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.

Now I think a very common attitude to this is that doing mathematics (that is, thinking of proofs) is something that you can’t really be taught directly: instead, you read your notes and do lots of carefully designed questions and find that proving results is a skill that you develop with practice, especially if you were born with a mysterious quality called mathematical ability. And undoubtedly there is some truth in the previous sentence — the method described is the method by which pretty well all mathematicians working today have learnt how to do maths. But there is a significant downside to this method, which is that there are also many people for whom it does not work. They go to university full of enthusiasm for mathematics and find that the subject at university level is much harder than they expected, and that they don’t know how to go about developing the skills that I’ve just been talking about. Gradually as the course proceeds, they fall further and further behind, while some of their contemporaries seem not to. It can be pretty demoralizing and also, given how hard it is to get into Cambridge, a real waste of talent. (I don’t think it is a total waste, by the way, since many people, myself included, have had the experience of understanding some mathematics much better a year or two later than they did when they were supposed to be learning it. I think that even people who get left behind by the sheer pace of the Cambridge mathematics course leave Cambridge having had their minds altered in a way that is very valuable in their working lives. I’d be very interested to hear from anyone in that position, to see whether I’m right about this.)

Another serious drawback with the attitude described above is that it underestimates the extent to which mathematical ability is something you acquire through hard work. It’s true that some people seem to find the subject easier than others. But nearly always you will find that these mysteriously clever people have spent a lot of time thinking about mathematics. In many cases, their ability is no more mysterious than the ability of a very good pianist who has practised for three hours a day for many years.

— From Dr. Gowers’ blog

Fortunately, you don’t have to be at Cambridge to learn from Dr. Gowers.  He’s posting everything on his blog.

At the moment he is discussing the various pieces of logic that mathematicians use every single day almost without thinking about it.  It’s a bit like if you wanted to teach someone to play baseball, then they first have to be able to stand, walk, throw a ball, etc. without even thinking about it.  Otherwise all their mental energy goes into the things which aren’t even the main point!

P.S. By the way, in the UK you start in on your major right away and don’t have a load of general education requirements, so they are quickly doing what is sophomore/junior math here at OU.   A little more precisely, the audience in his mind is the students taking the Cambridge Mathematical Tripos, which is the three year sequence of courses leading to an undergrad degree in math at Cambridge.