High expectation does make a huge difference to children who have the potential to learn rapidly (I say that deliberately, because e.g. my DD1 is in year 9 and is just getting to grips with some times tables as she has LDs). DD3 went to a small primary school. She was in year 4 and the school approach was to test the children on a table, then when they got 100% twice, they moved to the next one. DD3 was the best in the class, so she set the pace and that meant that she felt 'fine' if she got a few wrong - the other kids hadn't caught her up.
Then, she moved school to a large junior school that used TT Rockstars. Kids there were answering all 60 questions within 3 minutes (they're much faster now, they were in year 4). She came home and said "Mum, I only answered 47 questions in 3 minutes and other people were finished the whole lot!" I said 'Well, baby girl, you've got some work to do, then.' She soon caught up.
Now, in year 6, a few of the kids in her class are getting too good at TT Rockstars, so as well as that, they do Table Mountain, which tests division facts, percentages, decimals, fractions, etc. She loves it.
You can buy TT Rockstars at home for about £6 per year, so you could control the tables she practices.