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I am a rising college junior in US with a major in mathematics. I recently noticed a problem in my note-taking skill in the mathematics, both from the textbooks and lectures. When I was a microbiology major, I wrote extensive amount of notes from my books and lectures since I had to memorize all of them. I followed this trend and took extensive notes from my math books and professors' lectures. The problem with my note-taking skill is that I basically copy down almost everything from my books. For example, I am currently studying Apostol's Mathematical Analysis; what I am doing is that I copy down the theorems, proofs (including my footnotes), authors's remarks & motivation, and examples. I tried to copy them on my own words, but the result is that I basically copy down verbatim with little change in the grammars. I started to realized that I should not take such exhaustive notes since the textbooks contain exact materials from my notebooks, but I cannot overcome the feeling that I need to make my own version of books, by taking notes from them, and the fear of memory loss. Also I tried to take as much notes as possible from the professors's lectures. What I realized is that I copy down the exact words from my professors, and I often lost the concentration to focus on lectures and absorb the materials. I also lost my concentration whenever I take the notes from my books since I need to divert my attention from books to notebooks.

I am currently thinking about various alternative strategies, which are followings:

1) Taking notes within the textbook using the blank space of books and the Post-It: Take notes about interesting remarks, confusion about the exposition and examples, and my own interesting ideas or approach to the proofs and examples.

2) Taking notes on a separate notebook but only copying the information from 1).

3) Do not take any notes and try to absorb the materials from the books: I noticed that I usually learn the best by reading. Use the lecture to supplement the textbook reading and to gain different perspective.

Could you help me out by commenting about my strategies and/or share your note-taking skills? I am quite embarrassed about my note-taking problem.

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As you have noticed, mathematical text is often quite concise and it can be difficult to squeeze it into tighter space or write in your own words in a nontrivial way. Therefore it is easy to end up copying things verbatim if you want to take notes. Taking useful notes in mathematics is quite different from many other subjects, and you are not alone with your problem. You should not be embarrassed; you recognized a problem and want to do something about it and far too few students actively do this. I have some suggestions for taking notes in a more useful way. (This is, of course, based on personal experience and opinion.)

Reading a book and following lectures are two different ball games. Let me start with the lectures. It often happens that lectures move on so fast that if you want to copy everything that happens on the blackboard, it can be difficult to follow the content and — most importantly — the ideas. In these days lectures should not be the sole means of delivering the content; the professor can scan his notebook and put the pdf file online for the students to download. If he does not do this, request that he does. Mindless copying is pointless and wastes a lot of time. (Arguing this to some professors can be a difficult task, but it should be tried.) If the lectures are based on slides, ask for them and bring them with you to the lectures, printed on paper. If no such material is available, try to read the textbook or other such material in advance, so that you will have the material in mental form.

I have found it very useful to have the lecturer's notes he is using (or very close to it) during the lectures. This does not mean that I sit down idly. I follow the lecture and try to read the material in advance (at least quickly). There are always some tricky points that are not perfectly explained in the written notes. Often the professor says helpful remarks and answers questions from the audience, explaining what is happening or why certain choices were made. It is those gems that I try to catch and write them among the lecture notes. The notes can be informal and do not attempt to make rigorous mathematics. Instead, I try to capture my momentary intuition in words and explain things to myself or relate them to other things. The kinds of notes I take depend heavily on my proficiency in the subject and the notes can be of any kind. (Sometimes I just keep reminding myself about the nature of each object by comments: this is a function, this is a vector, this is a set of sequences…) The notes are rarely so lucid and well explained that no notes are needed, but this can happen, too. Most of the time I just follow the professor's train of thoughts and try to understand why everything happens as it does. Only a small portion of time is spent on making notes. If I have to take notes to copy the content to my notebook (this is sometimes the only way to get it, unfortunately), I have much harder time understanding.

If you are reading a book, the situation is different, as there is no hurry. Making supplementary notes works here like during the lectures. If you don't want to damage a book with your own writings, I would suggest post-it notes — I at least give great value to having notes right next to the content they comment. My notes are mostly comments. I find copying theorems and proofs verbatim quite useless. I would suggest reading the theorem and its proof in detail and then trying to figure out what it is about, what it means. Make sure you understand the overall plan of the proof, not just the technical details. If you want to take notes, put the book aside and write the theorem. If you understood the statement and the underlying machinery well, you should be able to write the statement. Then write a proof, at least a sketchy one. If you get stuck, look at the book; it doesn't matter if you have to look every three lines, as long as you really try to do it yourself first. Now you have notes about the theorem, and you have written them down in your own words and as your own thoughts. It might be almost exactly the same proof as in the book, and that's ok. It came from someone else, but it's your proof now.

If I have to choose one of your strategies, I would go with 3. Do what feels comfortable and natural to yourself, but make sure there is a point to the notes you are taking.

There is one more thing that I must add. Doing exercises is very important in learning mathematics. (I don't know about microbiology.) Do them, and if there are none, make some up and do them. I think the most important notes I have taken in my years of studying mathematics and theoretical physics are solutions to exercises. Exercises give you hand-on experience of the power of your new theorems and make you comfortable with the machinery and language of the theory you are studying.

Something of a summary: Take notes only when there is your own thought in them. If you find yourself copying without thinking, there should be another path to the bulk of content. Use notes to explain things to yourself, not to build all of the theory.

Addendum: Let me elaborate on making up your own exercises. (I did this in the comments, but it deserves to be in the actual answer.) If you are curious enough, you don't need to produce problems artificially. Try to dig a little deeper than your course. How are different things related? For example, is the sum of injective functions injective, or is a connected set always measurable, or is every square root of a continuous operator continuous? The questions can be anything, and they can be silly. Is a smooth function a Banach space, or is a finite group measurable? Learning what things are completely unrelated and what things don't even mean anything is useful. I find this kind of mental gymnastics entertaining and it gives me a better touch to the subject.

If no questions occur, another good source of questions are the main results of your material. Can you write a list of them without looking? Can you prove the things on your list? Can you define all relevant objects? Do the theorems remain true if you assume a little less? Why not, why yes, or why is it hard to see with the tools you have?

In my opinion, informal exercises are important. The problems in exercise sets in books, courses and exams tend to be quite formal, but that doesn't mean that only formal mathematics is good mathematics. Play with your new tools and you'll learn to work with them more efficiently.

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  • $\begingroup$ I'd add trying to find alternative proofs (or parts of the proof). See if you can extend the result (i.e., it is proved for positive $x$, what about negative ones? Zero?). Poking at the limits helps in understanding why they are there. $\endgroup$ – vonbrand Aug 1 '15 at 0:38
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    $\begingroup$ @MathWanderer, your course of action sounds like a good one. For me little exercises come up naturally when I think about the different objects and what they could do. Are there relations between different properties and operations? A possible question of idle curiosity could ask if the sum of invertible matrices is invertible. You can find counterexamples, but there also seem to be cases when you can guarantee invertibility of the sum... Ask yourself questions, however stupid, and answer them. Try to see structures and places where there is no structure. Dig deeper. $\endgroup$ – Joonas Ilmavirta Aug 1 '15 at 10:09
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    $\begingroup$ @MathWanderer: Questions should stem from curiosity. If you learn a new thing, some questions will surely arise. Ask them. If the teacher or the book can't answer, answer yourself! If you can get old exam questions, solve them. They are often slightly different than exercises during the course. After a while of studying the subject, sit back and think what are the most important objects and theorems about them. Make a big exercise: Define the key objects, write down the key theorems and prove them. Make an overview. I often do this mentally, but writing down is not a bad idea either. $\endgroup$ – Joonas Ilmavirta Aug 1 '15 at 10:16
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    $\begingroup$ @MathWanderer, I added a note at the end of my answer, because I felt that making your own exercises is an important thing and shouldn't be hidden in the comments. $\endgroup$ – Joonas Ilmavirta Aug 5 '15 at 20:24
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    $\begingroup$ concerning: "In these days lectures should not be the sole means of delivering the content; the professor can scan his notebook and put the pdf file online for the students to download. If he does not do this, request that he does." I would advise caution on this. There are certainly professors (not me) who would be offended by this. For some, the expectation is that it is your job to take notes and the expectation that they do it for you is lazy. Of course, you are not being lazy, but, still, beware this action has consequences in certain cases. $\endgroup$ – James S. Cook Aug 6 '15 at 3:24
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What works for me may not work for you. But here goes with what worked well for me:

  1. I used the textbook as my notes. It had an index and was better written (and prettier!) than my handwriting. Where the textbook was too dense or omitted steps in proofs that I needed to make explicit, I added notes in the margins. This sounds like your option #1.

  2. What's more important than "how to take notes" is this: Read the text. Read everything from titles, to captions and footnotes, to the example problems. Don't let ONE WORD get past you. If there is ANYTHNG you don't understand, keep at it until you do understand it or ask for help. This sounds a little like your option #3. And it's not necessarily mutually exclusive with your option #1.

  3. What's more important than ABSORBING the text is working the problems. Even if you completely understand everything in the text, you need the practice--the fluency--that working the problems gives you. In college, I spent about 4 hrs a night 3 nights a week doing calculus homework. But it paid off.

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  • $\begingroup$ Don't just do the exercises, work over the derivations, the proofs, and the examples. $\endgroup$ – vonbrand Jan 22 '16 at 15:08
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For many courses, note taking is concerned with the organization of data for the purposes of recall. Let's call such organized data "information". In mathematics, the purpose of writing is more, borrowing the language of John Dewey, to marshal resources toward a purpose more specific than recall. Let's call information aimed at a specific purpose "knowledge". Of course, one person's knowledge is another's information. Knowledge is subjective to the kinds of problems or actions an individual is motivated to solve or do (again taking from Dewey, a purpose forms when someone plans their way around an obstacle to satisfying an initial impulse). You can see this in the many mathematical blog posts and textbook treatments of the same basic proofs in mathematics. Writing about these things allows the author to arrange certain facts for ready use for the problems the author is interested in.

To risk sounding overly male or violent, this perspective of knowledge as "weaponized information" seems to be a good motivation for mathematical writing. You should have a purpose for your note taking…some problems you (or perhaps the community) want to see solved, or ideas that you want to have a use for, and that the information collected marshals facts aiming to the solution to these problems, or the use of these ideas. The necessary selectivity for good note taking comes from having such purposes, and a sure sign that you don't have them is that you feel like you are copying everything down. (I speak from personal experience!) The thing to do under these circumstances is to ask your own questions in response to the text, and to the instructor, to stimulate purpose. A question is such a seed of an impulse that, since you cannot immediately see an answer, yields a purpose as understood above. Just about any question you genuinely ask will help with this. A good many of the concrete suggestions in the above answer of Joonas will help, as well.

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  • $\begingroup$ While I agree that it can be good to learn math and take notes for a specific purpose, I find assessing usefulness somewhat counterproductive. Reasons: (1) Assessment takes time away from focusing on content. (2) One can learn math just for the fun of it, with no aim to solve specific problems. (3) Interesting applications can call for unexpected theory. The student is not always mature enough to filter information in a way that eventually serves them. [That said, you have a good point. I just wouldn't take it as a main principle when studying.] $\endgroup$ – Joonas Ilmavirta Jan 25 '16 at 8:42
  • $\begingroup$ @Joonas Ilmavirta: It is not necessary that the purposes I refer to have to be specific mathematical problems. Actually, your answer is a more detailed version of mine. You have to do something with the notes you are taking, or why are you taking notes? Why not just keep the textbook? Why reorganize the information? Thanks for the comment, though! I think it may not have been clear that I was not referring to specific mathematical problems. $\endgroup$ – Jon Bannon Jan 25 '16 at 11:47
  • $\begingroup$ I've made some edits to my answer's language to clarify things a bit. $\endgroup$ – Jon Bannon Jan 25 '16 at 12:40
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    $\begingroup$ Thanks for the elaboration! My comment came out more harsh than intended, but it's good if it triggered a small added explanation. (For the record, the answer is good and earned my immediate upvote.) $\endgroup$ – Joonas Ilmavirta Jan 25 '16 at 14:12
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    $\begingroup$ @Joonas Ilmaravirta: I think your (3) above is important to address, so I will, at risk of lengthening this thread a bit too much. While I agree that interesting applications can call for an unexpected theory, one can use up a LOT of energy and time collecting facts with the risk of never gaining any deep proficiency or the ability to evaluate when to apply an idea. Somehow the point of mathematical training should be to develop the ability to filter information as above. If the study purpose is to "look smart for friends", that's fine, but we owe it to students to state community expectations $\endgroup$ – Jon Bannon Jan 26 '16 at 15:03
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When in lectures, as mentioned above, if you have access to the slides or whatever, that is great! I'd suggest bringing a tape recorder with you, and then when you take notes, either take them on the slides itself, or in your notes write

Slide 1: lsdkfjslkfjslkfjsldkfjsljfsdlkfs

Slide 2: sdlkfjsldfkjslfkj

and only take what you feel at the time is relevant. If you feel you missed something later, you have the voice recording (as you get better at taking notes, and you feel more confident, you can skip the voice recording)

You can do the same with your own thoughts when reading a book, and take the page numbers, or since you mentioned that you learn best by reading, read with a highlighter as it may help keep you focused.

Creating your own problems can definitely be useful - if you find it hard to do that first, then google whatever type of problems you are struggling with and find something with solutions that you can check after. There are a ton of such problems up on the internet from various courses.

As a side note: Most universities have some sort of learning center, study skills center to help students learn various study skills. My suggestion is that you go and take a course on note taking, but also on one that could target your learning style. Although you seem to know your learning style, such courses also usually give you tips on how to best learn, and study etc, keeping your learning style in mind.

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    $\begingroup$ As a side-note: It is probably best to ask the instructor beforehand if it is okay to use a recorder (video or audio) in class. Somewhat related is ASE 48940. $\endgroup$ – Benjamin Dickman Aug 5 '15 at 23:23

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