I don't think the movies explained this.
The comics, specifically the one Domino series seems particularly relevant to this though, through events shown through flashbacks.
In this continuity, Domino doesn't go to the Essex School. She was created as part of a secret government breeding program and was the only one to survive it (probably due to her power). We see that she is able to use her powers against them at least once in Domino #3, right on her 10th birthday. The doctors there (who she describes as being the closest thing she has to family although they treat her like a "cosmetics rat") taser her (on what seems to be the lowest albeit still painful setting) and she doesn't do anything to react.
They then give her a cat, let her get attached to it, then were going to taser it. Just before this happened, the one doctor who was allergic to cats sneezed and accidentally tasered the other doctor, which was obviously caused by Domino's powers. Because of this they brand her with a black tattoo over her eye so that she remembers what she did every time she looks in a mirror. The operation is briefly shown in Domino #1:
As you can see, she's restrained on the table, and it doesn't seem like luck is helping her.
Based on this I would say that it's likely that:
- The school used restraints, both physical and perhaps also drugs. (Drugs are used on her in this other comic when Domino is taken prisoner.)
- The school threatened her with punishing others if she acted up. (I went to a regular elementary school and they would often punish everyone with something like no recess for the actions of one individual if that person didn't own up to the crime.)
- She wasn't as good with her powers then as she is now. (Like almost anything, you need practice to get better, which is why there are places like Xavier's school.)