- BetterExplained - https://betterexplained.com -

Math Learning Tip: Experience Ideas Directly

Math Learning Tip: Experience Ideas Directly

Let's say an alien visits Earth and wants to understand our Earthly ways. After a few hours of questioning, we resort to "Uh... just Google it." Back to playing Factorio.

The alien starts looking up random words. Blowgun, aquatic, heist (Uh... buddy?) and finally:

define color red


"Red (adjective): of a color at the end of a spectrum next to orange..."

We Earthers know the dictionary is missing a huge caveat:

Dear Reader: You can't truly understand 'red' by reading about it. You need to see it for yourself. The dictionary definition is an attempt with dry words. Even better is a metaphor: red is the sound of a blaring trumpet, the taste of a chili pepper, the feeling of stepping barefoot on a lego. But please, stop reading and find yourself a strawberry.

We know reading gives a limited understanding of the topic. But if we weren't paying attention, the alien would have claimed mastery of the official definition, and gone back to teach generations of students about it.

Whoops, I'm getting late for math class. What was this week's topic again? Oh right, imaginary numbers:

define imaginary number

The facts and the feeling

There's a missing caveat to every math lesson: The goal of this lesson is for you to truly feel an insight in your bones. The words are just hints about how to get there.

Let's take the concept of imaginary numbers. The abstract definition trotted out in countless lessons is something like: Imaginary numbers are the square root of negative numbers. We label the square root of -1 "i". Time for practice problems.

Ugh. There's no acknowledgement that the words "square root of a negative" are baffling limited, and no way to truly understand the idea. Here's the missing "see the color" caveat (full article):

Hey. That technical definition is frustratingly lifeless. You're probably wondering how negatives can have square roots. Picture imaginary numbers as rotations, like this:

Whoa. The "square root of a negative" is really "halfway negative", or something pointing vertically. If positives are East, negatives are West, imaginary numbers let us go North/South. Now, back to your dictionary definition.

There's a pernicious objection that getting an intuition for a concept is a "baby version" of the real thing. ("Aw, you weren't smart enough to rely solely on the technical description, here's a diagram.") That's like claiming seeing a color is the "baby version" of reading about it!

Experiencing an idea is our goal all along. If thermodynamics can be truly understood via an interpretative dance in a hula skirt... well, I'll bring the coconuts. I want an intuition.

We aren't hard drives trying to store the text of a novel without its meaning. (And for what it's worth: progress with imaginary numbers truly began after the 2d rotation interpretation was discovered.)

Applying the analogy

Ok. This color analogy helps us look for an experience beyond the lesson. What can we do with this mindset?

Words and symbols have their place: they're compact, precise, and easily expressed. But they should come after the experience (show, then tell). Once the experience is understood, and enthusiasm fired up, words can act as placeholders for concepts in our mind's eye ("red sports car").

Ultimately, I don't learn because I want more entries in my mental dictionary. I want to see new colors.

Appendix: A strategy to experience an idea

How do you uncover the experience behind an idea? In the best case, your teacher had one which they can share, saving you the trouble of looking.

But many times you're on your own. I use the ADEPT method as a checklist for what helps a concept click:

If a lesson isn't clicking, I run through that checklist: Do I have an analogy in place? A diagram? An example? A plain-English version? Can I find someone who as the above?

These pieces aren't always easy to find, and it can take years. But I never want to stop looking. Just because I haven't had an experience yet doesn't mean it's not possible.

Happy math.