Which Upper Level Courses Should I Take?

Linear Algebra

Linear algebra underlies almost every other topic within mathematics, and is needed in virtually every field that uses mathematics. Potential math majors and minors should take Linear Algebra (Math 355) or Honors Linear Algebra (Math 354) as early as possible. (Note that Math 211 only covers a small amount of linear algebra and should not be viewed as sufficient.)

Math 355 is more computational, though it does require students to work some with concepts and definitions, and to make mathematical arguments. Math 354 is more theoretical, and provides an introduction to mathematical proof-writing in addition to its linear algebra content. Math 354 is strongly recommended for anyone thinking about majoring in math.

Math 354 and 355 are offered in both semesters.


The formal language of mathematics is one of mathematical proofs. Every student who studies mathematics should take a course which prepares them to read and write proofs. There are several courses which specifically aim to introduce students to proofs; among them are Math 220, Math 302, and Math 354.

Students enrolling in their first proofs course should also consider concurrently enrolling in Math 290 (Mathematical Writing Seminar), which is a 1-credit, half-semester course that aims to provide additional targeted feedback on students' mathematical writing.

Analysis and Algebra

Real analysis is the study of functions on the real line. It closely examines the properties of the real line and gives rigorous justifications for all of the results that one takes for granted when learning calculus. Abstract Algebra studies abstract algebraic structures such as groups, rings, and fields, which model algebraic operations such as addition and multiplication. Both subjects are at the core of higher mathematics and are strongly encouraged for all students.

Introductory analysis courses are Math 302 and Math 321. Students who already have familiarity with proofs are able to skip Math 302 and enroll in Math 321. Starting in 2023, Math 321 is offered in both fall and spring semesters.

The introductory algebra course is Math 356; it is offered in the fall. Math 357 (Algebra II) is a continuation of Math 356 in the spring semester, but one can take Math 356 without committing to taking Math 357. However, taking just Math 356 will not give a sense of the full scope and power of abstract algebra.

Other Electives

There are many other 300- and 400-level electives offered. Some of them are advanced and should only be taken after one has had courses that provide preparation for writing proofs. Some involve almost no proof-writing whatsoever, and can often be taken by students who just have Math 211 and/or Math 212 as preparation. Some fall in the middle, and might vary from year to year.

The following table is a rough categorization of some of the electives offered, although students should check with the individual instructors to be sure about what level each course will be offered at. See the course catalog for descriptions of these courses.

Useful courses for students interested in math education, computer science, or economics/finance

One Possible Degree Path for a B.A.

Possible Degree Paths for a B.S.

Some Mathematics Electives

Few/No Proofs Involved
Fall Spring
Math 381 (Intro to PDEs) Math 382 (Computational Complex Analysis)
Some Proofs Involved (can vary)
Fall Spring
Math 304 (Knot Theory)
Math 368 (Combinatorics)
Math 365 (Number Theory)
Math 366 (Geometry)
Advanced Electives
Fall Spring
Math 370 (Calculus on Manifolds)
Math 374 (Representation Theory)
Math 423 (PDEs)
Math 425 (Integration Theory)

Math 401 (Curves and Surfaces)
Math 427 (Complex Analysis)
Math 443 (Topology)
Math 451 (Differentiable Manifolds)

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Rice University
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Rice University
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Houston, TX 77005