Most of our descriptions of how our brains work are fundamentally vague. We speak of our brains performing verbs like “think”, “realize”, “forget”, or “hope” but we aren’t talking about what’s going on mechanically to result in those qualities.

Sure, these can all be assigned truth values, in the sense that if everyone generally agrees that someone ‘realized’ something, we might define their brain to have performed the objective act of ‘realization’. But this gives no technical understanding of what the process of realization is – beyond, perhaps, some hand-wavey story about connections being bridged between neurons.

So, sometime in the last few years the English-speaking Internet became aware of the condition called aphantasia. Aphantasia is when a person is unable to picture images in their thoughts – they don’t have a “mind’s eye” at all.

This is interesting because, in contrast to the above, aphantasia is a concrete description of how the brain works. Some people see an image in their head when they draw or recall something; others don’t. Their brains work in materially different ways. I would have no idea how to figure out if two people “realize” something via different mechanisms, but I can be sure that two people’s brains operate differently, if one sees pictures and the others don’t.

Knowing about aphantasia is fun because lots of people who can see images are surprised that anyone can’t, and lots of people who can’t are flabbergasted that anyone can – they thought that talking about “picturing a person” was always just a figure of speech. It’s amusing when it comes up in conversation because you tend to get a mix of surprised people on both sides.

I happen to be aphantasiac (if that’s the word for it). I guess I wish I wasn’t; it sounds fun and useful to be able to picture things, and it always feels like I’m missing out on some aspect of human experience by not getting to. (Although I do get glimmers of images when half asleep, I think, and I seem to remember it happening at other times also when I was younger.)

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Mental Math

Here’s another variation in how human thought works that can be described in concrete terms: how we perform mental math.

First, an example. Try to multiply these two numbers in your head. Don’t look at them while you do it– read the problem, close your eyes, and multiply in your head:

\[45 \times 27\]

This paragraph is going to be random filler text so that the answer isn’t right below the question. If you’re reading it, start thinking about how you do the multiplication in your head. Some large percentage of people will just not do it because they don’t care what a blog post asks them to do. Another large percentage won’t do it because they have a strong nervous aversion to doing math of any sort (these are usually the people who calculate tips on their phone). Another category will do it and not have any idea if the answer they got is correct… – okay, that’s probably enough filler. Moving on.

What’d you get? Here’s the answer, written out in english words so you don’t scan to it automatically: one-thousand, two-hundred and fifteen. Did you get it right? Now try to describe how you computed it. If you can see images in your head, do you imagine doing it on paper, by the grade-school algorithm of stacking the numbers up and multiplying each term? You know, this one. If you can’t, what do you do instead?

I have asked a lot of people to do things like this in person (probably… 40 or so? okay, not that many). It’s anything but a rigorous study, but I’ll tell you what I’ve seen. See if it matches your experience.

First, how you do math in your head has a strong dependence on whether you are aphantasiac. People who can see images (‘phantasiacs’?) overwhelmingly tendency to picture performing the grade-school algorithm. I have only encountered a couple of people who can easily see images in their heads but don’t do math that way.

People who can’t see images obviously can’t picture doing the equation on paper. So how do they do it? There’s a bit more variation here, and it tends to be harder to explain.

I can tell you how I do it: I use my ‘verbal’ brain as something like short-term memory. Verbal memory seems to serve as a small amount of ‘scratch space’, so if I say something in my mind it bounces around for a bit and I can summon it back after thinking about something else. It’s similar to repeating something someone said while you weren’t paying attention; it floats in the background for a few seconds and you can summon it back during that time. There is also some amount of pattern-matching going on, where certain numbers look familiar and comfortable, like the \(2 \times 45\) in the above equation, which immediately feels familiar.

So the process of doing the multiplication above goes something like this in my internal dialogue¹:

45 times 27… let’s see…
[subconcious realization that it’s going to be easier to multiply \(45(20 + 7)\) than \((40 + 5)(27)\) because \(2 \times 45\) looks ‘pleasant’]
45 times 2 is 90, moved over one, so that’s 900
…7 times 45 is… let’s see..
7 times 4 is 28, so that’s 280
7 and 5 is 35, so those give
[verbally remembering the 280, and recognizing 28 + 3 = 31] … 315
[now I can still hear the 900 from a second ago too, so grab that back]
and so 900 plus 315 is …
[9 and 3 summons 12, and the rest feels like it can be copied over]
…1215

The text here is slightly subvocalized, and I can feel how it invokes the muscles that would do the speaking if I said it out loud. I guess I compute math by talking to myself. The realizations in []s are things that happen automatically, seemingly through instant pattern recognition – I’ve done a lot of math in my life, and a lot of math in my head, and I guess my brain comfortably knows that 2 times 45 is a comfortable calculation that won’t take a lot of effort. I could do 4 times 27 if I needed to, but my brain seems to prefer smaller numbers.

I’m curious to hear if there are any other methods, besides ‘verbal’ and ‘visual’, and if they work well. Or if there are any other notable counterexamples, of visual-math people who are very accurate or quick. I’m not even sure what I would search to find this, because it isn’t talked about that much!

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Why does this matter?

Well, the other thing I’ve noticed while asking around about this:

There is a huge correlation between doing math in your head via the visual algorithm and considering yourself ‘not that good at math’.

In fact there is a correlation (in my entirely unscientific survey over the years) between doing math visually and:

• resisting doing the problem at all
• being visibly anxious that you were asked to
• not liking math, and especially not liking math in college
• getting the wrong answer
• not being confident that you got the right answer

Although there are notable exceptions. I’ve met math majors who excel at some types of math, but are terrible at numeric calculations, and do math via the visual algorithm. This makes sense; the skills involved in logical proofs are quite different from those involved in raw calculation. And I’ve found one person, if I recall correctly, who does math visually and considers themself very good at it. But it’s rare.

I don’t think any of this is because “visual people are more likely to be bad at math”. I suspect it’s because doing it visually is just a crappy method, and so if you learned it early on in your math education, mental math was always hard, and everything else followed from there. I got lucky by learning to do math a different way early on and so math was always easy for me. Although, it’s hard to be sure whether math was easy for me because I learned to do it in a good way, or if I do it in a good way because math was always going to be easy for me.

And I’m not that good at mental math, not compared to people who are into that kind of thing. But I’m decent, and clearly have enjoyed math enough in my life to, like, blog about it. I’d be curious to hear from someone who has done competition-level mental math about what methods they use, and if they do it visually. I bet I know what the answer will be.

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These are all my unfounded hypotheses, of course. But: let’s speculate. Why would doing math visually be less effective than doing it through some other system?

First, I wonder (of course I haven’t experienced this!) if maybe the visual brain is less good at error-correction. Perhaps when you see images you capture the ‘gestalt’ of them, rather than careful details, and so if you look away and look back at a pile of numbers you see… another pile of numbers, but not the same one necessarily. It would be easy to confuse digits without realizing it, so you would make mistakes and maybe be unconfident in your result.

For comparison my step of ‘resummoning a term from a few seconds ago’ feels accurate. I am quite sure I remembered the right value, like how I would be sure that I had correctly heard something you said.

Second, I suspect that using the visual algorithm restricts you to doing the steps in a certain order. When I start a problem there’s a planning phase where I figure out where to start: which of these numbers will distribute better? Which makes fewer terms? If you only know how to do right-to-left, second number under the first, then there’s no planning, so there’s no change to make the problem easier.

More problematically, that means that you might not have a step of invoking intuition at all, since the algorithm is rote and formulaic. I suspect that what it means to be ‘good at math’ in general is to have a lot of intuition about it – and if your algorithm since a young age has been formulaic, there would be no opportunities to produce intuition, so you might not have trained that ‘muscle’ at all.

Suffice to say, I suspect that the American trend of deferring menial computation to a calculator is a really terrible practice, and I suspect that a way of teaching people to do mental math via a non-visual algorithm would be far more valuable in math education than just teaching mental math in general. Maybe by learning to multiply in their heads before learning the ‘long multiplication’ algorithm, if that’s possible.

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So what?

It is strange that all it takes to realize that aphantasia is a real thing is to ask people about it – and yet there are all of these people who recently learned about it, myself included, who had just… never asked. What other qualities of our thinking are concretely describable, but haven’t been concretely described, just because we never think to ask?

This feels important. For instance, I happen to think that knowing about aphantasia is important to understanding why some people are good at mental math. But if you didn’t know about aphantasia you would never guess that a skill like mental math might be governed by a variable like that. You might instead chock the variation up to … lack of discipline, or general unintelligence, or bad education, or something.

Since aphantasia is easily missed, this leads to the question: what other variations in our brain’s workings are actually due to some discrete difference in underlying machinery? If someone is, say, depressive, is that because of a complicated combination of variables, or because they have a single binary switch that’s in a different state?