boring Saturday morning maths ramblings

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.

So yesterday I read about DragonBox, a game for learning algebra. I haven’t played it myself yet, just seen some videos, but it looks like you learn the rules by manipulating a bunch of pictures on a touch screen, and only later see the usual symbols and numbers.

Anyway I sent it to my sister, and it sounds like my 7-year-old nephew was happily playing it most of the evening. So that’s nice.

I read about it on Hacker News, and today there are a load of comments there probably duplicating a lot of the content of this one. I can’t be bothered to read them all right now, I’m in a writing mood more than a reading mood.

What interests me about this game is its complete focus on teaching the algorithmic component of algebra – cancelling factors, ‘throwing things over the equals sign’, multiplying both sides, whatever. Algebra as game mechanics. And that’s likely to be a good thing, as games are generally somewhat more fun than algebra classes. Learning the rules by playing around and seeing what happens is likely to be more successful than being told explicitly what the rules are, and being afraid to experiment too much in case you ‘get it wrong’.

Of course, the downside is that it’s completely divorced from a conceptual understanding of what you’re doing, and how it relates to the maths you already know. I’ve always been kind of annoyed at how late algebra is taught in schools, and how its separated so distinctly from arithmetic. I feel like a good start on the way to algebra is made right at the beginning of school, where you get those worksheets with the boxes to fill in:

1 + 3 = [ ]
2 + 4 = [ ]

and then for a bit of variety you might get

2 + [ ] = 5

later in the sheet. No big deal. Then somehow later in school the first two examples become ‘arithmetic’ and the third is some abstruse topic called ‘algebra’.

I feel like if it was introduced alongside arithmetic it might be easier to take in. E.g. when you first learn multiplication:

2 * 3 = [ ]

then why not also learn

2 * [ ] = 6 ?

It seems unfair to save all this stuff up for a few years and then intimidate you with the likes of

2 * [ ] + 3 = 7,

under the threatening new title of ‘algebra’, along with an array of confusing new algorithmic techniques for ‘solving’ an equation.

I’m not trying to criticise DragonBox. I think it’s a great idea. I guess what I’m wondering is what DragonBox’s twin looks like. The game that teaches conceptual understanding of algebra divorced from algorithmic understanding, with the same emphasis on playing around and not worrying too much about whether you’re doing the right thing. E.g. in the equation above you could just try some numbers and find that 2 works, or notice that 4 + 3 is 7 and work backwards, or anything else that helps. It would be nice to be able to use a bunch of examples like these to work towards finding a general algorithm for solving the equation, but one you’re using because it makes intuitive sense to you rather than because some teacher or some game told you to do it.


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.

Currently flicking through Seymour Papert’s brilliant ‘Mindstorms: Children, Computers and Powerful Ideas’. I’ve read it before but the whole book is popping with insights and I should probably do this many more times. Plus maybe this time it’ll actually teach me how to juggle too?

In the epilogue on mathematical thinking, he discusses his experiences getting people “with little mathematical knowledge” (enough to rearrange an equation, though) to work towards a proof that the square root of 2 is irrational. In the process, he gives two standard proofs, the second of which I’d never seen before:

Of these I shall contrast two which differ along a dimension one might call ‘gestalt versus atomistic’ or ‘aha-single-flash-insight versus step-by-step reasoning’.

They both start with the usual proof by contradiction: let sqrt(2)= p/q, a fraction expressed in its lowest terms. This is then rearranged to get

p^2 = 2 q^2,

and then you can go down the route of “well p must be even, so p=2r, so q^2=2r^2 WHY IS q EVEN TOO WHEN WE CANCELLED THE TWOS AT THE START?” Which is pretty compelling stuff.

But I really like the second proof, what he calls the ‘flash’ version:

Think of p as a product of its prime factors, e.g. 6=23. Then p^2 will have an even number of each prime factor, e.g. 36=2233. But then our equation p^2=2q^2 is saying that an even set of prime factors equals another even set multiplied by a 2 on its own, which makes no sense at all. Done.

Papert makes the point that if you have the right idea (decomposition into prime factors) ‘pre-loaded’ into your head, the equation is directly seen as absurd just by looking at it. In fact it now surprises me that it didn’t look wrong before! The first, more algorithmic proof gets you there, but without the insight flash.

Anyway, this is a lot of set-up just to say that reading this made me understand more clearly that that dopamine hit of insight is a large part of what I’m attracted to maths for. For me, much of the purpose of studying maths is to pre-load a pile of the necessary structure into my head to enable these insights to occur. Differential geometry is great for this, and this is probably why I like learning it so much! In the process I’m willing to put up with a load of chains of logic (which I am annoyingly poor at) but I’m always really hoping that they will be the means to an aha-single-flash insight.

OK time to go practice juggling (= ‘throwing all the balls on the floor’) again.

‘two types of mathematician’ linkdump

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.

This is great, and something I thought was missing from the SSC discussion (pity I’m too slow to ever get round to commenting). I only really got a tumblr to read some stuff, but I am fascinated by this topic so here is some related crap…

I think there’s a lot to this ‘different ways of understanding maths’ idea, sometimes it seems that you can pretty much give a mathematician a pen and they will start writing an essay on two types of mathematician. The clusters seem to be roughly ‘algebra/problem-solving/analysis/logic/precision’ vs. geometry/theorising/synthesis/intuition/hand-waving’ but there is plenty of variation.

I keep meaning to collect together a set of all of these I can find, so this has motivated me to make a first attempt:

“It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. 

The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.”

  • Gian-Carlo Rota made a division into ‘problem solvers and theorizers’ (in ‘Indiscrete Thoughts’, excerpt here)
  • Timothy Gowers makes a very similar division in his ‘Two Cultures of Mathematics’ (discussion and link to pdf here)
  • Freeman Dyson calls his groups ‘Birds and Frogs’ (this one’s more physics-focussed)
  • Vladimir Arnold turns the whole thing into a massive ideological war in his wonderful rant ‘On Teaching Mathematics’
  • Michael Atiyah makes the distinction in ‘What is Geometry?’:

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in real mathematical problems.

There’s also his famous quote:

Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’

  • Finally, I think something similar is at the heart of William Thurston’s debate with Jaffe and Quinn over the necessity of rigour in mathematics – see Thurston’s ‘On proof and progress in mathematics’. There is also a wonderful list of ways of understanding the concept of a derivative in Section 2.

OK hopefully I know how to write a post now! Hope this is interesting to someone.