Mega Man was one of my favorite video games. You're a little cyborg, running through levels and fighting end bosses:

Here's the trick: every boss has a weakness. After you beat Fire-man with your regular gun, you earn a fire weapon. This makes your upcoming fight with Ice-man easier, which helps defeat the next boss, and so on.

So why's this special?

*In Mega Man, you look forward to encountering more bosses.*

Every level is a chance to permanently upgrade your abilities, not a grind you're trying to survive.

## The Tetris Mindset

Think about Tetris: would you look forward to a variety of new shapes appearing?

Heck no. Tetris can be fun in a "survive hordes of incoming zombies" sort of way, but in terms of learning, it's a frustrating, Sisyphean task. Every new piece is something to move beyond, not a learning opportunity. It's a test to find your breaking point.

In Mega Man, the game gets easier the more bosses you have. It's specifically designed for you to improve over time. Guess which game has 10+ sequels?

## Conquering Euler's Formula

Math learning can follow the Mega Man pattern. If we want to beat "Dr. Euler", we need to beat his henchmen and master their weapons:

- Rad-man, once defeated, lets you think with radians, not just degrees
- Power-man lets you understand the base and power of an exponent
- I-man lets you unlock the rotation of imaginary numbers
- Pi-man lets you think in cyclic patterns
- Multi-man lets you understand how quantities transform each other
- E-man helps you visualize continuous change
- And finally, Dr. Euler lets you understand the role of imaginary exponents: e
^{ix}= cos(x) + isin(x)

After defeating the henchmen — truly understanding them and mimicking their abilities— Dr. Euler becomes defeatable. And after that, Boss Fourier. Then Captain Convolution.

There many more challenges on the horizon… and that's great! Every formula, once mastered, is a power to use.

When learning, I ask: "Did I internalize the concept so much I look *forward* to seeing it?". Learned ideas become allies, a decoder key to help unlock future equations.

## The Mindset Shift

I constantly seek analogies for learning because my understanding has improved most from perspective shifts, not from studying specific concepts.

Despite years of math classes, I lacked intuitions on e, i, pi, radians, logs, exponents… (the list goes on). Math class involved grinding through Tetris levels, moving *past* the concepts and not absorbing them fully. Imaginary numbers were *not* friends I looked forward to seeing in an equation.

An Aha! moment helped me see learning as a set of additive skills we could internalize, and the process and challenges became something to look forward to. I hope the same shift happens for you. Wouldn't it be great to look forward to adding new abilities to your arsenal?

Happy math.

## Leave a Reply

11 Comments on "Learning Math (Mega Man vs. Tetris)"

Hi Kalid,

What was the aha moment you had that allowed you to view these skills as additive, not a grind? What were you doing when you had the aha moment? Where were you? Who were you with? I’m just curious to know what prompted the aha moment, and how I can help encourage that same aha moment in my students. It does seem to be a grind for a lot of students and that may be why they struggle—not seeing these math things as skill “upgrades”.

Thanks. Matt

Hi Matt, thanks for the question. I was a freshman in college, cramming in my dorm room for the final of my vector calculus. I’d been struggling to remember the formulas all semester.

8 or 9pm on a Thursday, a set of analogies struck me. I can’t remember the order, but it was something like:

* Flux is the amount of something crossing a surface.

* Divergence is simply flux density (flux, reduced to a certain point).

* Circulation was the amount of spin along a path (put a hula hoop in the water, see if it spins)

* Curl was “circulation density” or the amount of spin at a point. (Shrink the hula hoop to a the size of a cheerio — does it still spin?)

* Gauss’ Law was about the conservation of flux. (If you put a 3d boundary around a faucet, no matter the shape, you should measure the same amount of water crossing the boundary. A smaller boundary will have more flux per surface area, sure, but the same total as a giant, loose boundary).

I think that was the rough order of things. I’m not sure what prompted the analogies, I remember being frustrated/despondent that things weren’t making sense, so I was willing to try anything. “What if it were like this? Like this? Could it be like this?”. Eventually an analogy stuck.

I went back through the formulas we’d learned all semester and reworded them with my analogies (“flux density”, “conservation of flux”, etc.). I no longer had to memorize them, they became statements of truth (like “a circle is round” — I couldn’t forget it).

(The vector calculus articles are some of the first on the site: https://betterexplained.com/articles/category/math/vector-calculus/)

I think the grinding feeling comes from never having had a real “Aha!” moment where things clicked deeply. I had conceptually understood ideas before (learned/memorized and moved on), but it wasn’t a natural-feeling fit. Once you have that first Aha! moment you realize how much you’ve been missing all along. (Like beating the boss without picking up the new weapon.)

Megaman generally is one of the best designed games in terms of teaching players and motivating tbem, with Megaman X often cited as the quintessential example of this. There’s a really good YouTube ( search megaman x sequalitis… warning there’s quite a lot of swearing, but it’s a legit analysis) on how it motivates the player and teaches without being didactic.

I love that game. That’s my entire childhood.

Thanks, I’ll check it out! Mega man was extremely motivating as a game.

Thank you Khalid I like the view of learning something to understand something. I used to use that concept in science and it made me make better connections and move a 34 to a 97! I never thought it would be possible in math because the text book looks like each idea is separate and just has to be learned, it’s hard to find progression or an this might connect to this. I like your mr Euler idea especially.

That’s awesome, I love hearing when someone overcomes the idea that math can’t make sense. With the right approach, anything can… keep it up! :)

Great insight, Kalid! I wish math teachers all over the world read Better Explained! :)

Thanks!

Thanks I finally beat Megaman after so many years

I agree that at times learning can be a grind when you want to fully understand it. Next think you know, teachers are trying to layer more knowledge upon a fragile foundation. I subscribe to the idea that building a interconnected web of knowledge, and making analogies/associations to existing knowledge really solidifies concepts. Could you explorer the idea of intuition being nothing more than a highly complex web of neurons that has associated various branches of subject matter? Whatever it may be, developing an intuition can take years as you say, and most people don’t look any deeper than the initial layer of understanding. Also learning happens at different rates for each person, but unfortunately our educational system is very regimented and structured which could prohibit some great minds entering those institutions. Enough of my ramblings, great post!

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