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This is a question specifically about mathematics.

All math people I've talked with like discussing problems (or math in general) with others, and most people claimed that they learned most from communication rather than from reading textbooks. My situation is, however, different. Most of the time I have math-related conversations, I find that I'm not comprehending what I'm being said, even if it's something not hard (unless I'm familiar with the topic being discussed very well). [In particular, this is true of lectures. However, I am able to understand my notes.] (I'm not sure what the reason is.) In contrast, I learn much more from books and, more importantly, from posting questions on math stackexchange. I was wondering whether this is normal, and whether this will be an obstacle in writing a potential PhD dissertation, since in that case my adviser would probably be the primary source of information, and I wouldn't be able to write a math stack exchange post asking for help.

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  • Everyone has a different way to learn. There is nothing wrong with you, but there is something wrong with an educational system that over relies on lectures. Lectures work for some, but not all. If you have found an effective way to learn, then use that and forget (for the most part) the other. I recommend the answer of Allure, here, especially looking at the learning model suggested.
    – Buffy
    Mar 7, 2019 at 11:26
  • And, in fact, the "problem" discussed goes beyond mathematics. It is a feature of human differences - learning modalities.
    – Buffy
    Mar 7, 2019 at 11:27
  • @Buffy to describe our education system as simply lectures is a bit unfair, as it implies only passive listening by students. When it comes to mathematics, the tradition is to 1. hear the instructor explain, 2. see them work the problems, 3. write it down for yourself, 4. get prompted for questions if you failed to follow the derivation or proof. This may not be the best teaching method but it isn't chopped liver. Many do certainly execute poorly, or unwisely try to use slides in math classes in recent years.
    – A Simple Algorithm
    Mar 7, 2019 at 15:34
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    I'm actually a mathematician @ASimpleAlgorithm, I didn't say "simply lectures", but over-reliance. But the four points you make are still incomplete. People learn from practice and reinforcement. Only point 3 really starts to achieve that.
    – Buffy
    Mar 7, 2019 at 15:39
  • @Buffy I don't quite understand what you object to about my statement. Are you "simply" calling those four steps a "lecture" or not? If so that's unfair. If you're only referring to one step and skipping the others that's unfair too. As for the segue into broader criticisms of teaching methods, I was only arguing for fairly describing the classroom portion. Students are also supposed to work homework problems and generally study for tests on their own, and get feedback via those. In my opinion the only real flaw is the over-reliance on student self-motivation.
    – A Simple Algorithm
    Mar 7, 2019 at 15:59

4 Answers 4

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I suspect I have similar issues as you: when others talk about complicated subjects, I find they're speaking in English, but they're also not speaking in English (because I don't understand them).

I've thought about this a bit and although I am by no means an expert, I think the cause of the issue can be understood with the VAK learning style model. (Note that this model has been heavily criticized; however my perspective is whether it's correct or not is unimportant if it works for your particular case). Basically, people can be very broadly divided into three groups: those that learn best by doing something, those that learn best by listening, and those that learn best by reading. You are good at learning by reading, but poor at learning by listening. If you go to a lecture where the lecturer does not prepare lecture slides, only stands there and talks, you find you struggle. Comparatively if you do get lecture slides, then everything makes sense and you learn smoothly. If you can have only the lecture slide or the talk, you would much prefer the slides.

The bad news is, being bad at learning by listening is of course detrimental. It's just so much more convenient to explain something instead of get out pen and paper. The good news is, if you know what the issue is, you'll able to compensate for it. You'll need to concentrate on these conversations because if you drift, you won't be able to recover easily. Don't be afraid to say "can you repeat that?", and don't be afraid to ask for clarification if something is not immediately obvious to you either. You can also try bringing around a notebook and immediately writing down something you hear, since you are liable to forget quickly otherwise.

If you are so bad at learning by listening that face-to-face meetings with your supervisor are unproductive, try telling him that and suggesting email instead.

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Understanding mathematics is, in general, hard.

For me, some methods of having reliably fruitful mathematical discussions are:

  1. Have someone else explain a problem, an object, an operator, or a proof to you. Ask stupid questions. Have the other person explain. Do not accept anything before it has been clarified to you, unless it is common background knowledge that one should take on faith to get to the main point of discussion. The one explaining will likely benefit, as the discussion forces them to sharpen their understanding and exposes things that they have glossed over as obvious, or simply not noticed.
  2. Explain a specific problem you have to a colleague. Have them act as per point 1.
  3. Try to understand a given operator or object together. Draw pictures, give examples and counter-examples, calculate an easy example explicitly, come up with a physical/biological interpretation. Here, doing the process together reduces the amount of mistakes and thinking time. If one of the people discussing is faster than the other, than this might turn into point 1 or 2.

In general, I would suggest asking more questions, even and especially if they seem simple.

I would also suggest reflecting on what the problem is.

  • Can you follow up to a point, but then drop? If so, ask more questions.
  • Do you think the others are following or are they simply nodding along, thinking that everyone else is following and afraid of disturbing the process? Maybe ask them after a discussion with 3+ people.
  • Are you incapable of concentrating on the mathematics and thus drop from the discussion? If so, I do not have contructive advice. It might or might not be something that can be diagnozed.
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I have trouble understanding something said out loud because I need time to process the information. I usually have to have someone write it down. It is usually possible to request that or to bring paper and pen with you. I also learn much better when I have a carefully written textbook, but these become rare after a certain point. I think the best thing to do in this case is to ask for references (if a speaker gives a lot of expository information, ask for where to read about it).

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I got an applied math PhD 'without' discussions. Like you, I much prefered reading papers and textbook to lectures, discussions, working together etc. I would skip class to read course material. I later found out that most people visualize math expressions in their mind's eye (I can't), which seems like a key enabler of discussions. When people 'talk equations' I lose interest pretty quickly. There are a lot of talks which one is supposed to attend. It was meaningless to me. Bring pen and paper so you can work on something else. I did not benefit much from the advisor-student relationship, which is a pity. If you can, try to find an advisor that will read what you write and give feedback on that. On the plus side, reading a lot helps you hone your writing skills. There are many routes to the PhD.

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