I keep thinking I should study group theory. I topped out at differential equations in college (except for a topology class I took for fun), so there are whole areas of math that I’m not familiar with. Thanks for the information.
Unlike differential equations, group theory doesn’t require much in the way of prerequisites, so it should be a good subject to dip one’s feet into.
It’s misleading to speak of “topping out” at a math class. Some math classes build on each other, but some don’t. It’s perfectly possible to learn group theory without any calculus, for instance. Group theory usually isn’t introduced until after calculus or diffEQ, but it could be, without problem. One’s not really “higher” than the other.
Speaking of which:
I’m sure this is controversial, but I actually think that algebra should be taught before arithmetic. All of that “number sense” that gets talked about as something that arithmetic students should be learning is actually algebra, and they have a hard time learning it because they haven’t been taught algebra yet. Of course you need arithmetic before you can learn algebra of numbers, which is what’s usually referred to as “algebra” in schools, but most of the same principles apply to all of the other algebras, like algebra of rotations.
You can teach a lot of quadratic equation stuff without getting into complex numbers at all. There’s a lot of circular thinking in discussing and evaluating how and in what order various concepts are taught, because everything depends on everything else. “Let’s keep doing it this way, with minor modifications” is much easier to deal with than “Yes, we currently use X later when teaching Y, but we could change or eliminate Y, which would require changing Z, yes, but …”
Just for the record, I will mention that I took a course in modern algebra before I had taken differential equations. And that course included Galois theory and the Wedderburn theorems. At the end of that year I knew I would become a mathematician.
I agree. I let my experience going through an engineering program, where math generally stops at diffeq, with everything having built up to that (algebra, trig and geometry to calculus, and calculus to diffeq) bias me. And I should know better, because the topology class I took, in spite of having upper-level math students in it (except for me) didn’t require anything above algebra.
My undergrad degree was dual-major Mathematics and Physics. I got the “practical” math (integrals, differentials, matrices, etc) first in physics classes generally a semester or more ahead of when I saw it first in math classes. Which actually made learning the formalism a lot simpler because I already had a working feel for what was going on. The “real” math (like geometry and algebra) was harder in some sense because I didn’t have the experience of using them yet, but the formalism was easier. Eventually the algebra and geometry math got applied in grad school (physics, but still very mathy).