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.

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Other Posts In This Series

  1. Developing Your Intuition For Math
  2. Why Do We Learn Math?
  3. How to Develop a Mindset for Math
  4. Learning math? Think like a cartoonist.
  5. Math As Language: Understanding the Equals Sign
  6. Avoiding The Adjective Fallacy
  7. Finding Unity in the Math Wars
  8. Brevity Is Beautiful
  9. Learn Difficult Concepts with the ADEPT Method
  10. Intuition, Details and the Bow/Arrow Metaphor
  11. Learning To Learn: Intuition Isn't Optional
  12. Learning To Learn: Embrace Analogies
  13. Learning To Learn: Pencil, Then Ink
  14. Learning to Learn: Math Abstraction
  15. Learning Tip: Fix the Limiting Factor
  16. Honest and Realistic Guides for Learning
  17. Empathy-Driven Mathematics
  18. Studying a Course (Machine Learning) with the ADEPT Method