Selecting Your First Math Course

Harvard’s math department offers a multitude of options for your first math course, so regardless of your math background or level of interest, you can find a course that appropriately challenges and engages you. Here is some advice to help you during your first shopping week:

  • Shop at least two math courses during your first week in order to get a sense of the different styles of different math courses. Switching classes is possible for the first five weeks of the class, but depending on the year and the professor, some students have reported that switching can be difficult due to making up weeks of homework and material.
  • You can concentrate in math or any other field regardless of your first math course. In fact, there have been a number of math professors who have started at Math Ma. As such, it is more important to choose a course in which you are learning than to try a more challenging course in which you overwhelm yourself and don’t learn anything.
  • Collaboration is key. No class expects you to complete its problem sets on your own. Try to find a problem set group as soon as possible; come to Math Night if you have trouble. This will help you gain a more accurate perception of the difficulty of the class. Make sure to check the collaboration policy for each class.

A note about prerequisites: All prerequisites listed here are guidelines, not mandatory. Prerequisites are material/knowledge that all students surveyed have agreed was necessary for the course. Useful Knowledge is material/knowledge that only some students surveyed agreed was necessary for the course.

The information here is taken from the course catalog and supplemented with information from students.

Math Ma/b: Introduction to Functions and Calculus I and II

Course Content: Math M reviews precalculus and covers differential and integral calculus. Math Ma (fall) covers differential calculus, emphasizing the study of modeling using algebraic, exponential, and logarithmic functions. Math Mb (spring) continues with differential calculus and modeling with trigonometric functions, followed by an introduction to integral calculus and differential equations. Together, both courses are preparation for Math 1b.
Prerequisites: None!
Expected Workload: 4 – 12 hours/week
Note: Taught in sections by graduate students or faculty in the Math Department

Math 1a: Introduction to Calculus

Course Content: Math 1a (both semesters) is an introduction to single-variable calculus, covering differential and an introduction to integral calculus, along with the Fundamental Theorem of Calculus.
Prerequisites: Background in precalculus.
Expected Workload: 4 – 12 hours/week
Note: Taught in sections by graduate students or faculty in the Math Department

Math 1b: Calculus, Series, and Differential Equations

Course Content: Math 1b (both semesters) begins with integration, including improper integrals, numerical integration, and applicatios of integrals, and then further explores single-variable calculus by discussing series, differential equations, and applications to modeling real-world systems.
Prerequisites: Basic differential and integral calculus (5 on AP Calculus AB, Math Ma/b, or Math 1a)
Expected Workload: 4 – 12 hours/week
Note: Taught in sections by graduate students or faculty in the Math Department

Math 18: Multivariable Calculus for Social Sciences

Course Content: Math 18 (fall) covers multivariable calculus with an emphasis on applications towards the social sciences, particularly economics. Topics such as multivariable functions, partial derivatives, directional derivatives, optimization, and the method of Lagrange multipliers are emphasized.
Prerequisites: Single-variable calculus including series (Math 1b or 5 on AP Calculus BC)
Expected Workload: 5 – 10 hours/week

Math 19a: Modeling and Differential Equations for the Life Sciences

Course Content: Math 19a (fall) focuses on the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. It introduces mathematics that include multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. A final project concludes the course. This course replaces Math 21a for students more interested in life sciences, but you can also take Math 21a afterwards for credit.
Prerequisites: Single-variable calculus including series (Math 1b or 5 on AP Calculus BC)
Expected Workload: 4 – 10 hours/week

Math 19b: Linear Algebra, Probability, and Statistics for the Life Sciences

Course Content: Math 19b (spring) discusses probability, statistics and linear algebra with applications to life sciences, chemistry, and environmental life sciences. The linear algebra includes matrices, eigenvalues, eigenvectors, and determinants, and the probability includes standard models, the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis. A final project concludes the course. This course can be taken with or without Math 19a and serves as a replacement for Math 21b for students more interested in the life sciences.
Prerequisites: Single-variable calculus including series (Math 1b or 5 on AP Calculus BC)
Expected Workload: 4 – 10 hours/week

Math 21a/b: Multivariable Calculus, Linear Algebra, and Differential Equations

Course Content: Math 21 is a non-proof-based introduction to math just beyond AP Calculus BC. Math 21a (both semesters) covers multivariable calculus, including multivariable functions, partial derivatives, multiple integrals, and generalized versions of the Fundamental Theorem of Calculus like Green’s theorem, Stokes’s theorem, and the Divergence theorem. Math 21b (both semesters) covers linear algebra, including matrices, vector spaces, and eigenvalues, as well as applications such as dynamical systems and differential equations.
Prerequisites: Single-variable calculus including series (Math 1b or 5 on AP Calculus BC)
Expected Workload: 5 – 12 hours/week
Note: Taught in sections by graduate students or faculty in the Math Department

Some students who take Math 21a may want to take Math 101 in addition to or alongside Math 21b; see here for more details.

Math 23a/b: Linear Algebra and Real Analysis I and II

Course Content: Math 23 is a fairly rigorous course in linear algebra and multivariable calculus, designed for students seriously interested in mathematics. It is intended for students who are interested in being able to prove the theorems that they use, but who are equally interested in the application of mathematics to fields like physics and economics. Math 23a (fall) covers linear algebra, single-variable real analysis, and multivariable analysis and calculus. Math 23b (spring) provides an integrated treatment of linear algebra and multivariable calculus, building up to Stokes’ theorem in the language of vector calculus and differential forms.
Prerequisites: Single-variable calculus including differentiation, integration, and series (5 on AP Calculus BC or Math 1b).
Useful Knowledge: Exposure to multivariable calculus, linear algebra, or proofs can help.
Expected Workload: 5 – 12 hours/week
Notes: Previous students in Math 23 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here are a representative sample of quotes (all from the Q guide):

  • “It’s quite useful if you want something more than 21a (more theory and proofs), but still not at a very high level. In other words, it’s great if you really want to learn maths but not spend most of your time on it.”
  • “This course is difficult but really worth it in the end. I came out of the class truly feeling like a mathematician and none of my other math classes had done that for me which was a great feeling.”
  • “Only take this class if you think you’re seriously interested in math but don’t have the time to commit to Math 25. I never meant for Math 23 to be my “main” class but it became that – it took me the most hours every week and caused me the most anxiety and frustration to really try to understand the material. This class requires a LOT more work than I realized to do the problem sets well and to try to grapple with the material.”
  • “I would only take this class if you are interested in proof-based mathematics but have never seen it before. Otherwise, I feel that math 21 would have been much more practical and math 25 would have been much more helpful.”
  • “This class is incredible if you’re interested in learning more math, but note that a strong math background is absolutely essential. There [are] a lot of very intelligent students in the class, so to keep up, it’s very important that you have a very strong calculus and linear algebra background, preferably with some proof experience.”
  • “If you’re on the fence about whether or not you like math enough to pursue it further, and you’re willing to put in the time commitment, this class is truly worthwhile.

Note that the usual professor is not teaching Math 23a/b this year, so there may be significant changes beyond what is listed above.

Math 25a/b: Honors Linear Algebra and Honors Real Analysis

Course Content: Math 25 is a rigorous introduction to linear algebra and real analysis for students interested in developing strong proof-writing skills. Topics normally include set theory, vector spaces, linear maps, dual spaces, determinants, Jordan normal form, and spectral theory in Math 25a (fall), while Math 25b (spring) may cover sequences and series, continuity, differentiation in one and multiple variables, and the Riemann integral. The pace, additional content, and workload of the course may vary with the professor. The course is focused on forming rigorous constructions and understanding the motivations behind different proof methods, so problem sets consist primarily of proofs.
Prerequisites: Single-variable calculus including differentiation, integration, and series (5 on AP Calculus BC, Math 1b, or equivalent) and some familiarity with writing proofs.
Useful Knowledge: Experience with proof-writing and computational linear algebra will provide a great foundation for the more abstract approach of the course.
Expected Workload: 5-25 hours/week, depending on the professor and on your background.
Notes: Previous students in Math 25 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here are a representative sample of quotes (all from the Q guide):

  • “This class is really solid. If you’re willing to put the work in, you can definitely do well and get a lot out of the class. It introduces some very beautiful topics and lays the fundamental parts to calculus which is cool. It will be hard though, so be willing to put in work, but you don’t have to do all-nighters.”
  • “If you’ve only taken BC Calculus, this course is doable but very difficult. I spent a ton of time with all nighters on psets and understanding the material and got lost at some point of every class. However, in the end I felt that I had learned and understood the material and had covered a lot of ground by taking the course.”
  • “The course may seem at first a daunting one, maybe just because of the sheer level of formality with which the topics are presented. However, the proof-based approach to the subject is an extremely rewarding one, at least in my opinion. I actually found that after taking just one semester of math 25 my confidence with abstract mathematical concepts improved so much that I was able to explore different fields on my own (e.g. group theory, exterior algebra) knowing that I had the adequate background necessary to understand the basics of them.”
  • “I would really recommend this course *to those with a good math background.* In particular, it helps to be proficient with proofs. You learn a lot in this course and it can come pretty quickly. Also, this is different from other linear algebra courses in that it’s not very computation-heavy.”
  • “If you don’t have much experience with proofs or theoretical math (like I didn’t), this course will give you a run for your money at first, and you may be tempted to drop it. Gradually though, you will get a lot more comfortable.”
  • “Math 25 is a great course, but you should really rethink taking it if you don’t plan on doing any higher math. I entered the course having done proof-based math but not linear algebra; most will have done linear algebra but not proofs. I’d recommend having done at least one or the other. Of course, there are some who enter without either, so go for it if you shop the course and feel up for the challenge / prepared enough!”

Math 55a/b: Honors Abstract Algebra and Honors Real and Complex Analysis

Course Content: Math 55 is a highly rigorous and fast-paced introduction to college-level math. The fall semester (Math 55a) is focused on algebra, beginning with an abstract introduction to linear algebra and group theory and then covering more advanced material, which varies from year to year but can include axiomatic set theory, basic ring theory, Galois theory, tensor products, representation theory, and category theory. The spring semester (Math 55b) is focused on analysis, beginning with proof-based real analysis, functional analysis, and single-variable calculus and then covering advanced material which may include basic differential geometry (Stokes’ theorem), complex analysis, and basic algebraic topology. In both semesters, the class moves extremely quickly, and students are routinely expected to prove major theorems on problem sets.
Prerequisites: Exposure to proofs, multivariable calculus, and computational linear algebra. Depending on the year, instructors may assume knowledge of more advanced math as well.
Expected Workload: 10 – 40 hours/week, depending on the year and your background.
Notes: Previous students in Math 55 have had a wide range of experiences, depending on the year, their background, and how much time they have to devote to the course. Here are a representative sample of quotes (all from the Q guide):

  • “This is an excellent class – definitely THE BEST one I have ever taken. My life was changed by this course […], and it’s fascinating to expect more to come as a continuation of this class. Though this class might be known as “hard” and “full of genius[es]” (the genius part is true), “normal” people ARE able to do keep up with the course contents and do quite well also – I am not an Olympiad competitor after all.”
  • “[T]his class was quite difficult, and I would only recommend taking it if you want to devote your freshman year solely to math–the envisioned gain has to be worth a LOT of pain. We pulled an all-nighter nearly every week, spending half the time trying to even decipher to problems, and many parts of the class called for extremely heavy computation.”
  • “[I]t’s a class that will challenge you, but it’s also a class where you meet some of your closest friends. Don’t be daunted by the course — it has a reputation that is overhyped, and you can always think of those hours spent completing problem sets as hours spent hanging out with other kids like you.”
  • “Don’t be scared by the reputation of Math 55. The course is hard and you will spend many hours on the problem sets, but it’s not impossible and will not eat your life away. In particular, prior experience with abstract math is not necessary.”
  • “[Y]ou do not learn as much here as you could in a class like [Math] 122, but you gain not only math skills but also a small community that can be very helpful for you (your problem solving skills also probably improve here more than they would in a class like 122).”
  • “The class was excellent in delivering a concrete foundation for the rest of college mathematics, but do not underestimate the amount of work it entails. This class will literally take up all of your time, so I would not recommend taking this class along with any other challenging classes.”

Since student experience in Math 55 varies dramatically from year to year, we strongly recommend that you shop at least one other course (either 25 or a 100-level course) at the same time. Coming into Harvard, you may have heard about the loaded culture associated with Math 55; the data do not back up any of the assertions associated with this culture. In particular, please remember that most math concentrators do not take Math 55 and it is perfectly possible to have a successful career in math (academia or otherwise) without ever having taken 55.

Beyond 55: Other First Year Options

Although not common, or necessarily encouraged, beginning in 100 or 200 level courses or taking such a course alongside an introductory math course is an option. If you have background in college level mathematics or a specific interest in a particular course, you may consider an 100 or 200 level course. The course you select will vary based on your background; see the next two sections for more details on upper level and graduate level courses. One strongly recommended option is Math 122, if you have not seen the material before; most students begin with Math 122 if they choose to not take a freshman-year course. Be sure to shop several courses, including Math 25 or 55, if you are considering this option, to find the best fit for your background and learning style. It is also recommended that you speak to Professor Jacob Lurie (the Director of Undergraduate Studies) and the professor of the math class you are taking to ensure you are fully prepared for that class.

Prerequisites: Familiarity with abstract linear algebra (Math 25a or 55a), real analysis, multivariable calculus (Math 25b or 55b), and exposure to proofs.

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