Who should go to graduate school? Where and why should they go? What is graduate school? This note is a biased answer to some of these questions. Keep in mind that the answers are provided by a person who believes that education in general is a wonderful thing, has no regrets about going to graduate school, and feels it led to an employment position that almost ideally suits his ambitions and temperament. This article addresses some of the preparations that should be taken by students planning to go to graduate school, but provides only the basic details. (See also www.cs.umd.edu/Grad/index.html.) I do not pretend that it is a substitute for personal advice, which I strongly suggest you obtain before making such an important decision.

## Who Should Go to Graduate School?

My answer to this is easy: anyone with an interest in furthering his or her education in the mathematical sciences. An undergraduate program in mathematics only introduces you to the basic areas of the subject; it is not intended to provide a working knowledge of mathematics any more than an undergraduate premedical program intends to give a person a working knowledge of medicine. The discipline of mathematics is so broad that you will be only able to sample it in the ten or so courses that are required beyond calculus. If you want to be a productive mathematician in education, research, or applications, you must continue your study. The same is true for statistics and to a slightly lesser extent for computer science. In the latter discipline you can usually find professional employment with only a Bachelor's degree, but you will likely be doing rather routine programming.

Graduate programs in the mathematical sciences (by this I mean to include pure mathematics, applied mathematics, statistics, and computer science) differ from undergraduate programs in at least two major ways. First, an undergraduate program is designed to provide you a well-rounded education in the arts and sciences as well as introduce you to your discipline. In graduate school the program concentrates only on subjects that are felt to be relevant to your discipline. The second difference involves finances. Students generally pay for their undergraduate education, but most graduate students in the mathematical sciences do not.

## Who Pays?

The common device for avoiding the cost of graduate training is the Graduate Assistantship. The assistantship requires the graduate student to perform tasks for the university in exchange for a tuition waiver and a modest salary. The tasks for the beginning graduate student usually involve either teaching or grading papers. A more advanced graduate student may receive an assistantship for helping a faculty member on a research project. The duties might also include computer programming or consultation, particularly for graduate students in applied mathematics or computer science. Whatever the task, the assistantship is generally designed to require about 12 to 15 hours per week of the graduate student's time. The stipend that is given to the student in addition to the tuition waiver varies from institution to institution, but is generally on the bare subsistence level, approximately 25-35%, for the academic year, of what the student could obtain from a full-time job in industry. Summer support is often available for students in PhD programs, but seldom in Master's programs. While this stipend may seem to be a pittance, suppose that the assistantship requires about 13 hours of work per week for the 9-month school year and that the stipend is $9,000. Then on a 40 hour per week basis this would translate to approximately $36,000 per year. In addition, the assistantship provides a tuition waiver. It isn't, then, that the assistantship does not pay well, it is rather that you are paid a reasonable salary for the amount you work, but you don't work much.

## How to Choose a Graduate School

I will assume that you have some broad idea of your area of interest, for example, pure mathematics (like real analysis or abstract algebra), applied mathematics (like differential equations or numerical analysis), statistics, or computer science. This is all that is expected of a beginning graduate student. Once you have a Master's degree you should be able to refine your interest within the category you have chosen so that you can determine programs and advisors that have strengths in your area of interest.

You should take the Graduate Record Examination (GRE). This examination comes in two parts. The first part tests general ability in three areas: verbal, quantitative, and analytical. This part is very much like the Scholastic Aptitude Test (SAT) that is given to high school students. The second part of the examination is in your chosen subject area, for example, mathematics or computer science. To see sample questions and get information about testing dates, sites and registration consult the GRE webpage at www.gre.org. You should plan to take the GRE no later than December in your senior year, since it takes about two months for schools to receive the results and some graduate programs make their original decisions by the middle of February. You must study if you are taking the subject examination in mathematics; it is difficult and can be intimidating, even for good students, especially if they have not done sufficient preparation. Particularly review all your calculus, the theory even more than the techniques, and talk to faculty about problems in other mathematical areas.

There are a number of sources that you should consult to obtain information about various schools and their programs. First and foremost is your undergraduate advisor. The person who guided you through your undergraduate mathematics program should know your abilities and interests. This person will likely be writing letters of reference for you and should be kept aware of your plans. If you don't have a faculty advisor in mathematics, quit reading at this point and find a faculty member who knows you, is interested in you, and is willing to discuss your plans for the future. This is priority one.

Each December the American Mathematical Society publishes a booklet entitled *Assistantships and Graduate Fellowships in the Mathematical Sciences. *This gives a summary of the graduate support that is available in the mathematical sciences in the United States and Canada. It also gives, in summary form, information about the programs such as the number of students awarded Bachelor's, Master's, and Doctorate degrees, the areas of research for the Doctoral students, the amount of work required of graduate assistants, and the duties that the assistants perform. The booklet lists virtually all the mathematics programs and many of the statistics and computer science programs as well. There are advertisements from many of the larger programs and a listing of stipends that are available for supported travel. In all, this is an excellent place to start your search for a graduate program. Any mathematics department should have a copy that you can look at and perhaps borrow, but it is so useful it is certainly worthwhile to have your own copy. It can be ordered from the American Mathematical Society, www.ams.org.

A similar resource, entitled *Graduate Assistantship Directory,* is published for Computer Science programs by the Association for Computing Machinery. Fittingly, it can be consulted online at www.acm.org.

There is a book called *The Gourman Report* that annually rates programs in all disciplines. There are several online ratings of graduate programs as well (e.g., www.usnews.com/usnews/edu/beyond/bcphd-htm).Take these ratings with a grain of salt and remember that a program that produces students who are excellent in research may or may not produce students who are excellent in teaching.

How do you use all the information that these sources provide? The best answer to this is to be insightful. You want to choose a graduate program that will educate you well, give you the confidence you need to succeed in your chosen field, and help you find a position that is commensurate with your training and abilities when they have decided to release you to the world. Additionally, and perhaps most importantly, you want to choose a program where the faculty will respect your abilities and treat you as the apprentice mathematician you are. You should be aware, however, that all too often students enter graduate school full of confidence, only to become filled with self-doubt when they hit some mathematical walls in their first graduate courses or when they take their first exams. Often it is the most talented undergraduates who go on to graduate school, so for many students this will be the first time they have been challenged by their peers and their classes. Always keep in mind that you have a lot to offer the world, and that if things get rough at times it may not be because of your abilities. A program in which the faculty does not have sufficient self-confidence and maturity can sometimes make the hurdles to the PhD unrealistically high.

To help pick a reasonable program you should probably do a little data analysis. For example, to determine the schools that have unrealistic graduate expectations, you might compare the relative percentages of graduates to students receiving support. To determine which schools do a good job of finding positions for their graduates, find out who the PhD graduates were in the past five years and then find out where they presently are. You can determine the graduates by looking in the library at the *Dissertation Abstracts.* The Combined Membership List of MAA, AMS, and SIAM ( www.ams.org/cml) should tell you where these graduates are presently located. An indication of the type of training a school is providing might come from looking at the research grants awarded to its faculty and recent graduates.

Finally, talk to as many of your professors as you can. Especially consult those who are actively involved in mathematics and who seem to be in contact with faculty at other institutions. Be sure to talk to those faculty who have recently completed graduate programs. Ask them about their experiences, and see what advice they have to offer.

Recognize that this is one of the biggest decisions of your life. When you have narrowed your choices to a few, make an appointment and visit them. Talk to the faculty to see if they are truly interested in students. If they are not interested in you at this time it is unlikely that they will be interested in you later. Talk to as many graduate students as possible, first year through dissertation. Try to determine if their experience has been rewarding, and if they know what is going on in the mathematical world beyond their own institution. The more-senior students should have had worthwhile teaching and seminar presentation experience, and the dissertation students should personally know some experts in their area in the world at large. If you are a woman, talk to female graduate students to learn about the environment for women. There are many other things that I am sure you can think of to help you make your decision. After all, you are in mathematics which is the discipline of rational problem solving.

## Requirements

A graduate student on an assistantship will usually complete a Master's program quite easily in two years . A full-time load for an assistant is generally three courses per term. In most programs there is also a thesis option that reduces the course work requirements by about 15-20%, but requires a substantial, but not necessarily publishable, paper that is written according to rather rigid guidelines. Almost all Master's programs also include the requirement of a comprehensive examination at the end of the program. This examination usually covers about half of the course work in the program and is used to determine whether the student has the comprehensive knowledge of the discipline that would justify the designation of "Master." Students generally dread the examination, but it is uncommon, although not unheard of, for a student to fail to receive a Master's degree because of the examination. There is almost always a provision for re-examination if the student fails to perform adequately on the first try.

With the Master's degree you can become a professional mathematician and usually expect to command a starting salary of up to 20% more than you would if you started with a Bachelor's degree. You might even be able to direct your own research projects, but a research position normally requires the PhD.

The only true requirement for the PhD is the demonstrated ability to conduct substantial research independently. There are hour requirements, examination requirements, and often even a foreign language reading requirement, but these are all designed to provide the student with the broad working knowledge of the subject area that will serve as a base for the research work. Let me not mislead you regarding the examinations, however. In order to begin research work the student must have a research advisor. To ensure that advisors spend their time productively, the student must be formally admitted to the PhD program. This generally requires passing one or more of a series of examinations that are designed to eliminate the unlikely candidates. The examinations may be written (perhaps 3 examinations each about 4 or more hours long), oral (commonly 2 or 3 hours with 3 or 4 faculty as inquisitors), or, not infrequently, a combination of the two. The examinations are designed to test the basic knowledge of the student and also to see if the student can attack a new problem in a reasonable way. In contrast to the Master's examination, it is not uncommon for even a good student to fail to pass the PhD examinations the first time. In fact, the first set (and at some schools the only set) of examinations is usually given after the first two years of graduate work so that a student who does not pass the examination on the first try can leave with a Master's degree. Students are commonly given two or more tries at the examinations, and, when the examination is written, they can sometimes repeat only those parts they previously failed. The precise examination procedure varies from school to school, however, so you should check carefully the procedures at any school that interests you. Whatever the specific procedure, there is generally great rejoicing when the candidacy examination has been passed, accompanied, quite often, by a party and restrained decadence.

## Choosing Your Advisor

Once you have been admitted to candidacy for the PhD you select a research advisor who will work with you in finding a research project that should culminate in your doctoral dissertation. Choose your research advisor carefully, since this choice will likely determine your area of mathematical expertise, and, at least theoretically, from this point on in your educational program you are at the mercy of your advisor. Ideally, the advisor will be someone who is in your area of interest, is well-known and currently active in research, has had considerable experience directing doctoral students, has time to take on a new student, and is one with whom you have rapport. The advisor should work with you in deciding which additional courses you will take, which research papers you will read, and when you have sufficiently matured to permit you to go into the world with the title Doctor of Philosophy. It is usually stated that your research must make a substantial contribution to the discipline. This is generally interpreted as being good enough so that some respectable research journal will publish it.

## Earning Your PhD

The variables in a PhD program are so broad that no reasonable time schedule can be put on the program. In a sense, it should be considered as an apprenticeship program. You must first show that you are worth the time of a master (your dissertation advisor). You must then demonstrate that you have the skills of a journeyman by producing an original piece of mathematical research. This probably sounds impossible to you, but you'll find that when a master leads you to the boundary of mathematical knowledge, there are lots of good questions to ask. How fast you proceed through the various stages depends on your abilities and probably even more upon your discipline. The best that can be said is that it almost always takes at least two years beyond the Master's degree and that it commonly takes twice that long.

With a PhD you are a certified member of your profession. It is the usual minimal requirement for a university teaching position or a professional position in a research laboratory. Monetarily, the degree will provide you with a substantially increased starting salary if you take an industrial position, but give you little increased financial reward if you decide to take the more attractive university pPosition. Why, you ask, is the university position more attractive if does not pay nearly as much as the industrial position? In a university position you have the freedom to do almost exactly what you want to do and someone will pay you a reasonable salary for doing it. If what you do attracts attention in the world at large you will be promoted, rewarded, and envied. But most importantly, you will be in the position of showing young students why education in general is great and mathematical education is the best of all possible worlds.

## Subliminal Mathematics

Locked out,

frustrations mount

papers strewn

all about

Conjecture that

I know is true,

yet coyly it avoids

a proof

Hours I've

invested here

a thousand times

I've felt so near

But this work's not

for government,

and "close" is not

worth one red cent

My sanity's

grown razor thin,

it's time I give

these hopes a fling

Go trash these

"lead and paper" wads,

and bow down

to the theorem gods

Hours pass,

I've let it be,

kicked the habit,

kicked the need

Washing dishes,

Mind at ease,

drifting empty,

obsession free

Then suddenly

while rinsing cups,

two wires arc,

a circuit shuts

Neurons fire,

something clicks

could that be?

is that it?

On my knees

in the trash,

pawing like

an alley cat

Where's the end

of that thread?

where's that trail

I left for dead?

Aha! Aha!

sheer ecstasy,

like a glove,

this puzzle piece

Oh, mind of mine,

I should have known,

you simply needed

time alone

â€”Monte Zerger, Adams State College

## Credits

**DOUG FAIRES **is Professor of Mathematics, Youngstown State University

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