To check if you're flexible, you don't need a battery of tests. Just bend down and touch your toes. Was it effortless?
If it's not (and it's not), you know you need to stretch more. The goal isn't the splits, just some self-determined level.
我的数学目标也类似。我不需要专家的熟练程度，只需要对我关心的话题“摸一摸你的脚趾头”的理解。
Example: Exponent Gutcheck
Here's an internal dialog I might have to verify my understand of exponents.
Gutcheck: Roughly speaking, what's $2^{100}$?
It's a large, even, positive number. (This intuition should appear almost instantly. If it takes 10 seconds of thinking to realize it's large, even, or positive, exponents aren't natural.)
Gutcheck: Roughly speaking, what's $2^{-100}$?
It's a tiny, almost undetectable positive decimal. Intuition: It's like going "back in time" by 100 doublings.
Gutcheck:粗略地说，$2^i$是多少?
哦哦。虚指数!有了足够的直觉，你会意识到:“它在单位圆上，大约在ln(2) ~ .693弧度。”
There's a few gutchecks here. The first is that an imaginary exponent puts you on the unit circle (no matter the base). The next level is a rough "important constant" gutcheck, where you remember ln(2) ~ .693. (Not as important, but good to remember. It helps with things like the Rule of 72)
Gutcheck: Roughly speaking, what's $i^i$?
Oh, here's a tricky one. Remember how we blurted out that $2^{100}$ was large, even, and positive? How proud we were of our quick thinking? Well, what can you say about $i^i$, hotshot?
Yikes. Realizing I couldn't instantly rattle of any properties of $i^i$ meant my intuition for exponents wasn't complete. After getting an intuition forimaginary exponents, the thought becomes:
$i^i$ starts as growth pointing sideways, whose direction is rotated again. It's a positive real number less than 1.0.
Phew. If I truly understand exponents, the gutcheck for $2^{100}$ and $i^i$ should be similar in speed and detail. A painful stretch means I need more understanding.
Additional Examples
gutcheck过程并不能完全转化为文本。这些内在的反复在口头上发生:我想到一个问题，然后迅速感觉/想象/记住一个类比。(这是一种勇气检查，而不是对每分钟的思考大声说出来。)
Here's a few examples I run through from time to time:
Imaginary numbers: What's the cube root of -1?
- Thought: Ok, $i^2 = -1$ means we go from 1 to -1 in two steps. Getting there in 3 steps means a 60-degree rotation (180/3). Oh, we can go the other way too (-60 degrees). Oh, we can flip 180 degrees (180 + 180 + 180 = 360 + 180 = net 180 degree rotation). So there's 3 cube roots of -1.
Fourier Transform: What's the transform of[1 0 0 0]?
- 我们想要4个同样强的频率(0Hz, 1Hz, 2Hz, 3Hz)。They split the strength "1" between them, so we have
[.25 .25 .25 .25](using the notation in the Fourier Transform article).
Trigonometry: What's the connection between the 6 major trig functions?
Thought: I think "dome, wall, ceiling" and visualize this trigonometry diagram:
Calculus: Explain the derivative of $x^3$
- Thought: $x^3$ is really $x\cdot x \cdot x$. We have 3 perspectives, each seeing a change of $x \cdot x$. The result is $x^2 + x^2 + x^2 = 3x^2$. I also visualize a cube with plates added to it.
Bayes Theorem: What's the plain-English description?
- Thought: chance evidence is real = true positive / (true positives + false positives)
Exponents: What does discrete vs. compound exponential growth look like?
- Thought: I see continuous exponential growth as "filling in the gaps" left by discrete growth.
The key element is speed: an intuitive response should bubble as you hear the question. Struggling for an hour to touch my toes, though admirable, still means I'm not flexible enough.
The goal isn't learning minutia, it's a working understanding of an idea, enough to solve a problem without tremendous effort. It's a diagnostic, not a value judgement. If I struggle, I simply need a better intuition.
Strangely enough, not everyone wants to keep math insights top-of-mind. But pick something that's important to you and occasionally try a 5 second gutcheck on the essentials.
快乐数学。