*Ronald G. Douglas*

*Executive Vice President and Provost Texas A &M University*

Contemplating changes, especially the possible change of a system that has worked so well for the people involved, can produce more anxiety than change itself. Despite evidence that something is wrong, that circumstances are different, our gut reaction is to stand fast, to defend the system and the values in which we believe. In a nutshell, I think this describes the current stance of the mathematical sciences community toward graduate education.

American mathematics has enjoyed a golden age during the last few decades. Students and researchers from around the world have flocked to the United States to study and work. Many of the very best mathematicians have been turned out by doctoral programs in the multitude of fine and outstanding mathematics departments in U.S. universities. If only the federal government provided more funding to mathematicians to pursue their research. If only U.S. high school students Were better prepared for collegiate mathematics. If only more and brighter American students studied mathematics at the undergraduate and graduate level. If only public appreciation and understanding of mathematics were greater. If only there were more university jobs for young mathematicians. But these are the *only* problems and they have all been brought about by forces external to the mathematical sciences, and the solutions rest outside the community also. Or so we want to believe.

In this paper, I discuss some of the choices facing graduate education in the mathematical sciences today, presenting arguments in each case on both sides of an issue. While my personal opinions may be discernible, it is not my intention to be prescriptive. However, I do intend to be provocative and help the community confront and grapple with some of the hard decisions it faces and strike at its core values.

Before getting started let me make a few historical comments. Most American mathematicians now belong to the post-Sputnik generation. The response to this event transformed U.S. universities and academic mathematics. When I was a graduate student, the system was already in flux, but I still had some experience with the way things had been and, with difficulty, I can recall those earlier circumstances. Most mathematicians today have no such memory at all. I state this not to pine for the "good old days" but to make certain we recall that, although mathematics has a history which goes back several millennia, the current system has been in place just a few decades. Thus the notion that change will bring down the temple is probably an overstatement.

Let me now address several fundamental issues. The first concerns the size of the graduate enterprise. In particular, are there too many graduate students in the mathematical sciences? Are too many people earning doctorates?

These questions can be analyzed from various points of view. First, does the number of students exceed the capacity of the system; that is, are there students with unqualified advisors or in marginal programs? I think most people would agree that the contrary is true. There are good mathematicians without doctoral students, and many strong departments have decreased the size of their graduate programs in recent years. Some people would argue that it might be better for

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Educating Mathematical Sciences Graduate Students
Ronald G. Douglas
Executive Vice President and Provost Texas A &M University
Contemplating changes, especially the possible change of a system that has worked so well for the people involved, can produce more anxiety than change itself. Despite evidence that something is wrong, that circumstances are different, our gut reaction is to stand fast, to defend the system and the values in which we believe. In a nutshell, I think this describes the current stance of the mathematical sciences community toward graduate education.
American mathematics has enjoyed a golden age during the last few decades. Students and researchers from around the world have flocked to the United States to study and work. Many of the very best mathematicians have been turned out by doctoral programs in the multitude of fine and outstanding mathematics departments in U.S. universities. If only the federal government provided more funding to mathematicians to pursue their research. If only U.S. high school students Were better prepared for collegiate mathematics. If only more and brighter American students studied mathematics at the undergraduate and graduate level. If only public appreciation and understanding of mathematics were greater. If only there were more university jobs for young mathematicians. But these are the only problems and they have all been brought about by forces external to the mathematical sciences, and the solutions rest outside the community also. Or so we want to believe.
In this paper, I discuss some of the choices facing graduate education in the mathematical sciences today, presenting arguments in each case on both sides of an issue. While my personal opinions may be discernible, it is not my intention to be prescriptive. However, I do intend to be provocative and help the community confront and grapple with some of the hard decisions it faces and strike at its core values.
Before getting started let me make a few historical comments. Most American mathematicians now belong to the post-Sputnik generation. The response to this event transformed U.S. universities and academic mathematics. When I was a graduate student, the system was already in flux, but I still had some experience with the way things had been and, with difficulty, I can recall those earlier circumstances. Most mathematicians today have no such memory at all. I state this not to pine for the "good old days" but to make certain we recall that, although mathematics has a history which goes back several millennia, the current system has been in place just a few decades. Thus the notion that change will bring down the temple is probably an overstatement.
Let me now address several fundamental issues. The first concerns the size of the graduate enterprise. In particular, are there too many graduate students in the mathematical sciences? Are too many people earning doctorates?
These questions can be analyzed from various points of view. First, does the number of students exceed the capacity of the system; that is, are there students with unqualified advisors or in marginal programs? I think most people would agree that the contrary is true. There are good mathematicians without doctoral students, and many strong departments have decreased the size of their graduate programs in recent years. Some people would argue that it might be better for

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students if some of the very small programs were eliminated, leaving only those with a critical mass in faculty and students. Although practices in some sciences tend to encourage such an outcome, few mathematicians are prepared to argue in favor of this approach, believing that such a system would be elitist; and, in any case, almost no one is willing to support steps that would bring it about.
One important reason departments cling to doctoral programs even though the programs might be marginal may be related to the different working conditions that usually prevail for departments with and without such programs. Although cogent arguments are made linking the presence of doctoral students to the mathematical vitality of a department, the small number of students and the vanishingly small number of successful students in some departments can make one wonder about the relevance of the arguments in some cases. A second reason often concerns the role graduate students have in teaching the lion's share of lower-division mathematics at universities. However, should we decide that we need young, energetic, nonpermanent instructors and then use graduate programs to provide them?
Usually when the suggestion is made that the graduate enterprise may be too big, one has in mind the great difficulty that young mathematicians have today in finding jobs. In making this connection, two assumptions are implicit: the first is that the purpose of doctoral training is to prepare students for a university, or at least a college, position and the second is that the system has a responsibility to new doctorates to guarantee them such a job. Neither assumption has always been held in mathematics, and they are certainly not held currently in all other fields. That is why, for example, most colleagues in the humanities have little sympathy with the current plight of young scientists, since few new doctorates in fields such as history or English can expect to get an academic position. However, it is depressing to see eager, talented, young mathematicians leave the field for lack of a suitable position. Let me make one other observation. Most mathematicians strongly agree that a doctorate should be required for teaching mathematics at 4-year colleges and that it is highly desirable for teaching mathematics at 2-year colleges. However, many faculty view doctoral graduates who have taken such teaching positions as having left the community of mathematicians.
An issue related to the view of the doctorate as preparation for a job concerns the completeness of the training provided. That is, if candidates are being prepared for teaching positions, then shouldn't their graduate education reflect that? Although some recognition of this possibility is now made in most departments, I wonder what message is being given at most places on the relative importance of teaching.
Let us return to the question of size and the purpose of graduate education. Since there seems little likelihood that the number of university and college positions in mathematics will increase, at least in the foreseeable future, the profession must make some choices. Either the community must somehow reduce the size of the doctoral system, or it must broaden its views on appropriate jobs for graduates and both stimulate demand and prepare students for these alternative positions, or students must be told that one studies mathematics because one likes it. It is then the students' responsibility to find employment, and if they are fortunate, they will make use of what they have learned. Most groups that have thought about this issue have opted for the middle course, but this has consequences for graduate training. It would have to be broader and more responsive to the needs and desires of business, industry, and other disciplines. In particular, students would need to learn how to communicate what they have learned to the educated public. The realization that graduate education would have to adapt is particularly hard

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for many mathematicians, especially those in core areas, to accept. Part of the problem is that of uncertainty—we know how to prepare students to follow our path, to be a copy of ourselves, but how do we educate them to do something else? I will not go into detail about how to broaden graduate education, since much has already been written on that subject. Moreover, I believe that if a department becomes convinced that change is necessary and identifies the goals for its doctoral program to achieve, then an approach can be devised.
There is one other dimension that merits our attention here: the increasing length of time that students are taking to complete a doctorate. Calls to broaden graduate training can only exacerbate this trend unless hard choices are made about what is essential. There are, however, other reasons. Both advisors and students see little reason for finishing and entering an unfriendly job market; also, both want a candidate's thesis to be as competitive as possible and the curriculum vitae to list several papers. Would postdoctoral apprenticeships available to a larger share of the new doctorates reverse the trend? Would young mathematicians be both helped and encouraged to pursue realistic career choices sooner in such positions? In any case, does a longer doctoral program really benefit students or the profession in the long run?
An issue related to doctoral education is the role of professional master's programs in the mathematical sciences. For several decades, most master's degrees have been awarded as consolation prizes to students not allowed to continue toward the doctorate. There are some thriving professional programs, however, almost all in applied areas. Anecdotal evidence suggests that much of the need of business and industry for mathematically trained professionals could be supplied by well-designed professional master's programs in the mathematical sciences. Might not the best of these students, or those who are sufficiently motivated, still go on to doctoral studies?
Some people believe that the fight solution to the size problem is to raise the bar, that is, to keep the same number of programs and about the same number of students, but to make it harder to complete the doctorate. This could be done by requiring stronger dissertations or broader training. For people who view the profession as a kind of priesthood, it is appealing to reduce numbers by keeping out all but the most worthy. However, there might be several negative consequences to such an approach.
First, there would be the terrible human waste of labeling a large group of our most talented people as failures and choking them out, ill prepared for anything. Moreover, since a good many such people often find their way into policy positions or positions in which they mediate mathematics to the public, they might turn the tables on the mathematics community, strongly affecting public funding for mathematics and universities.
Second, while Darwinian selection appeals to many mathematicians as a fair way to choose who succeeds, the playing field is often not as level as many would like to believe. In many cases, it's as though someone taught some of the animals how to use weapons and then accepted the outcome of which animals survived as having been dictated by nature. That is, success in graduate school does not always go to the fittest or ablest, but often to those who are best prepared. And such preparation may have had little to do with an individual's abilities. For example, it might depend on where one was born in the United States or the economic status of one's parents. It can depend on how you were encouraged in school and whether or not your teachers expected you to excel in mathematics. There is also the possibility that students with reasonable preparation in U.S. colleges and universities might find themselves competing with students from abroad who have several years of postbaccalaureate experience.

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As an aside, let me ask if graduate students in any other country must compete with the brightest and best from around the world. This policy may work to maintain the strongest mathematical sciences community in the United States, but there may be a cost. Perhaps some American students choose, as a consequence, not to compete or to drop out of graduate school early in the process. The perception may be developing in schools, colleges, and universities that studying mathematics and the sciences is for someone else. Let me acknowledge that the presence of international students and researchers has been a source of strength and vitality for American mathematics in this century and that there would be a real loss if that were changed. However, what has changed during the past two decades is the proportion of students from abroad. It is appropriate to ask whether this change has negatively affected the nature of graduate programs in many cases.
About 5 years ago I chaired a study of doctoral education for the Board on Mathematical Sciences. The findings of our committee were rather dramatic and surprising to many people. Agreement on purpose and the general environment had a substantial effect on the success of students in a graduate program. This was particularly true for women and minority students. These groups are disproportionally affected in programs that adopt a hands-off attitude and subscribe to a "survival of the fittest" philosophy.
In summary, the approach of raising the bar or allowing Darwin to prevail might have several consequences as to who studies mathematics. I think the community must decide how much the composition of the U.S. mathematical sciences community matters. Will it affect mathematics education in the schools, colleges, and universities? Will it affect the public -perception of mathematics? Will it affect public funding of the mathematical sciences?
As I indicated, my goal at the outset was to raise questions, to point out problems. Mathematicians pride themselves on rational thinking and avoiding contradictions. I believe there are some contradictions in our current policies, or at least in our actions. I hope we will work to resolve them.