### Patrick Stevens

Former mathematics student at the University of Cambridge; now a software engineer.

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# Thinking styles

All the way back into primary school (ages 4 to 11 years old, in case a non-Brit is reading this), we have been told repeatedly that “people learn things in different ways”. There were two years in primary school when I had a teacher who was very into Six Thinking Hats (leading to the worst outbreak of headlice I’ve ever encountered) and mind maps. I never understood mind maps, and whenever we were told to create a mind map, I’d make mine as linear and boxy as possible, out of simple frustration with the pointless task of making a picture of something that I already had perfectly well-set-out in my mind. I quickly learnt to correlate “making a mind map” with “being slow and inefficient at thinking”. (This was back when my memory was still exceptionally good, so I wasn’t really learning much at school - having read, and therefore memorised, a good children’s encyclopaedia was enough for me - and hence relative to me, pretty much everyone else was slow and inefficient, because I’d already learnt the material.)

It’s only now that I’ve realised that perhaps some people actually do think in a way that makes mind maps helpful. I’m not bad at spatial visualisation (not great, but not totally inept), but I don’t think in pictures at all. Apparently, about 3% of people [sorry, the source for the statistic wasn’t given on that page] simply do not have mental images - I don’t fall into that 3%, but a close family member tells me ey does - ey cannot make sense of pictures at all without translating them into words. (Possibly a genetic bias? At least another two close family members are very visual indeed.) Ey told me that the world is really not set up for people who can’t visualise: whenever you say you don’t understand something, the default response is apparently to say exactly the same thing again, but accompanied by a picture - completely useless for a non-visualiser. I’ve never noticed this before, and a quick memory trawl is inconclusive, but I will certainly keep a look out for it and against it.

A prime example (no pun intended) of an extraneous visual approach to something was the multiplication of two two-digit numbers by using the fact that $(a+b)(c+d)=ac+bc+ad+bd$. As an example, I’ll take the numbers 35 and 27. The method involved drawing a box of (nominal) side lengths 27x35, and drawing two lines to divide the sides (nominally) into 20+7 and 30+5. Then in each of the four sub-boxes thus created, you had to write the area of that sub-box (that is, calculate $30 \times 20$, $30 \times 7$, $5 \times 20$, $5 \times 7$) and then add them all up to get the total area. This method seemed like an enormous waste of time and space to me; I had already learnt to multiply arbitrary numbers together through the Kumon program by using the standard long multiplication, and to have to learn a method that was about ten times slower and used four times more paper seemed immensely wasteful. I formed the opinion that the reason people were bad at multiplication was that they were being told to use these useless methods that no-one in their right mind could possibly understand. The Generalising from One Example LessWrong post contains an extremely relevant passage:

I only really discovered this in my last job as a school teacher. There’s a lot of data on teaching methods that students enjoy and learn from. I had some of these methods…inflicted…on me during my school days, and I had no intention of abusing my own students in the same way. And when I tried the sorts of really creative stuff I would have loved as a student…it fell completely flat. What ended up working? Something pretty close to the teaching methods I’d hated as a kid. Oh. Well. Now I know why people use them so much. And here I’d gone through life thinking my teachers were just inexplicably bad at what they did, never figuring out that I was just the odd outlier who couldn’t be reached by this sort of stuff.

And it’s only very recently that it occurred to me that this is quite possibly exactly my experience. The visual techniques simply work for other people.

Another example (again from arithmetic) is the number line (and the closely related and suggestively named real line). A large chunk of the first few years at primary school was devoted to learning to count and add (pretty tedious stuff, especially if you already knew how to count and add!). One of the key methods used was the number line - so, for instance, to work out $8-3$, you had to count forward 8 and go back 3. I hated this method - again, it wasted time (why not just go forward 5?) and space (draw out a line? no thanks!). Apparently there was a study done on an untouched-by-society tribe, and it turns out that viewing numbers spatially is not inbuilt in humans. 1 Maybe I was just unusually unable to learn this view of numbers.

Over the last few years, however, most noticeably as I have come to learn more maths, I have started to rely on pictures considerably more than I used to. I discovered the memory technique of “imagine a picture, the more ridiculous the better” to link two concepts (that’s how I’m learning the capitals of the world - Luanda is the capital of Angola, which I remember as a footballer scoring a GOAL [Angola] by kicking the ball into a LOO which is sitting between the goalposts), and I have used it to learn a variety of things. In the topic of analysis, I rely on pictures as a guide to intuition - the statement that “for every $\epsilon > 0$, there is a $\delta > 0$ such that for all $y$ where $% $, we have $% $ is pretty horrific to grok, but a simple picture is enough for me to understand it. However, I don’t use pictures except as intuition-pumps - they’re often too unreliable to use as proofs, and I don’t (yet) think sufficiently pictorially to do this.

I also have an unusually wide vocabulary. While I used to find it facile to augment the erudition of my verbiage, I am now afflicted by a slowness of thought that makes it something of an effort. It feels perhaps as if my facility with words has decreased as my mental use of pictures has increased (though this is hard to gauge, because I also know that my mental faculties in general have declined precipitously since perhaps the age of 13). It is possible that I was forced by my brain to become excellent with words, as a substitute for thinking in pictures (since words are the most general means we have of representing thought-symbols, though pictures may be higher-fidelity).

I also wonder whether gesturing during speech has an association with picture-representation. As far as I am aware, I don’t gesture very much when speaking; I know that the close-family-member-who-doesn’t-think-in-pictures does not gesture at all. However, the greatest gesturer I know certainly thinks in pictures; and Tom Körner gestures vigorously while lecturing proofs, while also using pictures very frequently as intuition guides. I don’t have very much data on this, though, and I would be interested to add more data points.

Anyway, the message to take away is that people don’t all think in the same way - if you find yourself trying to communicate something to someone who seems to be refusing obstinately to understand, do consider the possibility that you’re not approaching the explanation in a way that the other person can comprehend.

1. I saw this study over a year ago, and I can’t find it now. I thought it was reported on Ars Technica, but apparently not.