I am a first year PhD student in math. I have a course right now that is pretty challenging as expected. I get high grades on all the homework. On our midterm the highest grade was a 35 percent. I was told this is how nearly all of the exams this professor gives turn out. Is this a common thing to happen? I suddenly feel inadequate to be a PhD student in math.
This depends very much on your university and your professor. There is a whole range of opportunities spanning a wide range of difficulties. This issue alone shouldn't be the reason to drop the idea of the PhD degree itself.
If you are really interested in the field and are willing to work for it, then by all means, continue; don't let a little test put you off. PhD is not about exam scores, it is about research quality and how much you could contribute to your field. If you are able to hold your ground on that, stay strong and move on.
My best wishes!
On our midterm the highest grade was a 35 percent. I was told this is how nearly all of the exams this professor gives turn out. Is this a common thing to happen? I suddenly feel inadequate to be a phd student in math.
As others have pointed out, most PhD students spend some of their time while doing a PhD worrying about whether they belong. (And this is not a feeling that vanishes upon getting a PhD.) There's nothing wrong or worrying about feeling that way.
But, to the extent that feelings are amenable to rational inquiry, this doesn't seem like a good reason to feel that way. Raw percentages on exams are fundamentally arbitrary: they can be arbitrarily altered, independently of how much the students know, simply be asking easier or harder questions, or more or fewer questions. The raw information "I got X% on the exam" tells you exactly nothing.
Undergrad classes tend to follow some conventions, in part because students react negatively to low percentages anyway. This involves writing exams with some fluff that almost everyone gets, which brings the average up. Grad classes do less of this, in part because there's a higher expectation that the students will be able to interpret the score with more maturity (though it sounds like your professor has taken this to an unusual extreme).
So, if you're worried about what the score means, by all means investigate what it means. But the question you should be asking (possibly of your professor) is "what scores are promising/acceptable/worrying?" And, even then, don't overreact to one score in one class, especially when you have other scores from the same class.
Since you get high homework grades, I guess that there is little reason to believe that you are entirely unfit for a PhD in math. To the contrary, doing a PhD is about learning to do research, and research is a process that takes place on time-scales of the order of weeks to years. So, getting low grades in an exam situation, where you are pressured into producing certain answers in a very short time span and unnatural setting, is hardly a tool that reliably assesses your ability to do research. Performance on homework, where you have more time, acccess to resources, and freedom to let your mind do its thing, is a much better metric.
The above is, I would claim, in almost all cases true. Given that the best student (and I assume there is more than just a couple) has achieved only 35 percent on the exam, I would worry even less about your performance on the exam. And given that it is said that this is a common occurrence with this professor, there is even less reason for you to worry. Some professors just give bad exams, i.e. ones that are not adapted to the course, reasonable expectations of ability of students, and the circumstances the exam is given in, time being an important example.
By the looks of it, this might well be one those professors. And since, I assume, he doesn't fail almost entire classes year after year, I would not worry that you would fail if you keep up your good work, and not worry about you being unable to pursue your PhD.
It's possible that this professor designs all of his exam questions to be at the level of qualifying exams. Eventually you will probably have to take several of these to become a "real" Ph.D. student and begin actual research in earnest. They usually require studying above and beyond the course requirements—not only do you need more depth of mastery, but you will likely need to know things the class may not have gotten to. Some professors prefer to throw you into the pool head first and see if you sink or swim.
Others just want you to know that there's a big difference between being good at undergraduate math and being good at graduate math. I had a (non-math) professor that felt this way, and everyone in his freshman honors class got C's or worse (most got D's) on their first writing assignment. We learned our lesson pretty quickly and started pursuing a higher standard, eventually earning the A's and B's we were accustomed to by the end of the course.
Consider that 35% as less of a "wow, we're all terrible" and more of a "wow, look how much more we're going to achieve in the next few months and years" and work towards that.
I think your question is, How do I maintain my belief in myself as a math grad student in these circumstances?
Here is a trick to reduce anxiety:
Do your best on an exam, but think of yourself as a guinea pig who is there to validate the exam and the quality of instruction. Looked at this way, student performance on the exam does not necessarily reflect on what you have learned, or what you are capable of, but rather a way of evaluating the exam and the effectiveness of the teaching.
Try to find other ways of judging whether you are cutting the mustard, and of bolstering your self-esteem. For example:
form a small study group; challenge each other in these sessions
select some professors that challenge you in a supportive way and visit their office hours
attend a student seminar (or found one if one doesn't exist yet) where you take turns presenting a paper you have read and studied
And don't forget to keep up with one or two hobbies that have a good effect on your self-esteem.
I guess that every PhD student goes to the phase of "am I good enough?". I do a PhD in mathematics too and this troubles me often but I am a scientist and there is a way to find out for sure: by doing my best possible and trying to make everything work out. I would rather fail because I am really not that good than to fail because I was afraid of failing. Don't you feel the same?
As about the exam: it may be that the professor generally overestimates students ability to solve the problems or that it is intended to be that way. The reason might be for example that you are expected to select the problems you solve out of several possibilities.
In Quantum mechanics 101 I took years ago we were given the best possible grade A just for scoring over 70% of the total amount of points. Normally this bound is set for 90% to get A but this professor just had a different idea, he wanted to make the homework/exams a bit harder but more forgiving so it would not be too easy for the best but others can still get a good grade as well.