In Kentucky, education officials had to reduce the amount of money paid out, after more schools than expected qualified for new performance incentives.

But the real fun began when teachers were asked to allocate their share of a $26.1m (pound;18m) kitty, according to Harvard University researchers who visited four of the 480 successful schools.

Some set aside cash for an “appreciation dinner” for the bus drivers, or a scholarship fund for the classes that had earned the school its reward.

But in all the schools visited, the process of sharing out the money was time-consuming and sometimes very divisive, with endless staff meetings and ballots

In one school, classroom assistants and secretaries got twice as much as canteen assistants and the caretaker, but the bus drivers got nothing.

In another, the teachers kept the cash to themselves, offending the librarian, who said: “We give this big public relations speech to the community about how it takes a whole village to educate children and we’re all in it together, but when money comes along the teachers get it.”

But in a third, the teachers were aggrieved when some of the bonus money was given to ancilliary staff.

One left a note on the staffroom board saying: “To all of you who felt free to give my money away: the next time your husband gets a bonus, I hope you can convince him to give 15 per cent away to the office boy and the canteen boy.”