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Often I find myself just taking notes during lectures and not really following what is said. This has always been an issue for me, I do not seem to learn anything in the classroom. Instead the learning process starts right after when going through the notes.

On the other hand, I do somewhat feel more involved when attending to seminars (I have not attended to many), even though the subject is more advanced and sometimes way above my level.

I guess this is kind of a vague question, but are there any good techniques on how to become a good listener (the question can be split in two: one where you need to take complete notes, such as during a class, and the other case, e.g. seminars)?

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Something that works on both occasions: Sit in the fron row. The closer you are to the speaker, the more natural it is to pay attention. –  Michael Greinecker Feb 20 '10 at 18:50
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See mathoverflow.net/questions/12638/… You may find some of the suggestions there, specially Anton's on live-TeXing, useful. –  Sonia Balagopalan Feb 20 '10 at 18:57
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Perhaps the question could be rephrased as: how does one get the most out of lectures? That seems like a slightly more focused version, and one to which more people might contribute answers –  Yemon Choi Feb 20 '10 at 19:23
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In my opinion any question tagged "soft-question" should be community wiki. –  Grétar Amazeen Feb 20 '10 at 19:39
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10 Answers 10

The best advice I ever received was from Jordan Ellenberg, I hope he doesn't mind my rephrasing of it here. When sitting in a seminar, try to figure out an interesting question to ask the speaker. This will ensure that you listen to enough of the talk that you learn something interesting (it doesn't have to be the main part of the talk, but anything you enjoy in the talk), and spend time focusing your thoughts around the subject.

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Ha, I do indeed consciously do this but forgot that I revealed that I consciously do this! –  JSE Feb 21 '10 at 3:38
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At the Midwest Number Theory Conference for Graduate Students a few years ago, you told us you do this (especially when you run a seminar). You also told us it is a good thing to do in a talk which is particularly bad so that someone has a question to ask the speaker at the end and make them feel welcome. –  Ben Weiss Feb 21 '10 at 4:34
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Vaughan Jones says that he got in the habit of doing this when running the Berkeley colloquium, in part to make sure that the speaker always got at least one question, and that it really helps you pay attention is you know you have to ask something. –  Scott Morrison Feb 21 '10 at 14:53
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@JSE: Whereas I think that I unconsciously do this. Exception: if I am either well and truly lost through no fault of the speaker, or if I don't like the talk, I generally hold my tongue. –  Pete L. Clark Feb 21 '10 at 18:29
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You could also ask yourself if do not have another type of benefit from lectures. For me it is often not the case that I understand much during a lecture, but just the fact that I see someone who knows the subject handling it live before my eyes changes a lot:

To see what parts of an unknown matter the speaker emphasizes or labels as deep, and what is standard facts, or useful knowledge imported from other areas (and thus not in the focus of the story being told) makes a big difference for me when trying to understand the stuff later. When I sit down to study, I then have the feeling that I am following a road which will lead somewhere, which is often not the case when I just have big text in front of me. Having made the experience that someone could give an outline within, say, one hour also just makes it less intimidating. Also, even if after a talk I could not repeat a single line of what the speaker was saying, I often find myself surprised by being able to come up with details from that talk a year later - I think just the fact that the mathematics comes with a voice, gestures, a handwriting and all other kinds of perceptions makes it stick better in my long term memory and thus gives it a better chance to fall into place later.

If taking notes distracts you from this kind of benefit, you might consider just copying someone else's notes, and paying attention to the whole situation during a talk...

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+1. That's exactly what I love about lectures, in particular those where the lecturer makes a lot of throwaway remarks, which start making perfect sense and give me perspective when I go home and read a text. –  Sonia Balagopalan Feb 20 '10 at 20:31
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  • Prepare in advance.

You will probably get the most out of classes if you read the text ahead of time. This varies by the lecturer's style, but try at least skimming the material (or notes from the last class if there is no text). You don't have to understand 100% of the material before the lecture. Try to identify material that you don't know, and pay special attention to that during class. If you still don't understand, then try to ask at least one question during class, and if that doesn't satisfy you, ask the instructor after class. When you do understand the text, look for differences between the text and the instructor's presentation, both in material and emphasis. If the material is easy, ask yourself how you would present it if you were teaching the class.

When you attend a research lecture, try to do your homework ahead of time, too! Try to find an expository article, or read a few reviews and abstracts of papers in related areas so you know what people in that field find interesting, what is hard, what the key examples are, what techniques seem effective, and what the connections are with other areas. The first 5-15 minutes of the talk may be similar, and they are critical. Understanding the details of a technical talk does little good if you do not know the context.

  • Listen actively.

Keep a few examples in mind. How do the results compare with the basic examples? How much progress is there toward the examples people want to understand? What are the differences between the later examples and basic examples?

Try to understand where you are on the road map for the talk/course/the book which will eventually be written about the theory being developed.

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You might be interested in reading the advice of Ravi Vakil -- see in particular the section "On seminars".

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This is perhaps not a particularly useful or practical suggestion, but here goes: Ask questions. Interrupt the lecturer every time you get lost. Of course, if you really do this, you'll get a lot more out of the lecture, but the lecturer, as well as at least some of the others in the audience, might get very annoyed.

Or try to attend lectures or seminars where there is someone else who will do the above.

I personally find any passive approach to learning mathematics (i.e., listening to a lecture or reading a paper or book) very difficult at best. I learn only when I get to interact with the other person or work stuff out on paper myself.

So for a class, I used to concentrate on taking notes and would study them later. I didn't worry too much about understanding everything during class. For seminars, I also don't worry too much about understanding things too precisely at a lecture; I just try to understand enough to decide whether I want to devote some effort later to learn about the topic more properly. Unless it's an exceptionally well presented seminar talk, I tend not to take notes and look for other written materials later, if I decide I want to learn more.

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As an instructor the one thing that annoys me is to have students not ask questions when they get lost, thereby letting me talk on and on when that effort is effectively useless. Of course, there is such a thing as "too many questions", but I've seen it get to that point extraordinarily rarely. –  Mariano Suárez-Alvarez Feb 21 '10 at 15:26
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Listening is a very broad topic. I will focus mainly on listening to a math talk with the hope of extracting something useful from it.

I find taking notes during a talk helps. I simply write some key words. If the speaker emphasizes a point, I write it down, even if it does not make sense at the time. I might later google some of the key terms. So on...

People who don't write fast enough, or who are new to attending math talks, might find useful a digital recorder running on a timer. Jot down the time when an important point or example was made and some of the ideas surrounding it, maybe also the notation; then go back later if needed. You may want to save both the recorded talk, and via a digital camera in text mode, digitize your notes and place them in a folder adding some key words easily searched later if needed.

As Ravi puts it in these Notes (Thanks Kevin Lin for the link), one is never sure where new ideas are coming from, or where the seed of a solution to a problem one has been working on is going to come from. Sometimes, a new idea may come from listening to a talk on something seemingly disjointed from one's work.

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Actually, people into digitizing notes might find the following technology interesting. en.wikipedia.org/wiki/Livescribe –  S Lir Feb 22 '10 at 0:20
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I have the same problem when I am listening to something challenging(i.e. not the undergraduate math club). For me, one of the most helpful things I can do is that once a definition is given, I construct both examples and constructions that almost fit the definition but do not fit one of the criteria. I then check how these work(or fail) with the theorems presented and the claims made. If you find a simple example and are having trouble following, the person lecturing will probably be happy explaining how the concepts presented relate to your example.

Another idea that helps is for me to read all the material beforehand, no matter how confused I am. Then I can pay attention to how it is structured in class. Even if I do not understand the lecture, contrasting the structures of the reading and of the lecture can give me deeper understanding of the material. In addition, since I (like to think I ) understand how to structure a lecture, it gives me something I can understand to pay attention to.

Lastly, I think it is important to be forgiving of oneself. Think of paying attention to a lecture like being focused when meditating. If you find your attention wandering, just refocus on the matter at hand. Try not to get caught up in a cycle of being frustrated that your attention wandered.

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This is how I take notes in lectures:

  • First listen to the lecturer
  • Either when a fair amount of things have been said and you still haven't figured it out; or when you got the point already, then start writing things down. In the first case it is safest to copy down everything, while in the second case it is better to write it down your own way (presuming you have a book to check back later for details).
  • Repeat the cycle.

Most of the time, I leave the lecture hall feeling like I get the general idea, which is helpful when I review the material later on. However, I think it really depends a lot on your writing speed: I write pretty fast, and stopping to take notes doesn't cause a problem. If I wrote only with half that speed, I'd probably use a different strategy.

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There is a previous discussion here on taking notes on lectures. In it some of the issues you raise are discussed namely that note taking can interfere with concentration on a lecture.

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