Math is like cheat codes.

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I’m not here to garner appreciation for mathematics. I’m not implying that math allows you to hack reality and make your life better. Quite the opposite: Math is like cheat codes *for a game you aren’t playing*.

Sure, sometimes you’re playing the right game and you know the right codes. Then math is great. But most of the time, nobody’s playing the game.

Math education is forcing students to memorize the cheat codes for a game that they don’t care about. Students – often smarter than we give them credit for – are aware that these boring equations are useless to them. But when they ask for purpose, teachers respond

Well, you might play this particular game one day. When you do, it’s important to know the cheat codes.

This is wrong in so many ways it’s difficult to enumerate them in a single narrative.

For starters, the knee-jerk response to this drivel is “well then I’ll never play your stupid game” – and that’s exactly what four out of five students do. They avoid science, engineering, math, and technology because they were forced to learn the boring cheat codes for those games, and they sounded repetitive, so the games must be dull.

This is a rational response to the atrocious education that we provide.

It gets worse. Imagine a situation where our math education works perfectly. Imagine a student who has memorized all of the right cheat codes and who’s stumbled into exactly the right game. When this happens, we measure success by how quickly the student cheats their way through the game without ever actually playing.

The ability to memorize and apply formulas is not mathematics! It allows you to solve math problems in specific scenarios when you recognize that your formula applies, but that’s all. Being able to do calculus doesn’t make you Newton.

A student hasn’t learned calculus when they can derive an equation. A student has learned calculus when I can walk into the school and erase calculus from their mind completely and *then they rebuild it*.

The big insight of mathematics is that certain types of problem show up everywhere – biology, statistics, music, economics – and if you can solve the problem once then you can use the same method to solve it anywhere. It doesn’t matter what method you use. It doesn’t matter what syntax or terminology you use. What matters is looking at a problem and saying “I’ve already solved something similar to this, can I reuse those techniques here?”

If you learn economics and then try to learn biology, it’s easier because of the shared patterns. Understanding how money grows offers insight into how populations grow. Understanding one helps you understand the other.

And yet we teach students calculus *first*. We tell them to memorize these abstract formulas that they don’t understand the need for and *then* we show them biology problems and say “just apply that formula you memorized”. Sure, it’s easier: they apply a stored formula and the “right answer” (as determined by how the teacher grades it) comes out. But they don’t understand *why*. If I erase the equations from their brain then they can’t solve the problems any more.

The way we *should* teach calculus is by teaching biology and chemistry and physics end throwing the same problem at students in each of their classes. Make them struggle with the same problem in ten guises until they start to see the pattern.

Solving it once will be hard. The second time will be easier. Once they’ve solved the problem thrice, they’ll be able to apply their techniques and solve it anywhere. *Now* they’re starting to learn calculus.

Throw the problem at them in ten new contexts to test their resolve. Once they start breezing through, throw it at them ten more ways. By the time they’re done they’ll know their technique so well that they can write it down. Take a look at what they wrote down. *That’s* calculus. It’s *their* calculus. If I erase the equations from their mind they’ll be able to reconstruct them with ease.

Once you know calculus, the equations don’t matter. You don’t know a subject until you can reconstruct the equations on your own.

Math is just a collection of techniques. It’s cheat codes for solving problems that show up all over the place. If you show students the cheat codes *first*, before they see the problems, then the codes are boring. Even if they memorize the cheat codes, applying cheat codes is not the same thing as skill.

Make them play the game first. Make them live through the repetition until they make up their own codes. When the students see the patterns they’ll exploit them on their own.

Math is a human reaction to boring problems. When faced with a repetitive task you’ll develop a way to manage it – that’s just human nature. You’ll look for techniques that let you jump to the answers without going through the same monotonous work every time.

Problem-solving techniques are named “mathematics”.

Math is cheat codes for problems that show up all over the place. If you show students the cheat codes *first*, before they know the problems, then the codes are boring. It’s like teaching students music by forcing them to read the sheet music for Beethoven’s $5^{th}$ before ever hearing an instrument.

When we teach ancient techniques before people have faced the problems, the techniques look contrived and pointless. More often than not, the techniques *are* contrived – my methods of problem-solving may not make any sense to you.

If we want students to learn mathematics we need to confront them with the plethora of similar problems found in the real world until they start to see the patterns. We need to throw the same problem at them in different guises until they develop their own techniques to avoid the monotony.

Then their techniques won’t be cheat codes. They’ll be shortcuts.