On the other hand, there appears to have been no pilot programme to identify and deal with possible hiccups. The necessary changes in teacher recruitment and retention would require 10 years or more, yet reforms requiring more qualified staff are being introduced at a time when the number of maths teachers is falling. Without a co-ordinating committee for school maths, such issues were simply ignored. (Recent incentives were designed to stem a collapse in initial teacher education applications, not to increase the supply of qualified teachers.) Moreover, the implications for recruitment and retention of A-level students have not been thought through:

* Suppose more students do enrol for AS-level maths. Where will the additional teachers come from? Which courses will close - or have a reduced time allocation - to release the staff needed? We may increase the number of students taking AS maths at the expense of numbers completing a full A-level - without considering the consequences.

* In the absence of effective planning, the impact of the “reforms” depends on peripheral factors - such as the Further Education Funding Council regimes, career advice from medical schools, and the inflxibility of timetables struggling to accommodate four (instead of three) unconstrained choices per student.

* Maths is hard at first - but gets easier as technique improves and habits of mind develop. Until GCSE makes stricter demands in key areas, AS-level maths students will face a quantum leap (for example, in the required fluency in algebra). The expectation that students take AS module exams during Year 12 leaves little time for them to adjust and exam pressures will make it hard for teachers to spend time laying necessary foundations. AS-level could degenerate into “skating fast over very thin ice”, an experience that may discourage many from continuing into Year 13.

Yet again we are faced with changes whose success will depend on that peculiarly English brand of alchemy, in which overstretched teachers are left to transform the leaden ideas of politicians into educational gold.

I hope this pessimism is misplaced. But whatever the outcome, we must learn that successful “reform” needs careful planning and debate. And, if difficulties arise in this instance, we must not be intimidated by claims that A-level and university maths has to adapt to changes in school maths 5-16.

It is England - not maths - that is out of line. Maths is a universal discipline that is cumulative and conservative: new ideas build on old ideas without supplanting them. Key ideas retain, or even increase, their significance: to master “functions” students need greater-than-ever fluency in arithmetic and algebra.

TONY GARDINER Tony Gardiner is reader in mathematics and mathematics education at the University of Birmingham