Archive for the ‘Mathematical pedagogy’ Category

Just-do-it proofs

August 16, 2008

This post is another sample Tricks Wiki article, which revisits a theme that I treated on my web page. Imre Leader pointed out to me that I hadn’t completely done justice to the idea, and that the notion of a “just-do-it proof” had some more specific features that I had not sufficiently emphasized. His opinion matters to me since he was the one who told me about the concept. He himself got it from Béla Bollobás: I don’t know whether it goes further back than that. This is a second attempt at explaining it. Imre, if there’s anything you don’t like about this one, then please edit it when it appears on the Tricks Wiki.

Title: Just-do-it proofs.

Quick description: If you are asked to prove that a sequence or a set exists with certain properties, then the best way of doing so may well be not to use any tricks but just to go ahead and do it: that is, you build the set/sequence up one element at a time, and however you do so you find that it is never difficult to continue building. (more…)

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How to use Zorn’s lemma

August 12, 2008

I am continuing my series of sample articles for the Tricks Wiki with one that is intended to represent a general class of such articles. It is common practice in lecture courses (at least if the ones I attended as an undergraduate are anything to go by) to state useful theorems, lemmas, propositions, etc., without going to much trouble to explain why they are useful. Of course, there are many ways to pick up this further understanding: taking note of where and how such results are used, doing carefully designed exercises, and so on. Nevertheless, it is often the case that more could be done to help people recognise the signs that indicate that a particular result can be applied.

This article, which is principally aimed at undergraduates early on in a mathematics degree, is inspired by an experience I myself had as an undergraduate. I had a sheet of challenging problems (set by Béla Bollobás) and one of them completely stumped me. (I’m sure several of them stumped me but this is the one that sticks in my mind.) I can’t remember exactly what the question was, but it was something of similar difficulty to that of determining whether an additive function from \mathbb{R} to \mathbb{R} was necessarily linear. My supervision partner solved the problem using Zorn’s lemma, which we had been told about in a lecture, and I just sat there in disbelief because it hadn’t even remotely occurred to me that Zorn’s lemma might be useful. At some point in the intervening years, I “got” Zorn’s lemma and now find it straightforward to see where it is needed. This article is intended to speed up that process for other people. (more…)

Dimension arguments in combinatorics

July 31, 2008

Here is another article that I hope to develop into an entry on the Tricks Wiki. It concerns the use of linear algebra to solve extremal problems in combinatorics. The method is quite easy to illustrate with some well-known examples, but what I find interesting is the question of how to recognise the kind of problem where the method is likely to apply. I have something to say about that, but I’d like to make clear that I didn’t think of it for myself. If I remember rightly, I read it in something that Noga Alon wrote. I’ll draw attention to it when I get there. (more…)

Recognising countable sets

July 30, 2008

As may be obvious from the sudden increase in my posting rate (which I don’t expect to be able to keep up) The Princeton Companion to Mathematics is now off my hands, which gives me the chance to devote a bit of attention to other projects, of which the Tricks Wiki is one. So in this post I’m going to discuss a relatively elementary piece of university mathematics, and will do so in the form of a sample article for that site. I’ll be a little careful about predicting when the site itself will be up and running, but let me just say that I’ve put some work into it recently and I don’t want to waste that work.

In what follows, I shall adhere to what I hope will be the basic format of an article on the site. The most important elements of that format are that there is a brief description, or “slogan”, that encapsulates the basic idea, and a general discussion of the idea that is illustrated by several clearly delineated examples. (more…)

Examples first II

October 24, 2007

It’s what blogging is all about I suppose, but I have been surprised in several different ways by the comments on my previous post. To begin with, I was so sure of the principle I was advocating that I thought that all I’d have to do was explain it briefly and then anybody who read it would instantly agree with it. That was clearly pretty naive of me, and I certainly didn’t expect that some people would be actively hostile to the idea (though I suspect that their real target was not precisely the same as what I was putting forward). But I was also surprised by the number of interesting further points and qualifications that were made, which I will now try to use to articulate a more nuanced version of the principle. (more…)

My favourite pedagogical principle: examples first!

October 19, 2007

This post is about a very simple idea that can dramatically improve the readability of just about anything, though I shall restrict my discussion to the question of how to write clearly about mathematics. The idea is more or less there in the title: present examples before you discuss general concepts. Before I go any further, I want to make very clear what the point is here. It is not the extremely obvious point that it is good to illustrate what you are saying with examples. Rather, it is to do with where those examples should appear in the exposition. So the emphasis is on the word “first” rather than on the word “examples”.

If this too seems pretty obvious, I invite you to consider how common it is to do the opposite. (more…)

How should vector spaces be introduced?

September 14, 2007

This question arose in the discussion of my previous post, but deserves a place to itself because (in my opinion, which I shall try to justify) it involves different issues. For example, how does one explain the point of the abstract notion of finite-dimensional vector spaces when, unlike with groups, you don’t seem to have an interesting collection of different spaces? Why not just use \mathbb{R}^n? I addressed this point on my home page here, so won’t discuss it further on this post. But another point, which was raised in the previous discussion, concerns the relationship between theory and computation. I think it’s pretty uncontroversial to say that if you don’t know how to invert a matrix, or extend a linearly independent subset of \mathbb{R}^n to a basis, then you don’t truly understand linear algebra, even if you can state and prove conditions for a linear map to be invertible and can prove that every linearly independent set extends to a basis. Equally, as was pointed out, if you can multiply matrices but don’t understand their connection with linear maps, then you don’t truly understand matrix multiplication. (For example, it won’t be obvious to you that it is associative.) But how does one get people to understand the theory, be able to carry out computations, and see the links between the two? This is another situation where my own experience was not completely satisfactory: I’d be taught the theory in lectures and given computational questions to do, as though once I knew the theory I’d immediately see its relevance. But in fact I found the computational questions pretty hard, and some of the links to the theory were things I didn’t appreciate until years later and I found myself having to explain the subject to others.

How should logarithms be taught?

September 13, 2007

Having a blog gives me a chance to defend myself against a number of people who took issue with a passage in Mathematics, A Very Short Introduction, where I made the tentative suggestion that an abstract approach to mathematics could sometimes be better, pedagogically speaking, than a concrete one — even at school level. This was part of a general discussion about why many people come to hate mathematics.

The example I chose was logarithms and exponentials. The traditional method of teaching them, I would suggest, is to explain what they mean and then derive their properties from this basic meaning. So, for example, to justify the rule that xa+b=xaxb one would say something like that if you have a xs followed by b xs and you multiply them all together then you are multiplying a+b xs all together. (more…)