well your teacher pick songs from the book, maybe she thinks your daughter's level doesnt need to start from the begenning.
about buying piano, why dont you rent a paino for about 6 months to 1 years to see is your daugher really likes it.
becoz you never know how long their interest could last
you could go to TOM LEE to check
also if you wants to buy second hand you should ask the teacher to help you to check it out
it is her job to do that too
hope it could help you
I rented it in Jenny's Music in Royal Ascot Fotan. If you rent it in Tom Lee, I think you need to wait about 3 months. Last time Tom Lee called me about 3 months later after I left my information for them. The price is up to what kind of piano they have.