In a joint response to February’s 14-19 white paper, the associations say that the proposal is “precisely the... model that the mathematical community has argued against so strongly”. Encouraging teenagers to take exams when they are ready, with high-ability youngsters opting for earlier assessment and low-attaining pupils perhaps taking longer over courses, is one of the central ideas of the white paper. Some schools, particularly in the private sector, have routinely entered pupils early for maths for years.

But subject experts say pupils benefit from having the time to master subjects in depth. The white paper, which sets out a 10-year plan on reform across all secondary subjects, proposes a new emphasis on functional maths, to be assessed separately at GCSE. League tables are also to be tightened to rank schools on the number of pupils achieving five good grades, including English and maths.

The response, from the Association of Teachers of Mathematics, the Mathematical Association and the National Association for Numeracy and Mathematics in Colleges, praises the white paper for its emphasis on the subject. It also endorses the establishment of a national centre for excellence in maths teaching.

But it says it is unclear how the proposals will reduce the assessment burden, “despite the overwhelming evidence of the damage to mathematical education that results”. Sir Mike Tomlinson’s inquiry into qualifications reform, which preceded the white paper, suggested a drastic reduction in testing through radical changes to GCSE assessment. But these ideas were rejected by ministers.

The associations add that AS and A-levels are failing to assess the analytical and problem-solving skills sought by universities. They are also sceptical of the Government’s plans to incorporate advanced extension award questions into A-levels.