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I'm currently a second year Ph.D. student studying pure math. I've recently come to the conclusion that I must be studying wrong. Actually, more to the point, I must be thinking about mathematics wrong. It's not that I'm failing, quite to the contrary, so far I have been very successful in the program, the problem is that I don't feel like I'm learning and absorbing the material the way I should be.

I want to become comfortable friends with the objects of mathematical study. I want to know their quirks and behaviour. I want to feel like I am part of the mathematics and not just an outside observer. But so far my experience is totally the opposite. I have become a competent symbol and logic manipulator but this has made my recent experience with mathematics cold and detached. Even though I can solve difficult problems, complete problem sets, and pass qualifying exams my knowledge is disjoint and compartmentalized, and my intuition is surface level.

At my school it is mandatory that we take three classes so having three problem sets due (or more commonly 2 will overlap at all times) from the three classes creates a constant stream of pressure to solve problems.

I'm curious to know if I am alone in feeling this way. For those of you who have been through graduate school (or anyone with relevant advice): Did you feel like you were getting a deep understanding of the material? Ultimately the question is this:

Does anyone have advice for how to deeply understand a new mathematical object efficiently?

Specifically how to become comfortable with a new object (definition, theorem, "chunk" of a theory) efficiently, be able to recognize the object from different perspectives, and be able to see how it fits into a larger picture.

As an example stolen from Thurston's "On Proof and Progress" we can view the derivative in several different ways

  1. As a limit
  2. Slope of the tangent line
  3. Linear approximation
  4. What the function looks like around a point under increasing magnification
  5. etc

Thank You!

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Yes, this is similar to my experience. Problem sets take long enough that it eats away time that could have been spent reflecting and absorbing. It seems the paradigm is either meant for a different kind of learner or possibly its just broken –  mike vaiana Nov 13 '13 at 16:20
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I don't mean to be glib but: "Young man, in mathematics you don't understand things. You just get used to them." John von Neumann. –  Shaun Nov 13 '13 at 16:24
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@shaun an interesting quote, but then the question becomes, how long does it take to get used to something, is there an efficient way to approach this? For example, I could get used to the layout of a new campus by going on several scavanger hunts to the more hidden places of campus, but it might take me a while to really understand the layout of the campus because I'm focused so intensely on the hunt (read solving problems) where as another approach would be to look at a map or just walk around and note the large picture layout and then fill in the details after. Which is more efficient? –  mike vaiana Nov 13 '13 at 16:30
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@Shaun Oh right, now I understand the point of the quote, and thank you, it is somewhat of a comfort to be in good company in that regard! –  mike vaiana Nov 13 '13 at 16:41
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One way to get around problem sets is to do reading courses. At my grad program, I think you are required to take some "official" courses, but not many. I did a reading course this semester and I got a lot out of it. You can still attend lectures by auditing. That way you'll have the benefit of being a student in the course, but you won't feel obligated to do homework. I've started to approach lectures as I approach books - get from them what you want but feel free to wander. This is much easier if you're auditing. I can't say that my professors would agree with me though. –  Dylon Chow Dec 10 '13 at 0:10
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You really want to give your world of math another dimension where math sheds a light on you differently? Go into teaching. Rather than aiming for your students to become proficient at your level, you need to go down to their level and together meet the goals of the course curriculum. It's that aim to think at your students' math level to go through the material while you teach, that puts the math knowledge you aquired through the years in a different perspective. You will be so surprised how much you learn your own subject when teaching it to others.

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I fully agree that teaching helps one learn the material in a completely different and often more thorough way. But first, I don't have an audience for such a thing, and second I think one has to have an understanding of the material at least 1 level deeper than a textbook before one can teach the material at all (otherwise one would just be orating the textbook) My question is really about how to get to that level, 1 level deeper, and then from there how do you go deeper than that. –  mike vaiana Nov 13 '13 at 16:23
    
You don't need to go into teaching at the level where you are with your current studies. You could teach an undergrad level course, say a calculus course. Eventhough we both are done with that in our own studies, I can reassure you that after having taught calculus for a decade or so, I am still learning undergrad material, through the experience of teaching (and teaching it differently as a "try out"), in fact, I am still learning undergrad stuff from this website. We have Master's but are we honestly fully competent in each and every undergrad field of math? –  imranfat Nov 13 '13 at 16:29
    
I understand your point. And in this case I fully agree with you. I suppose I am not so interested in getting a better understanding of calculus as I feel that I understand that particular subject well (not that I fully understand it) I was more interested in a way to understand $new$ material that I haven't encountered before. –  mike vaiana Nov 13 '13 at 16:36
    
I was referring to your reference of Thurston's which are calculus based. But as far as new material goes, how well do you do in self study of new material? Because I think that's where your challenge lies... –  imranfat Nov 13 '13 at 16:59
    
I actually prefer self study, it gives me the opportunity to read, think, re-read, question, come up with examples and finally attempt to solve problems. I guess part of this whole post is that I'm frustrated that I don't have time to do this because I am constantly being forced to turn in problem sets which eat up a large chunk of every week. –  mike vaiana Nov 13 '13 at 17:59
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