DRMacIver's Notebook

How you should have learned mental arithmetic

How you should have learned mental arithmetic

Draft bankruptcy is over, but I got about a quarter of the way into writing this post before I realised that it was going to be much longer than I expected and also that I had no fucking idea why I was writing this post, so this is the point I stopped at.

I had a conversation the other day with a small child and her father, in which I opined that the eight times tables was one of the easy ones because you could just multiply by two three times.

This was apparently confusing. “How would you do 7 times 8 by multiplying by two three times?” “14 28 56, done.” “Huh I wouldn’t have thought of that”.

The father in question is certainly decent at mathematics, although not a mathematician, so this is probably a relatively common point of confusion.

7 times 8 is pretty easy in general. For example 7 times 7 is 49, so you can just add 7 to that to get 56 (because 6 is 7 - 1). Also really sometimes when multiplying by 8 it’s easier to go the other way and remember that 8 is 10 - 2, so 7 times 8 is 70 - 14, which is 60 - 4, which is 56.

But no, you’re not supposed to do any of that, you’re supposed to just know that 7 times 8 is 56. If you work it out you’re not doing the task of learning your times tables properly. That’s cheating that is.

Desire lines in the mind

For developmental reasons it probably does make sense to start kids off with learning their times tables. I’m not sure, I’m not actually very good at teaching small children, but the fact that people don’t then go on to learn how to do mental arithmetic properly is, I think, one of the injustices at the core of how we teach mathematics. Not because mental arithmetic is particularly important, but because it’s one of the easiest places to learn to think about mathematics like a mathematician does - rather than trying to memorise a bunch of information, you see how it all fits together.

I’ve talked about this before in the context of learning ROT13, but I think the non-serious nature of that post hid this somewhat (mental arithmetic is of course very serious). There is a specific technique of learning that works very well for mathematics in particular (but is, I think, general): Rather than trying to memorise the information, you instead learn how to work it out from first principles or other remembered information. This, in turn, will usually allow you to memorise a bunch of information, but it will be the right information - that which you actually use.

The way this works is simple: Being able to work out the information means that you have a reliable feedback loop for learning it, because every time you use it you will access it by the fastest method readily available to you - whether that’s simply remembering it, or working it out from scratch, or skipping steps that you remember the results of… Each time you use it, your access to it will get faster and more fluent, until the things you use most commonly are readily to hand. But even if you don’t memorise it, you’ll be able to get there.

You can think of this approach to memorisation as that of building desire lines. Desire lines are those lines in the grass that you see on parks where people have clearly decided it’s more efficient to walk across the diagonal than follow the paved path the long way round. People build a path by walking it.

In addition, by giving people the ability to build desire lines, you give them the ability to explore. Someone who has learned a large body of information this way has not only learned that information, they’ve learned how to learn new information like it. Memorising your 8 times table by brute force is not easier for having memorised your 7 times table, but the more mental arithmetic you learn to do “properly”, the more you start to understand how numbers work, the more other types of mental arithmetic and related mathematics are accessible to you.

How to do addition

I’m now going to teach you arithmetic from scratch. This will start out with me spelling out some things you might find either confusing or painfully obvious. Sorry. It’s important I promise.

If you start from the absolute basics of how mathematicians define addition (of natural numbers. If you don't understand this aside, please ignore this aside), it comes down to the following rules:

  1. m + 0 = m
  2. m + 1 is the next biggest number after m
  3. m + (n + 1) = (m + 1) + n

Or, in words:

  1. If you take any number (which we will call m), and you add zero to it, you get the same number (m) back. So e.g. 7 + 0 is 7.
  2. Add 1 is just counting upwards to the next number. So e.g. 2 + 1 is 3, because 3 comes after 2.
  3. If you take any number (which we will call m), and you add some number to it that is bigger than 0 (which we will call n + 1, which we can do because there’s at least one rock in it, so it’s just the number before it plus one), then this is the same as adding the number right before the second number (n) to the number right after the first number (m + 1). e.g. 8 + 6 = 9 + 5. We “move” a single object from the second number to the first.

If you think of numbers as representing counting objects (which you should), you can think of addition as saying “If you have this many objects on the left, and this many objects on the right, how many are in the combined pile?”. I find it’s particularly helpful to think in terms of counting piles of rocks if you want a concrete example.

Our rules combined says that you can count the combined pile by moving rocks from the right pile to the left pile one at a time (which always preserves the number of rocks), and then when the rightmost pile is empty you’ll have all the rocks in the pile on the left.

Or, to see this with numbers, suppose we want to to calculate 9 + 7. We can do this by moving one from the right, one step at a time. So 9 + 7 = 10 + 6 = 11 + 5 = 12 + 4 = 13 + 3 = 14 + 2 = 15 + 1 = 16 + 0 = 16. Done.

Using this method you can now calculate any addition. Great! So easy, right?

I will admit, there are one or two downsides. Mostly, it’s extremely slow if the number on the right is large. So we probably don’t want to use this as our main method of calculation, but nevertheless, it is our starting point, and we now have our first reliable slow method for doing mental arithmetic.

The first thing to notice is that even through we don’t always want to follow it the whole way, it’s often a pretty good tactic to deploy.

For example, what’s 99 + 37? Well, it’s 100 + 36 (moved one from the right to the left), and 100 + 36 is easy it’s just 136.

This leads to the most important thing to understand about how to do mental arithmetic (and mathematics in general): Mental arithmetic is done by figuring out tactics to change your problem into an easier one.

There is a more general version of this tactic, which we can represent as m + (n + k) = (m + k) + n. That is, we if we have our left pile (m) and our right pile (n), we can always take away some number of rocks (k) that is no larger than the pile on the right, and move that to the pile on the left, we’ve now got k more rocks on the left and k fewer rocks on the right.

Often this is useful for making there be more multiples of 10 or other nice numbers to add together. For example, suppose you wanted to figure out 96 + 13. You could do this by noting that 13 = 9 + 4, so 96 + 13 = 96 + (9 + 4) = (96 + 4) + 9 = 109.

Another tactic we could use here that looks very similar is this: We could write this as 96 + 13 = 90 + 6 + 10 + 3 = (90 + 10) + (6 + 3) = 100 + 9 = 109.