I am a graduate student in mathematics and from time to time when I need to learn new concepts either in a research paper or just courses, I feel that just reading the words (and just writing the mathematics) does not help me to understand the new concepts properly. So to really understand the concepts, I usually take notes with a pen and paper and this method really helps me to understand the concepts. By this, I mean that I both write down words and mathematics. I feel that this is not efficient as I have to write a lot and it is slow and also my fingers hurt after a lot of writing.

I wonder if I am doing it completely wrong or if there is a more efficient and easier way to learn new material.

On a related note, what kind of pen is suitable for long handwriting? I use a Staedtler triplus fineliner pen and was wondering if my choice of pen is ok for long hand-writing?

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    If you feel that the physical writing is an issue, you can try combining it with things: typing (slower for most people), taking notes more selectively (mathematical paraphrasing), working out examples (in your head or with writing), using a black/white/glass board sometimes, alternating your writing hand, ... – Kimball Jan 20 '19 at 15:28
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    (1) A pen with a thicker barrel and ergonomic grip might help a lot with fatigue. (2) Typing mathematics notes for yourself is often more efficient using Unicode symbols rather than LaTeX. – Elizabeth Henning Jan 20 '19 at 22:22

Don't confuse efficient with effective. The method you have chosen is well known to be effective. But I don't know of any particularly efficient method of learning other than to do it.

There is research, actually, that shows that taking notes by hand engages the brain in a way that other things does not. This means writing by hand, not typing on a keyboard. See this answer to a possibly related question.

But mathematics is special. You not only want to learn existing mathematics, you want to learn, as a doctoral student, to be able to actually do it. This involves the creation of new mathematics. For that, you need to go a bit deeper, though it won't be especially efficient. As with most mental things, you need to practice the art so that you change the neural connections, the mental pathways. One way is to get some practice by trying to reconstruct proofs of theorems. Find a theorem new to you. You trust that the statement is true since someone proved it (though mistakes occur, of course). Try to prove it without reference to an existing proof. If that is too big of a step, initially, try, when reading an existing proof, to figure out the next steps that would lead to the final result.

Mathematics is more, of course. It is knowing what is likely to be worth exploring - what might be true, so that you can then try to develop a proof of it. But you have to start with smaller steps.

But back to note taking. If you want to remember things you have learned you can/should also summarize your previous notes after a period of leaving them untouched. Also do this with hand writing. The summarizations may actually be enough for you to save for later.

One additional advantage of writing notes is that they will have a long lifetime if you use decent materials (pen, paper). Notes taken on computers will almost certainly disappear with technological change unless you take periodic steps to update the media. Don't depend on the cloud to preserve your notes. Companies can change their business models and simply wipe out what you have saved. Only fire and flood will destroy your paper notes. We can still read things written by ancient Egyptians, since they used materials with a long life.

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