International comparisons of maths achievement provide plenty of examples of England 'lagging behind' Korea or Germany - but also of areas where English children perform relatively well. If we simply look at the rankings produced by standardised international testing, English performance looks fairly unimpressive, but hardly uniquely bad.
However, rankings are not really very informative. If everyone's scores are very similar, coming near the bottom may mean nothing at all. Coming two-thirds of the way up, but with a big gap between you and higher scorers, may tell you something far more important. Ultimately what matters isn't being 'top nation' but whether your students are properly equipped. On this measure we are doing very badly indeed.
Many countries monitor student achievement systematically, but England's system, run by the 'Assessment of Performance Units', was shut down in 1989 by the Government, on the grounds that national curriculum testing made it redundant. As a result, we have no reliable national information about recent trends in maths achievement.
However, we do know that many people cannot carry out everyday mathematical operations correctly. High proportions of students in further education have serious problems with the mathematics their courses require: and many universities have simplified their first-year maths exams to take account of students' entry levels. It is ultimately irrelevant whether this reflects rising enrolments, changing courses, or actual decline in standards. Mathematics education is clearly inadequate; and our current system of education and teacher training guarantees that there will be no improvement.
England is unique in the opportunities it offers students to avoid mathematics. The process starts well before A-level. GCSE offers three alternative syllabuses and papers. To get an A or B grade, you have to sit the 'top' paper. However, with a far more restricted maths of the middle option, you can still get a C: and for many people, that is the crucial grade. With a C in maths they will be expected to study three A-levels, or the more prestigious vocational awards. A C grade is also decisive for much of higher education - including teacher training.
Certainly, one can argue that it is better to master a more restricted curriculum than to half-digest a more advanced one. But in every other industrialised country students who have followed a limited syllabus continue with mathematics after 16. The vast bulk of these English students will not. Among them are many of our future primary school teachers.
England is also unique in the final choices it imposes at 16. Increasingly, students choose to drop maths and science, and from their perspective, this is entirely rational. Most people at this age have two desires: to keep their options open, and to look better than their competitors for jobs or higher education. On both counts, maths loses out.
First, maths and science tend to restrict your choices more than arts subjects or the increasingly popular arts/social sciences mixes. Only a few avenues are closed to you if you don't take maths A-level: and they don't include most of the prestigious ones. This country, unlike France or Germany, is run and ruled by arts graduates: and bright 16-year-olds can see that for themselves.
Second, maths A-levels - again like science - are relatively difficult. Students with comparable GCSE results at 16 tend to have lower A-level grades if they choose science and maths than if they choose arts or social studies.
This won't always matter. If you are applying for a maths degree, you won't be competing with A-level English candidates anyway: and entry grades for science, maths and engineering degrees are correspondingly lower. Moreover, no one really believes all A-levels are equal. A grade B or C in further maths will impress people a lot more than a B (or even an A) in business or media studies. However, if you are a marginal student, then taking a difficult A-level can be a very big risk. You may end up not passing at all. Far better to go for a safer option. League table pressures will make schools happy to agree.
The result is that very few people do any serious mathematics after the age of 16. Many of our future graduates will have completed only the limited 'middle' GCSE. And many of them will teach maths - not only to schoolchildren but in further education colleges, where many courses require fully 'integrated' mathematics, to be taught in the course of other lessons.
In other countries this would be unthinkable. All students following academic pre-university courses automatically continue to study mathematics. The same is true for technical and vocational students. They are offered relevant but clearly structured mathematics courses taught by specialists.
In England, the decline in mathematics A-level entries creates problems recruiting maths teachers. Large amounts of upper secondary-level teaching are carried out by non-specialists and many schools simply cannot offer further maths A-level. At lower levels, the result is simply bad teaching. No one can teach well who is working at the limits of their own competence and expertise. Far too many teachers are in this situation.
We need properly structured maths teaching for all our 16- to 18-year-olds. We need to realise that trainee teachers should study the subjects they will teach and not just how to teach them. We also need, without delay, intensive additional training for large numbers of our current teachers. The alternative is a spiral: further reductions in A- level maths entries, even less competent teaching, and declining standards that really will set England apart.
The author is reader in the department of mathematics, statistics and computing, Institute of Education, University of London.