The Feynman Technique
Richard Feynman, a Nobel Prize-winning physicist known for his extremely lucid style of exposition, once said:
"I learned very early the difference between knowing the name of something and knowing something."
How can we tell whether we truly know something or merely know its name? The difference, I think, is whether you can offer explanations as opposed to trivia. Richard Feynman developed what has come to be called the Feynman Technique for studying, which efficiently builds deep understanding through a structured explanation process—forgoing the (unfortunately) common-place but ineffective applications of rote memorization.
This is intended to be a guide to using the Feynman Technique to study and understand mathematics. We'll use the following adage (dubiously attributed to Albert Einstein) as inspiration:
If you can't explain it simply, then you don't understand it well enough.
Grab a piece of paper and let's get started!
Identify your topic. Write the name of the concept, skill, or learning objective that you'd like to master at the top of your page. Keep it focused! You should only work on one concept at a time.
Write an explanation. Once you have a topic, begin writing an explanation as if you're teaching it to someone who's never seen it before. Assume as little knowledge as possible! This will force you to build your understanding from the ground up.
Pinpoint the gaps. As you're writing your explanation, you'll undoubtedly get stuck at certain points. These points are the gaps in your understanding. When you get stuck, go back to the textbook or source material to relearn what you're missing. Continue writing your explanation once you can fill in the gap with your own words. Repeat this step as many times as necessary!
Revise! Once you have a complete explanation down, mercilessly revise it for conciseness and clarity. Keep it simple! Trim wordy explanations and remove paraphrased jargon. Use metaphor or analogy whenever possible. Try to make your explanation fit on one side of the paper. If that's too easy, try to make it fit on half the paper.