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
- As a limit
- Slope of the tangent line
- Linear approximation
- What the function looks like around a point under increasing magnification