Learn Difficult Concepts with the ADEPT Method – BetterExplained

经过十年的解释，我简化了获取新概念的策略。

Make explanations ADEPT: Use an Analogy, Diagram, Example, Plain-English description, andthena Technical description.

Here’s how to teach yourself a difficult idea, or explain one to others.

Contents

类比:它还像什么?
Diagram: Engage That Half Of Your Brain
Example: Let Me Experience The Idea
Plain-English Description: Use Your Own Words
Technical Description: Learn The Formalities
The Mental Checklist
Modifying the Learning Order
The Goal: Explanations That Actually Work
Bonus: BE ADEPT
Appendix: ADEPT Summaries
Other Posts In This Series

类比:它还像什么?

Most new concepts are variations, extensions, or combinations of what we already know. So start there!

在我们几十年的生命中，我们遇到了成千上万的物体和经历。Surelyone其中有一些与这个新课题大致相似，可以作为起点。

Here’s an example: Imaginary numbers. Most lessons introduce them in a void, simply saying “negative numbers can have square roots too.”

Argh. How about this:

Negative numbers were distrusted until the 1700s: How could you havelessthan nothing?
We overcame this by realizing numbers could exist on a number line, allowing us to move forward or backward from zero.
Imaginary numbers express the idea that we can move upwards and downwards, orrotatearound the number line.

Instead of just going East/West, we can go North/South too – or even spin around in a circle. Neat!

Analogies are fuzzy, not 100% accurate, and yet astoundingly useful. They’re a raft to get across the river, and leave behind once you’ve crossed.

Diagram: Engage That Half Of Your Brain

我们经常认为图表是拐杖，如果你没有足够大的男子气概，直接解释符号。你猜怎么着?Academic progress on imaginary numbers took off onlyafterthe diagrams were made!

Favor the easiest-to-absorb explanation, whether that comes from text, diagram, or interpretative dance. From there, we can work to untangle the symbols.

So, here’s a visualization:

Imaginary numbers let us rotate around the number line, not just move side-to-side.

Starting to get a visceral sense for what they cando, right?

Half our brain is dedicated to vision processing, so let’s use it. (And hey, maybe for this topic, twirling around in an interpretative dance would help.)

Example: Let Me Experience The Idea

Oh, now’s our chance to hit the student with the fancy terminology, right?

Nope. Don’t tell someone the way things are: let them experience it. (How fun is hearing about the great dinner I had last night? The movie you didn’t get to see?)

But that’s what we do for math. “Someone smarter than you thought this through, found out all the cool connections, and labeled the pieces. Memorize what they discovered.”

That’s no fun: let people make progress themselves. Using the rotation analogy, what happens after 4 turns?

转两圈怎么样?4 turns clockwise?

Plain-English Description: Use Your Own Words

If you genuinely experienced an idea, you should be excited to describe it:

虚数似乎指向北方，我们顺时针转一圈就能到达。
Oh! I guess they can point South too, by turning the other way.
4 turns gets us pointing in the positive direction again
It seems like two turns points us backwards

These are all correct conclusions, just not yet written in the language of math. But you can still reason in plain English!

Technical Description: Learn The Formalities

The final step is to convert our personal understanding to the formal notation. It’s like sharing a song you’ve made up: you can hum it to yourself, but need sheet music for other people to use.

Math is the sheet music we’ve agreed upon to share ideas. So, here’s the technical terminology:

We sayi(lowercase) is 1.0 in the imaginary dimension
Multiplying byiis a 90-degree counter-clockwise turn, to face “up” (here’s why). Multiplying by-ipoints us South
It’s true that starting at 1.0 and taking 4 turns puts us at our starting point:

$\displaystyle{1 * i * i * i * i = 1 }$

And two turns points us negative:

$\displaystyle{1 * i * i = -1 }$

which simplifies to:

$\displaystyle{i^2 = -1}$

$\displaystyle{i = \sqrt{-1}}$

In other words,iis “halfway” to -1. (Square roots find the halfway point when using multiplication.)

Starting to get a feel for it? Just spitting out “i is the square root of -1” isn’t helpful. It’s not explaining, it’stelling. Nothing was experienced, nothing was internalized.

Give people the chance to make an idea their own.

The Mental Checklist

I used to be satisfied with a technical description and practice problem. Not anymore.

ADEPT is a checklist of what I need to feel comfortable with an idea. I don’t think I’ve actually learned a topic unless I have a metaphor that ties everything together. Here’s a few places to look:

Analogy – ?
Diagram – Google Images
Example –Khan Academyfor practice problems
Plain-English – Forums like/r/mathorMath Overflow
Technical –WikipediaorMathWorld

不幸的是，没有太多的资源专注于类比，特别是在数学方面，所以你必须自己做。(这个网站的存在就是为了分享我的。)

Modifying the Learning Order

假设我们可以按顺序呈现事实，就像向计算机传输数据一样，这似乎是合乎逻辑的。但谁会这样学习呢?

I prefer the blurry-to-sharp approach to teaching:

Start with a rough analogy and sharpen it until you’re covering the technical details.

Sometimes, you need to untangle a technical description on your own, so must work backwards to the analogy.

Starting with the technical details:

你能用自己的话解释一下吗?
Can you solve an example problem, describing the steps in your own words?
你能创建一个图表来表示这个概念是如何结合在一起的吗?
Can you relate the concept to what you already know?

With this initial analogy, layer in new details and examples, and see if it holds up. (It doesn’t need to be perfect, but iterate.)

If we’re honest, we’ll admit that we forget 95% of what we learn in a class. What sticks? A scattered analogy or diagram. So, make them for yourself, to bootstrap the rest of the understanding as needed.

In a year, you probably won’t remember much about imaginary numbers. But the quick analogy of “rotation” or “spinning” might trigger a flurry of recognition.

The Goal: Explanations That Actually Work

I’m wary of making acontrivedacronym, but ADEPT does capture what I need to internalize a new concept. Let’s stop being shy about thinking out loud: does a fact-only presentation really work for you? What other components do you need? I have a soft, squishy brain that needs the connecting glue, not just data.

Scott Young uses theFeynman Techniqueto explain concepts in everyday words and work backwards to an analogy and diagram. (Richard Feynman was a world-class expositor and physicist, and one of my teaching heroes.)

Tom Roth wrote anice summary如ADEPT、费曼技术等。

在任何技巧之外，提高你的标准，找到(或创造)真正适合你的解释。这是让概念保持不变的唯一方法。

Happy math.

Bonus: BE ADEPT

“BE” is a nice prefix for the style to use when teaching:

Brevityis beautiful.
Empathy makes us human. Use your natural style, relate to common experience, and anticipate questions in your explanation.

I’ve yet to complain that a lesson respected my time too much, or related too well to how I thought.

Appendix: ADEPT Summaries

ADEPT就像一个营养标签，用来说明:关键成分是什么?

Concept	Euler’s Formula
Analogy	Imaginary numbers spin exponential growth into a circle.
Diagram
Example	Let’s figure out the value of`3^i`. (It’s on the unit circle.)
Plain-English	Raising an exponent to an imaginary power spins you on the unit circle. The same destination can be written with polar (distance and angle) or rectangular coordinates (real part and imaginary part).
Technical	$\displaystyle{e^{ix} = \cos(x) + i\sin(x)}$

Concept	Fourier Transform
Analogy	Like filtering a smoothie into ingredients, the Fourier Transform extracts the circular paths within a pattern.
Diagram	Smoothie being filtered:
Example	Split the sequence`(4 0 0 0)`into circular components:
Plain-English / Technical

Concept	Distributed Version Control
Analogy	Distributed Version Control is like sharing changes to a group shopping list with your friends.
Diagram / Example
Plain-English	We check out, check in, branch, and share differences (“diffs”).
Technical	`git checkout -b branchname` `git diff branchname`

Combine ingredients with your own style. Steps might merge, but shouldn’t be skipped without a good reason (“Zombies coming, no time for biochem, use this serum for the cure.”). Thesite cheatsheethas a large collection of analogies.