What's a tough concept you finally figured out? (For me, it was the imaginary numbers. I'll never forget that *Aha!* moment.)

Ok. For that difficult concept, what *finally* made it click? It's usually:

- An analogy
- A diagram
- An example
- A friendly, plain-English description

Rarely is it because we're lacking:

- A technical definition
- A new technology
- A gamified incentive
- "more time" (I lack motivation, not time)

The limiting factor -- the thing holding me back -- is how I approach a concept.

## The Roman Numeral Problem

Imagine you're teleported to a Roman classroom. Kids -- heck, the adults -- are struggling with multiplication. (*IV times VII is really hard!*)

What do you fix: Flip the classroom? Gamify things? Invent a printing press to distribute more worksheets?

Helpful, in time. But the first fix should be a simple discussion:

Hey, I'm from the future. Yes, it's pretty nice. But first, we need to fix your concept of a number. Individual lines for digits is cumbersome. Instead, think in groups of ones, tens, hundreds, and so on. Now multiplication is built into your numbers, and arithmetic gets a lot easier. Let me show you...

Boom. The "Roman Numeral Problem" is not fixed with better tech. Just a better understanding.

Ok. Imagine a time traveler (you, 6 months from now) is going to tutor you today. What would they suggest?

**Imaginary numbers:**Don't try to conceptualize "the square root of a negative number". That's really clunky. Think about rotations instead.

**Trigonometry:**Don't try to memorize SOH-CAH-TOA and a mess of equations. Visualize a single diagram and the connections jump out.

**Calculus:**Don't force yourself through epsilon-delta proofs. Practice breaking things apart, putting them together, and get a feel for the patterns.

**Fourier Transform:**Don't simply memorize the formula. Internalize the notion of a*cycle recipe*and practice going from a pattern, to the recipe, and back.

Focus on the problems a time traveler would fix first.

While drafting this post, a comment came in:

So, I just started learning about imaginary numbers in math class, and I was so confused. I understood the idea, but not the practical application or really what i was. I am a person who needs to understand a concept fully, I have trouble accepting that i=the square root of -1. I was googling it and I only got more confused. Then, I found your article on imaginary numbers, and all of a sudden, I got it. I could visualize it, even though I have no specific examples of their importance, I can understand why and how they could be important. It clicked. It doesn't make me want to go do my worksheet on adding and subtracting them, but in math tomorrow, I will be a much happier camper. -Abby

It drives me crazy to see endless tutorials on imaginary numbers that don't address the *fundamental confusion* of how a negative number can have a square root. You can give me all the videos and interactive quizzes you want, I'm not truly learning until you explain the notion of a rotation.

This misprioritization shows up everywhere:

Fix the plot, then worry about special effects.

Fix the recipe, then worry about decor.

Fix the melody, then worry about the instruments.

Fix the analogy, then worry about the presentation format.

Identify what's held you back and fix that first.

Happy math.

## Appendix: Technology As The Education Fix

Technology helps with certain limitations (access, distribution, cost). But the quality of the source material is still up to us. I'd prefer handwritten letters with Socrates to a HD video conference with Carrot Top.

Veritasium has a great video on these lines ("This Will Revolutionize Education"):

If we think the limiting factor in education is still distribution, we'll focus on technical solutions.

But you know what? We've had Shakespeare online for a few decades now. Modern kids must be poetry experts because of free access to quality literature, right?

It's not an access problem any more. It's a motivation, interest, enthusiasm, understanding-what's-actually-going-on problem. Let's fix that first.

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## Other Posts In This Series

- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Learning to Learn: Math Abstraction
- Learning Tip: Fix the Limiting Factor
- Honest and Realistic Guides for Learning
- Empathy-Driven Mathematics
- Studying a Course (Machine Learning) with the ADEPT Method
- Math and Analogies
- Colorized Math Equations
- Analogy: Math and Cooking
- Learning Math (Mega Man vs. Tetris)

JoshMay 5, 2016 at 4:45 pmGreat post, Kalid! Wish I’d seen this site when I was in high school slogging through trig!

kalidMay 5, 2016 at 6:27 pmThanks Josh!

Jules MansonMay 5, 2016 at 6:01 pmAlbert Einstein once said, “If you can’t explain it simply, you don’t understand it well enough.”

I think you understand mathematics more than most math teachers/professors. Well maybe not most math professors. Perhaps some rely too much on textbooks or are too pressed for time to give us an adequate intuitive explanation/demonstration like you do.

I am a mechanical engineer which means that I studied math up to linear algebra and higher calculus (differential equations) and you have really helped me gain some insight into what I consider difficult or complex (not the imaginary kind) topics in math.

Thank you for all you do. Keep up the great work.

kalidMay 6, 2016 at 10:47 amThanks Jules, really appreciate it. I think many teachers have a deep understanding but the problem is converting that into something the rest of us can understand. There’s the “curse of knowledge” where once you learn something you forget what it’s like to not know it. I try to write things down as soon as I figure them out so I don’t lose the memory of of confusion. Thanks so much.

FionaMay 26, 2016 at 10:18 pmYes, I think Kalid seems to understand how human beings minds work alot better that some of the maths teachers I’ve come across. He seems to understand that we are visual creatures firstly, that’s how our subconscious responds to information and maybe we find it easier to assimilate and digest information ( mathematical concepts ) this way. I have found your calculus book very informative. My brain couldn’t make the leap from what pi actually was, to seeing its divisions used in graphs. The circle diagrams were the missing link! Thanks alot for these wonderful newsletters!

Dave XMay 5, 2016 at 8:10 pmThe imaginary number link goes to the intuitive trig post.

kalidMay 6, 2016 at 10:46 amWhoops, thanks! Just fixed.

Krishnamoorthy.A.NMay 5, 2016 at 10:15 pmMathematics has understood you more than you have understood mathematics – you are the right person to propagate the fragrance of mathematics and to inspire Mathematics learner.Good luck in your mission!

kalidMay 6, 2016 at 10:47 amThank you!

ashviniMay 5, 2016 at 10:30 pmhi khalid

Comp Sc. grad writing from india, – your insight that “education not a distribution problem anymore” is worth a TON of Gold. This is the central fallacy polluting all of the internet. No one is sweating about issues that are “fundamental confusion” issues. Those are too tough. Real hard nuts. So people are bypassing them.

I have long told everyone [who cared to listen] that most maths teachers are focused on the “grammar” of maths, and not on the prose. It is the prose where the real joy of discovery lies !

But prose is hard. You have to be creative, and imaginative. You have to ‘think’. So teachers avoid it, Besides, ‘grammar-oriented-maths’ is easy to grade. So all teachers follow the ‘memorize the rules and apply them mechanically’ approach. The grammar of maths,

Sompne once said “the joy maths is not in proof, but in discovery”. I wish that approach was more widespread. Your blog is one of the few light-houses, one of the few beacons of hope in a mostly dark world. Thanks for doing the heavy lifting for all of us. Thanks for imagining a new approach to maths. Thanks for spreading light.

kalidMay 12, 2016 at 6:00 amThanks so much for the note. I completely agree, it’s way to easy to focus on grammar (the rules of math) instead of the creative element (what message are we trying to convey?). We focus so much on distributing existing lessons that don’t seem to have a full understanding of the material (not that I have it either — I’m always looking to improve my intuition. Why don’t lessons constantly get better?).

Really glad you’re enjoying the site, thank you.

Matei MiricaMay 6, 2016 at 7:06 amSuper and happy math! Kalid, you can do a full explanation fo Integrals? ‘In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data” . https://en.wikipedia.org/wiki/Integral

kalidMay 6, 2016 at 10:48 amYep, I have a quick intro here: http://betterexplained.com/articles/a-calculus-analogy-integrals-as-multiplication/

The Calculus series goes into more details: http://betterexplained.com/guides/calculus/

M MunnMay 6, 2016 at 1:36 pmThis is great – I have to stare at it and let my subconscious absorb the info!

Math education needs more of this type of explanation.

I had once proposed a learning tool that essentially was a box. On the outside were point and line definitions. Then the dimensions of the box in a unit of 1 starting in one plane with a variety of planar geometric concepts;. area of the square, the Pythagorean theorem, as you show here.

And as you are aware, just plane geometry can go on infinitely!

Then, on to the 3 dimensional…

…then inside the box, the intro to chemistry, physics, biology!

wuyiMay 9, 2016 at 12:03 amthank you ~

LeonMay 16, 2016 at 8:02 pmKalid, you are one smart man. Do you have a link to your story and how you found your passion for explaining math in a different, much more helpful way? I ask because I’m amazed by your sheer drive.

KasperJuly 18, 2016 at 12:15 amI really like your approach to explaining concepts. I think you did a great job in your calculus course where you described it as starting with a blurry picture and slowly increasing the resolution instead of going through the picture line by line.

Rob VNovember 14, 2016 at 4:18 pmgreat piece, yet maybe i’m missing something out the roman numeral problem, didn’t they already have their number system ordered by tens ( X )? In fact they may have been closer to a gentle geometric language of mnemonics I for 1, II for 2, III for 3, then like the chinese, the realisation they needed to develop sophistication to represent higher values.

In the arabic 3 x 8 bears nothing visually resembling the amplification of those curves about a waistline to what we know as a squiggle next to a scramble as 24.

In a parallel universe they’re probably blogging about “those uncouth viking-bloods of yore and their unrelated incomprehensible glyphs..”

In the end, we’re both in the marketplace of practicality, snapping up that unbeatable deal of seed, and arriving at our struck total based on our three donkeys carrying eight each as a maximum.

as said, great post, and enjoyably so!

TrihorusJuly 19, 2018 at 5:22 amHey!

I do love physics a lot because I keep having my ‘Aha!’ moments. I crave understanding and it really pushes me ahead but when it came to mathematics, I was always a dud. I’m about to begin my bachelors in physics and had to rev up my mathematics and I just happened to find what I needed—your site! Now I’m having these mathematical Aha! moments with every post of yours and I’m naturally reading all of your posts and feel like pulling my sleeves up to do some math. Thanks a lot! My words can’t really describe or convey the excitement that’s coming off by rediscovering those mathematical “Aha!’ moments. Keep up the good work!