How to digest new math more quickly?

Question

Do you have any advices, tips and tricks on it? I have a feeling that there is never enough time to read every math book I find interesting, to learn about a new area etc. etc.

Answer 1

I suggest the following points:

1. Try to find the general (universal) pattern common in several concepts. For example, a homomorphism between two groups preserves the group multiplication and group multiplications are the only composition rule in groups. Then it is easy to learn a homomorphism should be in between two algebras, rings, vector spaces, etc. Maybe the names change but the underlying concepts are similar.
2. Take notes during reading new concepts, theorems, etc.
3. Try to generalize the statements.
4. Find the relationship between different concepts, definitions, statements, etc.
5. Try to solve some exercises, especially those which make you to review the content of the course (book, lecture, paper, etc).
6. Give a presentation or teach a course.

Answer 2

Here is great explanation by Scott Young how to learn faster. Post explain techniques to learn faster, which he used for his MIT Challenge.

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Following are some important points which was quoted from the post:

Demystify the process. Getting insights to deepen your understanding largely amounts to two things:

1. Making connections
2. Debugging errors

THE DRILLDOWN METHOD: A STRATEGY FOR LEARNING FASTER

Here’s the basic structure of the method:

1. Coverage
2. Practice
3. Insight

THE FEYNMAN TECHNIQUE

1. Get a piece of paper

2. Write at the top the idea or process you want to understand

3. Explain the idea, as if you were teaching it to someone else

For Ideas You Don’t Get At All

The way I handle this is to go through the technique but have the textbook open to the chapter explaining that concept. Then I go through and meticulously copy both the author’s explanation, but also try to elaborate and clarify it for myself. This “guided” Feynman can be useful when trying to write anything on your own would be impossible.

For Procedures

You can also use the method to fully understand a process you need to use. Go through all the steps and explain not only what they do, but how they execute it. I would often go through proof techniques by carefully explaining all the steps.

For Formulas

Formulas should be understood, not just memorized. So when you see a formula, but can’t understand how it works, try walking through each part with a Feynman. Here’s an example I used for the Fourier analysis equation.

DEVELOPING A DEEPER INTUITION

1. Analogies – You understand an idea by correctly recognizing an important similarity between it and an easier-to-understand idea.
2. Visualizations – Abstract ideas often become useful intuitions when we can form a mental picture of them. Even if the picture is just an incomplete representation of a larger, and more varied, idea.
3. Simplifications – A famous scientist once said that if you couldn’t explain something to your grandmother, you don’t fully understand it. Simplification is the art of strengthening those connections between basic components and complex ideas.

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The Strategy to Learn Faster

Learning faster doesn’t need to be a trick to work well. It simply means recognizing what is actually going on when we reach a new level of insight and finding tools to help us reach those stages consistently.