Schools shouldn’t have math classes.
Math classes are detrimental to our students. Hear me out.
First and foremost, you’ve got to understand that math isn’t a thing. ‘Math’ is a word that means too much. Some parts of it are socially universal: everyone needs to know what $7$ means. Other parts of it are pretty useless: practically nobody needs to know how to take the derivative of the product of two functions.
If we lump all of these terms under the same umbrella of “mathematics” then it’s hard to talk about education. Here’s a challenge for you: Get a pencil and paper and write down the things that math class should teach without using any of the following words:
What we want out of math class isn’t specific formulas or equations. Rather, we want our students to learn problem solving skills and general reasoning techniques. Math class is a poor way to fulfil that goal.
Math classes are not without value, but the word “mathematics” is a confusion that hides what we’re trying to teach.
Teaching math is like teaching door-opening. Opening doors is surely a life skill. Everyone needs to know how to open front doors and car doors.
Some people need to know how to open space shuttle doors or military bunker doors, and some people will need to pick locks when they lock themselves out of the house. This doesn’t mean we should teach door-opening class.
Advanced door-opening can teach important skills. Students who are good at picking locks will learn patience and a subtle precision that can be invaluable later in life, but that still doesn’t mean we should teach a door-opening class.
Instead, we should teach reasoning and dexterity. With reasoning and dexterity you can open almost any door, but more importantly, you can do anything else that requires reasoning and dexterity. You aren’t limited to doors.
Math works the same way. We shouldn’t have “math” classes. There are parts of math that everyone needs to know, but we shouldn’t have math classes. There are subtle skills and invaluable lessons that math teaches you, but we shouldn’t have math classes. Instead, we should teach reasoning, abstraction, and generality.
If you want students to know math, teach them everything else. Teach them how big things work and how small things work. Teach them how populations act and how governments react. Teach them how muscles move and how molecules form and where the rain comes from. Don’t just make them memorize these things: make them predict. Give them problems to solve.
Synchronize classes so that each class gives students the same problem in different guises. Have them calculate population growth in one class and monetary growth in another. Have them balance utilities in one class and balance energies in another. Show them the same problem again and again in different clothing.
Do this, and mathematics is what falls out of the process. It’s is everything students take from one class and use in another.
Math should be taught indirectly.
Math is just another name for problem-solving techniques. Modern math classes force students to memorize specific techniques. That’s not the same thing as teaching students to solve problems.
Math equations are like cheat codes to a game that most people don’t play. Forcing students to memorize formulas is not mathematics. It helps them through standardized exams, but it isn’t mathematics.
Math isn’t about the equations. Mathematics is the realization that techniques used to solve a problem once can solve the problem anywhere. If we want students to realize that, we need to keep giving them similar problems until they start to see the patterns and realize that they can invent shortcuts on their own.
It used to be enough for students to memorize and apply techniques handed down by the ancients. It used to be acceptable to memorize a bunch of derivative laws instead of really knowing calculus.
Students who memorized the rules could turn the cranks and solve calculus problems. Schools pumped out graduates who knew just enough to turn the cranks. Only the best and brightest ever invented new techniques, and that was fine – whenever one person invented a new technique, hundreds were necessary to calculate the results. For every Maxwell we needed thousands of people to apply Maxwell’s equations and get real answers. Calculus problems weren’t going to solve themselves.
Those days are behind us. In this day and age we teach computers to apply formulas. Calculus problems can solve themselves. One computer programmer isn’t worth ten number crankers, nor even one hundred: one computer programmer is worth all of the number crankers.
When you program a computer to solve a problem once, it can solve that problem a billion times per second forever.
We don’t need people who can apply calculus equations anymore. We need people who can invent calculus.
It is not important that our students know algebra – it is important that our students have the reasoning skills to invent algebra. It doesn’t matter whether they’ve memorized a particular algebraic syntax. What matters is being able to confront new algebra-level problems (in any domain) and solve them without help. What matters is teaching our students to solve problems on their own.
We shouldn’t be satisfied until our students have invented their own techniques akin to algebra in response to the problems that we throw at them.
This skill is hard to teach, and yet this single skill is more important than algebra, trigonometry, and calculus combined.
We don’t need everyone to know calculus. We don’t need everyone to know any specific techniques. Instead, we need everyone to know that there are techniques. We need people to realize that math isn’t about solving problems: math is about eliminating problems.
Math is about solving a problem so hard than next time you come across something that looks even remotely similar, it’s already solved.
What matters is understanding that problems have patterns to them. If you develop a method to solve one, you can solve them all. Newton, Maxwell, and Einstein weren’t people who were especially good at remembering the product rule. They didn’t discover laws of the universe by being really good at memorizing trigonometry. They advanced humanity by inventing new ways to solve problems.
We can’t teach students how to handle every problem that they will ever face. Instead, we must teach them how to stare a problem in the eyes and invent their own solution. This lesson is the single most important lesson that we can teach schoolchildren.
After students understand how to make their own techniques, once they prove they can confront problems and find their own ways to solve them, then we can show them all of the ancient formulas. Then we can rush them through the problems we’ve already solved and bring their minds to the forefront of our understanding.
We must not do this until we’re convinced they can invent mathematics on their own. That skill, not the specific equations, is the important part of math.
We have no shortage of equation solvers. What we need are people who can walk out to the very edge of human knowledge, stare into the wild unknown, and say
I can invent ways to understand this.
Everyone else is just cranking numbers.