New York last week began what is thought to be the biggest American experiment yet in tweaking the traditional timetable to boost achievement - a 37-and-a-half minute extension to the school day, four days a week, for a third of the city’s 1.1 million students.

Some 305,000 struggling students in America’s largest education authority must now stay after regular lessons for intensive tutoring, with a further 45,000 students getting the option of doing so.

New York’s Schools Chancellor Joel Klein said in a statement that the extra-curricular classes, capped at 10 pupils each, represented “an historic opportunity that is well worth any challenges”.

But to lengthen the day for 350,000 students, officials have had to lop ten minutes off the day for most pupils - provoking outcry from parents. Among that number are many of the very struggling students the scheme is meant to help, as their sheer numbers mean they cannot all fit in the extra classes.

Staff serving concentrations of under-achieving pupils, which is typically how such students are distributed, have been told to give priority to “the most struggling”, while schools with few challenging pupils can enrol non-strugglers in after-school lessons.

Daniel Weisberg, director of labour policy at the city’s education department said the lost time for non-participating pupils was “imperceptible” and that the scheme would reach a “high percentage” of struggling students.

But staggering final bell times for students has also entailed massive logistical upheaval. Drawing up a new bus timetable cost $24 million, Mr Weisberg said.

Meanwhile, Ron Davis, spokesman for the United Federation of Teachers, New York’s teachers union, reported “chaos” at some schools as parents complained that the changes had upset their arrangements for picking up children from school.

Mr Davis said some staff also felt ill-prepared for the changes, controversially agreed to by union bosses in November, as part of new pay terms for New York staff.

Union president Randi Weingarten said some schools had “no rosters of kids to tutor or materials to assist teachers”. “In one place some physical education teachers were told to teach foreign languages,” she added.

Douglas Wood, executive director of the National Academy for Excellent Teaching at Teachers College, Columbia University, said research showed extra-curricular tutoring could work but only if focused on pupils’ weaknesses and taught by specially-trained staff. “I’m not convinced that’s the case here,” he said.

Mr Weisberg said many teachers had received training for their new roles and defended bringing in the changes mid-school year. “We have hundreds of thousands of students struggling right now.”