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8 Gracluate Eclucation The mathematical sciences have shared in the general accelerated growth that has marked graduate education in all the sciences since the late 1940's. Especially at the level of training for the doctorate, the United States has made impressive gains in the quality of its mathematical education. This has been reflected in increased inter- national recognition of the research accomplishments of American- trained American mathematicians (see the section on The Position of the United States in Mathematics, page 10~. It would also be widely agreed by professionals in the field that the number of truly distinguished university centers for mathematical research in the United States has roughly tripled during the postwar period, from approximately three to approximately nine (see Appendix E). While U.S. graduate mathematical education has been growing in quality, it has also been "Towing in breadth. This is reflected in a broader span of courses and areas for research, as well as in the increasing election of graduate courses in the mathematical sciences by majors from other fields. It is also reflected in the creation and growth of new academically oriented professional societies in the mathematical sciences. In the early 1930's, there were just three such professional societies: The American Mathematical Society at the graduate and research level, the Mathematical Association of America at the collegiate level, and the National Council of Teachers of Mathematics at the secondary and elementary school levels. Increased interest in mathematical logic, a border area be- tween mathematics and philosophy, led to the founding of the Association for Symbolic Logic in 1934. The Institute of Mathemati 135

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136 1 ~ . applications The Mathematical Sciences in Education cal Statistics followed in 1935. During the postwar period, new of mathematics and new mathematical tools and tech- niques (see Chapters 5 and 6) resulted in the formation of four new professional societies in the mathematical sciences: The Society for Industrial and Applied :Mathematics, the Association for Comput- ing Machinery, the Operations Research Society of America, and The Institute of Management Sciences. In addition, today there exists an organization, the Conference Board of the Mathematical Sciences, which comprises most of the professional societies in the field and constitutes a medium through which their representatives can meet for exchange of information and discussion of problems of common concern. THE MASTER S DEGREE Corresponding to Table 1 (see page 123) for bachelor's degrees, we present in Table 9 the comparative figures of the U.S. Office of Educations ~ for numbers of master's degrees awarded in mathe- matics and statistics, engineering, the physical sciences, and the biological sciences during the period 1955-1966, with projections to 1976. As in the case of bachelor's degrees, the annual number of master's degrees granted shows a much greater factor of gain over the years 1955-1965 for mathematics and statistics (5.6) than for engineering (2.7), the physical sciences (1.9), or the biological ~ ;~ ~ ,~ ON ~' . . ~ . Amp `~.=J. alla Grille IS lrUe at tne factors of gain projected for 1965-1975: for mathematics and statistics this is 3.5, while for engi- neering, the physical sciences, and the biological sciences these projected factors are 2.6, 2.1, and 2.3, respectively. The projected factor ot increase that Table 9 yields for mathematics and statistics may be high, because the assumption on which it is based, that the percentage distribution of degrees by field will continue the 1955- 1965 trends, may not be completely fulfilled. On the other hand, the category "mathematics and statistics," which for the period 1955-1965 included virtually all the degrees of that period in the mathematical sciences, may miss significant numbers of these de- grees in the period 1965-1975, especially degrees granted in the rapidly proliferating graduate departments of computer science. It is clear that the standards and requirements for the master's r . r _ # Rounded by us to two significant figures. See footnote on page 122.

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Graduate Education 137 TABLE 9 Earned Master's Degrees, 1955-1966, with Projections to 1976 MATHEMATICS YEAR EARNED, 1955 - 1966 PHYSICAL BIOLOGICAL AND STATISTICS ENGINEERING SCIENCES SCIENCES 1954-1955 761 4,484 2,544 1,609 1955-1956 892 4,724 2,653 1,754 1956-1957 965 5,233 2,704 1,801 1957-1958 1,234 5,788 3,034 1,852 1958-1959 1,509 6,753 3,202 2,007 1959-1960 1,765 7,159 3,387 2,154 1960-1961 2,238 8,178 3,799 2,358 1961-1962 2,680 8,909 3,929 2,642 1962-1963 3,323 9,635 4,132 2,921 1963-1964 3,603 10,827 4,567 3,297 1964-1965 4,294 12,056 4,918 3,604 1965-1966 5,220 13,990 5,470 4,390 PROJECTED TO 1976 1966-1967 5,900 15,000 5,800 4,600 1967-1968 6,400 16,000 6,000 4,700 1968-1969 7,400 18,000 6,500 5,200 1969-1970 9,300 22,000 7,800 6,300 1970-1971 11,000 25,000 8,800 6,900 1971-1972 12,000 26,000 8,900 7,100 1972-1973 13,000 27,000 9,200 7,400 1973-1974 14,000 29,000 9,800 7,900 1974-1975 15,000 31,000 10,000 8,200 1975-1976 17,000 34,000 11,000 8,800 degree vary widely, though too little is known about this in a com- prehensive way. In some departments, a master's degree is awarded simply upon the completion of a specified amount of course work beyond the bachelor's degree; sometimes the passing of comprehen- sive examinations (written, oral, or both) is required; in some pro- grams a master's thesis is required; and in some departments a master's degree is awarded as a kind of "consolation prize" to those who try and fail to complete the requirements for a doctorate. For the mathematical sciences, about 60 percent of the master's degrees currently being awarded are granted by departments of mathe- matics that also offer a doctorate, and for these departments the CBMS survey furnishes a bit more information. Of the students ad- mitted to study toward graduate degrees in such departments in the fall of 1965, the department expected about 51 percent to get only

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138 The Mathematical Sciences in Education mathematics master's degrees, about 12 percent to get only master's degrees in the teaching of mathematics, and about 37 percent to get doctoral degrees. Of the last group, roughly half already had master's degrees. At about one fourth of these departments a thesis is definitely required for a master's degree; at about one half there are two alternative master's programs, one involving a thesis and one not; at the others no thesis is required, the requirement tending to consist of course work plus various kinds of comprehensive ex aminations. On the average, about one full year of course work is required for a master's degree in the mathematical sciences, and the median elapsed time from bachelor's to master's degree is be- tween one and two years. Many of those studying only for the master's degree are preparing to be mathematics teachers at the high school level. Though high school teachers usually teach more than one subject, it has been estimated that those high school teachers whose primary respon- sibility is mathematics teaching currently number approximately 120,000; and of these perhaps as many as 20 percent have master's degrees either in mathematics or in the teaching of mathematics. In addition, some mathematics MA's will certainly continue to find teaching positions in four-year colleges and universities. As we have seen (Table 3, page 128), in 1965 some 4,650, or 43 percent, of the [till-time faculty of 10,750 in such institutions had the master's degree as their highest earned degree. Junior colleges offer a still small but fast-growing field of opportunity for those with mathemati- cal training through the master's level. In the fall of 1966, according to the CBMS survey, there were about 2,700 mathematics teachers in junior colleges, of whom 84 percent, or 2,300, had master's degrees. (About 4 percent had doctorates, and the other 12 percent had bachelor's degrees.) Those employed by industry or government whose major educational field was mathematics now probably num- ber well over 30,000 (see references 30 and 31~; and of these approxi- mately one quarter have master's degrees as their highest earned degree (see reference 31, page 22~. THE DOCTORATE IN THE MATHEMATICAL SCIENCES As we have seen, the annual number of bachelor's degrees and the annual number of master's degrees showed much stronger gains

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Graduate Education 139 TABLE 10 Earned Doctor's Degrees, 1955-1966, with Projections to 1976 Y EAR MATHEMATICS PHYSICAL BIOLOGICAL AND STATISTICS ENGINEERING SCIENCES SCIENCE EARNED, 1955 - 1966 19501955 250 599 1,713994 1955-1956 235 610 1,6671,025 195~1957 249 596 1,6741,103 1957-1958 247 647 1,6551,125 1958-1959 282 714 1,8121,045 1959-1960 303 786 1,8381,205 196~1961 344 943 1,9911,193 1961-1962 396 1,207 2,1221,338 1962-1963 490 1,378 ,3801,455 1963-1964 596 1,693 2,4551,625 196~1965 688 2,124 2,8291,928 1965-1966 770 2,350 2,9602,030 PROJECTED TO 1976 1966-1967 860 2,700 3,1009,100 1967-1968 900 3,100 3,5002,400 1968-1969 1,100 3,600 3,9002,700 1969-1970 1,200 4,000 4,0002,700 1970-1971 1,300 4,100 4,0002,700 1971-1972 1,400 4,600 4,2002,900 1972-1973 1,800 5,700 5,0003,400 1973-1974 2,000 6,600 5,6003,800 19701975 2,100 6,900 5,5003,800 1975-1976 2,200 7,300 5,6003,800 over the period 1955-1965 for mathematics and statistics than for engineering, the physical sciences, or the biological sciences; and the same was true to a more moderate extent of the projected gain factors for the period 1965-1975. For doctoral degrees the corre- sponding statistics and projections of the U.S. Once of Educa- tion,~7 shown in Table 10, do not show any such decisively stronger gains for mathematics and statistics. The factors of gain in these numbers of doctoral degrees for the period 1955-1965 are: mathe- matics and statistics, 2.8; engineering, 3.6; physical sciences, 1.7; and biological sciences, 1.9. For the period 1965-1975 the corresponding ~ Rounded by us to two significant figures. See footnote on page 122.

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140 The Mathematical Sciences in Education projected factors are: mathematics and statistics, 3.0; engineering, 3.2; physical sciences, 1.9; and biological sciences, 2.0. For doctoral degrees, then, it is engineering, and not mathematics and statistics, that shows the highest factors of gain. While these gain factors are higher for mathematics and statistics than for the physi- cal or biological sciences, the "conversion factor" from master's de- gree to doctorate is lower. Allowing for a time lag of two to four years between receipt of the master's and doctoral degrees, we can compare the total masters production over a period (Table 9) with the total PhD production over a similar period (Table 10) three years later. When this is done, the following conversion factors are obtained, representing the approximate percentage of masters in a ten-year period who go on to get a doctorate three years later: Mathematics and statistics Physical sciences Biological sciences 30% 80% ~2= ' ~ /o Without better knowledge than we have of the flow of students through various degree levels and into various occupations, we can account only speculatively and tentatively for this relatively small conversion factor for mathematics and statistics. One reason for it appears to be the very considerable proportion-quite a large ma- jority of mathematics graduate students working only for the mas- ter's degree, many of them studying to be teachers. As noted above in our discussion of the master's degree, even in mathematics depart- ments offering the PhD degree this currently amounts to approxi- mately 63 percent of those studying for graduate degrees; and presumably it includes a much larger percentage of those studying in graduate mathematics departments in which only a master's degree is offered. Another reason may be simply that doing research ac- ceptable for the PhD degree is harder in mathematics than in other fields, leading fewer to try for this degree in mathematics and lead- ing to a higher attrition rate among those who do. Research for the PhD degree in mathematics has traditionally meant the discovery of new mathematics, as contrasted with the scholarly synthesis of pre- vious work often found in the humanities or the obtaining of new experimental results by established techniques often found in the laboratory sciences.

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Graduate Education THE DOCTORATE OF PHILOSOPHY AND INTERMEDIATE DEGREES 141 For academic fields generally, questions have been raised concerning the relevance of the PhD degree for college teaching tsee, e.g., E. Walters and F. W. Ness, "The Ph.D.: New Demands, Same Old Response," Saturday Review, 49, 62 (January 15, 1966~. Alternative programs and doctorates have from time to time been proposed (notably, in the mathematical sciences, an expository-thesis program leading to a proposed "Doctor of Arts" degree), but these have re- ceived little support. Our Panel on Graduate Education has given careful consideration to the PhD and intermediate degrees in mathe- matics as degrees for college teachers. The following discussion is based closely on their report. The PhD degree in the mathematical sciences has a unique value for university and college teachers in this field. Mathematics and its applications cannot be learned passively that is, not merely by listening, reading, and studying. It is only by doing substantial prob- lems, that is, by a genuine apprenticeship in research, that a grad- uate student can really absorb the mathematical way of thinking. The writing of a PhD thesis is an exciting and important phase in the development of a mathematician. He is at last functioning as a professional and not merely as a student. He realizes the difficulties, frustrations, and sheer hard work that the creation of mathematics requires and so will view with deeper insight the efforts of his col- leagues to do mathematics. His sense of participation as a contrib- utor to his subject will give him a greater authority and involvement as a teacher. The attainment of a PhD degree is an excellent scheme for making students gain the insight that comes from doing research. This insight is more important than ever now that purely routine use of mathematics is becoming totally inadequate for applications in other fields. For all these reasons it is highly desirable that, where- ever possible, mathematical-science faculty members in universities and colleges should have the PhD degree in their field. (Compare the statement on the role of the PhD as discussed in reference 29.) There are, of course, quite a few universities in which all, or vir- tually all, the full-time faculty members of the mathematics depart- ment do have PhD degrees. These would certainly include the 25 universities with mathematics departments rated as "excellent" or "strong" in over-all faculty quality listed on page 66 of Cartter's

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142 The Mathematical Sciences in Education study.32 As far as such universities are concerned, there have quite recently been indications of "saturation at the top," reflected in the fact that young mathematics PhD's of high ability and research promise are, in increasing numbers, finding positions in a broader range of universities. This means that the time is ripe for systemati- cally developing excellence in more mathematics departments through such programs as the National Science Foundation's Sci- ence Development Programs (see the discussion of Developmental Block and Area Grants on page 174~. In the country's colleges and universities in general, however, the problems of providing adequate numbers of PhD faculty in the mathematical sciences seem overwhelmingly severe. The CBMS survey has determined that, as of 1966, over 70 percent of the liberal arts colleges and teachers' colleges institutions that together carry nearly half of the undergraduate teaching load had mathematical- science faculties with at most one PhD member, while over 40 percent had mathematical-science faculties with no PhD members. In the face of this situation, the Committee on the Undergraduate Program in Mathematics has discussed and evaluated several lower levels of preparation in its report, Qualifications for a College Faculty in Mathematics.29 It concludes that a teacher with what it calls the "Advanced Graduate Component" has the subject-matter background to teach all the undergraduate mathematics except spe- cialized courses intended for students who are going to be research mathematicians. This Advanced Graduate Component is essentially the present-day training for the PhD degree except for the thesis. A student so trained at a university with a strong doctoral program will have participated for several years in the life of a research de- partment; he is likely to have absorbed some of the research at- mosphere and will therefore be less narrow in his interests and ultimately a better teacher than a student who has taken only a minimal master's degree. Moreover, he could complete this level of training and enter teaching in less time than he would require to complete his PhD degree. Discussions with representatives of four-year colleges (see, e.g., reports of CUPM regional conferences33) show that many colleges would welcome teachers with the Advanced Graduate Component. Indeed, there have been suggestions even from the universities that, for undergraduates, the teacher who is excited about mathematics and teaches creatively, even if he does not have the PhD degree, is sometimes to be preferred to a teacher who holds a PhD degree.

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Graduate Education 143 For example, R. L. Wilder, in his paper "The Role of Intuition" iScience, 156, 605 (May 5, 1967) i, says: As the student goes on to more advanced work, the intuitive component of his training begins to assume more importance. At this stage of his career it may be assumed that he is possibly going on to do some kind of creative work, if not in mathematics, then in some other science. And it is desirable that his teachers have had some experience with creative work. This does not mean that the teacher must have a Ph.D. degree; this is a fetish I wish we could get rid of. I would much prefer a teacher without a Ph.D. who is excited about mathematics and can teach creatively, than a teacher with a Ph.D. who is neither enthusiastic about mathematics nor capable of inspir- ing his students. Two questions now need to be answered. First, how many grad- uate students in the mathematical sciences stop their training at the level of the Advanced Graduate Component? Second, should this level of training be recognized in some formal way; and if so, in what way? There are no clear-cut answers to either question, but some reasonable guesses and comments can be made. Estimates based on the general study, Attrition of Gradual;e Stu- dents at the Ph.D. Level in the Traditional Arts and Sciences,34 sug- gest that, on the average, for every 100 PhD's produced there are per- haps 15 dropouts who will have reached the Advanced Graduate Component stage. Thus, corresponding to the approximately 800 PhD's peryear currently produced in the mathematical sciences, there are perhaps 120 who stop at the Advanced Graduate Component level. It seems likely that this number could be considerably in- creased, if it seems desirable to do so, by offering intermediate de- grees to students who are enthusiastic about mathematics but either lack the talent for original research or, possessing this talent only to a moderate extent, are not lucky enough to find supervisors who can help them develop it to the research-thesis level. On the other hand, it may be guessed that the present number of students (perhaps 120 a year) who complete PhD programs except for the thesis would not be greatly increased if an intermediate degree were generally avail- able; it might rise to 200 or 300 a year at the most. As for the question of formal recognition of the Advanced Grad- uate Component level of training, there is widespread agreement in the mathematical community that training to this level should be formally recognized in some way beyond the mere award of the master's degree. According to the CBMS survey, about 70 percent of the mathematics departments that grant the PhD degree are of this

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144 The Mathematical Sciences in Education opinion. Indeed, several distinguished universities (for example, Yale and Michigan) have recently instituted such degrees. The ex- istence of an intermediate degree would certainly make it easier to assess the quality of a prospective teacher or of a college faculty. It is a much more controversial question whether an intermediate degree should be called a doctorate of some kind. The research thesis has been a traditional requirement for a doctorate in the mathe- matical sciences in the United States. Many people feel that it is dishonest to let a degree without such a thesis be called a doctorate, and thus to acquire a measure of the prestige that is justly associated with the present PhD degree in the mathematical sciences. A number of colleges have difficulty in giving permanent employment to teachers if they do not have doctorates. From one point of view, of course, this is a good thing, since it tends to prevent the development of a large body of college teachers who permanently occupy positions that might otherwise come to be occupied by younger and more highly qualified ones. It has also been argued that it would be dan- gerous to provide an easy alternative to the PhD degree, since many students might then be satisfied with it and so would miss the valu- able experience of writing a thesis. On the other hand, there are also strong arguments for recognizing the Advanced Graduate Component as a doctorate. The most sig- nificant one is precisely that college administrations are reluctant to grant tenure and status to teachers who do not have doctorates. As pointed out earlier, the best available projections indicate that there is not going to be either a surplus or an adequacy of PhD's in the mathematical sciences, at least for quite a few years, and conse- quently that many college teachers will not have the PhD degree. Many people feel that there is little more reason to suppose that the Advanced Graduate Component teachers, more than the holders of the PhD degree, will become mathematically fossilized. The general effectiveness of the mathematical teaching in colleges would be in- creased if demonstrably competent people without PhD degrees could be retained on faculties instead of being dismissed in order to satisfy technical requirements about the numbers of doctorate hold- ers on faculties. If an intermediate degree at the Advanced Graduate Component level in the mathematical sciences were to be given only by depart- ments that already have strong PhI) programs (and want to give it), the majority of our Panel on Graduate Education would be in favor

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Graduate Education 145 of calling it a doctorate. While concurring that this level of training should receive some formal recognition, we tend to favor Professor Leonard Gillman's suggestion of the title, "Associate PhD," by analogy with Associate Professor, and with emphasis on the possibil- ity of eventual completion of a full PhD degree. GR\DUATE-STUDENT PARTICIPATION IN UNDERGRADUATE TEACHING There has been some criticism of the quality of mathematics teach- ing by graduate assistants in universities. Where this criticism is justified (and we believe that it often is not), some have adduced as a main cause the fact that teaching assistants get little or no ex- plicit guidance as teachers. While there may be some justice in this, we feel that the major cause lies rather in an unbalanced situation, in which a fellowship or traineeship student does no teaching at all, while a teaching assistant may do so much that he cannot do it well and still carry on his own program of graduate studies. We propose changes below. Universities use graduate assistants in freshman-sophomore mathematics teaching in two ways, mainly. Where elementary courses are taught in a number of separate sections of moderate size (20 to 3() studentsy, graduate assistants may teach some of these sec tions. Where SUCh courses are taught in large lecture sections (up- wards of 100 students), graduate assistants tonically conduct small ,, , recitation groups supplementing the lectures. The CBMS survey found that in academic year 1965-1966 universities with PhD programs in the mathematical sciences used large lecture sections in elementary calculus to varying degrees: approximately 55 percent of these uni- versities used no large lecture sections at all; approximately 18 per cent made some use of large lecture sections but for fewer than three fourths of their students; and approximately 27 percent used large lecture sections for three fourths or more of their students. Our Panel on Undergraduate Education has pointed out advan- tages of large lecture sections, both for the students and for the graduate assistants, and has recommended that more teaching be done in this way (see the section on Methods Used to Relieve the Shortage of Teachers in Chapter 4 of reference 1~. While recognizing that there are also arguments in favor of small sections and that the

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146 The Mathematical Sciences in Education mathematical-science departments at many universities will not find it possible or advantageous to use large lecture sections, we believe that this recommendation deserves serious consideration. Whatever the method a university employs for teaching its elementary courses, we firmly believe that it is important for all graduate students in the mathematical sciences to have had the ex- perience of participating in such teaching at some time during their graduate years. Currently about 70 percent of those who obtain the PhD degree in the mathematical sciences enter academic works Certainly for these, some teaching experience during the graduate years (and during their early postdoctoral years as well!) is a highly practical part of career training. We feel it is best for a graduate student in the mathematical sci- ences to acquire this teaching experience during his middle grad- uate-study years. He should not be burdened with teaching during his initial year of graduate study, when he must adjust to especially demanding and intensive course work; and during his final year of studying for his PhD degree, he may need to devote his full energies to research and thesis writing. Even during the middle graduate- study years, a graduate student's teaching load should be such that he can do justice to both his teaching and his own studies. Spe- cifically, three to fire hours a week of classroom work is as much as a graduate assistant should have to carry. We believe that if participation in teaching is spread in this way, so that all or most graduate students do a limited amount of teach- ing during their middle graduate-study years, then no graduate stu- dent will have to do an excessive amount of such teaching, and the over-all quality of the teaching will improve. Arranging to spread graduate-student teaching participation in this way will plainly in- volve some revision in the rules and administrative procedures gov- erniIlg fellowships, traineeships, arid teaching assistants. We believe, however, that such revision should not be overwhelmingly difficult. and that the result would be worth the effort. What seems to be needed is a larger number of fellowships and traineeships, but with a provision for limited participation in teaching during the middle graduate-study years. # It is not correct to conclude that therefore approximately 30 percent go into industry or government. Actually, about 15 percent enter industry or government and the remaining 15 percent are simply lost track of (see reference 16). It is a fair guess that the majority of these latter come from abroad for graduate study and return to their home countries after obtaining PhD degrees, usually to enter academic work there.