Launchings from the CUPM Curriculum Guide:

What has happened to Modern Algebra and Real Analysis?

David M. Bressoud February, 2007

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D.3: Majors preparing for post-baccalaureate study in the mathematical sciences and allied disciplines

Mathematical sciences departments should ensure that

• A core set of faculty members are familiar with the master’s, doctoral and professional programs open to mathematical sciences majors, the employment opportunities to which they can lead, and the realities of preparing for them;
• Majors intending to pursue doctoral work in the mathematical sciences are aware of the advanced mathematics courses and the degree of mastery of this mathematics that will be required for admission to universities to which they might apply. Departments that cannot provide this coursework or prepare their students for this degree of mastery should direct students to programs that can supplement their own offerings.

The preliminary report of the CBMS Survey, Fall 2005 [1] shows a distressingly large number of colleges, especially BA-only colleges but also across the board, that do not offer courses in modern algebra or real analysis/advanced calculus. Within the 2-year window Fall 2004 through Spring 2006, only 61% of mathematics departments offered Modern Algebra I and only 66% offered Advanced Calculus or Real Analysis I.[2] For Modern Algebra I, this represents a decrease from at least 71% only five years ago.[3] This should be an issue of concern for the entire mathematical community.

Since 1980, CUPM has wrestled with the question or whether modern algebra and real analysis should be required of all mathematics majors. Consistently, CUPM has decided that, no, not all majors need both or even one of these courses. At most colleges and universities, students who intend to pursue graduate study in the mathematical sciences constitute a minority of the mathematics majors. Those who will seek a PhD in Mathematics are often a very small fraction. What all majors need in their junior/senior years is an experience of mathematics in depth, an opportunity to explore conceptually complex and challenging mathematical ideas. CUPM recommends that majors take a two-course sequence in modern algebra or analysis or even applied mathematics and modeling rather than a wide variety of survey courses.

But the fact remains that modern algebra and real analysis hold a special place within the mathematics curriculum. Between them, these courses encode the bulk of the language and methodology that was discovered and developed in the 19th century. They are the gateway to almost all of modern mathematics. Most careers for mathematics majors need little more mathematics than linear algebra, statistics, perhaps some differential equations, but to be a mathematician with a broad understanding of the field and a recognition of where the powerful tools lie, one must be conversant with both modern algebra and analysis. These courses must be available to mathematics majors, and those with the potential to pursue doctoral-level work in the mathematical sciences should be advised to take them. Undergraduate programs that do not offer modern algebra or real analysis effectively close out the possibility of pursuing graduate work in theoretical as well as many applied fields of mathematics.

Behind the drop in the number of colleges and universities offering modern algebra or real analysis lies what I consider to be an even more disturbing figure. Since 1990, while total numbers have stayed roughly constant, there has been a shift of mathematics majors away from the comprehensive universities and the BA-only colleges toward PhD-granting universities.

Bachelor’s degrees in Math, 1990 (includes Act Sci, Oper Res, and joint degrees) [4]

	Univ (PhD)	Univ (MA)	Coll (BA)	total
men	3696	1933	2893	8522
women	1970 (35%)	1672 (46%)	2663 (48%)	6305 (43%)
total	5666	3605	5556	14827

14827 = 1.41% of total bachelor’s degrees for 1990 [9]

Bachelor’s degrees in Math, 1995 (includes Act Sci, Oper Res, and joint degrees) [5]

	Univ (PhD)	Univ (MA)	Coll (BA)	total
men	2867	2235	2956	8058
women	1933 (40%)	1833 (45%)	2470 (46%)	6236 (44%)
total	4800	4068	5426	14294

14294 = 1.23% of total bachelor’s degrees for 1995 [9]

Bachelor’s degrees in Math, 2000 (includes Act Sci, Oper Res, and joint degrees) [6]

	Univ (PhD)	Univ (MA)	Coll (BA)	total
men	2851	1340	3742	7933
women	1703 (37%)	886 (40%)	3142 (46%)	5731 (42%)
total	4554	2226	6884	13664

13664 = 1.10% of total bachelor’s degrees for 2000 [10]

Bachelor’s degrees in Math, 2005 (includes Act Sci, Oper Res, and joint degrees) [7]

	Univ (PhD)	Univ (MA)	Coll (BA)	total
men	4112	1350	3358	8820
women	2282 (36%)	1027 (43%)	2482 (43%)	5791 (40%)
total	6393	2377	5839	14610

14610 = 1.02% of all bachelor’s degrees for 2005 [11]

The recent 15% drop in the number of mathematics majors at BA colleges must be affecting the ability to offer upper division courses. In fact, within the two-year window of Fall 2004 through Spring 2006, only 52% of BA colleges offered Modern Algebra and only 57% offered either Advanced Calculus or Real Analysis. In contrast, ten years ago, in the one-year window 1995–1996, 71% of BA colleges offered Modern Algebra and 64% offered either Advanced Calculus or Real Analysis.[8] The effect has been to make it more difficult for many of our students to continue their study of mathematics into the graduate level.

The effect on women in mathematics is particularly pernicious. Women make up a much smaller fraction of mathematics majors at research and PhD granting universities than they do at other types of colleges. Women majoring in mathematics are still more likely to attend BA colleges than either MA or PhD universities, and it is precisely those institutions that they are most likely to attend that are dropping the courses they need if they are to pursue doctoral degrees. The fact that, over the past ten years, the percentage of mathematics majors who are women has been steadily dropping should be of concern to our profession.

I would appreciate hearing from readers at colleges or universities that do not offer modern algebra or advanced calculus/real analysis for their undergraduates at least every other year. What has happened that you do not offer them? How do you accommodate those students who need or want these courses? I would also like to hear from those colleges that have addressed this trend and reversed it. What has worked for you?

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[1] Preliminary tables of the CBMS Survey, Fall 2005 are available at www.math.wm.edu/~lutzer/cbms2005/

[2] CBMS Survey, Fall 2005, preliminary table SP.22 at www.math.wm.edu/~lutzer/cbms2005/Ch2Pdf/05Ch2TableSP.22.pdf.

[3] Note that 71% of all departments offered Modern Algebra in 2000–01. The 61% represents a marked drop at the same time that the window has been expanded to two years.

[4] Albers, Donald J. Don O. Loftsgaarden, Donald C. Rung, and Ann E. Watkins, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences and Computer Science in the United States, 1990–91 CBMS Survey, MAA Notes, No. 23, Table E.6, page 30.

[5] Loftsgaarden, Don O., Donald C. Rung, and Ann E. Watkins, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 1995 CBMS Survey, MAA Reports, No. 2, Table E.1, page 42.

[6] Lutzer, David J., James W. Maxwell, and Stephen B. Rodi, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 2000 CBMS Survey, American Mathematical Society, Table E.1, page 71.

[7] CBMS Survey, Fall 2005, preliminary table E.1 at www.math.wm.edu/~lutzer/cbms2005/Ch3pdf/05Ch3TableE.1.pdf.

[8] Loftsgaarden, Don O., Donald C. Rung, and Ann E. Watkins, Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States, Fall 1995 CBMS Survey, MAA Reports, No. 2, Table SE.5, page 10.

[9] NCES data, nces.ed.gov/programs/digest/d98/d98t264.asp

[10] NCES data, nces.ed.gov/programs/digest/d04/tables/dt04_250.asp

[11] NCES data, nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2007167

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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, he was one of the writers for the Curriculum Guide, and he currently serves as Chair of the CUPM. He wrote this column with help from his colleagues in CUPM, but it does not reflect an official position of the committee. You can reach him at bressoud@macalester.edu.