My topology curriculum is an example of How To: Learn a New Discipline.

0. Build the map.

The map is a high level representation of important concepts and connections — the threads that compose the subject.

I started by working through these two:

Where “work through” (\equiv) read carefully and annotate the text, take notes on important concepts, and complete a few exercises from each chapter.

“Catherine, you said you’d teach me math. Why are you drawing squiggles?”

“Math is formalized art! I get to draw these squiggles on a page, describe them with symbols, and call it math.”

1. Learn by working through complementary perspectives in parallel.

• Hatcher: Intuitive & Geometric Perspective: (\rightarrow) understand it.
• May: Category Theoretical Perspective: (\rightarrow) believe that you understand it.

2. Give weekly lectures on what you’ve learned.

I’m giving informal lectures to a small group of people on what I’m learning. These people ask questions and tolerate pauses to look things up.

Don’t forget to ask for help when you get stuck!

If you can’t find a satisfactory explanation online:

• talk to professors/PhDs/postdocs at your local university (they’re people too!)
• post your question to a website such as MathOverflow or Quora.