Asaf Karagila
I don't have much choice...

7 Easy Hacks to Improve Your Math Skills

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Everybody wants to improve their mathematical skills! And quickly, too! Since it's so hard to do just that, I've written down some quick and dirty hacks for quickly improving your mathematical skills!

1. Get a graduate-level degree in mathematics!

Getting a PhD in mathematics is not really about getting the PhD itself. It's more about getting much better at learning mathematics. So if you get a PhD in mathematics it will help you better your ability to study more mathematics and improve your skills.

2. Spend time, a lot of time, thinking about concepts.

If you don't spend a lot of time thinking about something, how do you expect to truly understand it? How would you improve yourself? Next time you're having a coffee, or tea, or beer, or just taking a long shower, spend some time thinking about whatever mathematical topic you're interested in. If you haven't spent that time understanding induction, I strongly recommend starting there, since almost all of mathematics is built on induction arguments.

3. Spend time writing proofs of things.

One of the key differences between mathematics and other subjects is that mathematics is deductive. You can't bend the data to your will. What is important is to get the right definitions and right proofs. When you sit down to try and prove something, if you're careful enough, you will find yourself learning how to spot gaps and how to tweak definitions to make things work. Nothing is absolutely true until you've written a proof, and preferably had someone proof read your proof.

4. Spend less time reading about quick hacks to improve whatever.

Quality is not obtained via "hacks", it is obtained by carefully and methodically crafting something. It takes about a full decade to become a master in something, and there are no shortcuts to that. Except, maybe, if you already have a PhD in mathematics, and you can apply your skills to mastering a new topics in mathematics a lot faster, but it will still take you a while.

5. Remember that failure is part of the learning process.

Success is a diamond, polished in the dust of previous failures. Learning from your mistakes, your misconceptions, and your bad naive intuitions is a good thing. Do not worry about wasting your time trying to prove false things. Do not worry about wasting your time failing to prove things. You cannot facilitate learning without failure and mistakes.

6. Teach others.

When you have to explain things to other people, you will end up learning from your own explanations. I only grokked induction when I taught it for the second time. Same for transfinite induction. And that's well after I was using both of these methods for a while. Use websites like and other online communities to explain what you try to understand to other people, and get feedback on those explanations as well.

7. Be patient.

Mathematics is much much slower than most sciences. Also learning it takes time. If you make it about the journey, rather than the end, you will enjoy the experience a lot more.

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