Hello everyone! As the school year wraps up, the peer advisors want to share some things we’ve learned in our time as math majors at U.C. Berkeley. Younger students and prospective math majors — we hope that this information answers some of your questions.

### What do you know now that you wish you knew when you first started at U.C. Berkeley?

**Antara**:

- Stay self motivated through the lower divisions. These classes are the foundation for everything to come, but it can be easy to not feel engaged with them because they tend to be bigger/less interactive.
- Make a few four-year-plan options (with different combinations of classes, different combinations of potential majors).
- I wish I worked together with classmates more.

**Grace**:

- Don’t be afraid to reach out to your classmates,
*especially*in your first classes at Cal. I’ve found my best friends and an amazing support system from reaching out to people in my classes. Take your academics seriously, but your first semester or two at Berkeley is an amazing opportunity to build connections with fellow students which will last throughout your college experience (and beyond!). - TAKE ADJUNCT CLASSES
- Check out the SLC–the tutors are amazing!
- Don’t let the MUSA office intimidate you; they have hot cocoa and Shin Ramyun for CHEAP.
- The 9th floor of Evans has a nice courtyard, and there are usually some really interesting people on it. A great place to relax in between classes 🙂

**Isabella**: There are so many underutilized resources on campus that are at your disposal. The SLC, res hall tutoring, even the math library!

**Jonas**: I came from a high school with great resources and helpful teachers, but where grades were very inflated and not much effort was necessary to do well. You need to be prepared to master everything and take all the material seriously. If there is a concept that you are even slightly unsure about, cover that and practice it until you know it by heart. If there is a type of problem which you are even slightly unsure about, keep practicing it until you know it like the back of your hands.

However, perhaps most importantly, do not forget to practice and work with other people. Sometimes, you don’t even know what you don’t know until one of your friends brings it up, or asks you for help about it. There are three steps towards mastery: understanding the material, applying the material, and being able to explain the material to others.

**Nick**: While it’s rewarding to dive deeply into a particular topic through advanced coursework, it’s important to expose yourself to a breadth of topics. You probably don’t want to graduate equipped with doctorate-level knowledge of number theory but unable to solve a first-order linear ODE.

### What course at Berkeley was the most formative in your decision to study math?

**Grace**: Math 54 (Linear Algebra). While Math 1B and 53 were interesting, and very focused on computation, this was my first introduction into ‘proofs’ and more abstract content. Math 54 teaches you how to really understand concepts, not just dropping formulas on you to accept as fact. You really look at some fundamental concepts and learn to understand them deeply, and this was what really hooked me.

Protip: Technically, you CAN take Math 54 and Math 53 at the same time (like I did), but this could really keep you from getting a good understanding of both classes. Math 54 was super engaging, and I focused on it more than Math 53. My grade in 53 suffered as a result…

**Isabella**: Math 55 (Discrete Mathematics). The course is sort of a brief introduction to a large array of interesting topics in math that aren’t taught in any of the other lower division courses. For me, it was the first math course that kept me interested from start to finish and left me wanting to learn more. I declared right after!

**Tianyu**: Math 105 (Second Course in Analysis). Although professor Charles Pugh will probably not teach this class anymore, I still recommend it and his textbook Real Mathematical Analysis. Hardest class I’ve ever taken, but definitely an eye-opening experience. It deals with multivariable calculus in R^n and Lebesgue measure theory. This class would be especially helpful if you want to take Math 202A (intro graduate course in topology and analysis) but don’t feel prepared enough out of 104.

**Nick**: Math 54 (Linear Algebra). In my opinion, the best post-calculus math class for a newcomer (as opposed to 53, 55). While 53 trained my visual intuition and 55 trained my proof-writing technique, 54 had a far more holistic positive impact on my mathematical development. Rather than present a bag of tricks lacking a unifying theme (e.g. 55), 54 develops the fundamentals of linear algebra, and common ideas are explored with increasing depth and breadth throughout the semester. And these ideas need not be accepted blindly as fact; nearly everything is proven from the outset, starting with the definition of a vector space. 54 was my introduction to the illuminating process of building, brick by brick, the house that is a mathematical theory. It was this process that clarified my interest in mathematics.

**Jonas**: It was actually Physics 205B, so not even a math course. I took Physics 105 the semester prior with a professor who is well-known for explaining things well and with great passion, but in a way which can be quite overwhelming at times, and his problem sets often feature very interesting problems requiring an infamous amount of algebraic computations to solve. I actually did not do that well in his Physics 105 class, but I fell in love with the section on phase space analysis and geometric mechanics.

It turns out that these topics fall under a greater, more general field of mathematics which I would grow to love, so much that it would pull me from physics and into applied mathematics: nonlinear dynamics and differential equations. I was not planning to take any graduate-level course at the time until he sent out an announcement inviting us to take Physics 205B with him in the spring. I took it, and while it was the most time-consuming course I had taken up to that point, it changed the course of my academic and research interests forever. (And yes, I did much better in it than in Physics 105.) Furthermore, I got to know this professor quite well, took Physics 205A with him the next spring, and he wrote several letters of recommendation for me that got me into a top REU program and a great graduate school. I am forever thankful for his inspiration, advice, and help, as well as for these courses I took with him.

I should note that, while Physics 205AB is technically a course for physics students, the material was very mathematical, and similar material can be found in nonlinear dynamics, mechanics, and ODE courses found in mathematics, MechE, and EE departments at other universities. (Side note: Even if you are convinced that mathematics is the only thing you ever want to learn about, there are very worthwhile courses that might be of interest for you in other departments, so don’t forget to give these a shot! After all, mathematicians often work alongside people in other fields.) Consequently, the course had enough mathematics to inspire me to take Math 104 and Math 128A the following summer; I had a great time in those, which confirmed that math would be the perfect subject for me to pursue.

### What are Berkeley math classes like?

**Antara**: Lower division classes are NOT representative of what the rest of the classes are like. Upper division classes have much more engagement and smaller class settings, and are more abstract and proof-based.

**Grace**: Lower divisions will introduce you to broad concepts, as well as a broad range of fellow students. Lower divs are extremely valuable for connecting with campus on a broader scale, as well as building solid foundations in mathematical thinking; however, they are very different from upper division classes. Upper divs are much more intimate, with about 30-45 students (likely primarily math majors) face-to-face with the professor. Many have no discussion sections, which can seem intimidating at first, but this also provides a great opportunity for you to build closer connections with faculty. Professors *want* students to come to their office hours and engage with them!

**Tianyu**: Some upper division classes don’t have discussion sessions and TAs that can help you understand / review the materials. Apart from spending lots of time absorbing and reinforcing what you’ve learned independently (which is important), definitely attend your professor’s office hours if you are confused about anything, or form study groups to discuss challenging homework problems.

**Nick**: Like other STEM classes, math classes at Berkeley can require a lot of commitment, effort, time, etc. But one key distinction I’ve observed that perhaps sets apart math classes (in particular, advanced math classes) from those of other STEM departments has to do with the dynamic between students. For instance, every computer science class (including upper division ones) I’ve taken at Berkeley has a sort of “you vs. your peers” type of dynamic, where the curve induces competition between you and the other students in the class. In contrast, in nearly every advanced (i.e. > lower division) math class I’ve taken at Berkeley, the dynamic is better described as “you vs. the material,” in the sense that your understanding of the mathematics is more important than your understanding of the mathematics *relative to that of your peers*. Math class is not a zero-sum game. This creates a highly inviting collaborative atmosphere where hard work translates more directly to better results (in terms of both understanding and grades).

**Jonas**: Berkeley math courses can vary greatly depending on the professor and the level of the course (lower-division, upper-division, or graduate-level/seminar). Several of the other peer advisors have essentially already covered what I would have noted about upper-division courses, so I will give more detail about lower-division courses and graduate-level courses first. Aside from Math 55 (which is more geared towards math majors and what they will see in upper-division courses), lower-division courses will hardly give you an image of what upper-division and graduate-level courses are like. The proofs you do (there aren’t many) are usually quite routine and involve definition-chasing, while depending on your professor, the computations could be quite tedious and meant to lower the curve. These courses in some ways are mainly structured to give other STEM majors the “tools” that they need to learn and apply mathematics in their disciplines. For these, the best way to succeed is to just practice, practice, practice. Understanding is not as important as learning how to do every problem in the textbook perfectly. The lectures for these courses also tend to be quite large, with smaller discussion sections that involve quizzes, the difficulty of which could vary greatly depending on your GSI. Finally, the curves could be quite brutal to okay, ranging from C- to B (with the rare B+-median curve). These can be quite competitive.

For *all* math courses, but especially lower-division ones, you should absolutely reach out for help and extra practice: the SLC and your peers will be especially helpful. At the same time, do not stress too much about how you do, because in the long-run, lower-division math courses are not that important. Graduate schools will hardly look at them, and I have actually found personally that your success in lower-division math courses does not correlate too well with your success in more advanced courses.

Upper-divs usually are graded primarily off of a few exams (one or two midterms and a final), with a minimal proportion of your grade (0-20%) for homework. They can vary greatly in size from ~10 to 450 students, depending on whether the course is a major requirement or an elective, or how “mainstream” the topic is. However, graduate-level courses generally are much smaller. You could have a course with as little as five people in it, but most graduate-level math courses (aside from the “mainstream,” introductory ones such as Math 202A, 228A, and 250A) have under 40 students, many hovering around 15 or less. These are usually graded only off of problem sets and maybe a final project/presentation, and generally give higher grades. But that does *not* mean that you should just speed into these, and they are *not* easier. You will probably have to spend much more time on them, because the material generally is 10x more condensed and deeper. The problem sets are generally much more involved. The material is hard. It really isn’t that hard to fall behind so much that you can’t even *do* the work in the class because you hardly understand a thing. Only jump to graduate-level courses once you know that you’re ready.

Finally, professors. The quality and way of instruction can vary significantly depending on the professor. I have often told people who have asked for advice on which courses to take and in what order that, while there are certainly some clear “no-nos” (e.g., do NOT take Math 185 before Math 104, avoid making Math 104 your first upper-div, etc.), you should prioritize the professors. What I mean is, look at who’s teaching the course in question next semester. Ask around about them. Look around on the Berkeley Mathematics Facebook group, Rate My Professors, or Reddit for prior students’ experiences. Take these all with a massive grain of salt, because some people are just bitter because they did not do well or overly positive because the course was graded easily.

### What has the math major taught you?

**Emma**: Being a math major has taught me to think in a more analytical way, even in regards to non-math related topics. I found that after taking more advanced math classes, my statistics and data science courses felt more intuitive because math at Berkeley provided me such a solid foundation. Having a strong background in math helped me understand the “why” behind certain topics, such as machine learning and data science, because math is the basis for many other fields.

**Nick**: For me, a major takeaway from the math major is the idea of approximation. In mathematics, on a small scale, you often approximate something complicated with something simple, e.g. linear regression, whereby you approximate a perhaps poorly-behaved function with a linear function. More generally, on a large scale, mathematics can serve as a more transparent approximation of an opaque reality. Often, a natural phenomenon is too complicated to understand explicitly through formulas and rules, in which case it is useful to place that phenomenon in a better understood framework that includes its own explicit formulas and rules, i.e. in a mathematical framework. In doing so, we gain an approximate, “mathematical” understanding of the phenomenon.

### What advice would you give incoming students who are considering a math major?

**Antara**: The idea of studying mathematics can be intimidating. It could be especially intimidating if you don’t feel like you see people like you in the major, and you don’t think you’re cut out for it. But I want you to know that it’s very possible, and that there are lots of resources to help you succeed. Don’t constantly compare yourself to others. Don’t feel like you’re not capable of being the quintessential math major that people think of when they hear the term. Just because you aren’t John Nash from A Beautiful Mind, seeing numbers all around you, doesn’t mean you can’t enjoy math and even be good at it. Have some confidence in yourself, try the classes out, and push yourself to be a math major if that’s what you want to do! If you need guidance, support or help, seek it.

**Isabella**: You might not love every math course you take, and that’s ok! You don’t have to know right away that you are going to major in math: take some courses in different departments, and see what you enjoy!

**Jonas**: Be ready to be skeptical. Challenge every assumption. Recognize the impressive, fundamental stature that the field of mathematics has as a foundation for many subjects, and perhaps, you will be motivated by the many connections that mathematics has. Learn what it means to truly master what you learn. Be curious, and I encourage you to think deeply. And if you don’t love mathematics that much, it’s totally fine. You don’t have to love all of math to do math and enjoy it. Find what you enjoy. Math is a very broad spectrum of topics, some of which overlap quite heavily with other fields. If you end up more in one of those other fields, that’s even fine too. I’ve heard of math majors ending up in CS, economics, finance, mechanical engineering, astrophysics, biophysics, and basically anywhere in the quantitative side of STEM. Contrary to what others might tell you, majoring in math is not restrictive; if anything, it’s too broad. That leads me to another piece of advice: remember to find what you enjoy and hone in on that. Absolutely don’t limit yourself and close yourself off from other opportunities, but specialization and focus gets you really far. Exploration is the initial stage that allows you to do that successfully, so please, have an open mind and don’t forget to explore courses both inside and outside of mathematics. Trust me, you absolutely have the time to do so!

### What can you do with the math major?

**Emma**: I really believe that math is one of the few majors that allows you to pursue a vast array of careers. Personally, I found the math behind quantitative finance to be the most interesting to me. I barely knew any finance before beginning my internships, but my background in math made it far easier to learn how to develop trading algorithms and perform quantitative research. After graduation, I will be working as a Software Engineer for JPMC, and I truly believe I wouldn’t have ended up on this path if it weren’t for choosing this major.

**Nick**: Math can be a standalone pursuit, in which case graduate school in mathematics might be a deeply rewarding path for you. On the other side, math can act as a sort of gateway drug into other parts of STEM, most commonly statistics, computer science, and physics. The math major exposes you to a breadth of foundational ideas early on, making it relatively easy to apply your knowledge to a new field (or to a job that requires technical work). Though I studied math (quite purely I might add), I am pursuing graduate studies in statistics next year.

**Jonas**: I don’t even know where to start with this question. People in other fields often act as if the only feasible outcome of a math major is teaching K-12 or becoming a professor/lecturer. There’s nothing wrong with those career fields, but that’s absolutely not true. In fact, math majors (esp. applied math majors, but you’ll find that there’s hardly a distinction here at Cal) very frequently end up in finance, software engineering, and data science, even at top places such as Facebook, Two Sigma, and Google. Cal just has really fantastic connections to tech and finance, especially in the SF Bay Area, and for a given field in industry, all that is required (and the rest could easily help) are a few key courses. The math majors at Cal (and at many other universities) are very versatile, and give you a rigorous, mathematical foundation which can be applied everywhere. You just have to make sure that you take the right courses, and with those, develop the connections and skill set that you need to get you to where you want to go. Aside from industry, I even know many graduate students at Cal in mechanical engineering, earth science, CS, etc. who actually majored in applied math, computational math, or something similar as an undergraduate. The mathematics major prepares you exceptionally well for STEM graduate school because the fields of STEM these days are increasingly becoming saturated with advanced, rigorous mathematics. For any of these cases, you have to do well, sometimes exceptionally well for the best positions, but that is sort of a given.

Anyways, majoring in mathematics doesn’t close off doors at all. In fact, majoring in mathematics opens a massive amount of doors, but too many. That’s why it’s important for math majors to have career/academic goals in mind and a plan to achieve them. Take the courses you need to achieve these goals. Of course, I am excluding the obvious possibility (perhaps, the one that math majors consider initially) of just going into math graduate school and hopefully academia. At that point, you can just take what you’re interested in, as long as you focus and show a demonstrated interest and success in mathematics.