If you are thinking about changing teachers, here are some things you should analyze about your/your child's current situation.
Suppose you decide the present teacher is not going to work out. When calling to find a new teacher, evaluate each teacher by the criteria you have just set up, as well your other basic criteria (credentials, location, cost, etc.).
Ask the prospective teacher for references - - people your/your child's age, sex, and level of advancement. Call these references. Ask the people how well they think the teacher has fulfilled the criteria you have set.
If you are still undecided, ask the prospective teacher if you may observe a lesson. Again, it should be someone of similar advancement. (I can't think why a teacher would not allow someone to observe a lesson, but there might be teachers who will not allow this. If it is not allowed, find out why and evaluate this!) It may take a couple of days to set up an observation lesson, as the prospective teacher will have to clear your visit with the student/student's parent. It may be a week or even two before you are able to observe.
Ask about the logistics of the visit: when should you arrive? should you open the door and come in? ring the bell?
While you observe, pay close attention to how the teacher interacts with the student and fulfills the criteria you have set. Say nothing during the observation lesson, except at the end when you thank both the teacher and student.
Do not tarry after the lesson unless the teacher has made prior arrangements with you to do so. Likely there will be another student arriving.
If there is still no obvious new choice after you have observed lessons and called references, you are better off keeping your present teacher and keeping your ear to the ground for other teachers to call. I most strongly advise you -not- to make a spur-of-the-moment decision. It's better to stay put than to jump out of the frying pan and into the fire.
First, handle it in person. To do otherwise makes you look like a coward. A phone call is also marks you this way. A letter is absolutely awful! It indicates you think so poorly of the teacher that you can't even look him in the eye with your news.
*Don't* send your child in with a note, either. -You- come in to talk to the teacher in person. Don't foist off on your child your own discomfort. You're the adult. You made the decision. You stand up and be counted!
Second, give your teacher two weeks' notice if possible; one is the minimum. Do not announce, with your hand on the doorknob, that "by the way, that was my last lesson." You are shooting yourself in the foot by not letting your teacher prepare the way for the transfer!
Similarly, don't send a letter or make a phone call prior to the next scheduled lesson saying that the last one was the final lesson.
If the teacher doesn't know there will be a transfer, he will be unable to talk to you/your child about the audition with the new teacher. You/your child should talk over what piece(s) should be played and why; and which pieces are best not played unless asked for specifically. You/your child should be able to answer questions intelligently when the new teacher asks whether diatonic scales or sight-reading has been done. You/your child should be able to list recital/competition pieces done in the last two years (last one year, as a minimum). You/your child want/s to present yourself/herself in the best possible light; let your teacher help you do this!
The teacher will be unable to prepare your child for the actual change if the final lesson took place without the teacher's knowing it. Obviously, if an adult makes the decision to change, the adult can handle the upheaval. A child may be upset. It's necessary for the child to know that -he- is not somehow "at fault" in the change. As unbelievable as it may seem, sometimes children feel they have done something wrong that somehow is causing them to be placed in a new situation, perhaps when the change was not their idea (or, even, not discussed with them prior to the final decision). Make it as easy as you can on your child by letting the teacher, who is experienced with sort of thing, lay the groundwork with your child and set up the expectations that the child will like the new teacher and will continue to do well at the piano. Allowing a final lesson to take place between teacher and your child allows closure, too, which is important for your child.
The teacher probably will conduct an exit interview with you/you plus your child, at the final lesson, in which the curriculum is summarized and the teacher places the student's achievement on a continuum, makes suggestions, and so forth.
If you leave town (or a teacher trims the roster), normally the teacher will prepare transfer papers. These help the new teacher know the content of the old teacher's curriculum, the student's strengths and weaknesses, the student's past recital pieces, the old teacher's plans for the student in the near future, and so on. If you are simply changing to another teacher in the area, do not expect such transfer materials, although if your old teacher does not offer, you might ask if the new teacher might call if there are questions.
Third, gather any lent materials and return them to the teacher at the time you give your two-week notice. Do not make the old teacher call you to ask for them.
Fourth, it is best to quit at the end of a fiscal period, which will depend on how the teacher does billing. This is tidy, financially, for both parties.
For a monthly billing, present notice at mid-month and have your exit lesson/exit interview at the last lesson of the month. Otherwise, make sure you are paid in full through the last lesson. It is incredibly poor manners to force the teacher to call you about your outstanding tuition balance.
It is unlikely the teacher will call you back. Make sure your account is paid in full and that the teacher gets the letter the next day (or 2nd day) after the lesson.
If you desire transfer materials, you will have to tell the teacher in person, and then you will have to give a reason ("I don't think my son is getting the curriculum he needs to have. As you know, I've spoken to you about this [insert number] times before" or "It's time for a change." As you are not to happy with this teacher's lesson content, I don't think transfer materials will be of much use.
Do it by mail. Again, make sure the account is paid in full, the change is made right after the lesson, and that you write a letter (don't let the teacher think you and/or your son is ill). Make sure to thank the teacher. Be considerate and courteous.