We don’t teach primarily facts in lectures. We have proofs, at least in pure maths lectures, and we need to teach our students what a mathematical argument is, by developing such an argument in front of them.

If you have ever been to a talk or lecture where a proof or argument is displayed all at once on a slide, you may see what I mean: I personally find a proof very hard to follow when it is presented all at once, and it makes it impossible to say things like “What should we do next?” “We’re actually trying to get to this, so this next step seems sensible” or similar. Obviously, if you want to show a lot of data, a picture/diagram, an animation, etc, blackboard or real-time writing on overheads is not the way forward. But for proofs, I think it is the best way to go, and I know a lot of mathematicians agree with me (though possibly not all, I haven’t asked all).

Should we be teaching new material in lectures? I have heard the argument that in the days of the printing press, lectures are really not needed any more for transfer of information from the lecturer to the students’ pieces of paper, but should be about understanding. Read more about this in “Should we give out lecture notes”. I am trying to compromise by telling students what will be in the next lecture (in my blog), so they can read ahead in one of the many good maths books that are around, so they don’t *have* to have it as new material if they don’t like that way.

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I think the idea of ‘developing arguments in front of someone’ is one of the most important parts of a lecture. You’re watching an expert develop a sustained argument in their specialist field. You can watch them as they suggest ways of thinking about the problem, connecting ideas etc. And you can do it live, so you can ask them questions. (Either during, or afterwards.)

One of the other areas is interactivity – not precisely through questions, but through a lecturer observing the attendees. The lecturer can observe whether they’re following OK (continue as before), keenly with it (maybe go a bit further) or completely baffled (maybe try previous point again, a bit differently?). This was one of the things that irritated me most about DVDs of lectures for maths – the lecturer would skim over a really hard bit as though it was totally obvious, and I’d be going ‘nope, no idea. What if I rewind 30s and try again? No, still completely baffled.’

As a pretty motivated and reasonably intelligent adult learner, I could not learn solely from books and DVDs. Lectures do give that explanation of *how* the lecturer is reasoning about the subject, so you can hope to go away and mimic/reproduce that behaviour yourself. That’s not really possible from a textbook – you can derive information about how to write up proofs, but you can’t derive the information about how to construct the proof yourself.

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Thanks, this is really helpful. I am just considering whether as a lecturer I do actually respond to my audience in terms of speed of explanation. I suppose some lecturers might not. But in fact today I believe I did: the material that my students told me had been in another lecture course I tried to cover slightly faster so they wouldn’t get bored, but when they said “this bit we haven’t seen yet”, I tried to go more slowly and explain a bit more. I hope that I succeeded in that 🙂

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