I do think the majority of this is boredom. I'd say she's probably about 90%, if not higher. She only has 3 chapters of the math book to finish and that was supposed to last the whole school year...she's not even finding the science experiments interesting anymore.
Is this because she's done them all before, because she knows the outcomes, or because she doesn't engage? Given that you see the problem in multiple settings, it is worth ferreting this out more closely.
If you think the instruction is poor, is it worth having? I know you are liking the music and art; can you get just those and pull her from science, and homeschool science another way?
If you are able to do that, though, I'd say to find situations where she must sit and attend.
But, even with boredom factored in or out, I agree that she needs to learn how to play the game...but HOW...HOW do you, not only teach them to do it, but make them WANT to do it?
You can't make someone want to do something right away, but you can acclimate them to doing it so much that it becomes a habit and they do it correctly without even wanting or not wanting to. All behavior therapy is based on this principle: you can change behavior by positively reinforcing the behavior you want to see.
Put her in a situation where she must pay attention to someone talking (not a video, a real person). Let her know your expectation-- start low. "You have to sit here for 5 minutes, look at the speaker more than twice a minute, and remember what she says." (Give her a quiz if you want, to test whether she listened.) Then, feedback: if she does it well, she gets a token reward of some kind (my DS used to work for points that would eventually buy a reward). You can ramp up expectations gradually over time (make it 8 minutes, or add "keep your hands still" or "write down three things she says" or whatever).
Personally, I feel like, if the teacher thinks this is a problem, the teacher should be addressing it in the classroom, but there don't seem to be any consequences for this...it's like at the last school, I feel like the teachers are tattling on her and expect me to do something about it, but I don't know what...
You could ask the teacher to make her expectations crystal clear to your DD ("I would like you to sit and watch me during the instructions without writing on anything") and then praise her after the lesson if she manages it. She may need personal cues (at that age my DS still did not recognize that an instruction given to a group applied to him). This is not much effort to make on the teacher's part.
Teachers in independent schools typically have much less experience in behavior management than public school teachers do. You may need to adjust your expectations accordingly. She probably has no idea what to do and is turning to you for help to manage because she can't.
HTH,
DeeDee
