Help transitioning into proof-based math?

[Axel Harper]

My proof-writing skills aren't quite as good as I'd like them to be, and I'm wondering if anyone can suggest particularly good books to help me develop better mathematical reasoning. I learned this the hard way by opening Spivak and struggling through the first few chapters. I understand the calculus concepts pretty well, but I'm having trouble following the algebra and understanding how he is able to come up with these proofs. I understand proof writing isn't supposed to be fast or easy, but I don't think I'm ready for Spivak yet. I've started going through Velleman's How to Prove It, and it's a decent book, but I feel that I'm going to need more practice over the summer. Does anyone know of good books at my level?

 

[micromass]

This is a very common question here, but not one that is easy to answer. Velleman is a good book, but isn't entirely satisfactory. It will teach you the different proof techniques and the foundations behind proofs, but it won't teach you how to actually do proofs. No books can really teach you that.

What you need is somebody to guide you through your first proofs. Somebody to help you identify the basic tricks and concepts. Somebody who will criticize your proofs and break them completely down to the ground. This person can be a professor in a university, or a tutor or whatever. But if you do not have such a person, then learning how to do proofs is very difficult and you won't know how to prove things the right way until somebody criticizes the hell out of your proofs.

Furthermore, every person is different, especially when it comes to proofs. Some people are really comfortable with it and are able to do it almost immediately, other people have immense difficulties. So you need to find the book right for you.

So going through Velleman is a good idea but it won't exactly get you where you want. Try to find a person willing to help you transition into proof-based math. I don't know any other way to learn this.

 

[Axel Harper]

When I matriculate this fall, I should be able to find good professors. I'll also attend office hours and find a study group for problem sets. I'm trying to decide if I should take the honors multivariable cal/linear algebra sequence or try the introductory analysis sequence intended for math majors. I want to challenge myself, but I don't want to take a class that I won't do well in. The bare minimum for the analysis class is a 750 math SAT and a 5 on the cal BC exam. I'm worried that the class will be way over my head.