Best way to learn maths - proofs or exercises?

Question

My question is regarding the most effective way to learn maths? Should one concentrate on churning through the exercises or is it better to concentrate on understanding and reproducing the proofs?

I understand that ideally one would do both however my time is limited, (father of two children under 3, run a small business, full course load). I am in the first year of doing a mathematics degree an am struggling.

I go to the lectures religiously and ask question, however more often than not I leave not fully understanding the subject matter. I usually end up poring over the textbooks, scouring the internet or harassing the teaching assistants till I understand, but this eats into the time I have left to do the exercises and review the proofs.

Any advice or insights gratefully accepted!

Answer 1

I would say that you should be able to do the exercises, and at least be able to understand the proofs. Of course, if you want to be really good, you will have to be able to write the proofs also, but the basic level of understanding would be doing the exercises and understanding the proofs.

Answer 2

Early math classes (especially calculus) are very often taught from an applied perspective, where proofs and understanding aren't very important. More advanced classes (should you take them) will be entirely focused on understanding and proof. So if you plan to continue in math, it's definitely worth learning the deeper nature of ideas.

Maybe you're just trying to learn some applied knowledge for, say, engineering. In that case, it's not necessary to learn anything well- as long as you can solve the problems. But it is often easier to learn ideas instead of algorithms: if you understand how to derive other differentiation rules from the chain rule, you have to memorize much less.

I support the proof approach for two reasons: firstly, I think it's a lot more enjoyable and interesting. But secondly, it's easy to forget details if you don't know why they matter.

Imagine someone teaches you how to make an origami crane. Then you make one on your own: you know if you made a mistake, because it doesn't look like a crane. You can try to back up and figure out your error, but you have an intuition about what should happen.

But suppose you learned how to make an origami crane without any paper, just by memorizing sequences of folds. If you had to write down the folds, it would be very easy to make a mistake: you don't really know what's going on, or what it leads to, and there's no way to differentiate "a pretty good crane" from "something really weird."

And then suppose you really got into origami, and understood exactly how certain folds would lead to the final result. You can even make your own constructions. Now, if you try to make a crane, you barely need to remember anything- just a general idea of what it should look like will be enough. It would take a while to get this level of knowledge, but it'd be really hard to make a bad crane.

Math is similar: you can memorize algorithms, you can learn when an argument makes sense, and you can learn how to make good arguments yourself. But the third option makes everything so much easier, once you get past the initial hurdle.