Advice for a newly minted math major I just recently decided to go back to school. I had previously majored in piano performance and dropped out to work as a software engineer. 
In high school, I did pretty well at math, I got a 5 on the Calc BC exam my sophomore year. I was also pretty young at the time. I've always been "good" at math.
Coming back to university, I've decided to major in math, as some of the software work I'd like to do requires mathematical chops that I don't currently have. Calc BC was the last calculus course I took, so now I'm in multi-variable, which is a cake-walk. On the other hand, I'm taking an upper-division number theory class which is seriously kicking my butt. Over the winter break (I didn't attend fall semester) I worked through about half of Velleman's how to prove it, to help prepare myself for "real" math classes. It feels totally insufficient. I wonder if I'm cut out for mathematical work. I can manage the "technical" aspect of doing math fairly well, but I can't help feel that I do a lot of bumbling where "real" math is involved.
I also feel like I have some mathematical gaps. While I've always been "good" at math, I think that my mathematical education has some definite holes in it. 
So, I'm trying to plug my mathematical holes, but I'm spending 3 or 4 hours a day easily studying for number theory alone. It's getting to be a little bit ridiculous. 
Does anyone have any advice? I wish there was a "SICP-like" book for Mathematics. 
 A: First of all, there are lots of different Number Theory courses around, and some are low-level, and some are high-level, so without knowing what goes on in your particular class there is no way to tell whether you're not cut out for math or whether you just took something too hard. 
Second, the people who are in the best position to answer your questions are not the people on math.stackexchange (with all due respect to them) but the faculty at the university where you are enrolled. It is (part of) their job to give good advice to people in your situation. Make them earn their pay!
A: As a current math major, I would say that it very much depends on the situation.
If it has been a while since you have been near a math course, you could be rusty in some areas, and that combined with trying to transition into upper-level math courses could be impairing you. Personally, starting so early with a number theory course was probably not the best choice to get in the groove, so to speak.
I do agree that speaking with the faculty at your school is a great strategy for addressing the concerns you have, but as far as my own experiences I think some of the "real" math, as you call it, does take some time to develop. Even as a junior I find I lack in certain areas of mathematics while I may have the technical skills necessary on a day-to-day basis.
Again, I don't know what type of classes your school offers, but some of my favorite early math courses that I feel really developed some of my more conceptual math chops were a course in Discrete Math and Combinatorics, a Linear Algebra course, and of course there are always the Real Analysis courses. For me these have served as good foundations as I continue into upper-level algebra courses, for instance. Of course each person is different and develops a deeper understanding in a unique way, but as you explore where to continue, these are fundamentals-type courses to seek out to fill in those holes.
Of course at some point you should also take a differential equations type course to expand upon the calculus you have taken as well.
As a last piece of advice, again I would just emphasize that you seek out help. Again I don't know where you go to school or what the availability is like of instructors, assistants, etc., but if you have ready access to someone with whom you can rack up some one-on-one time reinforcing concepts, that truly is invaluable.
A: I wasn't strong at math as a kid.  I didn't start teaching myself algebra till I was thirty.  Currently I am taking a senior level number theory course.  Its the course I've been waiting for and it is kicking my butt.  
I find asking questions in class helps.  Going to office hours helps too.  Still I find it best to ask while its on the board.  I've had students thank me for asking questions, because they feel uncomfortable doing it.  Sure sometimes I ask a question and realize what should have been obvious after I've opened my mouth.  Its better than falling behind during the lecture.
My instructor allows us to work on homework in groups which I find helps.  Sometimes one person will have an insight into a problem that helps to get the ball rolling.  I avoided working in groups for math structures and advanced calc (your school might call it real analysis) and wish I had now.  Its easy to get in a rut or 'go blind'.  A different perspective can be invaluable.  Just make sure everyone in the study group is making an honest effort.
Enough of this.  I have homework to do.
