Why do you want to do research?

When I was first looking into opportinities, I had VERY LITTLE knowledge of what formal mathematical research looked like. In truth, it's a very different experience from the typical undergraduate experience of "going into a lab to run small tests for a graduate student." Thus, most people don't go into their first project with a ton of ideas about what "type of question" they want to work on. However, you can definitely think about "why" you want to be there.

Are you simply curious about the experience and want to know what math research looks like?

Do you want to work with a certain professor and learn more about their area of interest?

Are you trying to prepare for graduate school admissions?

Knowing the core reason you're looking for a project will help you do craft a more compelling outreach message, so definitely take some time thinking about this step!

What areas are you interested in?

As an undergraduate student, you likely won't have extremely specifc interests at the start (I definitely don't). However, you should consider your previous math course and which ones you enjoyed the most. Then, tailor your mentor search to professors primarly working in that area. Making sure you enjoy the "type" of math you're working with is very important, as research can get frustrating at times, and you'll have a much better experience with the project if you have passion pushing you to continue.

However, if you truly have broad interests in mathematics, remain open to areas you have less familiarity with. I'd never explored matrix analysis or numerical analysis before my current project, and I love it nonetheless. That being said, there may be more self-study work required on your part (which you may enjoy!)

How should you find a mentor?

The first thing I did during my search was to explore the faculty pages of the local universty mathematics departments. (I'm fortunate that these were Wellesley, MIT, and Harvard, but you DON'T need to go to one of these schools to have a great research expereince!) I'd always recommend keeping your initial search within commuter distance, especially if starting during the acadmeic year; being able to meet in person will make people much more willing to work with you, and will likely be a better overall experience.

Next, take a look at a few professor (or graduate student!) websites. You can typically tell whether they've worked with undergraduates by their CV, but even if past work isn't immediately apparent, it never hurts to ask. You can also get a good sense of the mentors "research personality from this page." Consider their previous papers, but also their general philosophies about math and research; you'll want to find a mentor you enjoy talking math with!

Once you find someone, you'll want to send them an email. I have a page talking about that here.

***Also, a graduate student friend I met recently told me that many people reach out for "directed reading" before asking for a research project with a mentor. Directed readings are when you and an instructor choose a textbook or paper to work through collaboriatively. You'll meet a few times a week/month usually to check in on progress. You might find a project question along the way, or just learn some cool math! (many mentors like this approach because it let's them gage your work ethic/ability to pick up required material before embarking on a full project.)

Intial Stages (what to expect):

Congrats! You're starting your research project. In many cases, your mentor will give you a paper or collection of papers explaining the premise of your question and previous progress. You should spend A LOT of time with this paper in the initial stages. One of the things I wish I had done earlier was printing out the paper and annotating every question/insight I had on specific pieces of proofs or other sections. You'll want to know the question well enough to easily explain the premise of what you're working on to someone else. However, this won't be the only time you'll look at your initial paper either. I still review mine almost three times each day when I'm working.

Also know that it's completely normal to feel like you're in a little over your head at the start. A big difference between undergraduate math courses and research is the way it forces you to make connections between topics. I'm doing a project primarily linked to linear algebra, but we also incorporate methods from analysis and (a very small amount) of number theory. If you don't know a term or concept, take the time to do some self-studying and don't be scared to ask your mentor and be honest about where you're at. Understanding will build as you spend time with the project.

In many cases, your mentor will also pose the initial strategy to approach your question. When you finish the "reading stage" and start working be sure to ask a ton of questions (I still do, and honestly wish I'd asked more at the start!)

Middle of your project:

This is the part where I'm going to tell you that getting stuck is inevitable. You may have stages where you feel discouraged because you couldn't bring as much to a meeting, and you may try an approach only to realize it didn't yeild as many results as you'd hoped. This is simply the nature of research. It sucks, but sticking with the project through these moments will be extremely rewarding in the end. Be prepared to stare at the chalkboard in confusion for awhile, spend hours going down rabbitholes to understand a concept, and at times... forget how to take a derivative. Whenever something isn't clicking, I take a walk around the hallways or outside, grab a snack, or ask others around me for advice. (a professor once told me they went on runs! I may need to try this one...)

Also, I cannot stress the importance of choosing a good work environment enough. You might be someone who enjoys intensely focusing in solitude, but I'd personally recommend spending at least some of your time in the department's common spaces. I've had the privilege to work in MIT's graduate lounge, and being surrounded by rich mathematical conversation has made me more productive than ever. Also, if you're not completely locked into your work, be sure to talk to others in the community! You can always learn more form talking to more experienced researchers, and even fellow undergraduates!

Finally, you may have a shared LaTeX document with your mentor. As much as you can, update the page with your work and a ton of context about your choices. This gives you much less to "explain" during your meetings and will help you a ton when you go back to review previous progress. (As your high school teachers said, "show your work")

End results:

I have not gotten there yet! Stay tuned :)