I.

Recently, an excited friend was telling me the story behind why we care about the mean, median and mode.

They explained that a straightforward idea for what you might want in an ‘average’ number, is something that minimises how far it is from all the other numbers in the dataset - so if your numbers are 1, 2 and 3, you want a number $x$ such that the sum of the distance to each datapoint is as small as possible. It turns out this number is 2.

However, if your numbers are 1, 2, and 4, the number that minimises the distance from all of them is also 2.

Huh?

When my friend told me this, the two other people I was with sort of said “Okay”. I said “What? No! I don’t believe you! It has to change when the data does - it’s a linear sum, so it has to change! It’s like you’re saying the sum of 1, 2 and 3 is the same as the sum of 1, 2 and 4. This is just wrong." Suffice to say, my friend’s claim wasn’t predicted by my understanding of math.

Now, did I really not believe my friend? The other two people with us were certainly fine with it. Isn’t this just bayesianism? That’s how the old joke goes:

Math teacher: Now I’m going to prove to you that X is true.

Bayesian: You just did.

Actually, no. You taught me a detail to memorise, but my models didn’t improve. I won’t be able to improve how I use averages, because I don’t understand how it fits in with everything else I understand - it doesn’t fit with the models I use everywhere else in math.

I mean, I could’ve nodded along. It’s only one fact, after all. But if I’m going to remember it in the long term, it should connect to my other models and be reinforced. The alternative is to be stored in the brain with all those other memorised facts that students learn for exams and forget immediately after.

If you’re trying to build new models of a domain, it’s important to choose to speak from the confusion, not from the rest of yourself. Don’t have conversations about whether you believe a thing. Instead talk about whether you understand it.

(The problem above was the definition of the median, and an explanation of the math for the curious can be found in this comment.)

II.

It can be really hard to feel your models. Qiaochu Yuan’s method of learning involves rampi...