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Do you find it a good idea to take lecture notes (even detailed lecture notes) in mathematical lectures?

Related question: Digital Pen for Math: Your Experiences?

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Community wiki, please! –  Anweshi Jan 22 '10 at 15:11
    
Done! This question was promised here: gilkalai.wordpress.com/2008/07/15/arrows-economics-1 –  Gil Kalai Jan 22 '10 at 15:32
    
Many interesting answers. Of course, this is a very individual matter without a correct answer. Anton's answer was especially surprising and useful. –  Gil Kalai Apr 27 '10 at 4:49
    
I once read Thomas Korner's take on the subject of listening to mathematics lectures and found it helpful - there's a section on taking notes, as well. dpmms.cam.ac.uk/~twk/Lecture.pdf –  j.c. Nov 5 '10 at 19:04
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23 Answers 23

up vote 51 down vote accepted

I usually bring a pad of paper with me to talks, but don't take notes. I do write down things I want to revisit later, and sometimes it turns into full blown note-taking. Taking notes reduces my ability to concentrate on the lecture, so if it's a really difficult lecture or if the material is totally new to me, taking notes is a sure way to not get much out of the lecture (unless I commit a large chunk of time to reviewing the notes afterwards).

If I know I'm going to review the material carefully (e.g. if I'm taking a class and I'm putting a lot of effort into it), then I live-TeX notes, which has a number of advantages:

  • It's much faster to strike a key than it is to write a character by hand, so I TeX much better notes than I can take by hand. I find that I can even write down lots of things the speaker says but doesn't write.
  • I can throw the notes up on my web page, which makes other people in the same class happy.
  • I can grep (or otherwise search) my notes.
  • If I have the time, I can edit the notes and end up with an awesome reference (e.g. I put a lot of effort into editing my Lie groups and Stacks notes). More often, I review my notes but not edit them very much, in which case they're still pretty nice to have later on.

If you're considering live-TeXing, try it! It's a lot easier than you might think. Also, look at this page of advice I wrote up when I found myself repeating it to many people.

Edit: By the way, I post all my source, and you're welcome to look at it if you want to see how I set up my headers or manage my files. If I have an svn repository for the notes, you can get them with the command svn checkout svn://sheafy.net/courses/halg_sp2008 (to get my homological algebra notes from Peter Teichner's course for example). If there's no svn repository, you can download the tgz (tar+gzip) file and uncompress it.

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As one of the people Anton converted to live-Texing notes I can attest to its superiority! Before trying it I thought "No way! It's going to be too difficult to Tex it". But amazingly he's right. You end up with EXTRA time when you type it out. In my typed notes I always have lots of comments and jokes that the speaker said that I wouldn't have time for in written notes. For the first 3-4 lectures it was hard. Best to practice with talks/subjects you are familiar with. After that it became easier. If I know there are going to be lots of pics, I combine this with the digital camera approach. –  Chris Schommer-Pries Jan 22 '10 at 18:44
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I agree with this -- after a little bit of practice, live-Texing is just as fast or faster than conventional note-taking, and so much more useful afterwards. I think I got the idea from Bjorn Poonen in 2006 when I learned that he live-Texes most or all of the talks he attends. In retrospect, this is really funny to me: it must be the only time I have had the thought, "Well, if Bjorn Poonen can do it, then surely I can too!" I guess I have a high opinion of myself as a typist. –  Pete L. Clark Jan 22 '10 at 19:13
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Someday I want to live-wave a lecture. Any takers? Perhaps we need a better wave-->tex workflow first. –  Scott Morrison Jan 22 '10 at 20:00
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Do you actually type up commutative diagrams and things of that nature, or only really inline text and some easier displays? –  Harry Gindi Jan 23 '10 at 0:21
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I do commutative diagrams, and sometimes even spectral sequences. At the top of my game I could even live-tex a TikZ diagram! There are a couple of tricks which help. I copied one of Anton's preambles (you can get it from his many source files). This has lots of predefined commands which are useful to have, like \R for \mathbb{R} or \cC for \mathcal{C}. Have the whole alphabet ready to go. I have a Mac, and use TexShop, which is great since I can make macros. This way I can do commutative diagrams on the fly. I type ctrl-command-C and all the code for a com. square pops. –  Chris Schommer-Pries Jan 23 '10 at 2:50
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A couple of years ago I adopted a new custom. I try to write down just one thing from every talk I go to.

Everywhere I go I carry a tiny notebook, about the size of a matchbox. The one thing has to fit on one page. If I want to take more copious notes on ordinary paper I do, but I still write down one thing. I was amazed at the difference this made. It forces me to try to frame one idea from the talk in concise terms, and every now again I flick back through the notebook to remind me of old talks.

Of course it's shocking that it does make a difference, because you'd hope that on average you'd retain much more than one thing from an hour-long talk. But in my case, apparently not.

Depressingly, there are a few talks at which I can't even find one thing to write down.

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Wow. This is a great idea. I never would have thought to try this. –  Matt Jun 5 '10 at 16:50
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It's very interesting to me that most people find it hard to concentrate on a lecture and take notes at the same time. I write notes on paper in every lecture, and find it hard to concentrate my attention on the talk if I don't. So yes: I find it a good idea, but it seems this might be because I'm abnormal.

I'll bet live-TeXing would have the same focusing effect; maybe I'll try it. I've certainly had good luck typing plaintext notes while watching online lectures from the Newton Institute.

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Yes and no. I used to do so religiously, but often my notes would digress. I recall a talk on Kirby-Siebenmann in the early 1980s in which I got lost early and drew a vacuum cleaner. More recently, I took notes of selected speakers in a couple of conferences because I wanted to learn the stuff. When the speaker is really uninteresting, it is helpful to take notes if only to stay awake.

On the other hand, there is a mystical phenomenon that occurs when notes are not taken. To give an example, suppose that a speaker is talking about a general Lie group and algebra, and the listener is thinking explicitly about SU(2). The speaker says a generality, and the listener checks the specific case. This continues for a while and the speaker goes on one track while the listener thinks about an entirely different problem. Towards the end of the hour, if both have done their jobs well, speaker and listener come to the same conclusion. Sometimes having a pad of paper handy is good if only to do the calculation that the speaker says is trivial, and the listener can't fathom in the 30 seconds that pass.

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I wish taking notes would keep me awake. My notes include odd marks where I nod off. I have to burn myself with tea to stay awake. Good suggestion of keeping track of an example. –  Douglas Zare Jan 22 '10 at 16:04
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Hmm. Usually, when the lecturer is talking about some problem, I am thinking about a more general case... which in 99% of the cases turns out boring. –  darij grinberg Jan 22 '10 at 22:58
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Note taking is a bit of a religious dogma for me.

As a chemistry major,I was trained by Dr.Robert Engel something I've found to be very true as a student and has been confirmed by educational psychologists: "There's a connection between your hand and your brain." i.e. writing something out in detail forces your brain to process it.If the notes are good and informative,I find this is very true. Indeed,a good measure of how instructive lecture notes on a subject are is how well you learned from them by taking them down in detail!

That's why to be honest,I'm a little shocked by the responses here to the effect that note taking distracts them from a lecture. How can you be distracted from what the speaker in a graduate level mathematics colloquia is saying if you're forcing yourself to take notes on it?!? Yes,speakers in real time go quickly and of course,we're usually not completely alert and awake-but doesn't that force the mind to pay attention more?

I take very detailed notes along with my commentary. And later-I dissect the notes exhaustively. Or to use Paul Halmos' words;"Don't just read it,FIGHT IT!" And I do. Fiercely. I can think of no better advice to give your students.

There's also a more personal reason for taking notes in a detailed way:Each set of notes is a living record of an experience in your life. It's also a documentation of a personal style of a lecturer-each set of notes is like a personal fingerprint of the author. That relic will remain with the note taker long after the lecture ends.Your memories of the experience will be forever intertwined with those notes.

Now a lot of people here-such as Anton and Theo-are making the case for detailed TeX-ing of notes.Looking at their creations at thier websites as well as the notes made up by my other fellow graduate students,I have to admit-they're making a very compelling case for it. I just wonder about whether or not that personalized element so conducive to learning and nostalgia will be lost once everyone does this.

But once again-the results are VERY impressive. So it's certainly worth a good hard thought.

Those are my 2 cents on the issue.

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I think the benefit gained by having notes that you can instantly put online for others to use far outweighs any possible benefit of taking notes by hand. –  Qiaochu Yuan Apr 27 '10 at 21:30
    
@Qiaochu: I hope this doesn't sound too flippant, but if that is the case why not just get the speaker to put his or her slides up on the web? I am inclined, admittedly for emotional/personal reasons rather than rational(ized) ones, to share Andrew's feeling on this particular point. –  Yemon Choi Apr 27 '10 at 22:00
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Slides by themselves rarely contain enough of the substance of a lecture to really be useful to anyone not actually familar with the work,Yemon.And that actually goes for class notes as well-the "take" on a particular subject,even one that's well known to everyone like linear algebra,for example-is very subjective and each lecturer brings something unique to it. –  Andrew L Apr 28 '10 at 0:28
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I once heard Raoul Bott say that when he was a student he teamed up with a friend: one took notes and the other did not, and later they went over the notes together.

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Another way of taking notes in lectures:

alt text

This is taken from The Mathematical Gazette Vol. 25, No. 267 (Dec., 1941), p. 287 . Taken from "Memoirs of Archbishop Fredrick Temple, I".

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If I do take notes, I find it hard to concentrate on the lecture.

If I don't take notes, I don't remember anything from the lecture because I'm usually thinking about something else (still math-related. I don't slack off in my head!). So I find it's better to take notes than not, for me personally, because at least I'll end up with notes at the end of the class rather than nothing.

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A good compromise is to take extremely minimalist notes that condense the essential meaning of the lecture to a few short sentances. That forces you to process the information you're given down to a few short keywords or phrases that you can come back to later to process. –  Ryan Budney Jan 22 '10 at 17:28
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I think my notes for the first Peter Oszvath lecture I attended went roughly like this "symmetric products!" –  Ryan Budney Jan 22 '10 at 17:30
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I find it depends on a) Your familiarity with the subject and b) How fast/technical the lecture is.

If for instance I'm seeing the topic for the first time and the lecture isn't going at a breakneck pace, I'll take notes for the "Abe Lincoln effect" that J.Polak describes (apparently Lincoln even read his newspaper aloud to make sure the day's news stuck in his head).

If I've seen the topic before I'll generally take notes, but only sparse notes on particular parts of the lecture I think are interesting.

One way around all of this that I've seen is taking pictures of the blackboard. There is one professor at uga who brings a camera with the flash turned off and the shutter set to silent who asks the speaker before the lecture if he can take pictures so he can avoid getting lost in his notes.

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Pictures are a good way to go, and often better than video except for interesting questions from the audience. In fact, I have had students in my calculus classes take pictures of each blackboard which I posted on the web. This cuts down on the verbatim blackboard copying. –  Douglas Zare Jan 22 '10 at 19:01
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I agree on the Abe Lincoln effect. It helps me that I can hear the lecture, see what's written on the board or presented in slides, in addition to writing the notes (often I write my own version of what I think the lecturer is saying - so some processing is going on) and then reading my own notes as I write them. I think this helps me remember the lecture better. In fact, though I sometimes take nice notes, I hadly ever revisit my own notes, which I take as further evidence that the purpose of my note-taking is not reference but an aid to acquiring the information by involving many channels... –  Miguel Apr 27 '10 at 8:01
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I do not take many notes in colloquia, seminars, etc., although I always have a pad of paper handy to jot down ideas or to doodle. But for roughly one course per semester, I take detailed notes and post them online. Mostly I just leave the uneditted live-TeX notes, but when studying for my quals I found the process of editing course notes invaluable.

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Your notes are outstanding,Theo-great job and thanks for making them available for everyone. Especially since a lot of these are on topics there isn't really good sourcebooks for yet. –  Andrew L Apr 27 '10 at 21:05
    
I think it is a great idea to retype the notes after the course or lecture is over in order to review the material and sometimes ask further questions. I usually take quick notes by hand (I like having margins to try out my own ideas, tangents, etc) and then if I am really excited about the course I take the time to covert the notes to TeX which really helps reinforce and further explore the ideas. –  Artem Kaznatcheev Sep 1 '10 at 2:17
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I ususally do take notes in lectures and I think that for me it makes it easier to concentrate and to remember later what was in the lecture. (Perhaps a main reason for taking notes in lectures is that this was the tradition around here.) There were rare cases that in order to try following some details I had to stop writing. It is also rare that I go back to these notes but it did happen several times. I am better at writing lecture notes for blackboard lectures. Presentations are usually too quick. Also with a blackboard lecture where the lecturer write everything on the blackboard it feels only fair to take a similar effort.

I saw some people using computer tablets so that everything they write is stored. I do not know how convenient it is.

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I guess it depends on the type of lecture. Some lectures are way too fast and impressionistic to be able to take notes, and in those I seldom do. I may jot down an idea or two so that I don't forget. It's fair to say that I seldom learn anything substantial from such a lecture, although it may point me in the direction of resources from which I may later learn.

If the subject interests me and the lecture is not too fast, then I do take notes (on paper; although I would eventually like to change that) which I then scan to keep in my computer. Since no matter how slow the lecture is, it is impossible to take everything down, taking notes forces me to summarise the information being presented and this, I find, helps the cognitive process.

So, in summary, yes I think that in some lectures it is a good idea to take detailed lecture notes.

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Yes. It's essential for me to take notes or else I learn almost nothing unless it's something I've been thinking about for a while. I've discovered that my best learning technique is seeing symbols on paper. I often see something that many people find intuitively obvious but I don't feel sure about it until I do the computations myself.

There are two things I've found useful in lectures: a general overview or intuition from the professor's perspective on a topic and the professor going through the proof in detail. However, if I had to choose one I'd choose the latter, because I benefit more from seeing the syntax than hearing the intuition or meaning, and I can remember that part anyway. So when I do write down notes I pretty much just take down the syntax and listen to the prof when I'm writing.

Note that this doesn't apply to lectures for which I already know the topic well, but in this case I still have paper to do extra calculations, since the symbols on paper are inspiring.

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That's a very vague question and depends of course on the type of person who is answering.

Personally I always learn the most by writing something down. Sometimes all it takes is just a single line and everything's clear, but if I don't write this single line down, I just don't see it.

If the lecture itself is very dense and/or chaotic, it may be a good idea to take notes so you don't get lost in the flood of information / chaos. Of course in this case your notes should be in your own style. Just copying the blackboard doesn't work if the confusion comes from there. ;-) A Professor told me just yesterday "The best lectures are bad lectures": One learns most if it is necessary to check everything and think it through for yourself. (Of course this is very time-consuming.) Lecture notes are a great help for this task.

But as I said: That's all a matter of personality. Other persons need to hear something if they want to memorize it. For such persons it could be better not to take notes and concentrate on the lecturer instead.

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I still think that the very best lectures are good lectures. –  Gil Kalai Jan 22 '10 at 15:53
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Tom Körner's essay "In Praise of Lectures" gives lots of tips on how to get the most out of lectures, and can be found here: http://www.dpmms.cam.ac.uk/~twk/Lecture.pdf (The issue of note taking in touched up on page 5.)

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I find it useful to write notes during a lecture of key formulae that I don't follow or understand completely, so that I may refer to it in detail and in context later. Sometimes, I end up understanding the formula better 10 or 15 minutes later during the talk, and I can refer to the formula clearly while the lecturer has moved on to another slide.

As an amateur and a beginner, I also wrote notes by hand because my brain did not always comprehend or parse the equations and symbols correctly. Now that I am more familiar with the symbols and notational conventions, I find it possible to type LaTeX during a lecture, but I tend to avoid doing it in the smaller classrooms as the typing noises can be a distraction to others.

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I find it easier to concentrate on the talk if I take notes during the talk. Writing the notes also seems to embed the information into my brain in a stronger fashion than if I just listen to the talk. Since different, but regionally neighboring, parts of the brain are involved in auditory perception and in lexicographic perception and in the motor activity of moving the extremities for handwriting, perhaps the conjoint activity of all of these areas leads to reinforced learning.

I wonder if typing the notes live stimulates the regions of the brain in the same way as handwriting the notes does. Obviously, some things will be different, but what exactly? And is there a benefit to one over the other? My opinion is that either handwriting or typing notes during a talk leads to better recall and better comprehension.

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Taking detailed notes is terribly inefficient. It can still be the best option if there will be no chance to look up the details later, and if you are a rare person who can write rapidly while thinking about something else. If the slides or paper or lecture notes will be available later, then what's the point of detailed notes? Write down key ideas, or when something is surprising.

I'd appreciate it if everyone were to make lecture notes or at least slides available on the web after a talk. It does take a little more effort, and it makes it more obvious when you give the same talk at conference after conference. However, if you really want to communicate, you can tell more to people who aren't writing at the same time.

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Unfortunately, when it becomes clear what is surprising and what the key ideas are, it is usually too late to write down anything. –  darij grinberg Jan 22 '10 at 22:56
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Unless the talk is really important to me I don't take notes, I'm too slow and lost easily.

By the way, for live TeXing you might want to check out some tips here also

http://poisson.phc.unipi.it/~maggiolo/index.php/2008/12/how-to-take-lecture-notes-with-latex/#more-356

and perhaps even here

http://poisson.phc.unipi.it/~maggiolo/index.php/2008/12/latex-class-for-lecture-notes/

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I usually find it hard to concentrate on the lecture if I take extended notes, and my thoughts tend to wander if I don't - So I start the lecture by writing the general ideas explained: Theorems, corollaries etc, and try to summarize the ideas behind the given proofs, but then I realize that not all proofs can be summarized the way I hoped, and I usually end up writing almost everything.

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I use a variation of the method of Tom Leinster : I take a large size (A4) notebook with blank pages (no lines so that I can write big and draw pictures more easily) and take notes, but no more than one page per hour. It enables you to write down any non-standard definition, the most important references, and the main results (or a summary of them if they are too intricate) but forces you to extract the core of the talk.

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I've often found that by TeXing my notes, I spend much less time writing because I type a lot faster than I write. As a result, I'm able to pay much better attention to what the teacher is saying rather than spend my time trying to write it all down.

In general, I've found that if I attend a math class and don't write anything, I learn the material much better, since I'm actually spending the time in class absorbing the material rather than writing it down. The only reason that I do write it down is because it allows me to go back over what happened in class. Whereas when I don't write anything down, I may learn it fairly well the first time through, but that doesn't mean I'm able to learn it perfectly, hence I do have to go over it at some point.

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I have not tried it myself, but you might want to look into th Livescribe Echo Smartpen. You could take rather minimal notes with it, and they would be synched with an a complete audio recording of what was said.

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Muli Safra showed me some system with a pen that record the lecture and allow to synchrinize the recording with the notes (you need a special pad). This looks rather nice... –  Gil Kalai Sep 1 '10 at 8:17
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