学习数学?像漫画家一样思考。– BetterExplained

What’s the essential skill of a cartoonist? Drawing ability? Humor? A deep well of childhood trauma?

I’d say it’s an eye for simplification, capturing the essence of an idea.

For example, let’s say we want to understandEd O’Neill:

A literal-minded artist might portray him like this:

While the technical skill is impressive, does it really capture the essence of the man? Look at his eyes in particular.

Wow! The cartoonist recognizes:

The unique shape of his head. Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect.
The wide-eyed bewilderment. The whites of his eyes, the raised brows, the pursed lips – the cartoonist saw and amplified the emotion inside.

So, who really “gets it”? It seems the technical artist worries more about the shading of his eyes than the message they contain.

Numbers Began With Cartoons

Think about the first numbers, the tally system:

I, II, III, IIII …

这些是……图纸!漫画!对一个想法的讽刺!

They capture the essence of “existing” or “having something” without the specifics of what it represents.

山顶洞人会计Og可能尝试过画单独的简笔画、水牛、树等等。最终他可能会发现一条捷径:画一条线，称它为水牛。这抓住了“有东西在那里”的精髓，剩下的就是我们的想象力。

Math is an ongoing process of simplifying ideas to their cartoon essence. Even the beloved equals sign (=) started as a drawing of two identical lines, and now we can write “3 + 5 = 8” instead of “three plus five is equal to eight”. Much better, right?

So let’s be cartoonists, seeing an idea — really capturing it — without getting trapped in technical mimicry. Perfect reproductions come inafterwe’ve seen the essence.

Technically Correct: The Worst Kind Of Correct

We agree that multiplication makes things bigger, right?

Ok. Pick your favorite number. Now, multiply it by a random number. What happens?

If that random number is negative, your number goes negative
If that random number is between 0 and 1, your number is destroyed or gets smaller
If that random number is greater than 1, your number will get larger

Hrm. It seems multiplication is more likely toreduce一个数字。也许我们应该教孩子“乘法一般会减少原来的数字”。It’ll save them from making mistakes later.

No! It’s a technically correct and real-life-ily horrible way to teach, and will confuse them more. If the technically correct behavior ofmultiplicationis misleading, can you imagine what happens when we study the formal definitions of more advanced math?

有一种担心是，如果没有预先说明每个细节，人们会产生错误的印象。I’d argue people get the wrong impressionbecause你提前提供每个细节。

As George Box wrote, “All models are wrong, but some are useful.”

A knowingly-limited understanding (“Multiplication makes things bigger”) is the foothold to reach a more nuanced understanding. (“People generally multiply positive numbers greater than 1, so multiplication makes things larger. Let’s practice. Later, we’ll explore what happens if numbers are negative, or less than one.”)

Takeaways

I wrap my head around math concepts by reducing them to their simplified essence:

Imaginary numbers让我们旋转数字。Don’t start by definingias the square root of -1. Show how if negative numbers represent a 180-degree rotation, imaginary numbers represent a 90-degree one.
The number eis a little machine that grows as fast as it can. Don’t start with some arcane technical definition based on limits. Show what happens when we compound interest with increasing frequency.
The Pythagorean Theorem解释了所有形状的行为(不仅仅是三角形)。不要给出特定于三角形的几何证明。看看圆形、正方形和三角形有什么共同之处，并表明这个想法适用于任何形状。
Euler’s Formulamakes a circular path. Don’t start by analyzing sine and cosine. See how exponents and imaginary numbers create “continuous rotation”, i.e. a circle.

Avoid the trap of the guilty expert, pushed to describe every detail with photorealism. Be the cartoonist who seeks the exaggerated, oversimplified, and yetaccuratetruth of the idea.

快乐数学。

PS. Here’s mycheatsheetfull of “cartoonified” descriptions of math ideas.

学习数学?像漫画家一样思考。

Numbers Began With Cartoons

Technically Correct: The Worst Kind Of Correct

Takeaways

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