A teacher should prescribe a piece which adequetly challenges the student and which improves particular technical aspect of their piano playing which the teacher sees fit to practice. No matter what we choose for our students it should be something that they enjoy playing thats for sure. The trick is finding something which they like AND which will improve their ability at the piano.
In the end the more pieces you learn no matter what it is, the better you get overall. Even if you are faced with technical feats you haven't attempted before you have the capability to "teach yourself" how to aquire mastery over it just as you have with previous conquests. We have to try to give our students that overall experience as to how to tackle problems they face at the keyboard. They must be able to solve for themselves their problems and have a methodolgy to their practice routine which is instilled by the teacher.
With all my students I am trying to get into their heads and understand the way in which they are learning from the sheet music. What is limiting their perception while reading, making their memorisation rate slower? Even when they watch their hands, is the position of their hands at the most relaxed state? Are they keeping the fingers which dont need to move still? Is their contact to the keyboard efficient? What are they exactly thinking while they are playing for you?
The more you drill these things and demonstrate the use of it with pieces the better a student gets at trying to resolve other problems in other pieces in the future the same way.
Some students take years before they understand this system of teaching themselves how to memorise the notes.
Once a student understand how to teach themselves the notes, then they can really learn something more musical from their teachers. Too many teachers spend their time teaching their students the notes. This is not a bad thing, it is something that must be done of course, but we might be teaching a student notes for the rest of their lives, it is like the: giving a man a fish instead of teaching him how to fish for himself, logic.
We have to first give students the ability to learn the notes for themselves, this in turn will improve their ability to play the piano. If a student understand how to learn the notes they can pay more attention to what their hands are feeling like. This is where the teacher can spend more time about impoving the technique of the students fingers instead of staying in that thinking part of piano, the memorisation.
Of course teaching how to memorise and how to learn the notes will improve a student but at a slower rate. If a student understands how to teach themselves the notes then the teacher can increase a students rate of improvement furthermore without wasting time on memorisng the notes.
Some students who are even the earliest beginners can teach themselves notes. Even if you write your own little melodies for them for 2 hands with very limited notes with single hand positions where they don't have to move their hands at all throughout the piece. Get them to read the notes and teach themselves the peice. Write down for beginners the exact way in which you can read the the notes and memorise them, the way to see a group of notes, the exact feeling you should get from the hands while playing those group of notes. It might seem very long winded but it is essential to get the student to understand the way in which we are to learn the notes. We should stop spoonfeeding their early development stages of memorisation.
In the end it really doesn't matter what you give your students so long it isn't the same thing pracitcing the same movements all the time. A student should try something differnt all the time and master movements in the pieces they learn. There is no need to give 10 peices practicing LH arpeggios, one is enough for a beginner to absorb then you move on to something else. The different technical movements used in pieces should be well understood by a teacher so that when they give a piece to the student, the teacher knows that 1) there are technical movements the student hasn't practiced enough 2) There are technical movements which the student has knowledge of. I have found a piece proves too difficult if it is full of movements a student has had no experience with before. I call it "Throwing a student in the deep end" when you give them a piece which is full of technical movements they have no experience of.
This doesn't mean giving.... Sorabji to a beginner not at all, it means even giving a grade 2 peice to a beginner! Each person has their own "throwing in the deep" spot. You don't have to throw many people far and they already start drowning. Sometimes its a mind set, especially when you see their face once you have played the piece and written all over it is, there is no way in the world I'll get this.
Topic: Preparing for student's lesson