How do I, as an undergraduate, find interesting, accessible questions to work on to see if I'd be interested in research mathematics? For a little more context: I'm currently an undergrad (sophomore) at a small liberal arts college with a (from my experience so far) solid math program. So far, I've taken Calc I, II, III, linear algebra, and an introduction to logic class. This semester I'm taking graph theory and differential equations, an intro to statistics class, and an intro to proofs class. And outside of class, I'm (very slowly) working through an abstract algebra textbook.
What are some ways I can find questions that are accessible to me as a pretty mathematically inexperienced undergraduate? I don't even mean questions that are open or unsolved, just any questions that would be accessible that I could play around with and get a taste of working through.
 A: Exercises in abstract textbooks (the kinds where you need to prove something, not so much calculation exercises) are probably not that far away from what research mathematicians do, at least from the vantage point of an undergraduate. Bang your head against some of those for a while.
Alternatively, the motivation-theorem-proof cycle in textbooks itself is useful. Your algebra textbook is presumably of this type. When you come to a new theorem, don't start reading the proof straight away. Read the theorem, then close the book and think about it. How would you go about proving it? Is there a simpler version you can prove easily, perhaps a special case or if you assume additional preconditions? Conversely, can you find a counterexample to the theorem if you remove one of the preconditions? Can you apply the (as-yet unproven) theorem to some concrete examples? (In abstract algebra, there are many fun groups and other structures. What does a theorem in group theory say about one particular group?) Then read the proof provided and revisit your thoughts. Would you have chosen a different angle of attack? If so, would it have worked? Where in the proof do the various preconditions come in?
One very nice book for something like this is Proofs from THE BOOK by Aigner, Ziegler and Hofmann - many quite different topics, most quite accessible to an undergraduate, and beautiful proofs.
A: Welcome to MathOverflow! I am a professor at a liberal arts college. The best place to start is by talking to your favorite math professor, or your advisor in the department. The graph theory course will have plenty of accessible problems that will give you a taste of research mathematics. Abstract algebra does, too, and I'll bet your independent study will be smoother if a faculty member is involved helping you see the "big picture", suggesting good exercises to work on, etc.
Most faculty at liberal arts colleges welcome the opportunity to work with undergraduates, because it's a lot of fun, and liberal arts colleges usually don't have graduate students. It also looks good to the university when a professor works with an undergraduate student, so don't be shy to ask your professors.
It may happen that when you talk to your favorite faculty member, they suggest doing a "directed study" with them in the spring, e.g., reading an interesting book covering material not normally taught in the curriculum (e.g., hypergraphs, or Galois theory). They might also have research problems in mind already, accessible to students. For example, I list a few on my webpage, in case students want to work with me. As a rising junior, you can also apply to REU programs (that stands for "research experience for undergraduates"), though you're more likely to get in as a rising senior. Such an experience would provide you housing and a stipend on the campus of some university, where you'd work alongside other undergraduates on a research problem suggested and supervised by a professor. It's a great way to get an idea of whether or not you like research and to improve your resume during the summer. Deadlines are often in February. The same faculty member you ask about directed studies can suggest REUs and write you a letter of recommendation. Based on your background, I think you have a good shot, even for this summer. Good luck!
