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At the moment I'm trying to teach myself Riemannian geometry by reading some books. I take notes by hand but I find that I forget it all and worse, either lose the sheets of paper or don't want to read the sheets of paper since my handwriting is not great and it does not look aesthetically pleasing to me. I am considering writing up notes in LaTeX but am not sure if it's worth it or if it's good as a learning tool. So I am curious as to how other mathematicians take notes when they're specifically learning a new topic from a book. I'm not talking about how to take notes in lectures which is a different topic. Obviously different people have different styles which is what I am interested in, so I can adapt something and maybe it will work.

If you do use LaTeX, how do you manage and organise your notes? What styles do you use?

Thanks!

(I saw a thread in Meta in which I read that Math.SE is the only SE site that allows questions like this (subjective questions), so hopefully this won't be closed. Also, I read every thread on this topic on this site and on academia.SE and believe that this is different enough to warrant a thread.)

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I find it as easy to misplace a file as it is to misplace the piece of paper. I would use bound (to avoid loosing individual sheets) notepad to work out problems from the book. Doing it in LaTeX would be too much of a distraction for me. –  Sasha Jul 24 '12 at 21:52
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However it is easier to find a misplaced file than to find a misplaced piece of paper if one still knows something about the content (like some characteristic words). That's because the computer can search files for you, but not paper. –  celtschk Jul 24 '12 at 22:42
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@celtschk: Right but I make notes on a piece of paper in a minute which will take 15 minutes or so to write in LaTeX. –  Gigili Jul 24 '12 at 22:58
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For me drawings are important, and I have found nothing that replaces pencil & paper (I have laptops, wacom digitizers, tablets up the wazoo, but nothing approaches pencil for a quick sketch). Of course, I have a growing stack of drawings waiting to be scanned... –  copper.hat Jul 25 '12 at 0:10
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(+1) This is a question close to my heart, I too am facing this dilemma, have been for years, and haven't come up with a satisfying answer yet. TeX-ifying your notes can be monstruous work, considering that writing them up by hand may only take you a fraction of the time, especially if you have to typeset commutative diagrams and drawings. On the other hand, you can't revise your hand-writen notes, nor expand them the way you'd like, or properly melt them into a Grand Unified Collection of Notes... I long for the day my pad or laptop will convert my handwriting into TeX automatically :'( –  Olivier Bégassat Jul 25 '12 at 3:35

8 Answers 8

up vote 32 down vote accepted

Riemannian geometry is a difficult subject to type in LaTeX. For example, there are concepts in Riemannian geometry which are best understood by pictures that would take no more than five seconds to draw by hand but much more time to incorporate into a LaTeX file. Also, the same can be said for long computations in Riemannian geometry (e.g., the computation of the Riemann curvature tensor, the Ricci tensor, or the scalar curvature of a Riemannian manifold). Of course, there will undoubtedly be some people who can type the contents of a subject such as Riemannian geometry very efficiently in an organized manner but even for such people writing down by hand the same contents would be more efficient.

I think that writing down your notes by hand is actually a better idea than typing your notes in LaTeX. The reason is twofold. Firstly, writing down your notes forces you to think about the material before you begin writing; you cannot simply hit the "backspace" or "cut, copy, or paste" when you write down by hand and you will have to plan out that which you are going to write. Secondly, I think writing down your notes by hand is a more involved process in that your memory of the material will be better if you do this (but there are probably people who are exceptions to this rule).

Of course, the advantage of typing your notes in LaTeX is that you have a neat and organized set of notes in a readily accessible location. However, I think that the textbook itself serves this purpose; your notes should distinguish themselves from the textbook if they are to have additional value.

My advice is actually not to take down too many notes when you study. The reason is that this slows you down sufficiently that the disadantages outweigh the advantages. I think taking notes in some form is definitely useful when learning a new topic but this depends on what you actually write down. Moreover, for practical purposes it might not be possible to write down everything.

I would advise you to write down enough that you have an understanding of the "big picture", i.e., conceptualize what you are studying. If you are learning Riemannian geometry (as is the case), then this does not involve writing down every theorem and every proof. You should rather write down those definitions, results, and examples that help you to see the intuitions in the subject. For example, the existence and uniqueness of geodesics in a Riemannian manifold (subject to two initial conditions) is essentially a "clever" application of the fundamental existence and uniqueness theorem for ordinary differential equations. If you attempt to write down the entire proof (line by line), then the real reason the result is true will become obscured.

Similarly, the understanding that the proof of the existence and uniqueness of the Levi-Civita connection on a Riemannian manifold is similar in approach to the construction of the differential operator in the de Rham complex of a smooth manifold is also an important conceptual idea. You should highlight these conceptual ideas as much as possible rather than the line by line logic and computations that occur in the proof. The key to really understanding and remembering a subject is to have a solid conceptual framework in your mind.

I should add that memorizing formulas in Riemannian geometry is best aided by explicit computations involving the formula. You should try to compute at least a few examples of geodesics, the Riemann curvature tensor, the Ricci tensor, scalar curvature etc. in order to best understand the mechanisms of the formulas. The connections in your brain between the conceptual framework of the subject and the explicit computations are strengthened by doing as many computations as possible.

In summary, my advice is to write down as much as possible your intuitions in the subject and why you believe certain results are true. In fact, you should be creative in this process; if you understand the proof of a result but if you feel that you are not sure why it is true, then try to search for a deeper understanding that will help you to realize this. For example, doing examples with the theorem in question is often helpful if you really wish to understand why the theorem is true; write down these examples and your conclusions based upon them. If you find that you forget what you are learning, then I think writing down everything does not in fact help as much as it seems it should; you would be making better use of your time conceptualizing the material and thinking about the bigger picture.

I hope this helps!

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Thanks for the reply Amitesh. Very helpful and useful answer, even though I really wanted some reasons for switching to TeX. –  arzela Jul 26 '12 at 15:24
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@arzela You are welcome. However, it is not clear from reading your question (at least to me) that you would like some reasons for taking notes using LaTeX rather than by hand. Instead, it seems that you would like to know which is the better alternative. You might like to edit your question to make this clear in order to obtain more tailored responses. –  Amitesh Datta Jul 29 '12 at 0:41
    
I definitely wanted a balanced view (as opposed to "give me reasons why I should use X") on which format I should use but I was secretly hoping LaTeX would win :P Anyhow, the reasons for hand writing are much more convincing. –  arzela Jul 31 '12 at 21:42

If you do all of your note-taking by reading and writing, then you're really only working with two of your senses - sight and touch. I would recommend bringing in a third sense, so your brain gets information from one more channel.

Take your notes by reading out loud into a small tape recorder, or some other recording device. A few days later, listen to the recording, and type it or write it up neatly. It doesn't really matter which. Use LaTeX if you're good at it; or even the Microsoft Equation Editor.

If you do this, you've already heard your notes twice; once when you first recorded them, and a second time when you typed or wrote them up. Keep the tapes, so you can listen to them again. And when you revise, alternate between listening to the tapes, reading the original book, and reading the written notes.

But always, when you're studying, try to think of ways to make use of more than one of your senses. Just using your sense of sight is seldom enough to make the material stick in your brain.

UPDATE

It's many years since I was researching this particular area, so I'd prefer not to refer you to any of my own work. One really good source of information on this particular topic is John Medina's work "Brain Rules", particularly chapter 9.

See Brain Rules: 12 Principles for Surviving and Thriving at Work, Home, and School and this on the book's website. His references for this work are here. There's also a really interesting website that discusses this phenomenon, and refers to Medina's work.

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+1 , I have always used recordings when learning foreign languages (since they are heavy on memorisation) but it never occurred to me to use the same technique for mathematics. –  Daniel Littlewood Jul 27 '12 at 9:03
    
+1, nice answer ! –  vanna Jul 27 '12 at 9:24
    
This is an interesting answer, but I would like some concrete evidence; do you know of any studies exploring that view? –  user5501 Aug 2 '12 at 9:50
    
@LovrePešut - OK, I've added some links to my answer. Hope this is helpful. –  user22805 Aug 3 '12 at 2:42
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Explaining your notes to someone else, as in group discussion or tutoring, is another channel for taking notes. I would say reproducing proofs from scratch is important especially if that is going to be your area of specialization. I think TeXing creates too much distraction and channels your energy in the wrong direction. –  Maesumi Jan 13 '13 at 2:51

I don't always do this out of sheer laziness, but what sometimes works for me is to have two sets of notes: the ones I go writing as I read and write the proofs, and another set where I only write the major definitions and theorems, for easier parsing.

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Thanks for replying. I think I am more likely to lose 2 sets of notes though. –  arzela Aug 5 '12 at 18:45

For me, the learning process while taking notes is 90% in the writing itself. Later on I might glance at what I've written to remember what I was thinking at the time, but I have a hard time reading it thoroughly.

If I want to review the content of the notes, I usually have to take the notes again :)

Disclaimer: my method might be crap and I might be doing things the wrong way.

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(+1) for "my method might be crap" ^^ –  Olivier Bégassat Jul 31 '12 at 23:55
    
:) Let no one say I fooled them. –  rschwieb Aug 1 '12 at 1:31
    
+1 Because that's equivalent to my method. :) –  user5501 Aug 2 '12 at 10:02

I'm reading the book which is, though presents lots of great ideas, is fairly poorly written, with lots of typos and only skimpy outlines of the proofs. I was making my notes (with detailed proofs) and at some point realized I'd like to keep my notes as something other than the bunch of pieces of paper. So, I'm typesetting everything in LaTeX.

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Not a canonical answer, I'm afraid, but I thought I'd mention making notes in the book itself.

Marginalia has a long and proud history. Recently I bought a second-hand text on QM. Described as clean inside, it was in fact riddled with notes from the previous owner so I was annoyed - at first.

Then it turned out that the previous owner (sadly deceased) was a Physics Professor and the notes proved invaluable.

As an autodidact of many years (as we all are ultimately), I've often seen the ideal way of learning to be developing a relationship with the textbook where notes customise the textbook to your learning; in a sense you're manufacturing a new, bespoke textbook.

It'll be nice when ebooks really let us do this.

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That's quite cool. Unfortunately I don't own most of the books I use so I can't really do that! –  arzela Aug 5 '12 at 18:46

While I am reading the book I will write questions or comments in the margin and underline key ideas. I use a pencil so I can edit my comments.

I also tend to read around the book - skipping to interesting chapters first, then moving back to fill in details. I want to get the big picture and motivation first. In someways doing this first stage on the web rather than a print book is easier because the articles may have hyperlinks to related info or at the very least I can google stuff I don't understand.

I usually create a mindmap summarizing the ideas and how they connect. I find that helps me see the big picture and connections better. I used to draw my mindmaps by hand with different colored pens and now sometimes use MindMapper to create them.

A summary page of definitions and theorems is something I used to create when studying for an exam on a book and I find it still helps me to get a better memory on a subject. I have used index cards to do this too - that way I can sort them different ways - or if I put the name of the term or theorem on one side of the card and the definition or statement of theorem on the back - then I can use them as flash cards to see if I remember what each of them is. I haven't extend this but I guess you could also list the key steps in the proof too.

Addressing the LaTex vs handwriting issue personally I find the time to write easy to read handwriting is much less than the time to typeset. It also depends is this WORM or WMRN notes (Write Once Read Many or Write Many times Read Never) to corrupt some computer terms for storage. However if you find that you don't want to spend the extra time writing to be readable and that hard to read notes are getting in your way then go for typesetting them in LaTeX. I guess this also would be a benefit if you want to improve your LaTeX skills for other reasons.

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Thanks for replying! –  arzela Aug 5 '12 at 18:46

Disclaimer: I realise that this is an old thread but I'm adding my thoughts should someone stumble here from Google (as I did)

I think the issue is not how you take your notes but what notes you are taking. The best approach I find is the so called Feynman technique. Essentially the technique involves writing your notes as a summary/small book/lecture notes for someone else. This forces you to think about the connections between ideas, and to make suitable analogies which help you to remember the material.

Better still, because what you produce is in essence a set of brief lecture notes, when you come back to them in 12 months/12 years time they should be ideal for reminding you of what you studied originally.

As a final comment, I would add that it doesn't matter if you never read your notes again - I never do (I hate my handwriting too), but putting the ideas together in a way that I could teach them to someone else is usually enough.

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