I'm currently a third-year mathematics student at a top-20 private research university in the US (though mostly known for the humanities). We have a small undergraduate mathematics department, but we were offered the option to take many graduate courses. I’m always passionate about studying pure math, specifically algebra, and I’m very seriously considering pursuing a PhD degree in mathematics. Nonetheless, I’ve not been able to participate in mathematical research of any kind so far – getting undergraduate research experience is impossible due to personal status issues, and my department has no resource committed to undergraduate research.
Indeed, as of my course load, I’ve finished the three-semester algebra sequence by my second year (i.e., group/representation, ring, field/Galois, category theory, commutative algebra, and homological algebra). Though I started late, I'm also working my way through the analysis and topology/geometry sequence, taking differentiable manifold and complex analysis now. I maintained all As in the graduate courses I’ve taken.
Regarding this, I do have one important confession to make, and it is in fact the reason why I’m asking this question. Math SE is a very robust community with respect to algebra. This worked very conveniently for me as there are very few peers at my school to discuss math with. As a combined consequence I actively seek ideas on SE whenever I get stuck on homework. (Please note that I'm not in violation of any collaboration rules set by my department: I understand and then proceed to write every proof myself.) This happens in about 30% of the assignments. I have no problem with exams since they are usually much easier than assignments.
Only after having recently talked to graduate students and professors at a conference, did I realize this is a terrible approach. I vividly remember one said something like: “unless you went through a textbook and attempted to prove every theorem yourself first you won’t truly understand the subject”, which is, the exact contrary of what I’ve been doing. I’m seriously in doubt about my aptitude over these subjects, fearing that I will be subpar on the level of understanding as well as the ability to conduct research, to approach open questions when I reached graduate school. I fear I never try/explore “hard enough” to come up with proofs like others have suggested. I managed to do most just by familiarity of common methods/tricks and theorems, but those things can be forgotten over time.
So here comes some specifics of my question:
- Is searching SE for homework problems common for math students?
- How will doing so affect a student’s understanding of the material?
- In what ways does doing so tie to one’s ability to do research?
- What are some possible ways to remedy this, besides completely re-learning the material?
- How much do I have to pay in the future for stack-exchanging through my courses?