Examples first

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

I finally got nerd-sniped by the Arbital thing, so here are some rambly thoughts.

I’m definitely intrigued by the idea, because there are a lot of topics in maths that can be understood at very different levels. It’s true that there is often a level of understanding below which you have very little hope (e.g. @nostalgebraist‘s example of reading the integration by parts page without calculus). But often there are many tiers of understanding above the first. E.g. Thurston talking about different concepts of the derivative:

image

So I can see some potential here. However I’m not at all taken with the current implementation. The biggest dud for me is the clunky, prescriptive questionnaire interface over the top of it. I’m normally pretty good at identifying whether I can follow an explanation once I can actually see it, the main advantage I can see to the site is having many such explanations in the same place for easy access. I don’t want to be tediously clicking through branching pathways, like some Choose Your Own Adventure book where every adventure is just Bayes’s theorem again.

To my mind that part of Arbital’s just plain bad, but another aspect of the site that I don’t like may improve with time. At the moment it’s heavily curated, giving it a very homogeneous textbook feel. It sounds like the idea is to allow users with enough karma to contribute themselves, and then it may become more diverse.

I think the thing I would like is more *styles* of explanation rather than particularly different *levels*. For example, here’s the earlier part of Thurston’s list [this is such a wonderful paper, if for any misguided reason you are reading all this rubbish you should just go and read that instead :)]:

image

These are very different ways of thinking of a derivative, but I wouldn’t say that they are at different levels. I think different ones will appeal to different people, and your ideal starting point will vary depending on that. (Eventually you need to learn the others, of course, but initial motivation is important. For me (1) and (4) feel the most natural, and motivate me to learn the incredibly necessary (2), and even to deal with the tedium of (3)). I guess my ideal maths-explanation site would have a variety of explanations at each level.

[At which point, is it even worth trying to collect all this disparate stuff on one site? I honestly don’t know.]

My final bloody obvious objection is that politically they should definitely not have gone for Bayes’s theorem as a nice uncontroversial starting example instead of basically any other topic in mathematics, but, well, Yudkowsky and doing the politically sensible thing rarely go together.

Still, after all that grumbling I do appreciate any attempts at providing better explanations for mathematical concepts online. I find this stuff really interesting for some reason, and the idea I personally like to think about is an approach I call ‘examples first’, after these two blog posts by Timothy Gowers. (The second one has an absolutely epic comment thread – reading that and following the links has taught me more maths than any single course I ever took at university.)

I always like to learn by following concrete worked examples. This may just be a personal preference, but it sounds from the blog post that it’s pretty common. In my case, out of the Arbital explanations the one I’d personally choose was the beginner-level one (so much for that questionnaire). I would always rather learn by doing problems about socks in a drawer than read an explanation in terms of some abstract variables A and B. If I’m learning maths from a textbook I always start by looking at the pictures and reading any waffly chunks of text, then look at the examples and exercises. I only grudgingly read the theory bit when I’m really stuck. 

I guess what I would like is something like a repository of worked examples, where you search for a topic and then get a bunch of problems to try. Wikipedia generally ends up with a formalism-heavy approach, whereas I would always prefer to look at some specific function, or a matrix with actual numbers in it or something.

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