Hi! I am going into high school this year, starting with BC Calculus. If I take one Math class each year, I will take Statistics, which I am not looking forward to, Linear Algebra, which I am looking forward to, and Calculus 3, which seems great. However, I want to major in Mathematics and a field of physics in college, so what classes do they offer after that? If I am running out of courses, is there anyway to NOT do a math for a year without looking bad on an application?
Students typically take differential equations (ordinary and partial) after that. However, I’d recommend abstract algebra instead if it’s available in your area.
DS took linear algebra and then logic, set theory, and proofs (dual enrollment) after BC calc. It seems to be a slightly unorthodox sequence, but he wanted a break from calculus, and he’s really enjoying the proofs class (found linear algebra kinda boring, but it was fine, and he’s hopeful he won’t have to take it again in college).
Colleges have their course catalogs online - take a read through a few to see the variety of topics you could be studying at that level. My D is a math major and she’s pretty taken with abstract algebra, but some people go in other directions.
There’s a much more rigorous proof-based version of linear algebra. If a student enjoys proof-based math, such as geometry, s/he should definitely take abstract algebra.
There are a LOT of math classes that come after calculus. Personally I very much enjoyed probability theory and then (much later) stochastic processes. Some people like statistics and some don’t. Linear algebra is something that I did use later in other courses (including econometrics).
The most obvious questions are what your high school offers, and what you do after you run out of high school classes. Once you get to university there will be plenty to choose from.
Also, you will want to learn calculus very well. It will be used a lot in future classes particularly if you are inclined towards math, physics, or engineering.
I will admit that I didn’t learn linear algebra all that well when I first took it. I finally did learn it well when I had to use it in future classes, but this made the future classes slightly harder.
What exactly does abstract algebra entail? I may sound whimsical, but I love math because I find it elegant, which means courses such as statistics do not appeal to me, though I will take them, for that reason. Abstract Algebra seems very interesting!
Abstract algebra is the study of abstract algebra structures such as groups, rings, fields, etc. These structures are abstractions of other more common mathematical concepts. By studying these structures abstractly, resulting theorems, etc. can be applied to a much wider set of objects universally and consistently. They’re highly useful and practical too (e.g. group theory is indispensable in many branches of physics and chemistry).
Discrete mathematics is useful for CS majors. It might be another option. I believe it is about the same level of difficulty as linear algebra.
There is also real analysis, but it is sometimes advised to take other proof heavy courses first (proof heavy linear algebra, abstract algebra, etc.) because real analysis is commonly seen as one of the more difficult upper level undergraduate math courses.
However, a student in high school taking college math courses may be limited to finishing up lower level ones like differential equations and discrete math. Calculus based introductory statistics, if available, may be of interest.
I checked our local college’s Dept. of Mathematics plan of study and there are 35 courses to choose from post-MVC/LA. If you run out of courses in college because you took 2 advanced courses in HS, I’d suggest it’s not a very solid college Math program.
Wait a couple years and it may become more clear to you which areas of math you like more.
Multivariable calculus and linear algebra tend to serve as gateway courses that can lead, with perhaps a linking class or two, to upper-level math courses such as real analysis, modern/abstract algebra, functional analysis, complex analysis and topology. For mathematically oriented physics electives, look into courses on topics such as general relativity and mathematical physics.