If you’re asking this question, then the answer is “Yes, go and work towards a PhD.”

Now for some details.

Every so often (about once a semester), I have a student, a friend, an acquaintance, a client, etc. ask me if they should “get a PhD in Mathematics”. But that’s actually not where the question ends nor is that really the question. Usually the back story is as follows:

• The person is thinking about graduate school in a technical discipline because they happen to be reasonably good with mathematics (and for the sciences, being reasonably good, at least mechanically, at mathematics is a prerequisite).
• They have a burning desire for something beyond what their undergraduate curriculum offers.
• They’ve been in the workforce for some time and now want to go back to graduate school — these folks often have a science, engineering, or math undergraduate degree and their work environment has been technical.

From here the following types of statements are said when the individual goes to seek out the opinions of friends and colleagues for getting a PhD in Mathematics:

• Don’t get a PhD, you’ll have no job prospects.
• All you can do with a PhD in Math is teach and you won’t make a lot of money.
• You’ll be overqualified for industry positions.
• Don’t get a PhD, you’ll be an expert in something that only 10 people know about.
• You have to publish papers all the time to keep your professor position.

Some half-truths, some full-blown lies.

Jobs

Zeroth, who says that there are no job prospects? What is their background? Where / how are they making this claim? Every time I ask these sets of questions, the demographics of the folks who discourage others from further academic pursuit of mathematics are folks who themselves have either failed out of a PhD program or had made their decision to stop further pursuit because of someone else who told them not to. It would be like me saying to someone, don’t pursue a career in Graphic Design, you’re just going to spend the whole time touching up other people’s crappy design work. I’m not a Graphic Designer! I don’t work in that capacity! And even if I did work in that capacity, then seek out a few other opinions! This guy is an awesome graphic designer (he’s my brother).

First, check out 101 Careers in Mathematics. I haven’t read it, but the title alone should be convincing.

Next, at a personal level. I work and have worked as a mathematician in industry. I also teach part-time. I regularly get messages or phone calls from recruiters about full-time opportunities for someone with a strong mathematical and technical background. Depending on the recruiter, it can be hit or miss. Some recruiters don’t really understand the expertise and just see math, math, math, everywhere on my resumé. The recruiters that don’t understand my background will send me full-time positions for Bachelor’s degree candidates. Technical recruiters tend to have a better time understanding the various backgrounds. I am overqualified for some positions, mis-qualified for others, underqualified still for others, or a perfect match. This qualification problem is going to happen no matter what degree one holds. The best thing I can do for myself is to see which positions are within my technical expertise, but for which I am underqualified and start learning those topics. I may not want to apply for that job, but it’s an indication of something I don’t know.

I ran a Math department for a few years in the corporate world with three US offices directly reporting to me and a total of eight global offices over which I had policy oversight. I didn’t do the math in the Math department — I hired people with math and computer science backgrounds to take care of the day-to-day mechanical work. What I had to do was set up sensible and rigorous math policy, field technical questions from clients, create efficient processes, explain the mathematical underpinnings of certain methods to non-math savvy regulators, etc. — in effect, I was a business manager with a very technical background. That is exactly what was needed for that position. The company needed someone who was external facing who could speak about mathematical topics with proven authority in the field without speaking in subscripts.

I now work in a consulting role. I work with organizations to help streamline business processes so that they are cost-efficient without sacrificing quality. How do I do that? I write programs. I analyze the entire product development life cycle and see where bottlenecks are. I track relevant and necessary statistics to measure how effective these processes are. I also run pre-analysis to give a sense of what the expected gains are of a new process or method. And it doesn’t stop there. I also look at existing technical code and see where things can be more efficient.

Some of my clients are also individuals who are working professionals. Some are going through a career change, others just feel that they are falling behind the times and want to get “caught up”. Career changes often requires going back to school, taking some type of graduate school entrance exam, taking a statistics course, etc. Those who feel they are falling behind really just want to learn how to program. I’ve worked with company vice presidents, small business owners, working professionals returning to school for an MBA (for example), medical doctors, finance professionals, and even aspiring math teachers. I have yet to work with a school — I would love to do that, but I’ve found that there’s an immense amount of red tape. Anyone know how I can go about working with a school — math training, professional development, software development, leadership training, classroom management, “cool math stuff seminars”, etc.?

Finally, it is true that there are many teaching positions available with a PhD in Mathematics. It is also true that it is not all fun and glamour, living the life of luxury of an esteemed professor’s life. But tell me, how many other career choices are that glamorous? And if you wanted glamour, go do something else! Either that or solve a popular and open problem in mathematics. Then you’ll have fame! People confuse the fact that there are many teaching positions with there are only teaching positions.

If you are thinking about working towards a PhD in Mathematics and do not want to teach, then absolutely, you must learn to program. Without a reasonably strong programming background, there are fewer industry positions available (“fewer” does not equal “none”). If you do want to teach mathematics, I’d still encourage you to learn to program. It will help with teaching. Students will want to see the theory in action and not just in an academic setting. This is how I teach many of the “applied” courses or units like Graph Theory, Probability, Statistics, Personal Finance, etc.

The point here is that the degree is the degree and in some sense it’s irrelevant, but, if you have the degree and can actually put it to use, there are opportunities beyond teaching. A PhD in Mathematics is not somehow making a person unemployable. It just means that the type of employment one will find is different.

“Getting” a PhD

One does not “get” a PhD by hanging around in the system for a bunch of years. Unlike Bachelor’s and Master’s degrees where the majority of the requirement is to complete coursework in an academically satisfactory way, a PhD is not quite the same. There are few courses that one actually takes as “requirement”. The majority of the time is spent doing research, writing papers, giving talks, and even perhaps, networking within the math community.

Several of the people who have been interested in a PhD were under the impression that the process works as follows: 4 years for a Bachelor’s, 2 years for a Master’s, and then 4 years for a PhD. That may be a reasonable average for the length of time that it takes, but the content is different. The Bachelor’s programs are coursework heavy, with perhaps a senior project that is professor-led. The Master’s degree also tends to be coursework heavy. But there are two types of Master’s degrees. There is the Master’s degree that is not research-oriented, and then there’s a Master’s degree that is research-oriented. The research-oriented degrees are typically set ups for entrance into the PhD program. The non-research-oriented degree is coursework heavy and does not often require the same focus on theory as the research-oriented Master’s degree does. The two degrees have a lot of overlap, but there are nuanced differences in coursework.

Getting into a PhD program often requires the completion of the Master’s level coursework. Some programs award a Master’s degree and then require that the student “reapply” for the PhD program by successfully passing a grueling set of exams known as “Qualifying Exams” (“quals”). Once a student has passed the quals, they are admitted into the PhD program and are now considered to be a PhD Candidate. Other programs have one of two tracks — a Master’s track or a PhD track. If one enters the PhD track, it’s do or die and there’s no Master’s degree awarded regardless of the outcome. If one enters the Master’s track, then they have to apply to a different university for their PhD.

Your Advisor And Your Research

As I said earlier, the PhD is not a degree that one just “gets”. There is a “Master / Apprentice” type relationship between the PhD Candidate and his/her Advisor. There are many, many horror stories of tyrannical advisors who work their apprentices into the ground. I’ve been witness to a few of these incidents. It’s a bit scary and terrifying to have one maniacal individual control several years of a person’s life. For this reason, if you are pursuing a PhD in Mathematics (or any field), choose your advisor very carefully. You will be working with him/her closely for many an hour.

I had the great fortune of having probably the best advisor a student could have. Not only was he a capable and energetic researcher, he was patient, thoughtful, kind, and knew how to work with a variety of personalities and work habits. He left me alone to fiddle. I did a lot of work under him. He pushed me to present and publish. But in all of this, he was never once a tyrant. This is the type of person you want as your advisor. So, when you are working on your Master’s degree towards your PhD, pay close attention to graduate and research faculty. Interact with them. See how they treat their graduate students.

Your advisor, if you are able to cultivate a healthy relationship with him/her, will be your mentor for your life. It really is a Master / Apprentice relationship. You just have to decide if your Master is a Sith Lord or a Jedi Master and then you have to decide what you will become.

Your research is your baby. Nurture it, grow it, cultivate it, and stay with it. That’s a mistake I’ve made. I haven’t kept up with research mostly because I am not working in academia full-time. Regardless, I still make it a point to read the latest in the literature and keep fresh in my mind the things I worked on. One day I will get back into it.

Your research trajectory isn’t necessarily going to go the way you planned. You may start out studying Partial Differential Equations, but before you know it, as you wind through the maze that is Mathematics research, you may find yourself deeply immersed in Stochastic Calculus. Don’t pigeonhole yourself from the outset. Let research take you where it takes you and be excited about everything.

Research is not easy business. There are going to be many, many failures. Some even soul-crushing. My first soul-crushing failure was when one of my research papers was already published in full-generality! I had done an extensive literature review and had found nothing on the topic. Every few months I would do searches and nothing. Then once I had all my results ready and the paper written, I did one last literature review and wouldn’t you know, someone had already published it!

Research is open-ended. There is no textbook solution to the problem you are working on. There is no readily applicable theory. There are no existing methods. This is what you are doing. You are crafting the theory. You are devising the method. You are working on an unsolved problem, however esoteric as it may seem.

Failures define us just as much as our successes do. The sooner we can learn from our mistakes, overcome the setbacks, and get back on our feet, the closer we are to success. Working towards a PhD is not for the weak-willed and it isn’t a degree one just gets.

As I said at the beginning, if you are contemplating working towards a PhD, then do it. Do your homework with respect to who your advisor is, what research specialty you may want to start with, what the university program looks like, what the Math department looks like, etc. Ignore all the silly reasons that people give you about job prospects and the like. Anyone who tells you “don’t learn anymore”, isn’t your friend. Jettison them from your life, immediately.