Examples first II

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.

My favourite pedagogical principle: examples first!

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.

How should vector spaces be introduced?

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 ? 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 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?

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 x^a+b=x^ax^b 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.