HI, I am currently a Spring semester sophomore and just changed my major to Mathematics. I have always had a niche for math and love everything about it. I took Calc 1 in high school, Discrete last semester and Calc 2 now. All these classes so far are easy for me. As I am preparing for future math courses, what courses are the hardest ones? And if so, what’s a good way to prepare for them?
As a current math/CS major, I can’t really give you a solid answer to this question - it largely depends on your background and what you want to study. For me, abstract algebra was one of my hardest courses, but largely because I was bad a linear algebra.
Make sure you have taken the prerequisite subjects or are otherwise ready.
Also, being able to write a solid proof is essential in any math class you take. Not the really dumbed-down “two-column proof” they teach in some HS geometry classes. You took discrete math; you should know.
Additionally, I suggest learning LaTeX (a typesetting language) if you haven’t already. The vast majority of mathematicians type their papers in LaTeX, and a lot of math majors (including myself) usually type our homework instead of handwrite it. The learning curve is somewhat steep if you haven’t programmed before, but it will be worth it.
Whichever class is the hardest depends both on the school and professor. At UT, Real Analysis and Algebraic Structures are considered to be the toughest courses. As far as preparing for those two, it’s usually recommended you take introductory proof courses first but that is usually done anyway to satisfy the pre-reqs anyway.
Thus, as far as pre-preparing goes, you are supposed to be ready for the course just by having the pre-reqs. However, sometimes it is true that the department will not do a good job with listing which pre-reqs are needed. For example, to take Number Theory here, I need to have completed either Linear Algebra or Discrete Math. I have already completed both and I am currently in Number Theory. So far, I would feel dead if I did not take Discrete Math because I did not feel like Linear Algebra alone would have been a good enough preparation. As a result, it is good that I chose to take Number Theory after taking the other two courses instead of taking it earlier at a point in time that I had only completed one of those two. You can probably talk to some faculty members or other math majors at your institution to see if the pre-reqs listed for each course you need are actually good enough or if there are some classes you would need to take before others (like I did taking Linear Algebra before Number Theory despite already having the pre-req for Number Theory after completing Discrete Math). What this means is that the order you take the classes in (despite having pre-reqs) could be important.
Thanks for input. Here at my college, you have to have completed all the pre-reqs anyways to take the course. Very rarely you see teachers waive you into a class unless you have a legitimate excuse. For example, my math advisor here told me that if I wanted to take Real Analysis next semester I need Calc 3 but haven’t take it yet. He said he could get the teacher to waive me in it just so I won’t have such a high workload in my senior year. What are views on this?
@nicolettean you might be able to waive pre-requisites, but be careful when doing so.
For example, the abstract algebra class I took at MIT (18.701) lists real analysis (18.100) as a pre-req, but a bunch of students and I didn’t take 18.100. However in the math department, a lot of classes are willing to override the pre-reqs if you can demonstrate you have the ability. For example, the “actual” pre-req for 18.701 was essentially, writing proofs.
However, some classes that are over-registered might require you to have taken the pre-requisite courses, to limit the number of students.
I think my hardest class was Complex Analysis, but that was because of the professor. I did not study very much for it, however. I think the most challenging class was Real Analysis, but I did very well in it. I was not good at Linear Algebra, but I was somehow great at Abstract Algebra. IMO, I think you’ll know whether you’ve got what it takes after seeing your grade on a first midterm in Real Analysis. It was super encouraging for me.
A kid I know at MIT took all lower-division math classes as preparation.
I found the intro to proofs class super useless. The teacher herself admitted she had messed up with us because she was new to the department and completely changed how she taught the course for the next quarters.
Junior and senior level math courses will focus heavily on proof-writing, particular real analysis and abstract algebra. You probably have been given instruction and practice in proof-writing in discrete math, or there may be an introduction to proofs course as well.
You school may want you to take calculus 3, differential equations, and/or linear algebra (soph level courses) as well. These tend to be more computation based, unless honors versions with more proof emphasis are offered.
As of right now, my classes for next semester at my school are Linear Algebra and Calculus 3. Then I have a choice of either deciding whether I want to take Differential Equations or another math elective. No matter what, I have to take Linear Algebra 1, 2, and Modern Algebra. Then from there, I have to take Real Analysis one semester and Complex analysis the other or vice versa. Which would you all recommend that I take first? Real Analysis or Complex?
The answer to your question is best addressed to the chair of the math department at your school. They will understand best the pre-reqs, and why. They will likely welcome your interest.
The rest of us can’t know how the focus on each course is based at your school.
But good for you, math rocks!
Does complex analysis list real analysis as a prerequisite? If so, then you need to take real analysis first.
No, all complex lists is Discrete and Calculus 3 for a Pre-Req. My advisor says it’s up to me to me but my current math teacher says take complex first. In my opinion, whatever teacher I like better I am going to take it with and hopefully everything comes together.