How can a beginner researcher or Ph.D. student efficiently and effectively learn new concepts while staying motivated?

I hope this question is appropriate for MSE.

The situation is that someone is reading a book (e.g. a monograph), possibly helpful in his/her research, and the content is sufficiently extensive or deep. At first glance it seems very frustrating and time consuming. At least, I feel a bit of frustration.

I want to know how mathematicians keep themselves motivated and achieve this task? How do they manage their time to learn new concepts, especially when someone is a beginner researcher or Ph.D. student?

Solution 1:

I will propose an answer just so there in one!

For me it helps to ignore all literature and go for a walk. I ask myself what bothers me, what do I want to know and what are my interests. Then I go back to the library and look for an answer; without trying to understand how or why. I merely try to connect two dots, even if the method is completely stupid. If I find that answer then my interest (see start of story) gives me the motivation to understand how the text got to the result. If at any point I am frustrated or bored I simply restart my process. I go for a walk...

Finally my algorithm is totally personal to me but it does work for me. Others may substitute walk for watch a film. Anything to re-discover your childish passion for learning. The key to success is having that passion (see numerous famous people who have said similar).

Solution 2:

Let me offer a few points, though I'm probably forgetting many things.

First, you typically need a reason for motivation. It could just be you find a subject beautiful and want to learn more, or it could be because you want to use the theory for something else. My approaches to these situations are somewhat different, but hopefully it's not merely because someone else told you you need to learn this. In any case, your motivation level needs to be sufficiently high to learn something difficult.

Second, to help maintain your motivation over a long period of time, which is needed for serious projects, it helps to set easily achievable for yourself and get frequent indication you're making progress. If you're going through a monograph, you can try to aim for something like going through a certain amount each day, either page-wise (say 2 pages/day) or time-wise (2 hours/day). It's easier to feel like you're making progress with page goals, though you can't be too strict as some parts are harder to understand. If the difficulty of the material varies greatly from part to part, then setting time goals is better. If possible, run a learning seminar with someone else, or give talks on the material, or meet regularly with an advisor/mentor. Having some sort of external accountability makes it much easier to stay on track.

Third, you really need to find your own understanding of the material. Ask yourself lots of questions (big picture questions and small detail questions). Try to work out your own examples. After you've read something, try to explain to yourself what was going on. Personally, I find it very helpful to type up my own notes to understand things in my own way. This is time consuming, but very helpful, and it also helps me feel like I'm making progress as I see my notes grow.

Another point, which may not apply in your case, but I will say it anyway: if I'm trying to understand something specific (say X) in a monograph, I find it much easier not to read it linearly. Maybe read the introduction, then read the part you want, and after not understanding it, go back and to see what is the first thing you need to understand. Repeat. Often with this approach it's easier to keep up your motivation, as you're constantly setting little goals (e.g., figure out what the definition of Y is) which you have directly related to your overall goal X. Of course, this isn't appropriate if you're just doing preliminary reading for research.