Broaden Math Studies: Tips for Undergraduate Students

[Opus_723]

I'm an undergraduate student, currently finishing up Differential Calculus and preparing to start a Physics major. It had been a couple of years since I'd done any math when I started this class, and I had to brush up on Algebra and Trig, but I'm currently at the top of my class in Calculus, and I'm consistently getting 100% on exams, so I'm feeling pretty confident.

However, I still feel hazy on a lot of the more obscure techniques we learned in high school, and lately I've gotten very interested in various sites around the internet filled with really awesome looking math, as well as some more conceptual stuff. Anyway, I've gotten very interested in improving my math skills. So I've decided to invest in a couple of high school textbooks so I can work through them and review everything I've forgotten.

My question is, if I want to really deepen my understanding of math, what should I do besides this? I feel like a lot of stuff gets skipped over in classes, especially proofs. (I've been getting more interested in proofs lately. I've never been good at them, and feel like I should be able to start doing some simple ones without being walked through, however, I have no idea where to start with this.)

What I'm looking for is some sort of road map that takes you into more interesting, obscure math, while still being at a level that a Calc student can understand. I realize textbooks are a good road map to essential concepts, but I feel they're lacking. I'm having a hard time expressing what I want, but I'm interested in more... I don't know, fun, engaging, tough math.

What would be ideal is if these also pertained to algebra and calculus, so I wouldn't be wandering too far off track study-wise.

I'm sorry if this sounds really vague, but hopefully someone can grasp what I'm trying to say.

Thanks.

 

[chiro]

I'm an undergraduate student, currently finishing up Differential Calculus and preparing to start a Physics major. It had been a couple of years since I'd done any math when I started this class, and I had to brush up on Algebra and Trig, but I'm currently at the top of my class in Calculus, and I'm consistently getting 100% on exams, so I'm feeling pretty confident.

However, I still feel hazy on a lot of the more obscure techniques we learned in high school, and lately I've gotten very interested in various sites around the internet filled with really awesome looking math, as well as some more conceptual stuff. Anyway, I've gotten very interested in improving my math skills. So I've decided to invest in a couple of high school textbooks so I can work through them and review everything I've forgotten.

My question is, if I want to really deepen my understanding of math, what should I do besides this? I feel like a lot of stuff gets skipped over in classes, especially proofs. (I've been getting more interested in proofs lately. I've never been good at them, and feel like I should be able to start doing some simple ones without being walked through, however, I have no idea where to start with this.)

What I'm looking for is some sort of road map that takes you into more interesting, obscure math, while still being at a level that a Calc student can understand. I realize textbooks are a good road map to essential concepts, but I feel they're lacking. I'm having a hard time expressing what I want, but I'm interested in more... I don't know, fun, engaging, tough math.

What would be ideal is if these also pertained to algebra and calculus, so I wouldn't be wandering too far off track study-wise.

I'm sorry if this sounds really vague, but hopefully someone can grasp what I'm trying to say.

Thanks.

What kind of understanding do you want to obtain?

Would you like to understand the motivation behind the developments done in mathematics? Would you like to understand how the inventors thought about the mathematics that they worked on?
Would you like a theorem proof monologue for some specialized subject?

It would help if you were a little more specific.

 

[micromass]

Buy a rigourous calc book and work through it. Spivak's calculus will certainly be a challenge. Or if you don't want to do thesame calculus again, try "calculus on manifolds" by Spivak, but I fear that the book will be a little to hard for somebody not familiar to proofs.

You could also try to teach yourself algebra. Try a linear algebra book and work through all the proofs. Or perhaps some abstract algebra can be useful! Try Pinter's "a book of abstract algebra", it's a very gentle introduction and certainly suited for somebody new to proofs. (in fact, I think it's the perfect book to learn proofs with!)