Yikes. Seems to me as though there are two aspects to this:
- help your DD to have enough self-confidence to shrug off teachers apparently being idiots and pick a solution that works for her;
- analyse possible solutions in order to be able to advise and support DD.
The first is the most urgent, I'd say: can you and she brainstorm (literally - write down all ideas no matter how silly, then consider them slowly afterwards) what might have been going through the teacher's mind as she decided to set this homework? It's a good habit, I think, to bear in mind all possibilities, from "this teacher is sadistic" to "this teacher thinks that handwriting the syllabus will fix it in people's minds" to "this teacher thinks she is obliged to set n hours of homework and couldn't think of anything better this time". That might help to see the issue as a practical, rather than as an existential, problem.
As for what to do: my guess is that complaining right now may not be sensible, as this may have been an atypical week; I think I'd hold fire until I saw what the general pattern was going to be, at least. After that, if busy work continued, I'd be tempted to suggest to her that she make her own judgement of which pieces of work are worthwhile and do only those, but I know it's hard. Does the history syllabus come with learning objectives? If so, a handy sentence to write some time might be "I decided not to do this as I didn't understand how it would support me in achieving any of the learning objectives of this course."
