Why do analogies work so well? They’re building blocks for our thoughts, written in the associative language of our brains.

At first, I thought analogies had to be perfect models of the idea they explained. Nope.

“All models are wrong, but some are useful” – George Box

Analogies are handles to grasp a larger, more slippery idea. They’re a raft to cross a river, and can be abandoned once on the other side. Unempathetic experts may think the raft is useless, since they no longer use it, or perhaps they were such marvelous swimmers it was never needed!

Analogies are perfectly fine. But why do they work so well?

Our brains are association machines. Connections, relationships, patterns — we need meaning! Yet we present topics as if we could be programmed with raw information.

Consider the typical language class:

- Here’s the grammar
- Here’s the vocabulary
- Put the vocab in the grammar and go!

(We know how well that works). The mistake is thinking direct study of the grammar and vocabulary will build fluency — it’s a tough slog. I suspect a class of 80% speaking, listening, making idioms, building pronunciation and 20% vocabulary/grammar does much better than the reverse.

Start with simple analogies you deeply understand, then attach extra details.

Here’s an example: I can casually describe i (the imaginary number) as the square root of -1 and you can blindly accept it.

But you won’t really believe me until I start down the path of “Hey, numbers can be 2 dimensional, and i is a rotation into the 2nd dimension”. The word “rotation” stretches our brain about what a number could be — the number line may not be the final step. We’re having a real discussion and can start learning!

See, you’re extremely fluent with the idea of a line, and the idea of a second dimension, and we can work “i is a rotation” into that framework. In computer terms: we are programming with the native language of the machine. Our brain thinks with connections, so explain new data in terms of existing connections!

Although a subject can be distilled into rules and facts, drinking this concentrated math isn’t the best way to enjoy it. It’s not how our brains work, and presenting raw data suffers from a painful translation step.

I don’t think of algebra, trig and other math as a table of equations. It’s a web of connections and insights. But why show facts and hope you recreate the mental model in my head, instead of describing it directly?

No, no — let’s have a brain-to-brain. Here’s the analogies in my head, I want you to have them too.

## Other Posts In This Series

- How to Develop a Mindset for Math
- Developing Your Intuition For Math
- Brevity Is Beautiful
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Intuition, Details and the Bow/Arrow Metaphor
- Finding Unity in the Math Wars
- Why Do We Learn Math?
- Math As Language: Understanding the Equals Sign

Where would the axis of Imaginary and Real numbers meet, and what “kind” of number would be at (0,0)?

You can think of the real/imaginary axes behaving like an ordinary x-y setup–real numbers on the x-axis, imaginary on the y. They meet at (0,0), just like on the xy plane.

Terminology-wise, the ‘real’ number axis and the ‘imaginary’ number axis together form the ‘complex plane,’ and in a manner of speaking, all numbers can be thought of as ‘complex:’ a real part plus an imaginary part. (0,0) would correspond to 0 + 0i, or just 0. (1,3) would be 1 + 3i, and so on. There is an excellent in-depth explanation elsewhere on this site if you’re interested.

Thanks Joe — James, there’s some more background on imaginary numbers here: http://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/

As Joe mentioned, (0,0) is where they two axes meet (at the origin) and is a bit like saying “0 degrees longitude, 0 degrees latitude”.

I’m so glad this site exists. Even your analogy for analogies made things clearer for me. I have a question, though: do you consider analogies that can be taken further, or used to cross more than one river, “better” than analogies that only take you across one?

@Josie: Thanks, glad it helped! Interesting question, I don’t really consider analogies “better” (is a hammer better than a screwdriver?) but rather, one is more useful in certain situations than another. An analogy that works on multiple rivers might be heavier / harder to understand up front than a smaller one that “gets the job done quickly”.

I would agree. As a public speaker, a simple analogy sets up the next idea and an whole new analogy. I have combined them, i.e. lashed the logs together and put wheels on them. BTW I consider a hypothetical analogy of an imaginary senario as good as a real event analogy. (If your listeners can grasp it)

Thanks Dennis — totally agree. Analogies and stories were part of the human oral tradition for thousands of years.