A crucial factor in any one-on-one learning is the relationship between the teacher and the student. Sometimes enough information is obtained from the very first interview to know if a good working relationship is possible. A two-way partnership happens where the student needs to be comfortable to talk about how they feel about the music they are learning, the teaching style and direction. The teacher needs to feel that the student is able to connect with the language the teacher uses, that at least some of the goals of teacher and student are achievable, and that the student looks forward to lessons, home practice and progression to new concepts, speed of progression, new genres and pieces.
Asking for an interview with the teacher to talk through your concerns and preferences is a great start. It is important to bring your child and meet together. You know your child best and know what has and hasn't worked in the past. This also gives you a chance to watch the interaction your child has with the teacher and you are likely to observe how the teacher communicates in words and actions and the responses your child gives. It is important to talk about how the interview seemed to you and what your child thought or felt about it - when you are at home where you both feel comfortable. For the student to learn most efficiently, he or she must feel comfortable to discuss all aspects of the lesson; the material, any difficulties, any problems with home rehearsing, anything he or she doesn't understand.
Many things need to connect and sometimes it takes a term or two to get to know the teacher, the pattern of lesson, practice, lesson. And it also takes time for the student to feel comfortable, confident and secure enough to let their true selves be seen and the teacher can get to know the real person inside. There are a lot of really new concepts to learn early on and it is important to work out if it is the actual 'learning piano' that is right for the student as much as if the student is with 'the right teacher'.
Very occasionally, despite the best wishes and interests of both student and teacher, the personalities just don't work. Then it is time to find another teacher. Sometimes the current teacher can help with ideas about what style of lessons or even what instrument might be more appropriate for the student. It might be that group lessons might be better, that a more 'social' instrument might appeal more, a single melody instrument or a school setting with a variety of teachers and band options could work better. Do talk to the teacher about any of these thoughts and ideas - any good teacher will be open to discussing 'how things are going'.