Master Class

Are you taking Dr. Roche’s Abstract Linear Algebra, Dr. Walschap’s Introduction to Analysis, Dr. Kujawa’s Introduction to Abstract Algebra, or Drs. Miller or Rafi’s Discrete Mathematical Structures?  Or maybe you just have trouble keeping track of the stuff you are already “supposed to know”?

Like that “for all x there exists y such that …” means something completely different than “there exists y such that for all x …”

Then you’re in luck!  Fellow math blogger Timothy Gowers and professor at Cambridge University has started a series of posts on his blog for first year Cambridge math majors.  Here is Dr. Gower’s first post to get you started.

If nothing else, you should read this:

Can mathematics be taught?
Let me pause right there and try to explain in more detail what I mean. The usual way of presenting pure mathematics (which is all I’ll be talking about in these posts) is this: you have some definitions and some results; you write out the definitions, you state and prove the results, and perhaps you set some exercises that test understanding of the definitions and results. End of story. Well, perhaps it’s not quite the end of the story: if you’re being conscientious then you usually follow each definition by a list of two or three key examples.

OK, what’s missing from that? Well, for a start it is very common for lecturers and authors of textbooks to take for granted that their topic is an interesting and important one. This isn’t completely unreasonable, as usually the topic is interesting and important. But if you’re trying to learn about it, it can be a huge help to have a clear idea why you are making this very significant effort. (“To do well in exams” is not the answer I’m looking for here.)

But perhaps the biggest thing that’s missing, and the thing I most want to get across, is how to go about proving results for yourself. There are plenty of books about how to solve competition-style maths problems, but what about proving the more bread-and-butter-ish results that you are shown in a typical maths course?

Before I discuss that further, let me explain why it matters. You might think it doesn’t, since if a lecturer explains how to prove a result in a course, or an author in a textbook, then you don’t have to work out the proof for yourself. But, and this is a huge but, if you are studying for a maths degree, then

(i) you do have to remember lots of proofs;

(ii) memorizing things requires significant effort;

(iii) if you can easily work out proofs instead, then you place a far smaller burden on your memory.

So it turns out that being able to work out how to prove things (perhaps with the help of one or two small hints) is hugely important, even if those proofs are there in your lecture notes already. Of course, it also goes without saying that being good at proving things will help you solve problems on examples sheets.