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?

  • Have a mental gutcheck for learning. When I'm in a lesson (video, textbook, lecture, etc.) I'm constantly asking "Am I seeing the color, or just getting the description?". A quick check is whether you can make analogies about what you're learning. Homer (the blind poet, that one) had red described as a blaring trumpet. That poetic description conveys more understanding than "the wavelength of light at 650 nanometers".

  • Be gentle with yourself. If a concept isn't clicking, the most likely cause is that the lesson wasn't helpful enough. It may be throwing words at you when an experience is needed. Yes, some people can get by with words alone, just like some can glance at sheet music and think "That sounds beautiful". I'm not one of those people, I need to find the play button.

  • Hold lessons to a higher standard. This mindset does lead to some potentially uncomfortable questions: Has the teacher viscerally experienced the idea being taught? Is that the goal of their lesson? Is this lesson part of a chain of dictionary definitions recited by aliens?

  • Be open to new experiences. The counterpoint to having higher standards is compassion: teachers don't want to miss the point on purpose. If we're open to treating a lesson as an exploration, a potential experience, we should welcome any chance to have a "I finally see the color!" moment (teacher and student alike).

  • Unlock motivation in learning. Are people naturally curious? Let's find out. Did you know a new shade of blue pigment has been discovered? No real-world object could have that color. Feel that growing itch of curiosity?

    Here you go.

    Interesting, right? Shiny and shimmering? Now, maybe you don't want to paint your house that color, but you wanted to see it for yourself. How frustrated would you be if the news article didn't have a photo?

    That's the curiosity we can unlock for new ideas if we know an experience is coming. Colors are natural enough that we're sure we'll experience something when we look. But with math, we may have forgotten (or never had) that eye-opening experience, so "a new math concept" is like having someone describe that awesome movie they saw. "Oh, there was this part, and this part. And then this happened. I loved it! Why don't you?". We have innate curiosity when a subject is approached with an experience in mind, built on the trust of having several previous Aha! moments.

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.

Join 450k Monthly Readers

Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.