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:

Hah.

"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:

## 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?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.

StevenMay 17, 2017 at 7:28 amI think something worse about that Google definition is that it talks about THE square root of -1, not mentioning that there are in fact two square roots.

kalidMay 19, 2017 at 11:01 amHi Steven, great point. Without the proper experience / visualization you don’t realize how there can be a single square root of -1, let alone two.

DeanMay 19, 2017 at 12:14 pmWhy is this google description-like style of teaching accepted as the one and only, the right way ? Is it maybe that the teachers themselves know no other way ? Maybe they were taught the same way and simply pass what was passed on to them ? Why is just knowing a fact acceptable and understanding it not important ? Why the complex is more appreciated than the simple ? Is this just my experience, or it happens everywhere ?

Kalid, you are an inspiration. I first read one of your articles a year and a half ago, my perspective on learning changed completely and i have you to thank for it. So, thank you.

kalidMay 19, 2017 at 4:32 pmThanks Dean, great questions. I think we’re used to seeing the dictionary definition as the “official answer” and aren’t accustomed to questioning whether it truly captures the concept. For certain ideas, like colors (or flavors, scents, etc.) we know that words can’t capture the experience and aren’t scared to confront the dictionary. But for many subjects (math, science, etc.) we aren’t so confident in our objections :).

Really glad the site is helping you.

Kirtane ChironMay 19, 2017 at 12:36 pmI love your blog. If I had been exposed to any of these meta concepts when I was in grade school my life would have been changed forever. Thank you so much for sharing :)

kalidMay 19, 2017 at 4:30 pmThanks Kirtane, really glad it resonated!

HowardMay 19, 2017 at 1:14 pmMy take-away:

show and tell –> experience and describe

kalidMay 19, 2017 at 4:30 pmGreat way to put it!

PaulMay 19, 2017 at 1:40 pmI got thoroughly inspired by reading your article, while waiting for a technical interview. It releases me from a defense mood to an adventure mood. Who care what the result would be, I want to have a fun time talking to the a.z guy. Better to impart the same excitement into him than to make both of us bore.

kalidMay 19, 2017 at 4:32 pmThanks Paul, that’s a great attitude to have! Best of luck.

anthimalanMay 19, 2017 at 8:23 pmDear Mr. Kalid,

I have been “teaching” Maths for the past 25 years for Engineering graduates. I admit that very rarely I (and possibly my students) have seen and experienced the “red color” all these years and I am ashamed of it even though I am ranked as the best teacher!!!

At least in the remaining part of my career, I want to see and experience it and make my students as well to experience the thrill!

Tomer Ben DavidMay 19, 2017 at 9:16 pmDon’t worry

Tomer Ben DavidMay 19, 2017 at 9:15 pmExcellent. Especially liked: “These pieces aren’t always easy to find, and it can take years.”

kalidMay 20, 2017 at 10:54 amThanks!

Susie MayhewMay 22, 2017 at 2:21 amAnother WOW!

Tripti PaulMay 23, 2017 at 12:50 amHey Kalid, First of all, awesome blog, and awesome visual insights. Thanks always for it.

One thing that’s bugging me for some time is how to think, work with and imagine polynomials, because most books just give the Google-ish definition: “A polynomial is f(x) = Σa_ix^i, now let’s solve problems. While pushing symbols, assume the Fundamental Theorem of Algebra is true, and use it, because you’re too dumb to prove it. “, and then goes on to solving problems with it, which is basically symbol pushing.

You should make an article on polynomials with some new insights (and maybe with the connections with number theory or geometry). (I asked a Math SE question, but got no good reply).

kalidMay 24, 2017 at 1:20 pmThanks for the suggestion! It’s been something I’ve been meaning to write about for a while.

Here’s a quick analogy. We currently have a base 10 system: 123 is really 100 + 20 + 3, or 1 * (10^2) + 2 * (10^1) + 3 * (10^0).

We can create arithmetic rules like 2 + 2 = 4, but it only works in certain bases. In binary, there’s no ‘2’ digit, so the rules don’t work. We can convert to binary digits (“10 + 10 = 100”) but now the result doesn’t make sense in base 10. Ack.

One intuition for polynomials is that we have an unknown base (x) and try to derive rules from there. And another way to see an equation like

x^2 + x = 4

is “in what base does “110” equal 4?” (x^2 is 1 in the first digit, x is 1 in the 2nd digit, the missing constant is 0 in the third digit).

A polynomial is basically a single “number” in this base-less universe. Depending on what actual base we plug in, we can get a specific value out.

Just the beginning of an insight, hope it helps.

Alex K ChenMay 25, 2017 at 1:10 amYeah, that insight is good. Things are simple and have good connection with number theory when you have polynomials with coefficients from the positive integers [0,1,…,n-1], so you can basically interpret them as numbers in base n or higher.

Things get completely out of intuition when coefficient go from positive integers to negative, then REALS, (rationals are same as integers, just multiply with lcm of denominators), then COMPLEX, and I have not idea then. I have also no idea what’s the application of factorizing polynomials with non-integer coefficients or finding zeroes.

Inspired by Tripti Paul’s above comment, I have asked this question: (https://math.stackexchange.com/questions/2295088/painless-introduction-to-polynomials) (you may see the comment section is pretty demotivating there) or this (https://math.stackexchange.com/questions/2286075/why-are-polynomials-interesting question).

Awaiting for a betterexplained article on polynomials soon. !

Alex K ChenMay 25, 2017 at 1:14 amI replied a comment mentioning two math stackexchange questions I have asked (intrigued by Tripti Paul’s original question) from an account with the same name (you can google the question names to get the link) “Painless introduction to polynomials” (This refers to this article) and “Why are polynomials interesting”, but the comment got deleted :(

Awaiting for the article on polynomials.

VladaMay 29, 2017 at 9:48 amHi Kalid ,how do you find “the pieces”???

GEORGE JOHNJune 22, 2017 at 4:12 pmSuperb insights about learning and teaching and well written. My perspective (not necessarily the correct one) is that short term memory involves electrical circuits in our brain (its rapid) and long term learning (insightful, felt concepts) needs to be embedded in our proteins. Some people can easily form the electrical circuits and regurgitate facts. Others need “deep, meaningful” learning to remember. The way much of educational evaluations are now structured, based on MCQs, test only the “superficial” component. I am a medical doctor who has been teaching for many years. I could see this in the students who score high on MCQs but are lost when faced with real life situations- there are others who love to dive into the experience and are very good at practical management but are lost when trying to convey it to admiring peers! I guess we need both…one for ourselves and the other to store and transmit experiences and insights for future generations.