From poor performance on the world stage and the ‘shameful’ state of GCSE, to ill-equiped undergraduates, Victoria Neumark updates the latest slips and slides in the standards debate. Tom and Kate are asked how old they are. They both answer ‘15’. What is the greatest possible difference in their ages?” Nothing wrong with that you may think. It is a question from the Northern Board GCSE maths paper for those taking the top tier (1994). The two marks pupils can earn for this question can contribute to their cherished A*.

Stephen Bush, Professor of Engineering at Manchester University’s Institute of Science and Technology (UMIST) may not be the only observer to characterise the level of such questions as “Banal. Demeaning. Shameful”.

Professor Bush and his colleague Michael Robinson have recently taken space in the Financial Times (August 20) and Sunday Telegraph (September 8) to dilate upon their views on the declining mathematical standards of entrants to university in maths-related subjects. Like many other institutes of higher education, UMIST is introducing four-year courses in place of the former three-year ones because “we need an essentially remedial year to cover weakness in maths”.

Professor Bush dates the decline in mathematical fluency at this high level to the past 10 years, during which time the actual marks achieved at A-level by new undergraduates have risen by two grades while their failure rate in the department’s own end of year exams has doubled. And this is despite introducing a whole range of new remedial measures and monitoring. Those who deny there is grade inflation masking falling standards are, says Professor Bush, “living in educational dreamland”.

But what of the frequently reiterated claim that even if there has been some slippage at the “elite” end of the scale, England and Wales are serving the lower achieving end of the spectrum more adequately?

Two pieces of important new research appear to give the lie to this. Worlds Apart?, an Office for Standards in Education report by Professor David Reynolds and Shaun Farrell of Newcastle University, reviews international surveys of maths standards around the world. The results made Professor Reynolds “even more depressed than I expected”. And the word is out that the Third International Maths Survey (TIMS), due out in November, will paint a disturbing picture of continuing steady decline in the effectiveness of our country’s maths teaching.

The First International Maths Survey (FIMS) in 1964 showed the traditional British picture of a high-achieving elite and an ignored “long tail”. The Second International Maths Survey (SIMS) in 1982-83 showed that, so far from improving, Britain had slipped further down the scale, producing less outstanding mathematicians at pre-university, but also having a longer, ragged tail. Particularly weak were number, the key to everyday arithmetic, and algebra, the core of advanced maths.

TIMS is widely reported as showing that, more than a decade on, a whole range of other maths subject areas is also on the slide and that other nations have overtaken England in educating their populations more widely and deeply in mathematics, with those on the Pacific Rim two years ahead by their early teens.

Yet, remarkably, TIMS will show that science learning in England has improved over the same period. Does this suggest that it’s not English culture that is to blame for poor performance in maths, but the actual system of delivery?

Worlds Apart? suggests why this should be - aside from what David Reynolds calls the “maths phobic culture” of the UK. Apart from such factors as TV-watching (worldwide associated with falling educational standards) and homework (more of which is, unsurprisingly, good for standards), the biggest problem in the UK appears to be the wide variation in “opportunity to learn” which is directly correlated with variation in outcome. Put simply, individual children and individual schools are being deprived of the chance to learn much mathematics.

The likely culprit is differentiation where pupils in the same class learn a different syllabus, depending on their ability. Contrast with Taiwan, where the class does not move on from a topic until every child has mastered it. Contrast as well with Switzerland and Germany, whose methods of teaching number are now being adopted by a pioneering project in Barking and Dagenham. On the other hand, as Margaret Brown, Professor of Mathematics Education at Kings College, London, says, are such conclusions dangerous ones since “you can’t just transplant from one culture to another”?

Interactive whole-class teaching of the Swiss sort is one of the remedies for Standards of Arithmetic proposed by Dr John Marks in his latest pamphlet for the Centre for Policy Studies. Other remedies are dropping the curriculum on data handling and using and applying mathematics at key stages 1 and 2; ceasing to require the use of calculators in primary schools and banning their use in national curriculum tests; and better and more standardised textbooks. Dr Marks feels strongly that “investigative maths should not be required so early before children have reasonable accuracy in basic mathematics”.

He cites European schools where the objective of teaching is always to keep the whole class together and insist that all members of the class pass all subjects: thus no one can be several years behind their classmates in maths. Above all, “the focus should be primarily on accuracy,” on teaching children that precision and rigour are essential to arithmetic and later, by extension, to algebra and other maths.

More contentiously, in a paper for the Social Market Foundation, Dr Marks looked at key stage 2 results and analysed them to show wide variations in similar schools.

With whole-class interactive teaching of number becoming more widely recognised as a good thing, the debate on teaching methods - particularly on the value of “discovery” versus “instruction” - is still hot. But the debate on what is taught, how school maths is constructed, is most fiercely focused at the top end of the age range, where Professor Bush is advocating a self-selecting stream of 14-plus who will be prepared for the old-style O-level still set by the Cambridge board in Singapore.

According to Professor Bush, perhaps 15 per cent of the relevant age group would opt for this course up to O-level with possibly half of these going on to take the associated A-level.

Such a stream would more than provide for the 3-4 per cent who wish to study mathematics, engineering and science at degree level, many of whom presently come to university “lacking mathematical fluency in the things we used to take for granted and which we cannot do without in engineering”.

Professor Bush’s department now routinely runs a test for incoming undergraduates. They are set A-level questions in test conditions and marked exactingly in order to determine who needs additional maths instruction. “We usually find that the average grade for the cohort is around two below that given by the examining boards though a minority of students do confirm their A-level grading.”

“It is not on the whole that the students are of lower ability,” thinks Professor Bush, “but it is my professional view that the standards asked of students in schools are now much too low”.

Professor Margaret Brown sees the situation differently: “If more people go on to do A-level that is good. If we need more people to do harder maths, then they can do further maths as well. You have to decide what your objectives are.”

Professor Bush and Michael Robinson have compared the situation with the Pacific Rim, whose “tiger economies” are demanding a lot more of their maths students. Some of the Singapore O-level questions are very similar to those in the English A-level set by the same board (and candidates are not allowed to use calculators, tables or slide rules in many of the exams). Moreover, the Singapore government is keen for its students to enter the second year of English university engineering courses. Hong Kong University downgrades applicants with English A-levels by one to two grades. “We are talking about serious consequences for our international competitiveness here.”

Professor Bush believes that the situation, though serious, is repairable. The current Further Maths A-level (taken by 0.7 per cent of pupils, mainly from the big independent day schools) is about the same as the Hong Kong A-level. The fast-track maths programme would effectively expand this tiny number to the 3-4 per cent required for university engineering, science and maths course. If schools were given double credit for it - and only one board ran it - it would counter the current inflationary tendency for league-table jostling driving schools to choose the easiest syllabuses.

There are problems with this proposal, says Professor Brown. How would the system be operated? How would the teachers fit their fast-track students in with their other groups? How would universities and schools agree on the value of these different qualifications?

Overall, says Margaret Brown, “If we’re not doing our best by the top group we should reform our GCSE and A-level instead of going somewhere else for add-on bits.”

What of those who see some dilution of standards as inevitable given the expansion of students into higher education? Stephen Bush’s solution to keeping maths standards high while expanding higher education is to follow the Hong Kong route and offer O and A-levels in all vocational subjects, from accounting to textile design to building technology, without interfering with the academic rigour of maths testing. In Hong Kong each new subject in the O-level system is left to find its esteem in the market without trying to pretend it is “equivalent” to anything else.

To many, these ideas may seem tantamount to excluding all but a very few from the temple of higher mathematics. Yet at the same time, as Margaret Brown agrees, “a lot of people have realised that the A-level core needs looking at, that the level of GCSE algebra has been too easy”.

With SATs and GCSEs more contentious than ever and the Engineering Council and the London Mathematical Society casting about for new ways out of the maths labyrinth, it looks as if Tom and Kate may have to go back to the old map for new directions.