John Teague’s Advice from a Senior

Originally published June 18, 2023

tags: Senior, advice, computer science, added major, changing paths, student to student, grand parent

John Teague is a Senior majoring in Computer science and Mathematics. He also happens to be a former co-president of the UT Math Club. He enjoys rock climbing, chess, and topology.

What made you choose the path you’re on?

When I was younger, I was convinced I wanted to do something involving computers. For a long time, that thing was cybersecurity, or ethical hacking. As time went on and I did more engineering-type activities (e.g. high school robotics), I started to really enjoy the process of software development. That’s why I decided to major in CS. I wasn’t actually a huge fan of math after taking geometry, but I started appreciating it again during calculus (btw, I primarily study geometry now). In addition to genuinely enjoying some of the material, I started watching videos from people like 3Blue1Brown to learn more advanced subjects in a way that showed off how beautiful math can be. It was just before I started college that I decided I wanted to double major, and I didn’t actually commit until sophomore year — technically I wasn’t a double major until the fall semester of junior year!

Do you want to stay on that path? Why or why not?

I had considered going the graduate school route just before college (potentially even in CS), but by the time I added the math major, I was sure I wanted to get a Ph.D. in pure mathematics. This was sort of cemented after I decided to experiment with some of the math graduate courses offered at UT. I initially did well, and it suddenly became possible in my mind that I could do this kind of math for a living.

I’m still getting the CS degree as a contingency, but if I get into a strong program, yes! I would love to work as a math professor one day. Math is so much fun, and I really enjoy research, so I have trouble picturing myself in a career that doesn’t involve both of those things in some significant way.

When you first started getting into math, how did you study for math courses? Has this changed? Why? 

Even in some of the early upper division math courses like discrete and linear algebra, it was more than sufficient to just do the problem sets and understand them. I rarely did outside studying. Real analysis was a slightly different ball game. Usually in addition to the problem sets, I would spend the few days before an exam reading Rudin or reviewing problems. It really helped me personally to have a study buddy during those times to keep motivated.

For classes that actually have exams, my approach is similar to the above. For grad courses, if you do the homework and do it well, you’ll be fine for the exams. This does take quite a bit more time than the earlier math courses, but the problems also increase in quality, so it’s a bit easier to maintain motivation.

If you boiled it down into a recipe for maximum math absorption, what would the recipe be? 

There’s a few pieces of advice I’d share to get the most out of your math courses. Likely the most important is, don’t overexert yourself. And by that, I don’t mean to not challenge yourself with difficult coursework. I just mean limit the inward flow of information that you’re hoping to retain. You shouldn’t be taking more than 2 hard or important math classes a semester (plus maybe a DRP). If you do, you simply won’t get as much out of any of your classes. I frequently don’t follow this advice and end up forgetting some really cool math.

On the other hand, challenge yourself. I strongly encourage people who enjoy math to take hard classes, even graduate classes. If you want to do pure mathematics specifically, the best thing you can do as an undergrad is to find your balance between taking the most advanced classes possible while still being successful grade-wise. You can’t find that balance without trying more advanced classes! This way, you can get a flavor of the math that you personally enjoy before applying to grad school, and you’ll be a strong candidate for the tougher programs.

Also, you should really sit with problem sets. Understanding something from start to finish is by far the best way to remember it, since you should be able to recreate it on the fly if need be. It builds problem solving skills useful in both math and everything else, and you gain intuition that will be imperative for future mathematical endeavors.

Although I said not to overload yourself, I would strongly recommend checking out the seminars that the math department offers. You can find a list with times and meeting locations at https://web.ma.utexas.edu/cgi-pub/seminar/calendar. These seminars are a fantastic way to meet cool people from the department, learn higher level and even research level mathematics, and get you thinking about what graduate school might look like. They are separated roughly into the “junior” and “regular” seminar sequences (they don’t like when you call them the “senior” seminars), and include things like geometry, topology, algebra, number theory, representation theory, algebraic geometry, geometric string theory, analysis, numerical analysis, and so on. Also, some, like the geometry seminar, get dinner after every lecture, and this is a great way to network with faculty and grad students.

In a similar vein, you should attend conferences! UT, on occasion, hosts an undergrad math conference that you should attend and even try to speak at. There’s also loads of other conferences that might interest you at other universities, and giving presentations at these conferences is a great way to stand out in grad school applications. You will probably want a bit of research experience before presenting at “grown-up” conferences, but if/when this happens, go for it! Basically, as a general rule, immerse yourself in communities focused around mathematics, and search out opportunities to do and learn cool math as much as possible.

What do you wish you could tell little-you about math? 

 It depends on how little. If we’re talking elementary or middle school, I would strongly encourage myself to not be so anxious about exploring ahead of the curriculum. America’s public math education is in a sorry state, and I had it in my head that things like calculus were way too advanced for me to grasp or understand. Then when I got there, I was sad to see that it was actually pretty easy and I could have been in a much better spot than I was then. This advice applies almost verbatim to underclassmen now. You are definitely smart enough to understand really cool stuff, you just have to try it.

If I were to give math advice to my high school self, I would probably give more specific coursework advice and encouragement towards pure math. All of the above about taking advanced coursework applies, but I would specifically mention that self-study should not be overlooked. I was only successful in certain classes because I worked through the textbooks beforehand to get an idea of what the course would cover. Unless you’re an incredibly intuitive person, I would strongly recommend doing this before classes like real analysis, algebra, and topology.

What advice would you give to someone considering changing majors to math? What about someone considering going into pure math?

If you’re changing majors to math or thinking about a career in math, know that you are not alone. Almost every single one of my friends thinking about going into math started somewhere else: physics, computer science, psychology, engineering, medicine, law, you name it. Very few of us actually started knowing 100% that we wanted to pursue mathematics specifically as a career (the school system rarely makes that even seem like an option). People switch to and from math ridiculously often, so you’ll be able to relate to more people than you think.

My other advice is, somewhat ironically, don’t forget that there are other career options. Everyone should have the opportunity to pursue pure math, but ultimately it isn’t for everyone. The career prospects in academia are low, the politics can be difficult to navigate, and it is some of the hardest work you’ll ever do. Math majors are strong candidates to jobs in software, finance, informatics, teaching, etc. Even if you don’t love research, you can be very successful with math (almost definitely more than your more pure” math peers). Unless you cannot see yourself ever doing anything else, you should really consider whether grad school is for you. On the other hand, if you love math, the work you do in grad school and beyond will be the most rewarding work you’ll ever do.

In a similar vein, the line between pure” and applied math is *extremely* blurry. I think most would consider me one of the people that do pure” math, but a lot of the outside research I do involves programming and engineering challenges. Even the REU I did on 4-manifolds involved me writing a program to help with certain computations. My friend Kyle does applied” math, but the research he does is some of the most abstract stuff I’ve seen (from what I understand, he does work in numerical methods, especially for solving PDEs and whatnot). Don’t get obsessed with the label, don’t let it hold you back from doing something that you would enjoy, and certainly don’t develop any feeling of superiority or inferiority for whatever label you choose to go by.