in the United States: Enrollment, Abilities, and Post High School Paths." Ph.D. d issertation, Claremont Graduate School, 1980. DAI 41(1981): 3464A.
Haven, Elizabeth W. "Selected Community, School, Teacher, and Personal Factors Associated with Girls Electing to Take Advanced Academic Mathematics Courses in High School." Ph.D. dissertation, University of Pennsylvania, 1971. DAI 32(1971): 1747A.
Helton, William B. "A Comparative Analysis of Selected Characteris tics of Intellectually Superior Male Students Who Persist and Those Who Do Not Persis in an Advanced Placement Program." Ed.D. d issertatio n , North Texas State University, 1964. DA 25(1964): 3394-95. 266
Hepp, Donald A. "The Effects of a Gifted Mathematics Program on the Attitudes and Achievement of Secondary School Students Identi fied as Academically Gifted." Ph.D. dissertation, University of Pittsburg, 1978. DAI 40(1979): 1326A.
Jacobs, Judith E. "A Comparison of the Relationships between the Level of Acceptance of Sex-Role Stereotyping and Achievement and Attitudes toward Mathematics of Seventh Graders and Eleventh Graders in a Suburban Metropolitan New York Community." Ph.D. d issertatio n , New York University, 1974. DAI 34(1974): 7585A.
Kaloger, James H. "Characteristics of the Gross Point High School Students in Advanced Placement Programs." Ph.D. dissertation, University of Michigan, 1970. DAI 31 (1971): 6440A.
Keller, Claudia M. "Sex D ifferentiated Attitudes toward Mathematics and Sex D ifferentiated Achievement in Mathematics on the Ninth Grade Level in Eight Schools in New Jersey." Ed.D. disser tation, Rutgers University, 1974. DAI 35(1974): 330A.
Kerr, Donald L. "Establishing an Advanced Placement Program in the Public High School: Guidelines for School Personnel." Ed.D. dissertation, Columbia University, 1964. DA 26(1965): 4788.
Knauss, Thomas L. "The Mathematics Placement Program for Five Fresh man Level Mathematics Courses at Northern Michigan Univer sity ." Ph.D. d issertatio n , University of Michigan, 1975. DAI 36(1976): 6257A.
Lefkowitz, Ruth S. "A Study of the Advanced Placement Program in Mathematics at a Large New York City Public High School." Ed.D. d issertation, Columbia University, 1966. DA 27 (1967): 4025B.
Linkhart, Bennie R. "Current Patterns in Selected Advanced Place ment Mathematics Program within the State of Arizona " Ph.D. dissertation, University of Arizona, 1968.
Maclay Jr., Charles W. "The Influence of Two Prerequisite Programs on Achievement in the High School Advanced Placement Calculus Course." Ed.D. dissertation, University of Virginia, 1968. DA 29(1969): 3917A.
McClure, Wesley C. "A M ultivariate Inventory of Attitudes toward Selected Components of Elementary School Mathematics." Ed.D. dissertation, University of Virginia, 1970. DAI 31(1971): 5941-42A. 267
McGregor, Warren M. “The Significance of the Present Advanced Place ment Program at Massapequa High School, Massapequa, New York— with Recommendations for its Future Development." Ed.D. dissertation, Columbia University, 1962. DA 23(1963): 4191-92.
McKillip, William D. "The Effects of Secondary School Analytic Geometry and Calculus on Students' First Semester Calculus Grades at the University of Virginia." Ph.D. d issertation, University of Virginia, 1965. DAI 26(1966): 5920-21.
Montgomery, Warren G. "An Analysis and Appraisal of the Sioux City, Iowa, Secondary School Accelerated Mathematics Program." Ed.D. d issertatio n , University of South Dakota, 1968. DA 29 (1969): 2489A.
Morris, Ruby P. "A Comparative Analysis of Selected Characteristics of Intellectually Superior Female Students Who Persisted and Those Who Did Not Persist in an Advanced Placement Program." Ed.D dissertation, North Texas State University, 1964. DA 25 (1964): 3402-3.
O'Neal, Larry D. "A Comparison of the Predictors of Success of University and Junior College Students in the Initial Calculus Course." Ph.D. dissertation, University of Mississippi, 1980. DAI 41(1980): 4620A.
Paul, Howard W. "The Relationship of Various High School Mathematics Programs to Achievement in the F irst Course in College Cal culus." Ph.D. dissertation, The Ohio State University, 1970.
Perl, Teri H. "Discriminating Factors and Sex Differences in Electing Mathematics." Ph.D. dissertation, Stanford University, 1979.
Pocock, Richard C. "Advanced Placement Calculus as a Factor in the Study of College Mathematics." Ed.D. d issertatio n , Columbia University, 1974.
Ralston, Nancy C. "A Study of the Advanced Placement Program in the Cincinnati Public Schools." Ph.D. d issertatio n , Indiana University, 1961. DA 22 (1961): 3074-75.
Roberts, Fannie M. "Relationships in Respect to Attitudes toward Mathematics, Degree of Authoritarianism, Vocational Interests, Sex Differences, and Scholastic Achievement of College Juniors." Ph.D. d issertatio n , New York University, 1970. DAI 31(1970): 2134A.
Robinson, William B. "The Effects of Two Semesters of Secondary School Calculus on Students' First and Second Quarter Calculus Grades at the University of Utah." Ed.D. dissertaion, Univer sity of Utah, 1968. DA 29(1968): 2990-91. 268
Sklar, Martha R. "A Study of Three D istrict Senior-Year High School Mathematics Programs for High Ability and Gifted Students Relating to the Further Study of Mathematics." Ph.D. disser tation, Northwestern University, 1980.
Sommers, Dean D. "A Study of Selected Factors Predictive of Success in Calculus at Hope College." Ph.D. d issertatio n , The Ohio State University, 1973.
Sprankel, Charlene M. "The Validity of Placement Tests and Other Predictor Variables in the Placement of Students in Beginning Mathematics Courses at Southern Illin o is University at Carbondale." Ph.D. dissertation, Southern Illinois University, 1976. DM 37(1976): 3575A.
Tillotson, Donald B. "The Relationship of an Introductory Study of Calculus in High School to Achievement in a University Calculus Course." Ph.D. d issertatio n , University of Kansas, 1962.
Turner, Jonathan. "Factors that Affect Attrition in an Accelerated Secondary Mathematics Program." Ph.D. d issertatio n , Univer sity of Iowa, 1980. DM 42(1981): 123A.
Turner, Veras D. "Prediction of Success as a Mathematics Major at the Minnesota State Colleges." Ph.D. d issertatio n , University of Oklahoma, 1968. DA 29(1969): 2099A.
Wick, Marshall E. "A Study of the Factors Associated with Achievement in First-Year College Mathematics." Ph.D. dissertation, University of Minnesota, 1963. DA 24(1963): 1891.
Wight, Theodore A. "An Analysis of the Advanced Placement Program in Mathematics in the State of Utah." (Ed.D. d issertatio n , University of Utah, 1969). DAI 30(1970): 4270B.
Wilcock, Jack A. "The Relationship between Teacher Characteristics and Student Success in the Advanced Placement Program." Ed.D. d issertatio n , Utah State University, 1972. DAI 33(1973): 5457A.
Williams, Raymond "A Study of Differences in Achievement in Pre- Calculus and Calculus Courses by Junior College Transfer and Non-Transfer Students at the University of Southern M ississippi." Ph.D. d issertatio n , University of Southern M ississippi, 1980. DM 41(1980): 1994A. 269
Letters
Cornog, William H. Former President of Central High School in Phila delphia and Executive Director of the School and College Study of Admission with Advanced Standing (The Kenyon Plan), to author, 5 September 1984. Personal Files.
Haag, Carl H. Director, College Board Placement Test Programs, Educa tional Testing Service to author, 28 August 1981. Personal file s .
______. To all principals, July 1982. Personal f ile s .
Hanson, Harlan P. Director, Advanced Placement Program, College Entrance Examination Board, to author, September 1982.
______. To author, 22 August 1984. Personal file s.
______. To author, 29 August 1984. Personal files.
______. To author, 5 September, 1984. Personal file s .
______. To author, 9 September 1984. Personal file s .
______. To author, 13 September 1984. Personal file s .
Reed, Mariette. Associate Program Director, College Board Programs, Educational Testing Service, to author, 17 August 1982. Personal files.
______. To author, 8 July 1983. Personal files.
. To author, 16 September 1983. Personal f ile s .
. To author, 10 September 1984. Personal file s . APPENDIXES 271
APPENDIX A
CALCULUS AB AND CALCULUS BC: TOPICAL COURSE DESCRIPTIONS, 1969-1984
Until 1975 the College Board's Advanced Placement course descrip tions for Calculus AB and Calculus BC were presented sequentially. The Board adopted a more revealing format that year which gave a complete list of topics for Calculus AB in Column I; and, in the adjacent Column II were listed the additional topics required for Calculus BC. Knowing that students in Calculus BC were responsible for all the topics in Column I and the additional topics in the adjacent Column II made it very apparent that Calculus BC was considerably more extensive than its sister course. Although there have been some topic changes in this sixteen year period, the following columnar listing quoting the May 1983, May 1984 published course description is representative of the period:
Column I . Topics covered Column II. Additional topics in Calculus AB covered in Calculus BC A. Elementary Functions (algebraic, trigonometric, exponen tial, and logarithmic) 1. Properties of functions 1*. Vector functions and para a. Definition, domain, and metrically defined functions range b. Sum, product, quotient, and composition c. Absolute value, e.g., |f(x)|, and f( |x|) d . Inverse e . Odd and even f . Periodicity g. Graphs; symmetry and asymp g'. Graphs in polar coordi totes nates h. Zeros of a function 2. Properties of particular functions a. Fundamental identities and addition formulas for trigo nometric functions. b. Amplitude and periodicity of A sin(bx + c) and A cos(bx + c) c. ax (a > 0, a ^ 1) and loga x (a > 0, a ^ 1, and x > 0) and their inverse relationship 3. Limits a. Statement and applications of a*. Epsilon-delta definition properties, e.g., limit of a constant, sum, product, and quotient b. The number e such that Limit (1 + i)n = e n oo n APPENDIX A— Continued 272
Column I . Topics covered Column II. Additional topics in Calculus AB covered in Calculus BC c . Limit sin x = l x -* 0 x d. Nonexistent limits, e.g., ' Limit — i- , Limit sin -i , x -► 0 x x -* 0 and Limit lxl are each, for x - 0 x different reasons, non existent . e . Continuity f . Statements and applications but not proofs of continuity theorems: if f is continuous on [a,b], then f has a maxi mum and a minimum on [a,b]; the intermediate value theorem B. Differential Calculus 1. The derivative a. Definitions of the derivative; e.g., f'(a) = limit f(x ) ~ f(a ) and x -► a x - a f'(x) = Limit f(x + h) - f(x) h - 0 h b. Derivatives of elementary Derivatives of vector functions functions and paramet c. Derivatives of sum, product, rically defined functions quotient d. Derivative of a composite function (chain rule) e . Derivative of an implicitly defined function f . Derivative of the inverse of a function (including Arcsin x and Arctan x) g . Logarithmic differentiation h. Derivatives of higher order i. Statement (without proof) of The Mean Value Theorem; appli cations and graphical illus trations j . Relation between differentia bility and^continuity k. Use of ^Hopital's rule (quo l'Hopital’s rule (exponen tient indeterminate forms) tial and other indetermi nate forms) APPENDIX A— Continued 273
Column I . Topics covered Column II. Additional topics in Calculus AB covered in Calculus BC 2. Applications of the derivative a. Slope of a curve; tangent Tangent lines to para and normal lines to a curve metrically defined curves (including linear approxi mations ) b. Curve sketchings increasing and decreasing functions; relative and absolute maxi mum and minimum points; con cavity; points of inflection c. Extreme value problems d. Velocity and acceleration of Velocity and acceleration a particle moving along a vectors for motion on a line plane curve e . Average and instantaneous rates of change f . Related rates of change Integral Calculus 1. Antiderivatives 2. Applications of antiderivatives a. Distance and velocity from acceleration with initial conditions b. Solutions of y' = ky and applications to growth and decay 3. Techniques of integration a . Basic integration formulas b. Integration by substitution Integration by trigono (use of identities, change metric substitution; of variable) trigonometric integrals c . Simple integration by parts, Repeated integration by e.g., /x cos x dx and J* In x dx parts, e.g., /excos x dx Integration by partial fractions 4. The definite integral a. Concept of the definite inte gral as an area b. Approximations to the definite b' Approximations: upper and integral using rectangles lower sums; trapezoidal rule c. Definition of the definite inte c 1 Recognition of limits of gral as the limit of a sum sums as definite integrals d. Properties of the definite integral e. Fundamental theorems: / Xf(t)dt and dx a S. f(x)dx = F(b) - F(a), where F ' ( x) - f( x) ) APPENDIX A— Continued 274 Column I . Topics covered Column I I . Additional topics in Calculus AB covered in Calculus BC f . Functions defined by integrals e.g., f(x) = 5. Applications of the integral a. Average (mean) value of a function on an interval b. Area between curves b'. Area bounded by polar curves c. Volume of a solid of c'. Volumes of solids with known revolution (disc, washer, cross sections and shell methods) about d. Length of a path the X- and Y-axes or e . Area of a surface of revolution lines parallel to the f . Improper integrals axes D. Sequences and Series 1. Sequences of real number and of functions; convergence 2. Series of real numbers a. Geometric series b. Tests for convergence: com parison (including limit com parison) , ratio, and integral tests c . Absolute convergence d. Alternating series and error approximat ion 3. Series of functions: power series a. Interval of convergence b. Taylor series and error analysis (including linear approximations) c. Manipulation of series, i.e., addition of series, substi tution, term-by-term differ entiation and integration E. Elementary Differential Equations 1. First order, variables separable 2. First order, linear 3. Second order, linear with constant coefficients (both homogeneous equations and nonhomogeneous equations that are most easily solvable by the method of undetermined coefficients) 4. Applications with initial conditions including simple harmonic motion^-
The College Board, Advanced Placement Course Description: Mathematics, Calculus AB, Calculus BC, May 1983, May 1984 (New York: College Entrance Examination Board, 1982), pp. 6-10. 275
APPENDIX B
ADVANCED PLACEMENT MATHEMATICS: COURSE TOPICS, 1954-1968
During this period the Advanced Placement course in mathematics was a single, full-year course in calculus and analytic geometry. The course content described by the School and College Study of Admission with Advanced Standing for the first two years remained largely unchanged under the College Board; topics not specifically designated by the Study are marked with asterisks. Unchanged also was the course objective: to give a substantial introduction to the differential and integral calculus with enough applications to bring out the subject's meaning and impor tance . The content of the course and the scope of the AP examination in mathematics are given by the topics in the following list:
1. Analytic geometry review and extension a. Rectangular and polar coordinates b. Equations and graphs c. Distance and slope d. Straight lines e. Conics: circle, parabola, ellipse, hyperbola 2. Differential calculus of algebraic functions a. The function concept *b. Absolute values and inequalities c. Definition and basic properties of limits *d. Fundamental ideas of continuity e. Slope of a curve, average and instantaneous rates of change f. Definition of the derivative g. Formal differentiation h. implicit functions and implicit differentiation i. Differentiation of composite functions and of parametrically defined functions j. Higher order derivatives k. The differential and its use in approximation 1. Rolle's theorem and the theorem of the mean 3. Integral calculus of algebraic functions a. The inverse of differentiation b. Integration of simple expressions, basic formulas c. Integration by substitution; integration by parts d. Simple differential equations with initial conditions e. intuitive development of the definite integral as the limit of a sum f. intuitive treatment of the fundamental theorem of the integral calculus g . Evaluation of simple definite integrals *h. Approximation of definite integrals by the trapezoidal rule 4. The calculus of elementary transcendental functions a. The exponential and logarithmic functions; the inverse rela tionship of these functions b. The derivatives of eu , In u, and logau •i c. Integrals of eu and — u 276 APPENDIX B— Continued
d. Trigonometric functions of real numbers e . Limit s:*-n x x -* 0 x f . The derivatives and integrals of sin u, cos u, and other trigo nometric functions g. The derivatives of arc sin u and arc tan u and the correspond ing integrals *h. Differential equations of growth and simple harmonic motion i . Parametric representations involving trigonometric functions for curves such as the ellipse, hyperbola, and cycloid *j. Simple trigonometric substitutions in integration 5. Applications of differential calculus a. Tangents and normals b. Curve tracing, maximum and minimum points, points of inflection, asymptotes c. Problems leading to maximum and minimum values, both relative and absolute d. Rates of change e . Velocity and acceleration of a particle along a straight line *f. Velocity and acceleration vectors of motion along a plane curve. 6. Geometric and physical applications of integration a. The area under a curve *b. The average (mean) value of a function c. Areas between curves *d. Arc length of a curve e . Volumes of revolution and other simple volumes *f. Surfaces of revolution g . Motion in a straight line and along a plane curve h. Work^-
College Entrance Examination Board, Advanced Placement Program: 1966-68 Course Descriptions (New Yorks College Entrance Examination Board, 1966), pp. 129-31; and College Admission with Advanced Standing: Announcement and Bulletin of Information ([Philadelphia]: The School and College Study of Admission with Advanced Standing, 1954), pp. 48-49. APPENDIX C
TABLE 46: ADVANCED PLACEMENT PARTICIPATION, 1954-1984
277 TABLE 46 278 ADVANCED PLACEMENT PARTICIPATION, 1954-1984
Number Number Number Number of Exams of Exams Number Year of of in All in Mathe of Schools Candidates Subjects matics Colleges
1953-54® 18 532 959 120 94 1954-55a 38 925 1,522 265 134 1955-56 104 1,229 2,199 386 130 1956-57 212 2,068 3,772 724 201 1957-58 355 3,715 6,800 1,177 279 1958-59 560 5,862 8,265 1,870 391 1959-60 890 10,531 14,158 2,908 567 1960-61 1,126 13,283 17,603 3,609 617 1961-62 1,358 16,255 21,451 4,190 683 1962-63 1,681 21,769 28,762 5,848 765 1963-64 2,086 28 , 874 37,829 7,710 888 1964-65 2,369 34,278 45,110 9,021 994 1965-66 2,518 38,178 50,104 9,630 1,076 1966-67 2,746 42,383 54,812 10,675 1,133 1967-68 2,863 46,917 60,674 11,623 1,193 . 1968-69 3,095 53,363 69,418 13,954 1,288 1969-70 3,186 55,442 71,495 14,379 1,368 1970-71 3,342 57,850 74,409 14,673 1,382 1971-72 3,397 58,828 75,199 15,186 1,483 1972-73 3,240 54,778 70,651 14,310 1,437 1973-74 3,357 60,863 79,036 16,038 1,507 1974-75 3,498 65,635 85,786 17,090 1,517 1975-76 3,937 75,651 98,898 19,065 1,580 1976-77 4,079 82,728 108,870 20,317 1,672 1977-78 4,323 93,313 122,561 22,510 1,735 1978-79 4,585 106,052 139,544 24,727 1,795 1979-80 4,950 119,918 160,214 27,879 1,868 1980-81 5,253 133,702 178,159 30,558 1,955 1981-82 5,525 141,626 188,933 31,918 1,976 1982-83 5,827 157,973 211,160 35,489 2,130 1983-84 6,273 177,406 239,666 39,962 2,153
SOURCES: [Advanced Placement Program of the College Board], ’•Annual Advanced Placement Program Participation," Princeton, 1984. (Mimeographed); "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1956-84. (Mimeographed single sheets, by year); [The College Board], Advanced Placement Program (New York: College Entrance Examination Board, 1956), p. 7; and Harlan P. Hanson, personal letters dated 22 August 1984, 29 August 1984, and 5 September 1984.
Conducted as the School and College Study of Admission with Advanced Standing. ^Numbers 1969-84 include both Calculus AB and Calculus BC exam;;. APPENDIX D
TABLE 47; GRACE DISTRIBUTIONS OP MATHEMATICS CANDIDATES, 1954-1984
279 TABLE 47
GRADE DISTRIBUTIONS OF MATHEMATICS CANDIDATES, 1954-1984
Number of AP Calculus Candidates (N) and Percentage Below Grade (%B), By Year AP Grade 1954a 1955a 1956 1957 1958 1959 1960 1961 N %B N %B N %B N %B N %B N %B N %B N %B
5 5 96 12 95 44 89 80 89 115 90 225 88 176 94 296 92 4 8 89 23 87 109 60 169 66 235 70 283 73 404 80 627 74 3 35 60 70 60 87 38 176 41 334 42 586 42 606 59 1,162 42 2 34 32 74 32 100 12 117 25 217 23 329 24 711 35 875 18 1 38 0 86 0 46 0 182 0 276 0 447 0 1,011 0 649 0
Total 120 265 386 724 1,177 1,870 2,908 3,609 Mean Grade 2.2 2.2 3.0 2.8 2.7 2.7 2.3 2.7 Stnd. Dev. 1.1 1.1 1.2 1.3 1.3 1.3 1.2 1.2
Number of AP Calculus Candidates (N) and Percentage Below Grade (%B), By Year AP Grade 1962 1963 1964 1965 1966 1967 1968 N %B N %B N %B N %B N %B N %B N %B
5 443 89 359 94 480 94 661 93 763 92 660 94 965 92 4 749 7 2 809 80 1,051 80 1,161 80 1,148 80 1,595 79 1,667 77 3 1,269 41 1,941 47 2,313 50 2,532 52 2,828 51 2,680 54 3,259 49 2 1,012 17 1,550 20 2,033 24 2,846 20 2,848 21 3,241 23 2,894 24 1 717 0 1,189 0 1,833 0 1,821 0 2,043 0 2,499 0 2,838 0
Total 4,190 5,848 7,710 9,021 9,630 10,675 11,623 Mean Grade 2.8 2 .6 2.5 2 .6 2.6 2. 5 2.6 Stnd. Dev. 1.2 1 .1 1. 2 1 .2 1 . 2 1.2 1.2 TABLE 47— Continued
AP Grade Number of Candidates (N) and Percentage Below Grade (%B), Calculus AB and Calculus BC, By Year
1969 1970 1971 1972 1973 1974 1975 1976 Calculus AB N %B N %B N %B N %B N %B N %B N %B N %B
5 931 91 935 91 1,241 88 1,114 90 1,289 87 1,045 91 1,431 88 1,648 87 4 1,738 74 1,848 73 1,926 70 1,838 73 1,734 69 1,974 73 2,593 66 2,828 66 3 2,726 48 2,565 48 2,345 47 3,355 41 2,814 41 3,727 40 3,783 34 4,241 33 2 2,974 19 2,924 20 3,281 16 2,533 17 2,798 13 2,273 20 2,558 12 2,604 13 1 1,911 0 2,001 0 1,675 0 1,771 0 1,236 0 2,194 0 1,439 0 1,755 0
Subtotal 10,280 10,273 10,468 10,611 9,871 11,213 11,804 13,076 Mean Grade 2 .7 2 .7 2 .8 2 .8 2 .9 2.8 3 .0 3.0 Stnd. Dev. 1 .2 1 .2 1 .3 1 .2 1 .2 1.2 1 .2 1.2
1969 1970 1971 1972 1973 1974 1975 1976 N %B N %B N %B N %B N %B N %B N %B N %B
5 554 85 603 85 900 79 765 83 1,232 72 814 83 1,306 75 1,345 78 4 770 64 822 65 899 58 946 63 910 52 1,062 61 1,255 52 1,327 55 3 943 38 965 42 1,109 31 1,085 39 1,001 29 1,209 36 1,329 26 1,539 30 2 989 11 1,191 13 747 13 816 21 903 9 69 4 22 888 10 1,085 12 1 418 0 525 0 550 0 963 0 393 0 1,046 0 508 0 693 0
Subtotal 3,674 4,106 4,205 4,575 4,439 4,825 5,286 5,989 Mean Grade 3 .0 •2.9 3 .2 2.9 3 .4 3.0 3 .4 3.3 Stnd. Dev. 1.2 1 .3 1 .3 1.4 1 .3 1.4 1.3 1.3
Total 13,954 14,379 14,673 15,186 14,310 16,038 17,090 19,065
CD TABLE 47— Continued
AP Grade Number of Candidates (N) and Percentage Below Grade (%B), Calculus AB and Calculus BC, By Year
1977 1978 1979 1980 1981 1982 1983 1984 Calculus AB N %B N %B N %B N %B N %B N %B N %B N %B
5 1,939 86 2,168 86 2,329 87 2,759 86 3,173 86 3,581 85 3,905 85 4,639 85 4 2,935 65 3,181 66 3,740 66 4,079 66 4,762 65 4,996 64 5,879 63 7,071 62 3 4,336 35 5,281 33 5,949 32 6,970 31 7,7 59 30 8,011 30 8,962 30 10,255 28 2 3,099 13 3,233 12 3,286 13 3,423 14 3,952 13 4,288 12 4,685 12 4,843 12 1 1,780 0 1,911 0 2,311 0 2,865 0 2,891 0 2,949 0 3,275 0 3,775 0
Subtotal 14,089 15,774 17,615 20,096 22,537 23,825 26,706 30,583 Mean Grade 3 .0 3.0 3.03 3 .02 3 .06 3 .08 3 .09 3 .13 Stnd. Dev. 1 .2 1 .2 1 .20 1 .22 1 .21 1 .22 1 .21 1 .21
1977 197 8 1979 1980 1981 1982 1983 1984 Calculus BC IN U D IN IN IN IN IN .D IN * 0
5 1,560 75 1,696 75 1,705 76 1,700 78 1,840 77 1,978 76 2,362 73 2,525 73 4 1,558 50 1,484 53 1,570 54 1,608 57 1,937 53 1,764 54 1,853 52 2,013 52 3 1,654 23 1,825 26 1,724 30 2,064 31 2,147 26 2,267 26 2,376 25 2,580 24 2 753 11 901 12 1,236 12 1,265 15 1,105 12 1,111 12 1,198 11 1,039 13 1 703 0 830 0 877 0 1,146 0 992 0 973 0 994 0 1,222 0
Subtotal 6,228 6,736 7,112 7,783 8,021 8,093 8,783 9,379 Mean Grade 3 .4 3.3 3 .28 3 .19 3 .32 3 .33 3 .39 3 .38 Stnd. Dev. 1 .3 1 . 3 1 .33 1 .34 1 .30 1 .31 1 .31 1 .33
Total 20,317 22,510 24,727 27,879 30,558 31,918 35,489 39,962
SOURCES: Paul H. Hazlett, Jr., "Advanced Placement Evaluation Report," ([New York: College Entrance Examination Board, 1961]), First Draft, Appendix B, Table 1, p. 7 and Table B-7. (Mimeographed)j and [APP of the College Board], "Distribution of Candidate Grades: Advanced Placement Examinations," Princeton, 1956-84. (Mimeographed single sheets, by year.) ro a Conducted as the School and College Study of Admission with Advanced Standing. 283 APPENDIX E
EXTRACTS FROM THE GROUP SESSION REPORT ON MATHEMATICS CURRICULUM DEVELOPMENT, ADVANCED PLACEMENT INSTITUTE— AUGUST, 1965-
ADVANCED PLACEMENT INSTITUTE Kuydendall Hall, University of Hawaii August 16-20, 1965 Report of the Group Session — Mathematics Curriculum Development
The first session on Monday, August 16 was spent in discussing the general topic of what is advanced placement mathematics and how do schools develop an advanced placement mathematics program. The following points and issues were raised and discussed. The advanced placement program is an activity of the College Entrance Examination Board and provides a way of strengthening American Education. It is national in scope, it encourages colleges and high schools to work together. It helps to eliminate duplication of studies in college which is a waste of time, energy, and money. The A. P. Pro gram stimulates students and teachers to higher achievement. The A.P. Program is of specific interest to at least these three groups: 1. The secondary students that are capable of doing college level work. 2. The secondary schools that are interested in seeing that each student works to his own capacity. 3. The colleges that welcome and reward the students' achievement in the A.P. Program. The Advanced Placement Program helps coordinate the schools and colleges effort to help students work to their capacity. In 1953/54, 18 schools were involved. In 1964/65, 2375 schools had A.P. Programs. 34,000 students took 45,000 A.P. exams. These stu dents are attending about 1000 colleges, and this was an increase of 17% over 1963/64. This indicates clearly that the A.P. Program is a going concern. The A.P. Program helps the schools look out for individual dif ferences and different rates of learning. The A.P. exam is open to any student wherever he may be. He may have achieved his knowledge and under standing through special courses, tutorial services, or through his own efforts. Before a high school offers an A.P. Course in Calculus and Ana lytic Geometry in its mathematics department certain conditions must be met by the school, the teachers, and the students. 1. The school must have a curriculum offering in mathematics that allows the student to complete the mathematical prerequisite for calculus by the end of the Junior Year. This means that the student must have completed the study of m o d e m up-to-date material in elementary, intermediate and "so-called" college algebra, plane and solid geometry, trigonometry and much APPENDIX E— Continued 284 analytic geometry by the end of the junior year. 2. There must be at least one teacher on the staff who can teach a bona fide college calculus course on the college level as outlined in the Advanced Placement Program of the College Entrance Examination Board. It is strongly recommended and indeed urged that before a teacher attempts to teach an A.P. course he should take a refresher course in the calculus or at least go through the complete course doing theory and solving problems in the book to be used as a text. Probably the best preparation is attendance at a simmer institute sponsored by NSF where they offer a calculus course for teachers of A.P. courses in high school. Adequate preparation of the teacher cannot be overemphasized. 3. The students must be adequately prepared, mathematically, for the course and willing to spend 8 to 10 hours a week on home work. He must be adequately motivated for taking the calculus. Why offer calculus in High School? Some high school students can do calculus and enjoy it. They may be going into the sci entific field and calculus is a necessary tool in many college physics courses that are taught in high school. In many col leges the Freshman physics course needs calculus for complete understanding before the calculus is learned by the student that waits and takes calculus and physics concurrently. Some of these top capable students are interested in other fields and the A.P. calculus course is an excellent terminal course in mathematics. We have sorely underestimated the mathematical powers of children in grades K-12. For students who are bright enough (and there are more of them than we think) these new mathe matics programs are getting rid of wasted time in the seventh and eighth grades as well as in other years. The new programs in mathematics are also raising the level of mathematical com petence of many students in grades 7 to 11, so that at the end of grade 11 many students will be prepared for calculus. Calculus in high school is successful. There are some failures, of course, that is to be expected. There are also failures in college calculus courses too. If there are many failures in the high school calculus course, these failures are probably due to ill-conceived calculus courses, taught by ill-prepared teachers and taken by ill-prepared students. Many times this situation has been brought about by pressure from the public combined with poor planning by the administration and the mathematics teacher. We are now beginning to find new ways of developing the mathe matical power of children beginning in the elementary schools. In a few short years many more students will have finished the regular four-year high school program at the end of their junior year. At the present, calculus and analytic geometry seems to be the best course for them. When calculus is introduced into the high school mathematics pro gram, the quality of the other courses picks up. The teachers are APPENDIX E— Continued 285 stimulated and so are the students in other courses. The calculus teacher does a better job in his other courses, too. With all the work being done to improve mathematics programs in America, we should not neglect the many students who will soon be ready for the calculus in the senior year as a result of these new programs. Teachers trained and capable of teaching the calculus will see that these capable students are not neglected. More teachers of mathematics today must get trained to teach calculus tomorrow. Many schools that have been successful in establishing excellent A.P. Programs have found the following boundary conditions must be ful filled by the Administration, and/or teachers, and/or students. 1. Homogeneous grouping is a must. Many schools start homogeneous honor sections in grade 7 and these lead to A.P. calculus in the twelfth grade. 2. Teachers that are enthusiastic about teaching an A.P. course should be relieved of part of their normal load in order to give them time to prepare and do extra reading. Many schools that have a normal 5 class load cut the load to 4 classes if the teacher teaches A.P. math. They also watch out to see that the A.P. teacher is not saddled with too many non-teaching duties. Classes must be small (10-25) and this automatically means more teachers if you expect to keep the class size reasonable in all other courses. 3. More and better guidance must be available to keep from getting students improperly placed. 4. A good A.P. Program does cost more because we need better library and reference works, more and better equipment for the laboratories, more teachers because classes must be smaller and the teacher load reduced one class. 5. Arrangements should be made so that at least one A.P. teacher from each school could attend an A.P. Conference in June. 6. The teacher must know the math in the A.P. course and know something about the next courses for students and the previous courses students have had. 7. The A.P. course must be a full year college course in calculus with analytical geometry; a shortened version is inadequate. 8. Even one A.P. course in a school tends to "beef up" the courses in all other areas. It forces all teachers to improve their subject matter knowledge and their teaching skills. The proper selection of students for honors and A.P. courses is a difficult task. The students selected must be highly motivated to do extra hard work and independent work. In the beginning only the top few students should be selected for the A.P. course. The students should be selected by the Chairman of the Department, and the teachers involved along with consultation with the Counsellors. The counsellors alone should not attempt to make the selection. Small classes like 6 or 8 to a class must be allowed. A class of 10 qualified people must not be filled up with 20 other non-capable people just to have a class of 30. 10 to 25 is the range for many A.P. math classes. Any student that takes an A.P. course has an obligation to take the A.P. exam. This is the one external, unbiased, exam that will assist the teacher in assessing his effectiveness in teaching the A.P. Program. APPENDIX E— Continued 286
Time for the A.P. teacher to plan and study must be made available. The number of A.P. courses that a student may take should depend upon the student. Some can handle only one, some two, some three and some four. The criteria for selection of students for Honors or A.P. Programs varies with the school. In general terms selection depends upon (1) pre vious teachers judgment, (2) previous grades, {3) faculty estimates of students willingness to work independently, (4) test scores and (5) rank in class. In general the Math Department Head, the Math Teachers and the Counsellors must work together to make the selection. A.P. is not easy and the results the first year or two may not be as satisfying as you would like so most schools are very conservative and let only the most able students in— that is, the ones with (1) High I.Q., 120 and up, (2) High grades, A's and few B's, (3) top teacher recommen dation, (4) High standardized test results. Parents in general are asked to sign statements agreeing to honors courses and paying for the A.P. exam. This is one place to develop some community support for the A.P. Program. The prestige factor sometimes plays too great a role in the pres sures brought to bear on a Department Head to put someone in an A.P. course. One cannot deny the prestige value of being in such a course, but at the same time it should be the right course for the student from a hard headed educational point of view. Probably the ideal way to have an A.P. Program is to have students (capable ones, well motivated etc.) take courses of the following types: 8th grade (Elem. Algebra) SMSG-like 9th grade (Geometry, Plane and Solid and some Analytic Geome try) SMSG-like 10th grade (intermediate Math) SMSG-like 11th grade (Advanced Math— Math Analysis)— something like Brixey and Andree— Fundamentals of College Mathematics 12th grade (A.P. Calculus with Analytic Geometry)— something like Thomas— Calculus and Analytical Geometry
Omitted at this point is a two-page list of possible textbooks that might be used at the various grade levels in an A.P. Program as well as regular and slow tracks.
In many schools today the student that is having difficulty with mathematics but is still college capable takes a "pre-algebra" course in grade 9 and Elementary Algebra in grade 10, Geometry in grade 11, and Intermediate Algebra in grade 12. "Pre Algebra" by Nichols, published by Holt, Rinehart and Winston. There are also some students that study in grades 9 and 10 the new Basic Math books that are on the market now. Then they take Elementary Algebra in grade 11 and Geometry in grade 12. Some schools have a General Mathematics Course available for any student in any year up through grade 12. APPENDIX E— Continued
Here is the Honors Program as we think it should be. MATHEMATICS HONORS PROGRAM
7 8 9 10 11 12
ARITHMETIC ELEMENTARY GEOMETRY INTERMEDIATE MATHEMATICAL AP CALCULUS & ALGEBRA MATHEMATICS ANALYSIS ANALYTIC GEOM or 2ND YEAR SMSG-like UICSM UICSM ALG. a TRIG. Pre-calculus Text like text Book 1, Book 2, course text Thomas, SMSG SMSG Text like like Brixey Calculus and Algebra Geometry Fehr, Carna & Andree, Analytic or a or a han a Beber- I ntroduction. Geometry, "modern" "modern" man, Algebra to College Book One, algebra Geometry with Trig., Mathematics 3rd Ed. text text 2nd Course
REGULAR TRACK MATHEMATICS PROGRAM
8 9 10 11 12 College
7th grade 8th grade 9th grade 10th grade 11th grade Take Calculus program program program program program first year as above as above as above as above as above in college
The following table shows how to take the present course offerings and within 3 years have an up-to-date A.P. program going in those schools that have capable teachers for the program. MATHEMATICS HONORS PROGRAM
Year 7 8 9 10 11 12
1965 ARITHMETIC ARITHMETIC ELEMENTARY GEOMETRY 2ND YEAR ALGEBRA a 1966 ALGEBRA ALGEBRA TRIGONOMETRY
Laidlaw Laidlaw W & K, I W a K Geom W & K Alg II Dolciani SMSG SMSG UICSM 1-4 UICSM 5,6 F, c, a B, UICSM 9-11 Scott Scott F. UICSM 4,5 W & A Trig, a Alg. Analytic Foresman UICSM 1-3 Weeks & A Geometry > N 1966 SMSG- ELEMENTARY ELEMENTARY GEOMETRY & ALGEBRA II PRE-CALCULUS 1967 like ALGEBRA ALGEBRA INT. MATH a TRIG. ANALYTICS V V V. N Si s 1967 SMSG- ELEMENTARY GEOMETRY GEOMETRY a PRE-CALCULUS ANALYTIC 1968 like ALGEBRA INT. MATH. Dolciani GEOMETRY A. a 0.
1968 SMSG- ELEMENTARY GEOMETRY INT. MATH. PRE-CALCULUS AP CALCULUS 1969 like ALGEBRA a ANAL GEOM APPENDIX E— Continued 288
Three years from this fall, a school could have an Advanced Placement Program operating very nicely in their school. The regular track students, that is the regular college preparatory students, would take the above program and do SMSG-like materials in grades 7 and 8, then Elementary Algebra in grade 9, Geometry in grade 10, Second Year Algebra and Trigonometry in grade 11, and Pre-Calculus in grade 12. The material in the above report has been discussed in the group sessions and has come from the group. Mr. Francis Bowers, Jr., Consultant, Head of the Mathematics Department, Punahou School, has given valuable guidance and information out of his rich and successful experience with the Advanced Placement Program at Punahou. Any omissions and errors in reporting is entirely my responsibility.
W. Eugene Ferguson, Consultant Head of Mathematics Department Newton High School ^ Newtonville, Massachusetts
[AUTHOR'S NOTE: Approximately 10 hours were devoted to curriculum development during the week 16-20 August 1965. Each registrant in the Advanced Placement institute was expected to continue in the same group during the curriculum development meetings. Discussion leaders of the group sessions are given in the following list;
GROUP SESSIONS— CURRICULUM DEVELOPMENT MEETING
Counselors, Registrars - Robert Cameron and Administrators - Harlan Hanson Director, Advanced Placement Program English - Fred Stocking - Winifred Post Dana Hall, Wellesley, Massachusetts Foreign Languages - Sister Jean Louise Parish Foreign Language Coordinator, Catholic Diocese, St. Anthony's Home History - Timothy Tomlinson Mathematics - Mr. Francis A. I. Bowers, Jr. Chairman, Mathematics Dept, Punahou - Dr. Eugene Ferguson Head, Mathematics Dept, Newton High School, Massachusetts Board of Examiners of the CEEB Science - Dr. L . Reed Brandtley Professor of Chemistry, University of Hawaii]
W. Eugene Ferguson, "Report of the Group Session— Mathematics Curriculum Development, Advanced Placement Institute, University of Hawaii, 16-20 August 1965," Honolulu, 1965. (Mimeographed.) APPENDIX F
TABLE 48; AP EXAMINATION PARTICIPATION BY SUBJECT FOR SELECTED YEARS
289 TABLE 48 290
AP EXAMINATION PARTICIPATION BY SUBJECT FOR SELECTED YEARS
Number of Candidates Schools Colleges Subject 1969 1973 1976 1980 1984 1984 1984
Art 533 961 1,697 2,536 History 270 508 785 1,264 199 246 Studio-General 263 453 695 987 256 296 -Drawing 217 285 116 133 Computer Science 4,262 915 475 English 23.691 22.422 29.503 49,125 71,263 Lit./Comp. 23.691 22.422 29.503 45,082 60,969 4,477 1,634 Lang./Comp. 4,043 10,294 1,512 937 History American 11,837 12,720 18,718 32,098 49,939 3,254 1,319 European 3,562 3,511 5,283 8,092 12,609 1,221 822 Language French-Lit. 2,711 1,381 1,515 1,541 1,571 298 218 -Language 1,195 1,859 3,379 5,272 1,216 491 German-Lit. 494 495 547 455 -Language 804 1,631 601 351 Latin-C 138 -Prose 165 130 -Vergil 67 5 400 546 772 1,285 250 189 -Lyric 230 175 199 350 398 95 110 Spanish-Lit. 1,611 1,481 1,749 1,714 1,958 301 337 -Language 3,421 7,032 1,393 672 Mathematics 13,954 14,310 19,065 27,879 39,962 Calculus AB 10,280 9,871 13,076 20,096 30,583 3,525 1,103 Calculus BC 3,674 4,439 5,989 7,783 9,379 1,136 482 Music Listening/Lit. 383 194 71 95 Theory 524 427 194 177 Science Biology 4,449 5,764 9,482 13,549 19,387 2,207 1,001 Chemistry 4,205 3,447 5,341 8,209 11,539 1,645 675 Physics B 850 756 1,267 2,411 3,682 653 400 Physics C 846 1,674 2,396 3,811 4,698 547 274
Total Exams Taken 69,418 70,651 98,898 160,214 239,666 Total Candidates 53,363 54,778 75,651 119,918 177,406 Schools Sending 6,273 Colleges Receiving 2,153
SOURCES: [APP of the College Board], "Advanced Placement Exami nation Volume Changes," Princeton, 1982-84. (Mimeographed); and "Summary Program Report for May 1984 Advanced Placement Examinations," Princeton, 1 9 8 4 . (Mimeographed.) APPENDIX G
TABLE 49: ADVANCED PLACEMENT PARTICIPATION BY STATE IN 1984
291 TABLE 49
ADVANCED PLACEMENT PARTICIPATION BY STATE IN 19843
AP SCHOOLS AP CANDIDATES AP EXAMINATIONS % » NON PER 100 K STATE N * IN STATE FEMALE WHITE POPULATION
Alabama 69 12 1,166 52 9 1,378 35 Alaska 19 15 586 56 9 799 200(3) Arizona 65 35 1,320 52 14 1,556 57 Arkansas 22 5 274 55 8 327 14 California 690 45(8) 27,612(2) 51 29 35,703 151 Colorado 111 33 3,306 54 11 4,698 163(10) Connecticut 155 56(2) 4,212 50 9 5,764 185(7) Delaware 26 42 786 55 7 1,066 179(8) Dlst, of Columbia 24 39 750 43 22 1,226 192(5) Florida 233 39 9 , 292(3) 52 20 12,613 129 Georgia 135 24 3,383 51 10 4,242 78 Hawaii 34 44(9) 1,089 56 75 1,645 170(9) Idaho 19 14 263 48 4 330 35 Illinois 251 25 8,609(4) 45 16 12,108 106 Indiana 72 15 961 49 11 1,176 21 Iowa 37 7 537 57 2 600 21 Kansas 40 9 754 48 8 913 39 Kentucky 67 17 973 51 4 1,156 32 Louisiana 57 10 827 56 10 1,045 25 Maine 50 30 397 53 3 486 43 Maryland 187 58(1) 5,849(10) 49 14 8,267 196(4) Massachusetts 235 47(7) 7,687(6) 45 9 10,970 191(6) Michigan 231 27 5,085 48 9 7,062 76 Minnesota 61 11 1,286 49 5 1,589 39 Mississippi 49 11 700 55 7 821 33 Missouri 53 8 1,221 42 9 1,791 36 Montana 18 9 179 63 2 189 24 Nebraska 31 8 495 46 6 623 40 Nevada 12 19 262 51 10 306 38 New Hampshire 48 42 954 41 8 1,316 143 New jersey 283 52(5) 7,384(7) 46 11 10,454 142 New Mexico 20 13 443 46 16 662 51 New York 691 50(6) 28,770(1) 49 14 40,564 231(2) North Carolina 198 37 3,978 53 7 5,081 86 North Dakota 4 1 60 55 0 65 10 Ohio 304 31 6,873(8) 50 8 8,924 83 Oklahoma 33 6 674 48 14 965 32 Oregon 105 34 2,420 51 6 3,013 114 Pennsylvania 367 36 8,413(5) 48 6 11,328 95 Rhode Island 37 54(3) 945 39 7 1,240 131 South Carolina 124 32 2,796 54 9 3,593 115 South Dakota 4 2 65 43 5 71 10 Tennessee 117 25 2,882 51 14 3,899 85 Texas 223 16 5,320 52 17 7,156 50 Utah 62 54(3) 3,367 48 4 4,695 321(1) Vermont 40 44(9) 468 54 4 612 120 Virginia 198 42 6,230(9) 54 10 8,178 153 Washington 134 33 2,296 53 13 2,817 66 West Virginia 15 7 194 44 3 235 12 Wisconsin 62 11 991 46 4 1,202 26 Wyoming 6 7 75 65 3 86 18
Total (US) 6,128 27 175,459 49.6 14.4 236,605 104 Non-US/Unk. 145 1,947 3,061 Grand Total 6,273 177,406 239,666
SOURCEi [APP of the college Board), "State summary Report of 1964 Advanced Placement Examinations," Princeton, 1984. (Mimeographed.)
aFigures within parentheses give top ten ranking within category. APPENDIX H
TABLE 50: MEANS, STANDARD DEVIATIONS, AND FREQUENCIES OF STUDY POPULATION SUBGROUPS
293 TABLE 50
MEANS, ST AN CARO DEVIATIONS, AND FREQUENCIES OF STUCK POPULATION SUBGROUPS
HOR API -TYPE— -TRKL. RAF EX HCALCNCOOP HAVKA HJCPA R P8A TV HP8ATH RBAT7 t e 982 3.949*8 9.*2138 46.7488 9.7273 3.8*83* 84.3147 68.*448 87.3921 2 i i 238 3.88294 8.64118 43.2279 9.4639 3.38238 8*.1137 63.1961 33.9*78 3 2 i 16? 3.87423 9.*2398 48.8227 9.621* 3.8891* 86.6887 64.8449 89.89B2 4 3 i 269 4.MM* 8.97212 46.1784 9.6448 3.49698 83.8242 64.1264 86.1747 5 4 i 291 4.2921* 9.39863 81.7477 1*.*938 3.88986 87.3814 67.8**7 6*.921* 6 2 623 3.98*4* 9.1176* 46.9944 9.7782 3.81763 83.6844 64.9882 86.7886 7 1 2 337 3.94678 8.85294 46.2928 9.6434 3.4837* 88.47*6 68.2*17 88.4838 8 • i 3 ■ 39 3.8*667 8.69667 43.*379 9.4247 3.39*47 48.94** 62.8333 81.84** 9 • a 3 91 3.89*11 9.11838 46.27*3 9.6769 3.88689 88.3816 63.23*8 88.86*4 19 • 3 3 194 4.*2*62 9.1*3*9 46.8876 9.71*3 3.8*881 83.4836 64.6443 86.2887 I I • 4 3 19* 4.28789 9.46879 81.8*88 I*.1683 3.6*837 86.7684 68.*1*8 6*.8842 12 1 1 3 163 3.619*8 8.8619* 43.83*1 9.82** 3.37*76 81.79*8 64. 1429 88.1*48 13 1 a 3 76 3.88826 8.91447 44.869* 9.8839 8.86211 88.2237 66.1184 61.3*** 1 4 I 3 3 73 3.94667 8.63333 43.12** 9.4827 3.46627 83.7*67 62.7867 88.88*9 13 1 4 3 161 4.38644 9.27228 81.641* 9.9884 3.888*8 88.8347 67.4*39 61.8343 ias m ATH RDSATB NACHL2 KALCI NCEOH H1KTK 1UDVM RTAP SHAPEX 8DCALC SDCOOP S&AVHA l 68.46*3 64.7841 73.88*3 9.8817 9.7342 9.7*91 9.6*39 2.88397 *.98689 1.49432 7.84741 9.91877 2 66.618? 62. *7*6 71.9723 9.6949 9.4243 9.4824 9.3196 1. ***** J.*I763 1.89891 8.**696 9.94163 3 67.2278 64.79*4 72.8263 9.8393 9.8389 9.67*7 9.4671 2.488*3 *.89966 1.41*14 6.827*6 9.99812 4 67.3643 63.6617 72.8883 9.8617 9.6387 9.883* 9.4891 2.82186 *.91*14 1.37412 6.99132 9.87394 8 71.7973 68.1989 76.9723 I*.*766 1*.2*79 1*.*722 1*.*683 3.7*79* *.94718 1.46094 6.97186 9.63283 6 68.2992 64.4782 73.B636 9.9661 9.7712 9.736* 9.6344 2.8936* *.9788* 1.47178 7.89383 9.89692 7 68.7423 68.3249 73.8233 9.7339 9.6698 9.6619 9.83*4 2.48489 1 .**279 1.82*49 8.28437 9.96191 8 68.6333 6*.94** 71.7267 9.72** 9.3233 9.4433 9.3*33 l.***M *.98816 1.88633 7.8688B 9.94719 9 66.6813 63.9869 72.2837 9.9723 9.8989 9.7*33 9.8278 2.43986 *.86217 1.4*846 8.73483 9.86962 19 67.8828 63.84*2 73.1398 9.9336 9.7*88 9.6*31 9.4974 2.8*412 *.91*27 1.32618 6.7998* 9.83163 11 71.91*8 68. 1632 77.9316 1*. 1884 18.2711 I*.1184 1*.*868 3.71*83 *.96899 1.47111 6.73877 9.78925 12 68.*19* 63.6887 72.3238 9.639* 9.8686 9.8381 9.3429 l.*«*** 1.*8988 1.61379 8.26148 9.93576 13 67.8816 63.7893 73.4737 9.7237 9.46*3 9.6316 9.3947 2.83947 *.948*4 1.42686 7.38828 9.96365 14 66. B*** 63.2*** 72.1867 9.67*7 9.4373 9.43*7 9.369* 2.86667 *.91376 1.44871 7.4*7*1 9.98959 15 71.3842 68.2374 76.B614 9.8663 1*. *891 9.9881 1*.*248 3.7*297 *.91197 1.46377 7.4*827 9.89789 IBS 8DJCPA 8DP8ATV 8DPSATH SDSATV 8D8ATH 8DD6ATS S0ACHL2 SDALC1 SDiirnt 8DADVH 8DTAP I 9.399867 8.69516 6.37273 8.69389 6. 17383 8.98988 8.38841 1.28981 1.14842 1. 18269 1.11882 2 9 . 3 2 8 9 6 8 8.95192 6.39494 8.98128 6.93*87 8.62789 8.29*31 1.3*882 1.28433 1.22662 1.14792 3 9.268916 8.41454 6.18267 8.55592 6.1*924 6.*6378 8.29376 1.3*44* 1.2328* 1.2*337 I.13427 4 9.269839 8.17811 6.46229 8.96991 8.89978 8.33372 8.34*86 1.22611 1.1*868 1.19471 1 .*448* 5 9.292993 7.91919 5.32687 7.86597 3.24*11 8.93773 3.98349 1.19884 *.B3B37 l.*322l 9.99872 6 9.299734 8.54563 6.49447 8.69675 6.28*26 6.*3248 8.38873 1.17998 I. *9843 I. 17393 1.1*792 7 9.316397 8.59889 6.32266 8.39816 6.936*1 8.77293 8.37*81 1.3779* 1.23466 1.198*6 1.13*64 8 9.399148 8.95294 6.16296 7.98678 6.13739 8.6*229 8.11334 1.2711* 1.23*4* 1.22137 1.14384 9 9.267217 8.92271 5.98169 8.44992 3.7482* 8.86498 8.21384 1.19341 1.1*86* 1.22922 1.18837 19 9.257849 B.37777 6.43439 8.27626 3.94673 8.43642 8.36929 1.1*218 1.96381 I. 1892* 1.92494 I 1 9.282689 7.67178 3.53193 7.57682 5.1862* 4.97978 4.*3843 1.14271 *.77422 *.98823 *.99421 12 9.356575 7.78646 6.41127 7.77992 8.61974 8.28298 8.83919 I. 36886 1.27982 1.23782 1.15882 13 9.266989 6.63579 6.98817 8.46877 6.47439 6. 18483 8.38932 1.42219 1.97*86 1. 17861 1.1118* 14 9.267727 7.67723 6.38261 7.59438 8.77749 3.96448 8.23681 I. 49167 1.2*642 1.2*797 1.99332 15 9.397196 8.28970 5.52481 8.38269 3.3896* 3. 18383 3.99136 1.2768* *.99*98 1.119*8 1.*1*88 4 9 2 TABU! 50— Continued
0B8 SEX h o r API -TYPE- -TREO. HAP EX HCALC HCOOP KAVKAKJGPA HPSATV HPSATH HSATV 16 1 4 349 3.84484 9.45424 48.8147 9.4944 3.52211 54.2493 63.6443 86.9164 17 2 4 413 4.44444 9.44143 47.8529 9.7498 3.49317 84.3844 63.9119 87.6786 18 1 0 114 3.42281 8.74784 43.8758 9.4026 3.44734 54.3474 62.5789 03.1228 19 1 2 8 99 3.82828 9.17477 45.4321 9.6212 3.57131 86.3232 63.8556 89.3436 2 6 t 3 8 119 4.44494 9.14387 48.8487 9.7311 3.53448 54.4844 62.9832 56.4262 21 1 4 8 37 4.27427 9.28474 34.8333 14.4148 3.39439 61.3784 68.8919 63.8148 22 2 8 141 3.49448 8.83941 42.9314 9.3518 3.32968 49.9574 63.6956 82.9149 23 2 2 8 48 3.94118 8.84147 45.4447 9.6246 3.34132 57.1471 63.9833 66.7647 24 2 3 8 184 4.44444 8.62444 44.4444 9.8844 3.47467 53.4844 68.4333 33.9844 25 2 4 6 284 4.29828 9.41929 01.8947 14.1435 3.38917 56.7992 67.6417 66.5444 2 6 1 • 4 288 3.88498 9.43922 45.4888 9.6783 3.81525 83.6314 63.2824 56.1961 2 7 1 4 114 3.89474 9.48772 43.4844 9.7228 3.53746 55.4316 64.3246 88.3263 2 8 2 • 4 374 4.41892 9. 17142 48.1441 9.8441 3.81927 83.6743 66.1481 57.1919 2 9 2 4 243 8.97119 8.74284 44.4237 9.6462 3.43848 55.3981 65.6132 88.4198 36 1 • 7 77 3.81948 8.49481 43.3443 9.5435 3.44491 49.1818 61.874! 31.9614 0B8 IBATH HDSAT9 HACHL2 TUOJat HCEOK niim i KADVH HTAP 8DAPEX BDCALC SDCOOP SDAVHA 16 66.7919 63.4589 72.4779 19.9694 9.6984 9.6459 9.4919 2.32791 6.94669 1.37714 7.54128 6.84238 17 69.5188 65.5824 74.6779 9.7741 9.8199 9.7476 9.6713 2.69445 1.61137 1.86138 7.97143 6.96181 18 65.8979 61.6965 71.8421 19.9944 9.5219 9.6369 9.3246 1.44444 6.92845 1.51128 8.62813 6.85367 19 66.9696 63.B981 72.2727 9.9798 9.4646 9.6616 9.4596 2.47475 6.66944 1.29818 6.29472 6.BS789 2 9 66.5378 63.1849 72.9588 19.1379 9.6639 9.5798 9.5219 2.94116 6.94779 1.28898 6.98344 6.86764 21 7 1 . 7 9 2 7 69.1351 76.3243 19.292? 19.9811 9.8378 19.9999 4.65445 6.96173 1.34178 8.36694 6.82421 22 67.2695 62.4539 72.9789 9.4447 9.3454 9.3582 9.3156 1.44444 1.68644 1.63947 B.61163 6.99639 23 68.9265 66.2296 73.6324 9.6838 9.6397 9.6838 9.4779 2.54444 6.94446 1.53536 6.94166 6.98323 2 4 68.9299 64.9499 73.5467 9.6433 9.6187 9.5353 9.4199 2.72667 6.88234 1.42446 7.63346 4.91973 2 3 71.8119 68.9591 77.9669 19.9583 19.2264 19.1963 19.9748 3.65748 6.94692 1.48789 6.73184 6,83563 20 66.3894 63.9999 72.2275 19.9333 9.6216 9.6353 9.4647 2.34962 6.94398 1.35392 7.43338 6.84791 27 67.4211 64.4825 73.9351 19.1211 9.5789 9.6667 9.5526 2.28476 6.93476 1.43324 7.69418 6.83263 2 8 69.6216 65.4919 74.9946 9.9197 9.8743 9.8954 9.7514 2.76216 6.99711 1.54723 7.82436 6.92464 29 69.3621 65.7292 74.1934 9.5523 9.7119 9.6597 9.5494 2.58428 1.63413 1.05636 8.57564 1.44161 36 64.8182 69.5714 71.3596 9.9416 9.4286 9.6939 9.2987 1.66444 6.92637 1.53862 8.68163 6.87262 088 BDJCFA 8DP8ATV SDPSATH SD6ATV 8DSATT1 SDDSAT9 SDACHL2 SDALCt SDIKTM SDADVH SDTAP 16 9.266734 8.5321 6.99353 8.79691 6.99796 5.36895 5.32278 1. 13423 1.13432 1. 16911 1.16824 17 9.318992 8.6556 6.38488 8.68995 6.93436 5.86973 5.21217 1.31894 1.15161 1.19667 1.11583 18 9.299141 8.3974 6.99279 8.37943 8.84592 5.48999 4.89699 1. 17461 1.22966 1.14966 1.49762 19 9.258729 7.9948 5.99783 .8.44122 6 . 199B8 5.96876 5.48461 1.21622 1.13895 1.21662 1.17949 2 6 9.233244 7.8389 6.99986 7.77954 5.87553 5.51265 8.44821 1.67376 1.67378 1.17528 1.44637 21 9.269512 6.1615 4.43962 7 18346 4.29228 4.96996 4.27566 4.96796 6.88895 1.22565 1.62662 22 9.349476 7.7B82 6.81585 7.86983 6.11892 5.73529 8.66748 1.36122 1.27327 1.36429 1.19133 23 9.277931 9.1444 6.33279 8.79772 5.72847 5.95717 4.93614 1.41396 1.36464 1. 19359 1.67354 24 9.278781 8.4325 6.68195 8.33895 5.85561 5.17464 8.17815 1.29576 1.13878 1.21347 1.44443 25 9.295628 7.9878 8.65769 7.88442 5.38147 5.15611 3.93919 1.22966 6.83456 6.99927 6.99719 26 9.263542 8.4993 6.93587 B.5B9B3 5.92476 5.89524 0.33773 1.13891 1.49442 1.16643 1.11346 27 9.274297 8.6988 6.18662 8.89588 6.18277 5.99679 5.26966 1.12639 1.22398 1. 17997 1.49982 28 9.398421 8.6557 6.49454 8.76421 6.13292 5.98337 0.67888 1.26681 1.68564 1. 17362 1.68962 29 9.331737 8.5687 6.35637 8.51716 5.89163 5.67835 8.38824 1.44863 1.23986 1.26886 1.14765 36 9.285524 8.2824 5.95675 8.23284 8.88425 5.37643 4.78695 1.24845 I.21589 1.11617 1. 16994 295 TABLE 50— Continued
OBS CRSE SEX h HOR o r API -TYPE- -FREQ— HAP EX NCALC HCOOP HAVMA KJCPA HP8ATV KP8ATH I6ATY 31 • 2 7 8 9 3.89831 9.17797 48.8983 9.6458 3.36831 58.5980 62.6619 58. 1186 32 • 3 7 91 4.91999 9.18681 48.8242 9.6906 3.81989 54.9119 63.2198 86.3497 3 3 9 4 7 2 8 4.14286 9.21429 49.9824 19.9286 8.89286 69.6786 68.6786 63.3214 3 4 1 7 37 3.83784 8.91892 44.2999 9.7216 3.46135 52.6486 64.9541 88.0495 35 t 2 7 49 3.72599 9 . 17099 44.7576 9.8889 3.57575 57.5289 64.8759 61.9599 3 6 1 3 7 2 8 3.96429 9.98929 48.9286 9.6464 3.56321 84.3214 62.2143 56.6786 3 7 1 4 7 9 4.66667 9.38889 82.8889 9.9586 3.69999 63.0856 69.5556 65.3333 3 8 6 7 7 3 3.49310 8.69863 42.7469 9.2973 3.33726 48.6849 63.2329 81.9909 39 • 2 7 32 3.87899 9.99999 46.9618 9.7344 3.83899 55.9625 64.2813 59.3759 46 • 3 7 193 4.92913 9.92913 47.2621 9.7233 3.49993 02.9612 65.9929 56.2427 41 • 4 7 162 4.27778 9.09926 52.1478 19.1889 3.61195 56.9926 67.8951 69.1111 4 2 1 7 6 8 3.09999 8.36768 43.1599 9.4193 3.82147 81.3238 64.1912 54.8676 43 1 2 7 36 4.99999 8.62899 44.3299 9.8194 3.84694 89.9999 67.5999 62.9999 4 4 1 3 7 47 3.93617 8.36179 44.6383 9.2669 3.49881 53.3494 63.1277 55.4943 43 1 4 7 92 4.32699 9.26987 51.4783 9.9887 3.85968 88.9435 67.1957 61. 1848 OBS HSATH HD6ATO HACHL2 11ALC1 HCEOH Hiirm HADVH ITTAP 8DAFEX 8DCALC 8DC00P SDAVKA 31 60.8136 63.2293 71.6192 9.9831 9.5999 9.7934 9.4831 2.49678 9.82419 1.21339 0.47816 9.84943 3 2 66.0824 63.1756 72.1319 19.9714 9.6813 9.5449 9.4725 2.91299 9.93683 1.21205 6.77183 9.81417 3 3 71.2143 68.6429 76.2399 19.2679 19.2143 9.6759 9.8571 4.19714 1.94491 1.43649 9.63956 9.84499 3 4 67.8649 63.7297 72.8649 19.1351 9.7162 9.7927 9.3784 1.99999 9.89795 1.46966 8.99693 9.81142 3 5 66.4259 64.6759 73.2399 9.9759 9.4125 9.6999 9.4259 2.57899 9.93336 1.43921 7.36134 9.89256 3 6 66.3929 63.2143 71.8214 19.3599 9.6971 9.6964 9.6786 3.93571 9.99934 1.53399 7.64405 9.78669 3 7 73.2222 76.6667 76.5056 19.9999 9.6667 9.7222 19.4444 3.68889 9.89999 1.95499 3.86739 9.89485 3 8 66.4932 61.3288 72.1233 9.4063 9.2123 9.2749 9.3982 1.99999 1.95872 1.64937 7.68318 1.91936 3 9 68.2813 65.3125 73.5313 9.9531 9.7813 9.7931 9.6994 2.59999 9.94186 1.72739 6.23295 9.93298 4 4 68.4669 64.4272 74.9689 9.8155 9.7339 9.6853 9.5194 2.79874 9.89965 1.42983 6.78516 9.85951 41 72.9399 68.9892 77.1667 19.1747 19.2899 19.1695 19.1265 3.64198 9.95391 1.47792 6.95855 9.7797! 4 2 68.1929 63.6618 72.9294 9.4999 9.4882 9.4485 9.3230 1.99999 1.12635 1.66531 8.45439 9.98829 4 3 69.0999 67.9276 73.7222 9.4444 9.8139 9.6667 9.3611 2.59999 9.95618 1.38501 7.49844 1.92896 4 4 67.9426 63.1913 72.4943 9.2669 9.3681 9.2723 9.1792 2.76596 9.86969 1.33393 7.39225 9.99461 4 5 71.4239 68.9217 76.8913 9.8533 19.1394 19.9199 9.9837 3.68478 9.93889 1.59179 7.75471 9.91949 OBS 8DJCPA SDPSATV SDP9ATH SDSATV SD6ATH BDD6A7B 6DACHL2 8DALC1 SD1RTK SDADVH SDTAP 3 1 9.241976 7.4493 5.43491 8.29867 5.60525 5.60491 0.34421 1.21707 1.97677 1.19669 1. 19253 3 2 9.238279 7.6995 0.95315 7.79457 5.89625 5.59271 0.05819 1.97992 1.91794 1.17296 1.94712 3 3 9.288616 6.2966 4.95292 6.88921 4.43352 4.99198 4.41964 9.96688 9.82134 1.25185 1.11279 3 4 9.393951 8.2579 5.90643 8.27178 5.26690 5.12979 0.91299 1.19333 1.25923 1.12988 1.98273 3 3 9.285889 8.4973 6.38583 8.57839 6.76297 6.37869 5.69563 1.22971 1.24493 1.25678 1.17424 3 6 9.216839 8.4418 6.23822 8. 11972 6.29238 5.64699 8.16436 1.95692 1.25725 1.19670 1.94717 3 7 9.226936 5.8119 2.29734 8.39662 3. 11359 4.12311 4.96544 1.99999 1.99999 1.29185 9.46398 3 8 9.328786 7.8898 6.33886 7.75522 6.32943 5.84298 5.44147 1.39164 1.24429 1.31599 1.18624 3 9 9.312771 9. 1191 6.84499 8.92893 5.65819 6.99393 4.89581 1.16646 1.17945 1.39668 1.99799 4 4 4.274771 8.9497 6.69736 8.78777 5.90796 5.33590 5.65123 1.11362 1.19661 1.29666 1.99987 41 4.282976 7.7145 5.63111 7.61988 5.39841 0.12572 3.96698 1.17203 9.76894 9.93359 9.97955 4 2 9.375521 7.5398 6.71211 7.58963 5.82619 5.49297 5.82963 1.43979 1.29771 1.29629 1.29569 4 3 9.245355 8.8834 5.59965 8.43891 5.89886 0.79895 0.11921 1.08913 1.81415 1.19190 1.95296 4 4 4.289326 7.2549 6.59958 7.31187 5.56369 4.74863 5.32288 1.57374 1.17974 1.19922 1.96986 4 5 9.314572 8.3454 5.79769 8.34263 5.51568 5.23718 3.99676 1.39413 9.98589 1.19438 1.94158 296 TABUS 50—Continued u 5 I y § O 0 + O tft«n « t f t O + 0 0 0 0 0 0 0 0 0 0 ♦ ▼ ♦ * • S »iB#r»C9**»»*~ae»co h _ < 0Mt«tDt»tteNo>«^n S 2 £ : £ £ S £ 2 ! S m S 2 S £ 2 -«N-«N- N « - N « - — - » t n n < £ ffln n » N "« sn « st st « sn "« N » n ffln £ < n n • C90®Ct0BO>r<--" + — + + 0 BtAonnonttnnttn QCIBQ'+*0«««ei9t**e> hh0O«f0M00«8f»(i O » Q O > t f > O f l O f i F F S » ( 0 8« * ? $*a««!09Nl«Bttl*S n?N 5SSS2£??8SSSJS25? * Sng ~ !*• ~S*nKg ^ C I 0 I O « « * < O “ - « o 5 « n b - }»n«gnK«0Nfti N 0 « K n g « n » t} n a n n a n n n j o n n a n o n o n tttt00»000tt^ O «IO 0M otnnN 9nr*9~«9fi**i) 9nr*9~«9fi**i) otnnN ®23S«n33i o i» 3 3 n « S 3 2 ?® $ « * 5 "£**;*£Oi-F-ef5f— 0 + N - + O 9 C - • -NNNM MN- —
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k
TABIZ 50--Conti nued
0B8 CRSE SEX XOR API -TYPE- -FREQ- HAP EX flCALC HCOOP HAVHA HJGPA HPSATV HPSATH IBATV 61 0 2 11 56 3.96600 9.46000 45.1600 9.2920 3.45666 53.2000 62.6200 55.9400 62 0 3 11 106 3.92453 9.31604 44.5566 9.2934 3.42972 51.6396 62.4434 54.4037 63 6 4 11 20 3.70000 8.85000 43.7500 8.9356 3.31250 56.8566 62.9500 53.7006 64 1 1 11 57 3.52632 6.56877 41.6146 9.0456 3.24066 56.2105 61.6667 52.8947 65 1 2 11 38 3.63158 9.00000 42.9737 8.9895 3.46842 54.5600 62.7368 57.6316 66 1 3 11 46 3.62609 8.65217 42.7391 9.1457 3.39543 52.5660 61.6876 54.3660 67 1 4 11 16 3.70000 8.60000 43.2000 8.6300 3.29506 66.7000 64 . 1 0 0 0 59.1000 68 2 0 1 11 47 3.82979 8.71277 47.4828 10.0681 3.56234 4 9 . 9 7 6 7 65.6596 53.6383 69 2 6 2 11 41 3.80488 8.69512 48.5833 10.1463 3.67854 57.9756 64.7673 61.7361 70 2 0 3 11 88 4.13636 8.84659 49.6341 16.2125 3.66469 55.3977 67.2955 58.5368 7! 2 6 4 11 170 4.32353 9.53824 53.1983 16.3100 3.64318 57.4647 68.6659 6 1.3941 72 2 1 1 11 48 8.72917 8.62500 47.7368 16.0633 3.51583 53.6667 67.6833 57.7292 73 2 1 2 11 38 4.67895 8.82895 47.6000 16.1184 3.65579 61.9474 69.8006 63.3684 74 2 1 3 11 29 4.13793 8.60343 48.8966 16.6172 3.57862 55.6207 63.4828 58.6696 75 2 1 4 11 91 4.42857 9.34615 52.8824 16.1644 3.58363 58.2967 67.7692 61.8242 088 F6ATH HD0A7B HAGHL2 KALC1 HCEON h i i t t h HADVH HTAP SDAPEX SDCALC SDCOOP 8DAVHA 61 64.5000 61.6200 70.1400 9.6400 9.1100 9.3800 9.0600 2.40000 0.78142 I.18597 5.3428 0.84003 62 65.8113 61.6792 71.0566 9.6368 9.3491 9.1038 9.0142 2.63208 0.94313 1.35809 7.6212 6.76149 63 66.6000 62.3000 71.8000 8.9750 9.4000 8.8256 8.7250 3.70000 1.17429 1.74793 8.8612 0.84497 64 65.5088 61.2632 69.6140 9.1439 9.1702 8.9828 8.6246 1.00000 1.01955 1.52799 8.2387 6.86380 65 63.7095 61.7895 70.5000 9.1447 8.8684 9.0395 8.8289 2.44737 6.91300 1.27672 6.8916 0.82717 66 64.9130 61.5217 70.2391 9.3843 9.0500 9.1261 8.9891 2.67391 6.92627 1.47524 7.3679 0.91231 67 66.6000 64.0000 73.3060 8.1000 9.1500 8.5600 8.9500 3.70000 1.25167 1.52388 9.9867 0.96269 68 70.1702 64.6596 74.5632 16.3085 16.0745 10.0426 9.9787 1.00000 1.00691 1.74376 6.5886 0.56148 69 69.3415 66.8049 74.9624 16.3700 10.1951 10.0976 10.6976 2.48780 0.95448 1.55273 5.9411 0.65272 70 70.3182 66.4432 75.6932 16.2955 10.1420 10.2045 16.6795 3.61136 6.86012 1.24651 5.6557 6.60623 71 72.8353 48.8529 77.6471 16.3312 10.3735 16.2706 16.2471 3.71176 6.92053 1.42360 5.2174 6.64311 72 71.0000 66.5625 75.5417 16.2700 10.0417 10.1979 9.9583 1.00000 1.10588 1.72436 6.6667 6.64026 73 71.9737 69.7895 76.4474 16.3026 10.6526 16.2237 9.9665 2.63188 6.94101 1.67376 7.4297 6.71648 74 69.7931 65.8621 75.2759 10.1724 10.1034 9.9130 9.9483 3.17241 6.67522 1.42296 5 . 8 0 2 4 0 . 7 8 2 8 8 75 72.1319 68.7253 77.2527 16.0664 16.1923 16. 1484 16.1429 3.76336 0.84515 1.44663 6 .1 2 2 0 0.76506 OBS BDJCPA SDPSATV SDP8ATH SDSATV 8D6ATH SDD8AT8 8DACHL2 8DALC1 SD11VTH SDADVH SDTAP 61 0.280572 7.9650 6.39990 7.9473 5.36903 5.13845 5.11504 1.28190 1.66565 1.36368 1.19369 62 0.239004 7.8675 5.95232 7.6405 6.62908 5.27807 5.53918 1.11384 1.62166 1.26656 6.97762 63 6.264672 4.3319 5.82621 6.6526 5.13399 3.92166 4.66008 1.31264 6.78807 1.40745 1.21909 44 0.352336 7.6011 6.68668 6.3743 5.36876 4.15557 5.25342 1.45994 1.29311 1.24636 1.07323 65 0.269638 7.6563 5.85262 6.7159 5.71937 4.68556 5.69239 1.67618 1.22366 1.21378 1.62190 66 0.255184 6.9817 5.51292 7.2992 6.37412 5.62766 4.89528 1.63377 1. 18116 1.21622 1.69283 67 0.397331 10.4568 8.26573 9.6200 4.76562 5.14242 3.56659 1.2649 i 1.29267 1.31233 1.44241 68 0.211580 8.4666 5.67627 7.7612 3.49116 3.54631 4.35794 0.93572 6.82738 6.76492 0.80033 69 0.191618 7.4448 5.12467 7.9931 5.06237 5.46909 4.64231 0.94061 6.84319 1.01378 0.80793 76 6.248488 8.5803 6.60665 8.4837 4.55249 4.40443 3.87849 0.97861 6.94974 0.84294 6.74234 71 6.264406 7.6863 5.19706 7.2563 4.83594 4.62136 3.56912 1.03531 6.70724 6.86513 6.83249 72 0.306329 7.6613 5.52685 8.5172 4.40019 5.66093 3.94683 6.93943 1.10645 6.84896 6.93873 73 0.230459 8.6537 4.17036 8.3388 4.22666 4.76563 3.73224 6.77581 1.26165 0.78383 6.89386 3.89802 4.22496 1.07965 6.94849 1.63279 0.81662 74 6.251917 8.4406 6.82234 7.6622 5.16436 6.88396 75 0.284126 8.6422 5.67297 8.2511 5.15581 5.66615 3.75231 1.12566 6.90299 6.96173 298 TABLE 50— Continued
OBS API -TYPE--TREQ- M A P EX HCALC HCOOF KAVHAKJGPA MP8ATV UPSATM FSATV 76 12 2 1 3 3 . 7 3 7 6 9 9.67812 43.1972 9.2961 3.43174 81.8481 61.8681 5 3 . 9 3 7 7 77 12 2 1 7 3 . 6 4 6 8 8 8.88418 43.1429 9.6860 3.31388 8 1 . 6 4 1 8 6 2 . 2 8 3 5 5 3 . 7 2 3 8 70 12 18 6 4 . 6 3 0 4 6 9.62864 49.7789 16.2389 3.64381 87.8321 66.3846 6 6 . 9 5 8 1 79 12 0 9 6 4. 1 9 6 9 7 9.18988 86.9228 16.1873 3.89366 86.1692 67.9167 8 9 . 8 4 6 0 86 i 13 7 8 3 . 4 8 8 8 8 8.74888 42.1067 9.2987 3.37167 49.1666 66.8867 5 1 . 4 9 3 3 at 2 13 8 8 3 . 8 4 4 8 3 9.31634 43.9820 9.2224 3.47293 84.1267 6 2 . 4 6 2 8 5 6 . 3 4 4 0 82 3 13 7 8 3 . 9 3 3 3 3 9.27333 43.9867 9 . 3 8 0 8 3 . 4 6 7 8 7 82.4933 61.5467 5 4 . 4 2 6 7 03 4 13 8 3 . 4 6 6 8 6 8.48888 30.6666 8 . 4 8 6 8 3.32286 36.8886 66.6686 3 6 . 2 0 6 6 04 1 13 8 8 3 . 3 6 4 7 1 8.82383 41.3882 8.9289 3.21671 49.6238 6 1 . 9 4 1 2 5 1 . 3 2 9 4 08 2 13 3 6 3.76667 9.16667 44.6667 9.8433 3.44886 63.8667 6 2 . 6 3 3 3 8 7 . 3 8 8 8 06 3 13 7 7 3.88714 8.96184 44.6266 9.1136 3.37286 81.8974 62.8668 84.4416 87 4 13 2 8 3.76888 8.84888 44.8660 8.9848 3 . 3 6 3 6 6 83.7688 62.8886 8 5 . 3 6 6 8 00 1 13 3 9 3 . 8 9 7 4 4 8.82881 48.1667 16. 1872 3.89388 82.8128 66.4183 56.2864 09 2 13 41 3.88488 8.98786 49.6876 16. 1 8 8 4 3.71849 89.4396 68.6732 63.4878 9 6 3 13 4 4 4.11364 8.97727 49.6227 1 6 . 3 1 8 9 3.63756 56.7955 65.431B 89.8102 OBS I6ATH MDSATS KACHL2 HALC1 MCEOHMIRTH HADVR HTAPSDAPEXSDCALC 8DC00P SDAVHA 76 64.1315 60.7559 76.4178 9.7362 9.1878 9.2371 9.6235 2.99399 9.96963 1.39541 7.1697 0.78178 77 68.3318 61.4362 76.7885 9.8447 9.6783 8.9766 8.6894 2.92765 9.97649 1.51524 7.7521 6.92527 78 76.2115 67.1474 75.2888 16.5832 18.1627 16.2819 19.1314 2.64744 9.8716? 1.35575 6.1297 6.57372 79 71.8131 67.8434 76.8861 16.1737 16.2189 16.1798 19.8997 3.95303 6.97648 1.37751 6.3319 6.70335 80 63.5467 59.5688 70.2667 9.7133 9.2133 9.3467 9.6989 1.98888 6.94954 1.47126 7.6327 6.81282 81 63.8966 61.3621 70.4316 9.6638 9.6776 9.2588 8.9483 2.41379 6.85433 1.24896 5.8534 0.76881 62 64.8867 61.2267 76.5733 9.8907 9.2667 9.2267 9.1299 2.81333 9.96329 1.31851 6.9971 6.74632 63 68.0888 64.6888 70.2886 8.6888 8.9888 7.6899 8.8999 4.98888 1.81659 2.13397 13.5388 6.68952 64 64.8624 60.3176 76.6333 9.6141 8.9024 8.8882 6.8768 1.88888 9.99818 1.36292 8.2815 0.97848 88 64.7667 62.3333 76.6333 8.9667 8.8667 9.2698 8.9833 2.43333 6.85836 1.24196 6.6068 6.97581 66 68.7792 62.0266 71.6396 9.2288 9.2586 0.9974 0.8961 2.48852 9.91386 1.51042 7.3466 0.84658 87 66.1688 62.8288 72.7288 8.7888 9.4890 8.9496 8.8999 3.64988 1.95198 1.58588 7.8797 0.92262 80 76.1538 65.5128 74.8718 10.5641 16.1164 10.1923 9.9467 1.98889 9.82862 1.66381 7.6253 0.88946 09 69.1228 67.2683 74.8788 16.4268 16.0122 16.2439 16.1829 2.86998 9.98854 1.35779 5.9842 6.63701 98 69.8636 66.5227 74.5989 16.5568 10.3489 16.1818 16.2945 3.18989 9.92946 1.22927 5.6630 0.81882 OBS SDJCFA 6DPSATV SDP8ATH SD8ATV 8D6ATH SDD6A7S 8DACHL2 SDALGt SDiirrn SDADVH SDTAP 76 0.264617 7.6484 8.77415 7.7468 5.60698 0.66863 8.14661 1. 19593 1.19635 1.22912 1.05929 77 6.314938 8.2772 6.31469 7.8296 5.69695 5.21831 6.97814 1.45839 1.19866 1.27387 1.12288 78 6.213859 8.6879 6.39238 8.3321 4.46782 4.79789 4.15352 6.86974 0.89810 9.79875 0.81738 79 6.274217 8.3235 6.47497 8.3626 4.87519 4.86353 3.81562 1.03921 0.98097 9.89477 6.84839 88 9.287499 8.1224 8.53147 8.8629 5.32788 4.91341 4.63341 1.21692 1.23893 1.18971 1.86225 81 9.263972 7.3834 6.14515 7.4846 0.99617 5.86345 8.85798 1.39917 1.09144 1.24693 1.14971 82 6.227957 6.6887 5.46887 6.8678 5.89896 0.65797 5.30479 1.97894 1.09449 1.22003 0.97551 83 6.322831 6.4428 7.41628 16.6188 3.96232 4.18933 6.72389 0.54772 0.65192 1.51638 1.33993 84 6.348173 7.8229 6.66765 6.9836 6.16138 8.41299 8.4866! 1.42479 1.25821 I.34823 1. 17499 88 6.297828 8.6301 6.23938 7.4689 4.43895 4.64857 4.97248 1.68632 1.22428 1.33649 1.88662 86 0.256877 8.4344 6.17641 8.1416 5.82333 0.31377 6.41791 1.48369 1.18024 1.10455 1.93662 07 6.311956 8.6616 6.47431 7.9471 0.68536 4.38862 3.12943 1.45774 1.01936 1.24432 1.29899 88 0.237333 8.6888 4.96683 8.1788 4.19736 4.26595 3.87429 9.85208 9.97644 0.77473 6.87955 09 0.177439 7.6846 8.26618 8.8866 4.85384 5.45998 4.26839 0.91215 6.97788 0.90223 6.76867 9 0 9.262887 8.9364 6.18837 8.1165 4.34335 4.61287 4.75699 0.94149 0.82660 0.80037 0.77997 299 TABLE 50— Continued
OBS CRSE SEX BOR API -TYPE— —FREQ— MAP EX HCALC HCOOF KAVHA HJCPA HP8ATV HPSATH IBATV 91 2 4 13 32 4.49625 9.39963 53.2899 19.2599 3.63719 62.2188 69.3438 65.9999 9 2 2 2 13 86 3.69643 8.56259 47.1613 9.9982 3.59197 51.3759 66.3571 56.3214 9 3 2 2 2 13 38 4.97895 8.51316 47.9952 19.9763 3.62132 69.3684 69.1953 63.5999 9 4 2 2 3 13 73 4.15968 8.67123 48.9863 19.9726 3.57384 54.6438 67.6986 57.6927 9 3 2 2 4 13 229 4.35371 9.48253 53.9893 19.2367 3.62935 57.1319 68.1793 61.9611 9 6 1 9 14 149 3.73826 9.19967 43.2148 9.2913 3.42463 51.6399 61.5579 53.6846 9 ? 1 14 64 3.73438 9.91563 43.1563 9.2875 3.44828 52.3438 61.5938 54.5938 9 8 1 2 6 14 139 3.66923 9.95999 44.9977 9.1115 3.33462 49.7769 62.3154 52.6846 9 9 1 2 14 87 3.59779 8.43678 41.8596 8.8483 3.28969 52.9319 62.1699 55.2759 190 2 9 14 196 4.99943 8.95283 49.6375 19.2151 3.64264 56.4434 65.7975 59.7264 191 2 14 89 4.19999 9.18999 59.9833 19.2899 3.63169 59.8499 67.8299 63.5699 1 92 2 2 9 14 249 4.29833 9.23769 51. 1751 19.2498 3.61929 55.7792 68.1625 59.6333 1 93 2 2 14 136 4. 17949 8.91346 59.5947 19.9286 3.55763 66.7692 67.5385 69.1731 194 1 9 13 36 3.41971 8.74197 42.1964 9.3143 3.38714 48.9821 69.6697 51.4464 193 I 9 2 15 33 4.99999 9.37879 44.5152 9.2515 3.45636 54.9393 62.2121 55.6979 OBS WATH MD6ATS HACHL2 MALC1 HCEOH HIICTH KADVH ITTAP 8DAPEX SDCALC 0DCOOP 80AVKA 91 72.1563 69.8438 77.2813 19.4331 19.2656 19.1875 19.1875 4.96259 9.71298 1.16905 4.9776 9.55533 9 2 79.8929 65.6964 75.1786 19.9982 19.9179 19.9714 9.9821 1.99999 1.18965 1.81174 5.6985 0.59924 9 3 72.2195 69.2895 76.4737 19.2399 19.2599 19.9658 9.8684 2.55263 9.99679 1.72614 7.2194 9.72165 94 79.3836 66.1644 76.1918 19.9899 19.9968 19.1927 9.9521 2.98639 9.82784 1.32982 5.7094 9.71497 9 3 72.4279 68.6638 77.8415 19.2966 19.3166 19.2336 19.2149 3.65939 0.91859 1.46663 5.7619 0.71313 9 6 64.9537 69.6949 79.3499 9.7215 9.2114 9.2459 9.9967 2.96711 0.98225 1.39978 7.1069 0.89992 9 7 64.3125 61.1994 79.5781 9.7793 9.1328 9.2188 9.9625 2.15625 9.94688 1.39434 7.3795 9.74141 9 8 65.2538 61.9538 71.1383 9.2923 9.1398 9.9577 8.9898 2.96154 9.95956 1.51331 7.4932 0.99611 9 9 65.4483 62.9578 79.9469 8.6747 9.9999 8.8849 6.7529 1.97791 1.99533 1.45933 7.9906 9.93633 199 69.6599 66.3679 74.8679 19.4717 19.1981 19.1849 19.1985 2.74528 9.86735 1.28828 6.1382 0.57583 191 71.4999 68.8999 76.1899 19.3799 19.1599 19.2499 19.1899 2.44999 9.88641 1.49965 6.1569 0.57259 192 71.9875 67.6958 77.9833 19.2596 19.2771 19.2194 19.1687 3.14167 0.96721 1.56446 5.9398 9.65296 193 71.5449 67.7628 76.5964 19.9417 19.1999 19.1999 9.9936 2.91667 9.99344 1.58233 6.9419 0.76995 194 63.1429 59.3936 79.4197 9.6964 9.1964 9.4196 9.9268 1.99999 0.94920 1.49846 7.6629 9.84995 193 64.2424 61.3333 69.8182 9.6364 9.9758 9.3485 8.9697 2.36364 9.75999 1.18695 4.9882 0.76735 OBS SDJCFA 8DPSATV SDPSATM S?6ATV SDSATH 8DDSATS 8DACHL2 8DALG1 SDiirm SDADVK SDTAP 91 9.238844 5.7739 3.77265 6.9161 4.11221 3.62882 2.89866 9.75519 9.77235 9.71561 0.84093 9 2 9.275877 8.9353 5.69638 8.5453 3.89255 4.69798 4.37897 9.94590 9.97218 9.83355 0.86846 9 3 9.233576 8.2999 4.41909 8.7325 4.35957 5.93988 3.51627 0.89329 1.14343 9.91669 0.89898 9 4 9.279881 8.1995 6.17946 8.2895 4.91269 4.90593 3.26878 1.99119 9.99475 9.96451 0.73676 9 3 9.277158 7.8691 5.31363 7.6889 5.05586 4.87255 3.72669 1.19822 9.7B723 0.88296 0.85459 96 9.269741 7.5636 5.89332 7.8163 5.79909 5.32319 5.24781 1.18933 1.07675 1.23939 1.98114 9 7 9.274898 7.8794 5.75969 7.6194 5.47976 4.49821 4.94629 1.24029 1.17933 1.21458 1.01379 9 8 9.392979 8.1167 6.39408 8.2194 5.94329 5.47484 5.59659 1.27557 1.14383 1.39538 1.11787 9 9 9.331164 8.1993 6.22985 6.9845 5.33692 4.77726 5.13517 1.63389 1.27899 1.22254 1.12289 199 9.219117 8.7498 5.32219 8.4924 4.37979 4.68995 4.24954 9.91763 9.83993 9.77866 9.79584 191 9.229633 7.9163 4.83164 7.6242 4.46399 4.64679 3.83161 0.76272 1.03633 9.84660 0.86756 192 9.262159 8.2999 5.40365 8.9720 4.79494 4.75174 3.71153 1.91992 9.89616 9.86339 0.81137 193 9.288677 8.4867 5.37921 8.8963 4.99989 5.99419 3.95702 1.95793 1.92437 9.94065 9.89440 1 9 4 9.287388 8.2781 5.69*37 8.2812 5.67197 5.42693 4.71661 1.22355 1.23832 1. 16298 1.99719 193 9.243282 6.9799 6.09691 7.7358 5.78898 5.20016 5.42249 1.31264 1.03169 1.22783 1.21796 300 TABLE 50— Continued
OBS CRSE SEX HOR API -TYPE- -FREQ- MAP EX HCALC HCOOP HAVMA HJCPA HP8ATV HP8ATH HSATV 196 1 • 3 19 57 3.96491 9.36842 44.9326 9.3474 3.40895 82.7368 61.9649 54.6842 197 t 9 4 15 3 2.66667 7.66667 32.9999 8.2333 3.12333 83.6667 63.3333 84.3333 198 1 13 19 3.68421 8.73684 41.8421 9.2826 3.32684 49.6842 69.3684 81.6316 199 I 2 13 25 3.64999 9.22999 43.2899 9. 1849 3.49489 54.2499 62.8499 57.2999 119 1 3 13 18 3.83333 8.97222 43.7778 9.5167 3.49278 01.7222 69.2222 53.6111 111 1 4 13 2 4.09999 9.09999 48.0999 8.8599 3.62999 09.8999 79.9999 59.9999 1 12 1 2 9 IS 47 3.29787 8.62766 41.2979 8.9128 3.22285 47.8511 61.6383 49.5532 113 1 2 • 2 15 17 3.88235 9.61765 46.4118 9.3796 3.40796 81.5882 61.6471 06.4118 114 1 2 9 3 13 49 3.87755 9.25519 48.1429 9.2396 3.39571 09.7909 63.9999 54.9816 115 1 2 9 4 IS 17 3.68235 9.95882 48.8230 9.9888 3.34068 59.3529 62.8824 03.5862 1 16 1 2 1 15 38 3.44737 8.39474 41.8999 6.9421 3.29947 09.4737 62.3188 83.0263 1 17 1 2 2 13 13 3.61038 6.57692 42.8846 8.6154 3.41769 85.9999 62.5385 58.4615 1 IB 1 2 3 15 28 3.82143 8.44643 42.9714 8.9971 3.33286 53.9999 61.6429 50.9714 119 1 2 1 4 15 8 3.89999 8.37599 41.8709 8.5759 3.21375 61.9999 62.6209 59.1209 129 2 1 • 1 13 21 3.89902 8.57143 48.9769 19.1619 3.88429 49.7143 65.9902 53.3333 OBS PBATH HDSATS HACHL2 KALC1 HCEOH MIlfTH HADVH HTAP SDAPEX SDGALC SDCOOP 0DAVKA 196 64.6491 61.3158 79.6667 9.8599 9.3158 9.1149 9.9614 2.84211 9.96297 1.2(253 6.7949 9.75024 197 67.6667 63.3333 69.9999 8.6667 9.9999 7.3333 8.9999 4.99999 2.98167 2.46644 14.1774 •.66083 198 64.7368 69.3158 69.8421 9.7632 9.2632 9. 1316 8.9211 1.99999 9.94591 1.42759 7.7479 •.71443 199 63.4499 61.4999 71.2499 9.7999 9.9899 9.1290 8.9299 2.48999 9.95219 1.34691 6.6779 •.78299 119 64.3889 69.9444 79.2778 19.9167 9. 1111 9.5833 9.3906 2.72222 9.98018 1.47999 8.9627 •.73184 t i l 79.5999 66.5999 72.9999 8.5999 8.7599 8.9999 19.9999 4.99999 9.79711 1.41421 2.1213 •.•7971 112 64.9638 09.1792 79.4681 9. 1596 8.7234 8.8723 8.9074 1.99999 9.95359 1.55593 7.9062 1.91159 113 65.9999 62.1765 79.7647 9.6471 9. 1765 9.4412 9.2353 2.47959 9.88749 1.29584 0.9273 •.98664 114 66.9816 62.1929 71.5192 9.3878 9.3878 9.9918 B.9092 2.98776 9.92729 1.47239 7.3993 •.77169 115 66.4118 62.1176 72.2941 9.9294 9.4796 9.9882 8.8529 3.64796 9.92702 1.59963 6.9336 •.82694 116 65.8947 61.7368 69.5999 8.8342 9.1237 8.9979 8.7763 1.99999 1.95772 1.58181 8.6915 •.94910 1 17 64.4615 62.0380 69.9769 8.9769 8.4610 8.8846 8.6538 2.38462 9.86972 1.98763 7.1594 •.89813 118 63.2599 61.8929 79.2143 8.9286 9.9197 8.8321 8.7887 2.64266 9.99487 1.46148 6.9545 •.94826 119 65.6259 63.3759 73.6239 8.9999 9.2599 8.6259 8.6870 3.62099 1.39931 1.55265 19.8422 1.98332 129 69.2857 63.9524 73.8571 19.5952 19.9476 19.9952 19.9238 1.99999 9.81358 1.66946 8.4919 •.69296 OBS 8DJCPA SDPSATV SDPSATH SD6ATV 8D6ATH SDD6ATS 8DACHL2 8DALC1 8D1RTH 8DADVH SDTAP 196 9.236535 6.5969 5.64994 6.9992 0.91756 5.19679 5.57294 1.96471 9.94789 1.2138a •.98239 197 9.142945 4.0992 8.62168 8.1445 5.13169 4.94140 8.71789 9.07730 9.86693 2.92973 1.99999 198 9.299824 7.8399 5.14469 7.5904 4.95391 2.99694 4.47049 1.22892 1.27332 1.11697 •.97058 199 9.299829 7.9754 6.31580 7.2953 6.30136 4.98331 5.73934 1.33973 1.18743 1.28517 1.97793 119 9.292421 7.9778 4.78423 6.5999 5.07431 4.72132 4.48272 1.14494 1.18266 1.29355 9.90799 111 9.282843 9. 1924 4.24264 15.5063 9.79711 4.94970 4.24264 9.79711 9.35355 9.79711 9.79711 112 9.327185 7.0543 6.58861 7.4799 6.34653 5.77219 3.21641 1.87067 1.19241 1.38888 1.21598 1 13 9.359299 9.4872 7. 13216 8.0664 4.06892 5.12634 4.04876 1.25952 1.15841 1.47778 1. 16979 114 9.239766 9.1241 6.39896 8.3969 6.12637 5.39384 5.52314 1.12854 1.19992 1.21902 •.97808 115 9.269799 4.2418 5.36644 6.8835 5.26852 3.99816 2.88887 1.49835 9.78994 1.16237 1.23446 116 9.376864 7.3721 6.47283 5.6747 0.84849 4.61899 0.68566 1.48183 1.31733 1.31445 1.13124 117 9.223128 7.2667 5.97634 5.8397 4.49862 4.13502 3.27774 1.89199 1.23207 1.19218 •.92161 118 9.268988 7.9926 5.95175 7.7838 5.31681 5.26582 0.22307 1.77281 1.29919 1.14919 1.14293 119 9.392298 11.3137 8.53439 9.1870 4.86799 5.31675 3.62284 1.38873 1.43925 1.43393 1.48650 129 9.239186 8.4743 3.32182 8.1384 3.78342 3.01392 4.19226 9.99394 •.92966 9.78452 9.78224 TABLE 50— Continued
OBS CRSE 8EX BOR API -TYPE- JFRE€L- HAPEX HCALC HCOOF KAVHA HJCFA KP8ATV KPSATH H9ATV 121 2 9 2 15 26 3.76923 8.92398 48.9333 19.1462 3.71938 57.3846 63.2398 61.1923 122 2 9 S IS 34 4.98824 8.88238 48.7941 19.2794 3.62296 56.1471 66.3236 89.1176 123 2 9 4 19 25 4.32999 9.49999 82.9444 19.2449 3.64929 61.5299 69.3299 64.4999 124 2 19 18 4.99999 9.11111 48.2727 19.2167 3.69333 95.7778 67.9444 69.6667 129 2 2 IS 15 3.86667 9.19999 49.3769 19.2833 3.71967 63.9999 68.2667 67.4667 126 2 9 15 19 4.29999 9.39999 49.8999 19.4499 3.69999 59.9999 68.8999 62.2999 127 2 4 13 7 4.71429 9.85714 84.1429 19.2714 3.89429 64.7143 69.4286 67.1429 126 2 2 9 IS 26 3.84615 8.82692 47.9999 9.9928 8.84462 59.1923 66.1154 53.8846 124 2 2 6 2 19 19 3.86667 8.39999 48.9999 19.1467 3.42333 59.9999 67.2667 62.7333 136 2 2 6 3 15 54 4.16667 8.82497 49.1652 19.1794 8.89278 84.9289 68.8379 68.2937 131 2 2 9 4 15 149 4.32414 9.96297 93.2449 19.3214 3.64214 56.7686 68.4828 69.8759 132 2 2 1 15 39 3.06667 8.33333 47.3333 19.9933 3.46333 52.4999 66.8667 56.5667 133 2 2 1 2 15 23 4.21739 8.65217 46.4167 19.9394 9.62999 61.2699 79.3943 64.9999 134 2 2 1 3 15 19 4.19526 8.23684 48.4211 9.7947 3.82999 63.8421 65.3188 50.8947 139 2 2 1 4 15 84 4.49476 9.34924 92.7377 19.9995 3.88274 87.7619 67.6319 61.3819 OBS H9ATH HD6ATB HAGHL2 HALG1 HCEOH M irra HADVH MAP 8DAPEX 8DCALC 8DCOOP 0DAVHA 121 67.8977 65.6154 73.8846 19.4231 19.9385 19.1838 19.1846 2.46154 9.99898 1.22223 8.43993 9.647899 122 69.8238 66.2941 74.5882 19.4412 19.2941 19.2647 19.1618 3.92941 9.99999 1.97389 8.68874 9.828481 123 71.6499 69.2899 77.1209 19.4699 19.8699 19.1899 19.9899 4.12999 9.74833 1.21621 4.37299 9.889821 124 71. 1667 67.3333 76.9556 19.5278 19.1944 19.3996 9.8611 1.99999 9.84917 1.52966 7.99139 9.599364 129 71.4999 79.1333 76.6999 19.4333 9.9667 19.4999 19.2667 2.73333 9.91548 1.69579 7.39626 9.634569 126 79.9999 67.3999 74.6999 19.9599 t6.5666 9.9999 19.3599 3.69999 1.93289 1.68655 8.18967 9.489796 127 74.9999 71.8571 77.8571 19.4286 9.9286 19.2143 19.8714 3.85714 9.46798 1.96994 3.33899 9.662278 128 79.8846 65.2398 79. 1154 19.9769 19.9962 19.9999 9.9423 1.99999 1.18859 1.63272 4.89698 9.526999 129 72.9999 68.8667 76.6667 19.3999 19.4667 19.9999 19.9333 2.53333 1.96919 1.98926 7.91783 9.684384 136 79.6296 66.5379 76.3889 19.2937 19.9463 19.1667 19.9278 3.99999 9.84116 1.35339 8.87646 9.651741 131 72.6897 68.7793 77.7379 19.3999 19.3759 19.2862 19.2789 3.64138 9.94929 1.45879 8.37666 9.659775 132 79.9999 66.1999 75.2333 19. 1167 9.9599 19.1333 19.9167 1.99999 1.22287 1.79238 6.61688 9.669133 133 72.3478 69.5692 76.3478 19.2174 19.1987 19.1987 9.7699 2.86522 9.95139 1.86259 7.88837 9.796448 134 69.6842 65.1953 75.6316 9.7632 9.8947 9.9211 9.7868 2.94737 9.89939 1.14798 6.19422 9.826908 139 71.9762 68.4643 77.2924 19.9298 19.2143 19.1429 19.1971 3.69948 9.86894 1.47872 6.36632 9.779499 OBS 8DJGFA 8DPSATV EDPSATH 6D6ATV SD8ATH 8DDSATB 8DACHL2 SDALC1 SDlimf SDADVH SDTAF 121 9.146765 7.72568 4.51939 7.88426 4.89996 5.37458 4.34812 9.93489 9.87999 1.99766 9.781763 122 9.296717 8.94133 9.94786 8. 14179 3.99393 4.85624 4.65226 1.91328 9.82689 9.63948 9.755992 123 9.243326 5.89576 4. 17933 6.93462 4.26196 3.56511 2.86249 9.81548 9.79999 9.71995 9.999212 124 9.281723 7.68923 3.82672 6.97896 4.34299 4.47214 3.31613 9.81299 1.95912 9.76962 9.997136 129 9.226919 6.38977 5.94928 6.73866 3.97851 4.43793 3.43927 9.99379 1.17219 9.68661 9.798899 126 9.189389 9.91899 7. 16163 7.94149 8.84998 4.96767 5.38828 9.49721 9.84984 1.22929 9.683491 127 9.234866 4.88925 1.98896 5.67164 3. 19913 3.33899 3. 18479 9.83482 9.97599 9.75893 9.345933 (2B 9.198116 6.62999 4.73563 7.39646 3.12828 3.52486 4.55479 9.91315 9.76183 9.76158 9.828976 129 9.246248 7.97197 9.23719 8.36295 4.37526 5.16674 2.76887 9.97834 9.74322 1.90221 9.875599 139 9.272787 8.39553 5.79311 8.74886 4.92799 4.34682 3.15282 9.95423 1.91539 9.90661 9.739989 131 9.268649 7.75802 5.35689 7.34979 4.92816 4.78637 3.67898 1.96946 9.71986 9.82929 9.818474 132 9.327618 7.48698 6.339 17 9.23518 4.99055 8.49338 4.29662 9.98888 1.13221 9.89958 9.914286 133 9.239872 9.95146 3.36337 9.11542 4.42728 5.90269 3.98416 9.68798 1.33958 9.83878 9.915393 134 9.265979 7.77648 6.B3173 6.73213 4.93348 3.96931 3.69879 1.98485 9.95139 9.98971 9.714397 139 9.289939 8.63999 5.23146 8.29124 0.27329 5.94967 3.89798 1.15735 9.89942 9.98945 9.998698 302 APPENDIX I
TABLE 51: FORTY COLLEGES AND UNIVERSITIES RECEIVING THE LARGEST NUMBER OF ADVANCED PLACEMENT EXAMINATION CANDIDATES— MAY 1984
303 304 TABLE 51
FORTY COLLEGES AND UNIVERSITIES RECEIVING THE LARGEST NUMBER OF ADVANCED PLACEMENT EXAMINATION CANDIDATES— MAY 1984
College or University Candidates Examinations University of California— Berkeley* 2,344 4,673 University of California— Los Angeles 1,935 3,548 University of Michigan— Ann Arbor* 1,875 3,773 University of Illinois— Urbana* 1,658 3,034 Brigham Young University* 1,491 2,361 Cornell University* 1,440 3,313 University of Virginia* 1,420 2,973 University of Texas at Austin* 1,270 2,058 University of North Carolina— Chapel Hill 1,182 1,946 Stanford University 1,158 3,027 University of Pennsylvania* 1,131 2,799 University of Florida 1,052 1,686 Duke University* 1,020 2,412 University of California— Davis 985 1,546 Harvard-Radcliffe* 984 3,130 University of Utah* 983 1,688 Virginia Polytechnic Inst. & State Univ. 953 1,634 University of Colorado— Boulder* 875 1,464 University of California— Irvine 845 1,399 University of California— San Diego 831 1,412 Northwestern University* 794 1,727 Pennsylvania State University* 780 1,215 Michigan State University* 778 1,113 University of Maryland* 773 1,279 Yale University 770 2,383 Princeton University* 753 2,322 Massachusetts Institute of Technology 723 2,095 Boston University 718 1,207 University of California— Santa Barbara 715 1,037 Brown University* 681 1,868 University of Southern California 677 1,107 Georgetown University 659 1,388 Texas A & M University— Main Campus 658 937 North Carolina State University— Raleigh 648 948 University of Washington 643 967 Miami University— Main Campus* 608 892 Georgia Institute of Technology* 601 1,013 Clemson University— Main Campus* 590 891 University of Notre Dame 582 1,147 Ohio State University* 575 824
SOURCE: AP Program of the College Board, "Two Hundred colleges and Universities Receiving the Largest Number of Advanced Placement Examinations from May 1984 Candidates." Princeton, 1984. (Mimeographed.)
* Institutions listed which participated in 1974-80 validity study. APPENDIX J
TABLE 52: PUNAHOU AP CALCULUS SCORE FREQUENCY BY GRADE DISTRIBUTION IN ADVM AND CALC BY COURSE, 1969-1984
305 TABLE 52
PUNAHOU AP CALCULUS SCORE FREQUENCY BY GRACE DISTRIBUTION IN ADVM AND CALC BY COURSE, 1969-1984
Grade Distribution of Advanced Math Grades Distribution of Punahou Calculus Grades
and Calculus AB Exam Calculus BC Exam Calculus AB Exam Calculus BC Exam
Grade interval 1 2 3 4 5 EAB 1 2 3 4 5 Ebc 1 2 3 4 5 EAB 1 2 3 4 5 EBC
A 10.6-12.0 5 10 6 21 9 32 122 163 4 7 30 41 1 3 77 81 A- 9.6-10.5 3 16 29 33 81 2 9 39 79 99 228 12 46 47 105 2 38 104 144 B+ 8.6- 9.5 3 5 46 68 36 158 11 36 42 28 117 41 60 15 116 13 60 48 121 B 7.6- 8.5 4 12 53 36 13 118 1 10 10 14 4 39 8 46 29 4 87 7 27 40 19 93 B- 6.6- 7.5 2 7 14 8 9 40 1 1 1 1 5 2 7 28 7 1 45 1 9 41 23 6 80 C+ 5.6- 6.5 1 3 5 1 10 3 10 7 2 22 7 8 3 18 C 4.6- 5.5 2 2 2 4 2 1 9 1 7 4 12 C- 3.6- 4.5 3 1 1 5 EH- 2.6- 3.5 1 1 2 D 1.6- 2.5 1 __1
Total 10 30 141 152 97 430 4 31 96 167 254 552 10 30 141 152 97 430 4 31 96 167 254 552 Mean Grade BB B B+ B+ 8.96 B+ B+ B+ A- A- 10.11 C C+ B B+ A- 8.94 C- C+ B- B+ A- 9.09 Stnd. Dev. 1.26 0.87 1.46 1.52
OJ o APPENDIX K
TABLE 53: PUNAHOU AP CALCULUS COURSE FREQUENCY BY GRADE DISTRIBUTION IN SATM AND ACHL2 BY COURSE, 1969-1984
307 TABLE 53
PUNAHOU AP CALCULUS COURSE FREQUENCY BY GRACE DISTRIBUTION IN SATM AND ACHL2 BY COURSE, 1969-1984
Test Distribution of SAT Mathematics Scores Distribution of Math Achievement Level II Scores
Score Calculus AB Exam Calculus BC Exam Calculus AB Exam Calculus BC Exam
Interval 1 2 3 4 5 Eab 1 2 3 4 5 Ebc 1 2 3 4 5 EAB 1 2 3 4 5 EBC
Max= 80 1 12 13 1 2 5 7 15 5 18 40 134 197 75 - 79 1 3 6 3 13 7 21 30 90 148 1 14 19 28 62 1 8 23 44 66 142 70 - 74 3 21 29 22 75 1 17 36 65 96 215 2 36 52 37 127 3 7 29 40 31 110 65 - 69 5 7 50 49 34 145 2 4 31 48 43 128 3 7 32 42 15 99 4 8 16 3 31 60 - 64 14 40 43 29 126 1 2 4 20 11 38 2 3 20 12 3 40 1 1 2 55 - 59 5 4 14 19 7 49 1 4 3 1 9 3 3 3 1 1 11 1 1 50 - 54 11 4 2 17 45 - 49 1 2 2 5 1 1
Total 10 30 141 152 97 430 4 31 96 167 254 552 8 17 107 131 91 354 4 26 78 140 235 483 Mean Score 64.7 71.4 70.2 76.3 Stnd. Dev. 5.9 5.0 5.4 4.1 Stnd. Err. 0.3 0.2 0.3 0.2
U) o 00