Two questions at the end of this synopsis.
A 41yo man has been teaching a 13yo girl on the piano for a year. No rude touching or sexual comments have ever occured, but he tends to show rather informal affection for her by chatting off-topic (such as her other hobbies) for 5 or 10 minutes every lesson (weekly), and he says little things like 'her hair looks nice today', and he has put a brief arm around her shoulder for a quick cuddle once or twice when she's done well. Basically, he behaves like a friendly Uncle (but we do not know if any other thoughts lay behind this on his part). He is well-liked and trusted by the parents, but they are not acquaintances beyond the 30-min weekly lesson at their house.
They are added on each other's Facebook with the girl's parents knowledge and consent. One night, at *11pm*, he notices that she is online and he sends her a private message: "Hey. Cu at the next lesson! :-) "
She replies: "hey. i've been practisin :-) "
He replies: "Ooo that makes me happy :-) "
She replies: "Good :-) "
He says nothing further. After 10 minutes she logs off.
Question 1a) She has kept these messages secret. If you were her parent, would you be alarmed to discover them?
Question 2a) As above, but the girl innocently and excitedly informed parents of teacher's messages.
Question 2) As an impartial, unrelated adult, would you feel this messaging 'incident' was 'innapropriate' behaviour from the teacher.