How does an advanced math course differ from calculus 1-3?

[MathWarrior]

I've been taking a lot of mathematics courses, and I am getting to where I will eventually be doing upper division mathematics. I was looking for someone to explain some insight on how upper division math courses differ from the traditional calculus sequence.

For example, number theory, abstract algebra, computational methods, graph theory, or optimization.

Will these be more or less the same style as the calculus sequence? Sometimes it seems like all we do in calculus is more or less use integration/derivatives in different ways or extend them to multiple dimensions/multiple equations.

How will the above classes be different is what I am wondering?

 

[MarneMath]

Depends. Linear Algebra (lower level) was basically like my calculus sequence, learn this technique, learn this theorem, prove this theorem, solve a bunch of problems with this technique prove some simple theorems, rinse and repeat.

After linear algebra it became pretty different. For example, you probably came across the word 'uniform convergences' someone may have talked about it, but odds are, you were never asked to prove anything about it on your own, but in an intro to real analysis course, a typical problem will go like this:

Assume that (f_n) converges uniformly to f on A that each f_n is uniformly continuous on A. Prove that f is uniformly continuous on A.

It isn't to terribly difficult to prove, but for a lot of students, they struggle at first. Going from "do technique A when you encounter problem X and use technique B when encounter problem Y" dwindles and eventually it becomes uses this 'theorem and definition to prove more theorems'. Leanring how things connect, and why certain things in math work is complicated and time consuming, but I think fun.

You might encounter some more calculus type classes (like a complex variable class vs complex analysis or PDE made for engineer not math majors).

 

[SophusLies]

You will have to prove theorems using concepts and techniques from math. An advanced math class will teach you think concise as well as logically. The thing I learn(ed) most from pure math classes is the ability to generalize. For instance, instead of working with things in R³ you'll learn to prove things in Rⁿ.

 

[ZombieFeynman]

I forget the name of the method (someone here will know I'm sure), but my analysis professor used a really cool way of teaching. You were given a list of axioms at the beginning of the semester. He gave a list of theorems too (unproven; some were false). Each day he called people to present their proofs (or counter examples) in order. You got points if you were right, you lost them if you were wrong, you stayed the same if you passed to the next person (but then weren't up for a proof until your name was called again). If you got a certain number of points, you got a certain grade.

Certainly most of my upper level courses weren't like that, but that was still my favorite math class.

Edit: What he used is a modified Moore Method