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.

You can add an aha! moment or question, and vote / discuss them individually. This extends aha.betterexplained.com, making mini-posts for key ideas in an article. Why?

Mine the gold in the comments

Several articles have awesome discussions, like understanding e. There’s gems, but unfortunately they’re buried inside hundreds of comments, which is tough to read through.

If we can extract and rate key insights, we can build a “living FAQ” of what’s most helpful. These are good branching points for new articles as well.

Better collaboration

I love chatting math and finding new learning approaches. A few examples:

• Audrey McGoldrick made an awesome animated slideshow explaining the introduction to calculus.

• Joshua Zucker helped me refine my recent “functions are plates, derivatives are breaking plates into shards, integrals are weighing the pieces” analogy, and corrected a huge misconception about integrals and anti-derivatives.

• Stan and YatharthROCK have been helping me develop ideas on aha.betterexplained.com (thanks guys).

The aha / question area is like a mini-forum to discuss analogies. (Teachers, please ransack these insights and whatever is useful — every article is free to use, print, mime, etc. for non-commercial use). The goal is to enable conversations about what is actually working.

Continual improvement

I have a nefarious plan for the widget: gather improvements for each article! Knowing exactly what parts of the article helped (or didn’t) makes it easier to keep revising: I want the analogies to sing.

Try it out

The aha section is new, there’ll be some bugs, but I’d love your feedback anyway:

1. Share what really worked. As you read articles, post what analogies helped the most.

2. Ask questions. Have a question that’s been bothering you? Add it!

3. Vote on what’s helping. Just click the heart to rate an insight or question.

The aha section isn’t a replacement for comments — it’s a way to organize the best parts. If there’s other types of “aha” items you’d like organized (Followup:, Example:, etc.) let me know!

If you’re looking for a digital gift this holiday season, I know just the ticket .

A lot of work was done to convert the original PDF but keep the image and equation formatting the same, and readable on eInk and color screens. Here’s the cover and what it looks like on a few different devices:

I’d like to thank everyone for their feedback and support! If you’ve already bought the PDF ebook, you should have received an update via email (I apologize for the improper formatting, and formatting, and formatting — I’ll stick to a blog post for today!).

Happy math,

-Kalid

• Be warm & friendly
• Be clean & readable (more whitespace & larger fonts)
• Be easy to skim & search (search results and posts have image previews)

BetterExplained isn’t an authority lecturing you on facts: it’s an excited friend sharing what actually helped when learning. I don’t want to give the impression of some all-knowing reference — it’s a journey of discovery.

The layout is new and kinks are likely — let me know if anything is off!

-Kalid