How can I convince my mom to take me to the doctor?!

Show it to your school nurse and let her send you where you have to go. Once another adult has seen your injury, your mother can't make believe it's not there.

Ask her if you can go tomorrow if it doesn't seem better. Try to get the special kind of tape so you can tape them together. If it doesn't get better ask if you can go to the doctor for x-rays. If she still says no depending on where you live ask if you can go to a cvs minute clinic or something like that and see if they can help you.
Good luck!

Having the same problem

Nothing can really be done for broken fingers...all they do is tape it to the one next to it (maybe put a pop sickle stick or something firm under it as well..)

Sounds like it's just a bruise, but maybe get it checked if it stays for more than a week or so (or gets worse)