Tell me about your approach to teaching.
How do you balance inspiring children and having fun with making technical progress?
What is your attitude to, or approach to, background musical skills such as aural training, rhythm work, singing etc? Do you just presume this is being done elsewhere, do you not care about it, or do you have ways of integrating it with the piano work?
How do you teach scales? Do you teach wider musicianship skills such as harmonising and improvising as part of scale knowledge, or just teach them by rote?
How do you teach reading music? What have you found to succeed or fail in various methods, with various age groups?
What's your attitude to playing by ear? Do you help develop it, not mind if they do it themselves "on the side" or are you against it?
What's your attitude to exams? How do you structure progression for children who don't want to do exams?
Do you teach music in other than classical styles? If so, how?
Can you describe some examples of how you've adapted your teaching to the musical life, attitudes or situation of individual children?
Give me a trial lesson (I'll pay if necessary) and demonstrate in it as much of the above as possible.
(Of course this depends on your being able to interpret and judge the answers given, but whether they seem to make sense to your kid would be a good start. Feel free to get back to me with them if you want help.)