I'm pretty sure she understands the expectations...I'm not sure what you mean about strategies? For dealing with the boredom? So far, what I'm hearing in all situations is that people expect her to just sit there. Which isn't really helpful. If you tell her to "entertain herself in her head" she gets in trouble because she's gone too far in there, kwim? No teacher has wanted to LET her have any sort of "fidget" or anything. I can't seem to get any kind of accomodations...
This may not be the strategy you’re looking for, but I challenged myself to see exactly what I could get away with in school without being noticed. Room for creativity inspires me far more than consequences. Maybe you could make it into a game. “No complaints from the teacher + good grades = winning,” not playing for a prize (unless that works for her) but just for the satisfaction of success. I don’t know how subtle/socially mature your daughter is yet (to notice if the teacher or other kids are noticing her and to keep the “game” to herself), but if it worked, it could be your private joke and finding quiet crazy things to do could possibly be something that brings you together. Brainstorm things she could try, decide which ones and how many she should try, and have her report how it went. As we all know, there are a lot of ways to cope with boredom, the key is having a repertoire and using the right one at the right time.
Is it physical activity that she needs? Gum chewing was a life saver for me (although not allowed in a lot of schools, I imagine). I also quietly tapped patterns with my feet, wiggled my toes, or wore lots of bracelets on my left arm (charm and beads worn loosely are fun, wear a sweatshirt so you can tuck one hand slightly in - voila! sitting at your desk with your hands nicely folded in front of you). If she can get her hand(s) out of sight so she’s not distracting the other kids, the possibilities could be endless (especially if it helps her make more eye contact with the teacher and appear focused). She could wear loose clothing with pockets perhaps if she can’t put her hand(s) in or under her desk, which might allow her to bend paper clips, tie knots in string, use silly putty or a stress ball, fold paper or tin foil, etc. If you think things like this might be helpful, I’m sure others have lots of coping strategies as well. It might be a very popular topic – how to look like you’re paying attention, retain content, and entertain yourself in school without getting into trouble?
I agree with the suggestions to find out what the teacher can tolerate and work within those limits. I also agree with teaching your daughter to check her work which will use up some time, since it’s a great strategy for later.
