okay, this is a little skit I wrote out for you. It's a little extreme, but you should get the idea
*looks down in the corner, and rubs hands in between legs in a nervous manner*
You: Hey umm, can I ask you something that has been on my mind for quite a while?
Teacher: yeah sure! Ask away!
You: I've been thinking, and you know... I've been thinking, and... I put a lot of though into this. You know, I've been putting a lot of thought thinking of thinking about thinking thoughts into thought thinking things recently and these are one of the things that I put a lot of thought in to because I've been doing a lot of thought thinking recently and... You know, I've been thinking, and I've put a lot of thought into this and... I've been thinking about this thought provoking idea that if, you know... I........
Teacher: I...?
You: I.......
Teacher: I... What?
You: I......... I've been thinking, and you know... I've been thinking, and... I put a lot of though into this. You know, I've been putting a lot of thought into things recently and these are one of the things that I put a lot of thought in to because I've been doing a lot of thought thinking recently and... You know, I've been thinking, and I've put a lot of thought into this and... I've been thinking about this thought provoking idea that if, you know... I've bee-
Teacher: Jesus Christ just spit it out!!!
You: I WANT TO GET A NEW TEACHER BECAUSE I WANT TO BE A PIANIST SOME DAY BECAUSE I REALLY ENJOY PLAYING THE PIANO AND I NEED TO GET A NEW TEACHER BECAUSE DESPITE THE FACT THAT I'VE KNOW YOU FOR A REALLY LONG TIME, I DON'T THINK THAT I'M PROGRESSING FAST ENOUGH WHICH IS WHY I NEED TO GET A NEW TEACHER BECAUSE IF I DON'T PROGRESS FAST ENOUGH, THEN I WON'T BE A PROFESSIONAL PIANIST AND THATS NOT GOOD BECAUSE I REALLY LIKE THE PIANO AND I WANT TO BE A PROFESSIONAL PIANIST AND DESPITE THE FACT THAT I HAVE A NEW TEACHER! I CAN ALWAYS COME OVER TO YOUR HOUSE (or wherever he teaches you) AND SCREW AROUND ON THE PIANO BECAUSE THATS FUN AND WE'LL ALWAYS BE CONNECTED TO MUSIC AND IM SORRY! SORRY! SORRY! SORRY! SORRY! SOR-
Teacher: okay okay okay shut up! Are you sure about this?
You: *breathing heavily*
Teacher: are you?!?!?!?!
You: yes!
Teacher: okay fair enough. I want what's best for you so, sure.
_____________________________________________________________________
Okay now you don't have to go that extreme, but you get the idea? Act all nervous and that this has been bothering you for a while, and then stall so your teacher can get the idea that you're really nervous and serious about this, and then she'll tell you to hurry up and say it, and you'll say it. Ad since she's expecting something a lot worse because of the dramatic scene, she won't be as surprised, and when she's not surprised, she'll be calmed down, and when he's calmed down, then he'll be easier to talk to, and when he's easier to talk to you it's easier to convince her to let you change teachers IF she's a stubborn girl and doesn't really listen to others. If she's not stubborn and she's open minded, then ignore everything I just said, and just approach her but act nervous. I hope she is stubborn though, because I wouldn't want to see this extra long post go to waste! I put a lot of effort in to typing this you know!!!
But yeah, seriously though, you're possibly thinking that I'm joking because I always joke around, but try it. This is similar to a term in psychology that I can't remember, but it works! I do it all the time with my parents! I take the more extreme route because I think it evokes more emotion. The goal is to make them anticipate something REALLY bad; worse than you wanting to switch teachers so they'll be almost disappointed that it wasn't THAT serious. I sincerely hopes this works out for you!