Solve Problems Like a Mathematician

What you've learned in school is more powerful than you think.

Posted Apr 27, 2020

There’s a question you hear in high schools all around the world, a question that marks yet another student giving up on maths once and for all: “Why on earth are we learning something that we’ll never even use in real life?”

The question actually has some merits, because indeed, a great deal of high school maths has little practical use. Computers will cover our bases for calculating percentages and plotting graphs, and the rest of the topics will only be useful for the physicist-to-be. So why do we study maths at all?

When kids pin teachers with this question, most replies come straight from a politician’s playbook. The teacher might act as if they didn’t hear the question—or answer a different one. From the more useful attempts, you might hear an answer along the lines of:

“Classes aren’t supposed to teach you formulas. Maths is supposed to teach you to think.

Which is an answer I can wholeheartedly agree with, but it’s one of those that’s hard to nail down exactly what we mean by. How does maths “teach you to think”?

Math’s real use

Watching Jurassic Park as a kid has a lot to do with me having a degree in mathematics now. But even in my younger years, I had to admit that the math wiz dressed in all black, together with his chaos theory blah blah, was nothing but kitsch. Mathematical kitsch is the kind of maths that sounds fancy and makes great TV, but has not much use other than getting kids inspired.

In fact, a lot of maths isn’t even meant to be useful.

Mathematics is extremely well gamified, in that you can always find a more difficult problem to solve. Even mathematicians on the highest levels, people with an Abel Prize in their pocket, will die with problems left unsolved.

This structured approach is what’s important. In class, the teacher will show a few key ideas, definitions and simple proofs. Our job is to use those building blocks and see what else follows from them, what other proofs we can unlock with their help. Solving smart puzzles is the way we’ll make use of maths in real life.

How to use math in daily life

They say that a smart person is wrong more intelligently than a dumb person is right.

It’s not enough to be right about something; the real deal is to be right in a way that makes sense to the person on the other side of the argument. In many ways, explaining how you got to the result is more important than the result itself.

Using mathematical logic to solve problems provides just the right tools for that. We can all steal a few tricks from mathematicians:

  1. Make sure you understand the problem before attempting to solve it. This one sounds so obvious that it’s painful to write down, but the devil is in the details. What assumptions do we need to make for the question to make sense? Are those justified at all? It’s hard to understand anything in general until you’ve seen a few specific examples first.
  2. If you can’t solve the problem, solve a simpler problem. When ancient Greeks needed to calculate the surface of a disk, they didn’t know anything about pi. They started by drawing a square around the circle, then another square inside it, easily calculating the squares’ surface areas and splitting the difference. Similar approaches provided a close enough result for centuries, until pi was introduced.
  3. Or, you can work backwards. When the job is to prove a given supposition, examining the result might help in finding the proof. You can often keep tracing steps back to earlier assumptions until you eventually reach the solution. In real life, setting goals can be a powerful tool for finding a path to reach them.

The next time you see a problem you don’t understand or can’t solve, use the tips above. Write the question down, and keep simplifying the problem space until the problem becomes clear.

Make your case

Mathematicians prefer their proofs to be elegant. “Elegant” means a proof that’s both accurate and creative, something with extra “smarts” in it. You’ll find that most mathematical proofs are structured into definitions and lemmas, which ensures that each step is simple and airtight. This makes communicating complicated ideas easier. The next time you need to make a case, structure the arguments in a way that’s easy for others to follow.

My grandma always told us grandkids to pursue the most generic studies available, because who knows what tomorrow brings. “Whenever you’re in doubt about what to do next, choose things that unlock more options after you’ve finished them,” she would say. Surely she would have approved of pursuing a degree in maths.