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7 Unclergraduate Education Mathematics departments in colleges and universities today serve a wide variety of student majors. They train future mathematical scientists, both for academic work and for work in industry and government, and, as it has for centuries, mathematical training plays a key role in the education of physical scientists and engineers. In recent years mathematical methods have proved to be of increasing use in the biological and social sciences as well, and so the mathe matician is now acquiring another large gTOUp of clients. Second- ary and elementary school teachers form yet another increasing contingent. Some knowledge of mathematics has traditionally been recognized as a significant part of the general education of a college student: today this knowledge is being broadened to include some appreciation of the widely useful tool of computers. Chapter 3 of the report of our Panel on Undergraduate Edu- cationt surveys in detail these various components of demand for mathematical training; and, on this basis, Chapter 4 of that report makes an assessment of the resulting needs for college and university staff in the mathematical sciences. We give a briefer discussion along similar lines in the sections below, on total mathematical-science course enrollments and on quality and distribution of mathematical- science faculty. The first thing to be emphasized about the recent history of undergraduate education in the mathematical sciences is the extent of its growth and change over the past 25 years. Our Panel on Undergraduate Education has documented this vividly in Chapter 121
122 The 1~lathematical Sciences in Education 2 of its report, presenting eight detailed case histories of changes at individual colleges and universities. While these institutions repre- sent considerable variety in their educational goals and in their clientele, they reveal many similar trends over the last quarter cen- tury with regard to the mathematical sciences. The following items occur repeatedly: very significant increases in mathematics-course enrollments; spectacular increases in numbers of mathematics majors; new undergraduate major programs in the mathematical sciences; many new advanced courses; increased undergraduate enrollments in graduate courses; impact of improved high school curricula on beginning college mathematics courses; special new courses for statistics, computing, and the social sciences; significant increases in staff size; difficulties in recruiting new staff. THE INCREASE IN MATHEMATICS MA TORS Perhaps the most striking item on the above list is the greatly increased number of mathematics majors. This is documented more precisely in Table 1, which shows comparative tabulations and projections by K. A. Simon and M. G. Fullam of the U.S. Office of Education.~7 These tabulations show that over the years 1955-1965, during which college enrollments roughly doubled, the annual number of hachelor's degrees granted in mathematics and statistics increased by a factor of 4.9, while for engineering, the physical sciences, and the biological sciences the corresponding factors were only 1.6, 1.7, and 2.8, respectively. The projections shown for the years 1965- 1975 call for the annual number of bachelor's degrees ire mathe- matics and statistics to increase by a factor of 3.0, while for engineer- ing, the physical sciences, and the biological sciences the correspond ~ We have rounded off these projections to two significant figures, though, no doubt, for a year as remote as 1976 it is at most the initial figure that is at all likely to be borne out. The projections are based on the assumption that the percentage distribution of degrees by field will continue the 1955-1966 trends. This is, we feel, a reasonable basis for projections; but should it turn out to be only indifferently fulfilled, the projections could, of course, be even more in error. For instance, projections from the 19~55-1959 trends on this basis would hardly have predicted the depression in numbers of engineering bachelor's de- grees that actually occurred in the early 1960's. as shown in Table 1.
Undergraduate Education i23 TABLE ~ Earned Bachelor's Degrees, 1955-1966, with Projections {0 1976 YEAR MATHEMATICS PHYSICAL BIOL~ICAL Ai!lD STATISTICS ENGINE~NG SCIENCES SCIENCE EARNED, 1955 - 1966 1954-1955 4,034 22,589 10,516 9,050 1955-1956 4,660 26,312 11,672 12,566 1956-1957 5,546 31,211 12,934 13,868 1957-1958 6,924 35,332 14,352 14,408 1958-1959 9,019 38,134 15,460 15,149 1959-1960 11,437 37,808 16,057 15,655 1960-1961 13,127 35,866 15,500 16,162 1961-1962 14,610 34,735 15,894 17,014 1962-1963 16,121 33,458 16,276 19,218 1963-1964 18,677 35,226 17,527 22,827 1964-1965 19,668 36,795 17,916 25,305 1965-1966 21,190 35,830 18,020 25,680 I'ROJEC~ TO 1976 1963-1967 24,000 37,000 19,000 28,000 1967-1968 31,000 43,000 23,000 33,000 1968-1969 36,000 45,000 24,000 37,000 1969-1970 38,000 44,000 26,000 38,000 1970-1971 41,000 43,000 27,000 39,000 1971-1972 45,000 44,000 28,000 41,000 1972-1973 49,000 44,000 30,000 44,000 1973-1974 55,000 45,000 31,000 47,000 1974-1975 60,000 46,000 33,000 50,000 1975-1976 65,000 46,000 35,000 53,000 ing projected factors of increase are 1.3, 1.9, and 2.0, respectively. Admittedly, projections can be no more than rough guides; and, in fact, discrepancies for 1965 between the above actual figures and the Once of Education's slightly earlier projected figuresl8 suggest that the above projected factors of increase for 1965-1975 might turn out to be somewhat high for mathematics and statistics and some- what low for the biological sciences. Even so, there seems little doubt that increased numbers of majors will contribute considerably to college staffing strains in the mathematical sciences over the next few years.
124 The Mathematical Sciences in Educations TOTAL ENROLLMENTS IN THE MATHEMATICAL SCIENCES The Lindquist and CBMS surveys together show that over the period Mom academic years 1960-1961 to 1965-1966 the total enrollment in undergraduate mathematical-science courses increased from 744,000 to 1,068,000, or 44 percent. Since the most closely compar- able general enrollment figures show a 48 percent increase over this same period, we conclude that for this period the growth in under- graduate mathematical-science enrollments has been roughly the same as the growth in college enrollments generally. (See reference 16.) For the succeeding period from 1965-1966 to 1970-1971, however, we have reason to believe that this situation will change consider- ably. If the projections below, which are a more conservative ver- sion of those of our Undergraduate Panel, are at all near the mark, then mathematical-science enrollments will increase by more than 70 percent over this period, while general enrollments will increase by only around 30 percent. The remainder of this section presents the case for these projections. For the academic year 1965-1966 the CBMS survey found that in the four-year colleges and universities of the United States there were some 1,068,000 undergraduate course enrollments in the mathe- matical sciences. For these colleges and universities, the U.S. Once of Education predicts a 29 percent rise in total undergraduate enrollments over the period 1965-1970. Thus, taking 29 percent of 1,068,000, one might predict a rise over this period of approximately 310,000 undergraduate course enrollments in the mathematical sci- ences. We believe this to be a serious underestimate because of two factors. The first of these factors is the projected faster-than-average growth in the number of majors in certain fields where it is custom- ary to take a greater-than-average amount of mathematical course work. The second is the fact that majors in several fields are no beginning to take more courses in the mathematical sciences than formerly and by 1970 will almost surely be taking significantly more such courses on the average than in 1965. Chapter 4 of the report of our Panel on Undergraduate Edu- cation~ has analyzed, for each of the principal fields concerned, the extent to which one or both of these factors may be expected to increase mathematical-science course enrollments over the 196.~- 1970 period, above and beyond the 310,000 attributable to the gen
Undergracluate Education 125 TABLE 2 Projected Mathematical-Science Course-Enrollment In- crease, 1965-1970 FACTOR General growth Mathematical-sciences excess Physical-sciences excess Engineering excess Biological-sciences excess Psychology excess Social-sciences excess Elementary-teacher excess Introductory-computing excess rotary INCREASE 310,000 81,300 37,000 37,000 9,500 19,500 67,500 60,000 172,000 796,800 evil 29 percent growth predicted for college enrollments over this period. This analysis is based on the following: (1) the U.S. Office of Education projections, 7 regarding the number of bachelor's de- grees to be expected in various fields in 1970; (2) information from the CAMS survey regarding the average number of mathematical- science courses taken by majors in these various fields as of 1965; and (3) the Committee on the Undergraduate Program in Mathe- matics (C~PM) panel recommendations,~9-23 as an indication of the number of mathematical-science courses to be reasonably antic- ipated for majors in these various fields by 1970. We and our Panel on Undergraduate Education are very much aware of the oversimplifications and possibilities for error involved in making, for several years ahead, any course-enrollment projec- tions whatever. For this reason, the Panel on Undergraduate Edu- cation has, in Chapter 4 of its report, considered the consequences of varying its projected data and its hypotheses in several reasonable ways. The prediction of intensification, for several years to come, of the shortage of qualified college teachers of the mathematical sci- ences is found to be stable under all these variations. Our own figures, shown in Table 2, provide a further variation. It too pre- dicts an intensified shortage in the supply of qualified teachers over the next few years. It is for these reasons that we feel some confidence in this prediction. With these qualifications and reservations, we present, in Table 2' our best effort to project mathematical-science course-enrollment
126 The Mathematical Sciences in Education increases from 1965 to 1970. The first item in this tabulation is the one mentioned earlier: the basic 310,000 increase in mathematical- science course enrollments to be anticipated simply from the U.S. Once of Education's prediction of a 29 percent general increase in college enrollments. For the mathematical sciences, the physical sciences, and engi- neering, the projected excesses shown in Table 2 are those computed by our Panel on Undergraduate Education. The mathematical- sciences excess arises from the more rapid growth in mathematical- science majors already discussed, together with the fact that such a major accounts for approximately six course enrollments in his field during his undergraduate years. The physical-sciences excess arises from two factors: first, a higher-than-average projected rise in the number of majors, who in 1965 typically took approximately three mathematical-science courses during their undergraduate years; and second, the prediction, based on reference 23, that by 1970 these majors will be taking four rather than three such courses. For engi- neering, the projected 1965-1970 increase in majors is at 21 percent instead of the general figure of 29 percent, which by itself would yield a deficiency rather than an excess in mathematical-science course enrollments; but this deficiency is more than offset by the tendency, already beginning to be seen, for the undergraduate engi- neering curriculum to shift from a five-semester to a seven-semester sequence in the mathematical sciences (see reference 24~. For the biological sciences, psychology, and the social sciences, the figures shown in Table 2 are just half as large as those in the report of the Panel on Undergraduate Education. Our figures correspond to the assumption that, on the average, by 1970 half of the majors in these fields will be taking, during their undergraduate careers, one more mathematical-science course than was typical for the majors in these fields in 1965. (The Panel on Undergraduate Education made the assumption that by 1970 all majors in these fields would on the average be taking one more such course.) With regard to elementary school teachers, we recognize that a strong movement is already under way to stimulate the upgrading of their preservice training in mathematics. The CUPM report23 and regional conferences of elementary teachers sponsored by the CUPM25 have done much in this respect. As a result, we feel that, on the average, perhaps two thirds of these teachers will by 1970 be taking one more mathematics course during their undergraduate years than in 1965 (whereas the Panel on Undergraduate Educa
Unclergraduate Education 127 lion felt it likely that, on the average, all elementary teachers would be taking one more such course by 1970~. It is particularly difficult to make predictions regarding the rapidly growing and changing field of computing. The recently published Pierce reports recommends that by academic year 1970- 1971 all college students should have an introductory course in computing. It may turn out that the ideal place for such an intro- ductory course is in high school rather than college; or it may turn out that introductory computing courses oriented toward various subject-matter fields will tend to be taught in various college de- partments, much as elementary applied statistics courses tend to be today. The most likely assumption, however, seems to be that the burden of teaching such courses will fall on the college faculty in computer science and the mathematical sciences generally. In any event, our Panel on Undergraduate Education has made its projec- tions on the assumption that half of the 5,500,000 undergraduates in four-year colleges in 1970 will take, at some time during their four-year undergraduate careers, a one-semester college course in computing, and that half of these are not already included else- where in the tabulation ire Table 2. (This last is certainly not un- reasonable in view of the present distribution of college majors.3 This yields the equivalent of about 172,000 (academic-year) course enrollments for 1970, and it is this figure that appears in Table 2. Table 2 yields a total projected course-enrollment increase of 796,800. Dividing this by the course-enrollment total of 1,068,000 for academic year 1965-1966, we find that, over the 1965-1970 period, mathematical-science course enrollments are projected to increase by approximately 74 percent, which is more than 2.5 times the 29 percent by which general college enrollments are projected to increase over this period. QUALITY AND DISTRIBUTION OF MATHEMATICAL-SCIENCE FACULTY The CBMS survey found that for the four-year colleges and univer- sitiLes of the United States the 1,068,000 undergraduate mathemati- cal-course enrollments of academic year 1965-1966 were handled by a full-time faculty of some 10,750, aided by a quite small part-time faculty and, in universities, graduate assistants. This yields a ratio of one full-time faculty member to approximately 100 course en
128 The Mathematical Sciences in Education TABLE 3 Highest Famed Degree, 1965-1966 DECREE NUMBER Doctorate (mathematical sciences) Doctorate (education) Doctorate (other fields) Masterts degree Bachelor's degree TOTAL 5,000 500 coo 400 10,750 emollients. Now the preceding section estimated some 796,800 mOre course enrollments in the mathematical sciences for academic year 1970-1971 than for academic year 1965-1966. Therefore, this course-enrollment increase will require some 8,000 more full-time faculty in the mathematical sciences for academic year 1970-1971 than for academic year 1965-1966. We now try to assess what the quality of the additional faculty Is likely to be, using as a rough measure of faculty quality the proportion of full-time faculty holding doctorates, as is done in Cartter's studies.26 ik For academic year 1965-1966, the CBMS survey found the 10,750 full-time faculty in the mathematical sciences to have highest earned degrees distributed as shown in Table 3. Thus in academic year 1965-1966 some 53 percent of the mathe- matical-science faculty held doctorates in some field, a little over 46 ~ Academic degree is, of course, only one dimension in the measure of quality. it is clear, for instance, that one is qualified to provide instruction in a core mathematics topic only if he has a thorough grasp of the substantive material of that topic, of its relation to mathematics at large, and of the manner in which others will build upon the foundation it provides. By the same token, one is qualified to provide instruction on a facet of the use of mathematics only if (again) he has a thorough grasp of the mathematical disciplines he must use, of the facets of science or technology under study, and of the attitudes and objec- tives that are appropriate to the questions with which he is coping. Thus, not only must there be an adequate level of competence as measured by academic degree acquisition, but there must also be a breadth of topical competence measured in terms of ranges of interest, attitudes, and scientific literacy. Clearly, in institutions that are only beginning to develop comprehensive prc>- grams, these requirements of quality cannot be met immediately, but they should be used as guidelines for that development. A minimum first step requires that the means be found to assist faculty members to maintain contact with the changes in relevant fields of }knowledge.
t~nclergraduate Education 129 percent being in the mathematical sciences. Shortly we shall dis- cuss the distribution of doctorate-holding faculty among colleges and universities, which is far from uniform. To see how the percentage of doctorates is likely to change over the next few years, we now need an estimate of the number of new PhD's who will enter mathematical-science teaching in four-year colleges and universities over the period 1965-1970. For this we take the estimate of our Panel on Undergraduate Education, which found this number to be approximately 3,300 (see Chapter 4 of their report)). Their analysis followed Cartter's modeler and used his figure of 2 percent per year for attrition (due to death, retirement, and net outflow to other professions) of doctorate holders from college and university teaching. Also, in accord with information from the csMs survey, they assumed that 70 percent of the newly produced mathematical-science PhD's will go into college and uni- versity teaching, as opposed to Cartter's value of approximately 33 percent for all academic fields combined. To collect our figures, we have estimated that over the period 1965-1970 some 8,000 new full-time staff members will be needed in n~athematical-science teaching in the four-year colleges and uni- versities. We also have the estimate of net inflow of some 3,300 PhD's into such teaching during this period. Thus only about 41 percent of the new faculty will have doctorates; hence, if these estimates are at all close, the percentage of doctorate holders on mathematical-science faculties, currently about 53 percent, will decline. The distribution of the doctorate-holding faculty is far from uni- form, the universities having more than twice the fraction of the total doctorate-holding faculty that liberal arts and teachers' col- leges [have, while the latter carry slightly more of the total under- ~aduate teaching load. Specifically, the scams survey reveals that for academic year 1965-1966 the distribution of doctorate-holding faculty was approximately that shown in Table 4, which should be read with the following comments in mind. [First, it is not surprising that the universities have a much higher ~ With the strong demand from other quarters, it is a distinct possibility that i the future fewer than 70 percent of the new PhD's will go into academic work. This could intensify the college teacher shortage predicted here. In any case, it should be emphasized that the prediction of a growing shortage is stable under any reasonable variation in the particular percentages used for attrition and for resew PhD's entering teaching.
130 The Mathematical Sciences in Education TABLE 4 Distribution of Doctorate-Holding Faculty, 1965-1966 PERCENTAGE OF THE PERCENTAGE OF THE UNDERGRADUATE DOCTORATE- HOMING TYPE OF INSTITUTION TEACHING LOW FACULTY Universities Liberal arts and teachers' colleges Technological institutes TOTALS 45 49 6 100 63 30 - 100 fraction of a limited doctorate-holding faculty, since it is univer- sities that carry the bulk of graduate instruction and the direction of doctoral and postdoctoral research. Second, as far as the first two undergraduate years are concerned, the proportion of PhD's teach- ing at universities is much more closely comparable with that at liberal arts colleges and teachers' colleges, because of the wide employment of graduate assistants in such teaching at universities. In this connection, the CBMS survey has found that, in the median university, 40 percent of the freshman-sophomore teaching load is carried by graduate assistants, and, in 38 percent of the universities, at least half of the freshman-sophomore load is carried in this way. In Chapter 8, we discuss (see page 145) Graduate Student Par- ticipation in Undergraduate Teaching in universities and suggest improvements for the future. In Chapter 9, in the discussion of The College Teacher (page 147), we consider what may be done to meet anticipated shortages in qualified faculty in the mathematical sci- ences, especially in the weaker colleges and in certain critical fields. THE JUNIOR COLLEGES There are now more entering freshmen in junior colleges than in universities, and over one third of all entering freshmen are junior college students. This statement is impressive even when it is under- stood that the term "junior colleges" here includes two-year techni- cal institutes, and that the data include part-time students and students in occupational or general studies programs not chiefly creditable toward a bachelor's degree. The actual figures, from the U.S. Once of Education,28 are shown in Table 5. Total enrollment
Undergraduate Education 131 TABLE ~ First-Time Enrollments in Fall 1966, Nondegree Credit and Part-Time Students Included NUMBER (THOUSANDS OF STUDENTS) PERCENTAGE OF TOTAL Universities 427 27 Other four-year institutions 591 38 Two-year institutions 547 35 TOTALS 1,565 100 figures with a breakdown into full-time and part-time students are also instructive and are given in Table 6. These figures show that in 1966 two-year institutions accounted for 21 percent of all under- graduate enrollments, 17 percent of the full-time ones, and 30 per- cent of the part-time ones. Enrollments in junior colleges are geographically heavily con- centrated in certain states, reflecting not only differences in popu- lation density but also differences in state policy regarding the establishment and expansion of these institutions. Thus 38 percent of all junior college students are in California, and slightly over 50 percent attend junior colleges in California, Florida, or Illinois. Against this background of general student enrollments we now give a few results from the 1966-1967 CBMS survey of the mathe TABLE 6 Total Enrollments in Fall 1966, Nondegree Credit and Part-Time Students Includeda THOUSANDS OF STUDENTS TOTAI~ FULL-TIME PART-TIME Universities 2,4821,789 693 Other four-year institutions 2,6261,941 685 Two-year institutions 1,331739 591 All institutions 6,4394,469 1,969 aFigures from reference 20.
132 The Mathematical Sciences in Education TABLE 7 Distribution of Mathematical-Science Course Enrollments in Fall 1966, by Size of Junior College INSTITUTIONAL SIZE NO. OF COURSE DISTRIBUTION' (NO. OF STUDENTS) ENROLLMENTS (%) 5,000 and over 107,000 31 2,000 to 4,999 104,000 30 1,000 to 1,999 55,000 16 Under 1,000 82,000 23 All institutions 348,000 100 matical sciences in junior colleges, the first such survey ever made. A much more thorough presentation and discussion appears in the CBMS Survey Committee's report, Volume I, Chapter V. Numbers of mathematical-science course enrollments, broken down according to the size of the institution, are given in Table 7. Enrollments of entering freshmen in mathematics courses tend to be at less advanced levels for junior colleges than for four-year colleges, as Table 8 shows. In the junior colleges of largest enroll- ment (over 5,000), 55 percent of all mathematics course enrollments by entering freshmen for the [all of 1966 were below the level of college algebra and trigonometry. While junior college freshmen tended to have a generally lower attainment level in high school mathematics, the differences shown in Table 8 appear to reflect not so much differences in ability as differences in goals. A considerable TABLE 8 Percentage Distribution of Entering Freshmen Enroll- ments at Three Mathematics Course Levels LEVEL TWO-YEAR FOUR-YEAR INSTITUTIONS, INSTITUTIONS, FALL 1966 FALL 1965 Below college algebra and trigonometry 42% 19% College algebra, trigonometry, and equivalent 44% 49% Analytic geometry, calculus, and above 14% 33%
Undergracluate Education 133 fraction- perhaps 30 percent of the junior college students aim for immediate occupations in business and technology rather than for careers in teaching or other professions. (Most of the remaining 70 percent are students intending to transfer to four-year colleges upon graduation.) Among junior college students, those in occupational curricula have mathematical needs and abilities somewhat different from those intending to transfer to four-year colleges. Some of the strongly occupation-oriented mathematics is, in fact, taught outside mathematics departments altogether. The most common example of this is business mathematics taught in a division of business. Other examples are precalculus technical mathematics and statistics. It is to be emphasized that industry needs technical aides who are graduates of strong two-year college programs. In fact, a spokesman for a prominent industrial laboratory has indicated that although salaries and opportunities for advancement are good, well-trained technical aides are often harder for industry to find in needed num- bers than are those with more advanced professional training. THE MATHEMATICS FACULTY IN JUNIOR COLLEGES In junior colleges in 1967, the full-time mathematical-science faculty numbered approximately 2,700; and taking into account part-time faculty members, the full-time equivalent faculty was approximately 3,100. Their training was overwhelmingly at the master's level: 84 percent had the master's as their highest degree, 12 percent the bachelor's, and about 4 percent the doctorate. The field of highest level of training was within the mathematical sci- ences for only 62 percent; for 24 percent it was mathematics edu- cation, and for 14 percent it was in some other field. To the question, "Do you have difficulty in recruiting and keep- ing an adequate mathematics faculty?" about 73 percent of the junior college mathematics departments responding to the CBMS questionnaire said, somewhat surprisingly, that they did not. Prob- ably the principal reason for this is that the better high school teachers form an enormous and highly available pool of supply. Another reason is that, even in comparison with the private four- year colleges, the public junior colleges can offer a better median salary to professors.
134 The Mathematical Sciences in Education Whether this faculty is indeed adequate, and whether it will prove adequate for the future, may be questioned. The Mathe- matical Association of America, in its resolutions to the Congress and the National Science Foundation,4 states that the junior col- leges appear to form the weakest link in the chain of higher educa- tion in mathematics. Certainly a junior college teacher now qualified to teach only the most elementary mathematics courses may in the future find nothing he can teach; for the preparation of entering freshmen will undoubtedly continue to improve, and remedial teaching will increasingly be done by such techniques as programmed instruction. In its recent report,29 an ad hoc CUPM Panel on the Qualification of College Teachers of Mathematics states that a strong mathematics master's degree (what it calls the "first graduate component") ". . . should represent adequate training for teaching transfer students in junior colleges, provided the teacher continues to remain intellectually alive." Although there are no firm percent- ages, many with experience gained in teaching-institute programs for college teachers feel that numerous junior college teachers with master's degrees fail to meet these criteria (see reference 16, Chapter 5) . In summary, the junior colleges form a fast-growing but geo- graphically highly nonhomogeneous component of higher educa- tion. In these colleges, a strong and continuing effort will be needed to raise and maintain faculty professional standards in the mathematical sciences. The university mathematical community can contribute to this in two ways, primarily: first, by producing new junior college faculty with strong mathematical training through the master's level; and, second, by providing opportunities for appropriately oriented continuing education for those already doing mathematical-science teaching in junior colleges. The first of these ways underlines the fact that there is need for federal support for graduate mathematical training that stops at the master's level, as well as for PhD training. The second is a part of the effort in continuing mathematical education for college teachers, both academic-year training and summer training. This too will require strong federal support.
