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.

My math goals are similar. I don't need expert proficiency, just a "touch your toes" understanding for the topics I care about.

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: Roughly speaking, what's $2^i$?

Uh oh. Imaginary exponents! With enough intuition, you realize: "It's on the unit circle, at about ln(2) ~ .693 radians."

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 for imaginary 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

The gutcheck process doesn't quite translate to text. These internal back-and-forths happen pre-verbally: I think of a question and quickly feel/visualize/remember an analogy. (It's a gutcheck, not a think-aloud-for-minutes check.)

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]?

• Thought: We want 4 equally strong frequencies (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.

Happy math.

"Argh, why couldn't they have explained it like this the first time?"

The difference between a semester of pain and instant understanding was one stupid, missing analogy. It still riles me up thinking about how close I came to missing the key concept (and disliking math).

In class there's the lesson about a specific formula, sure, but the meta-lesson is how well the experience went.

What worked? What didn't? How can we get more of the first and less of the second?

Over time, I realized individual topics were chances to explore what truly worked when learning. Not what a learning theorist or book said (Flashcards! Mnemonics! Just study harder!), but what actually worked for you.

A few of my scattered meta-lessons:

• Analogies, while imperfect, are a huge jump start. It's motivating to get the ball rolling and course correct along the way, vs. waiting to line things up perfectly.
• Nearly every explanation is improved with a visual or diagram.
• Humor and empathy put the reader at ease so they can tell you when they're actually confused (vs. mindless head nodding).
• Share the gotchas. The Wise Teacher hiding the 14 mistakes he made when learning the topic does students a disservice.

Every lesson is a chance to silently wonder "How well did that work?".

These days, I use ADEPT as a running checklist for how I get things to click:

This was pulled from actual frustrations (Why can't they share a plain-English version first? A diagram?) and I'm sure you'll have modifications of your own.

Don't just take a single lesson away from a lecture, article, or video. Think How well did that work for me? and build your learning approach around the best parts.

Happy math.

P.S. My buddy Nasos runs the excellent MetaLearn podcast and we have several chats about learning, a previous interview is below:

Einstein’s Theory of Special Relativity has many consequences, including the famous equation E = MC² (Energy = Mass x Speed of Light²) and the notion that measurements of space and time are relative to the observer.

In this 30-minute episode we discuss:

• Einstein's backstory and how it influenced his work
• The mechanics of the equation in a way that you can understand
• The implications of the equation for our view of the Universe

It was a lot of fun -- hope you enjoy it.

Multiplication can be repeated addition, scaling, rotating (via imaginary numbers), and more.

What about division? Let's take a look.

The permutation formula lets us pick 3 items out of 10, in a specific order. To order 3 items from 10, we have 10 options for the first choice, 9 options for the second, and 8 for the third, giving us 10 _ 9 _ 8 = 720 possibilities.

But how is this process written in most math books?

What's going on?

Well, we just want a portion of the factorial. 10! gives the full sequence (10 _ 9 _ 8 _ 7 _ 6 _ 5 _ 4 _ 3 _ 2 _ 1), but we want to stop it after we hit 8. That means we divide by 7!. The only part that's not removed is 10 _ 9 * 8.

In this case, division is acting like a brake/boundary/filter that stops the factorial from running hog-wild. (They get out of control, you know.)

Ah! If writing a software program, you wouldn't actually compute 10!, 7! and do a division. What if we needed 3 choices from 1000 options? (1000 factorial has 2568 digits and will make your computer cry. I told you this would happen!)

If we realize the role of division as a boundary marker, we can just compute 1000 _ 999 _ 998 = 997,002,000 and call it a day.

Let's keep going.

Suppose we don't care about the order of the items we pick: ABC is the same as CBA. What to do?

Well, we can apply another division! This time, we don't want a boundary, but want to merge/consolidate/group up similar items. Everything that looks like ABC (e.g., ACB, BAC, BCA, CAB, CBA) should be counted once.

With 3 items there are 3! rearrangements, so the final count is 720/3! = 720/6 = 120 choices.

As a formula:

Neat, right? The division in the permutation formula acts as a boundary, and the division in the combination formula is a type of "group up". I imagine the variations being merged into a single option:

The words we pick frame how we think about an equation. "Divide" implies we're splitting things apart. If we know alternate meanings (repeated subtraction, boundaries, consolidation), we may pick a better description. Saying "divide by k!" doesn't have the same intuition as "consolidate the reorderings".

I think math concepts are fundamentally simple but their written description may not be. (Ever try to describe how to put on a shirt?) The goal is finding the words to make the idea click.

Happy math.