This sounds like Bernie (yes?) and also like my brother in law's brother in law, who is a wealthy Silicon valley entrpreneur born and raised in India. My B-I-L's B-I-L told him there will be no Americans creating the next generation of computer products, because Americans are too wealthy to be motivated to do the severely difficult work it would take to learn the math.

He could be right.

These are serious questions – provided they are not merely rhetorical. Resolving the present crisis will not be easy; but, as we shall try to indicate, there is no essential contradiction in the analysis.

First, one has to understand that the long term challenge of ensuring a natural flow of home-grown mathematically competent graduates is quite different form the short term goal of selling off an unfashionable product simply by “dropping the price”.

Second, one has to recognise that a modern economy is mathematical in so many ways that we really have no choice but to find ways of producing a reliable flow of mathematically competent graduates – unless, that is, we are content to become a dependency of those countries that do appreciate the essentially “mathematical” character of a modern economy.

Third, we need to remember that the number taking A level Mathematics as recently as 1989 was more than 50% larger than at present, so there is no obvious logical reason why the goal is unrealistic.

....an important contributory factor in the decline after 1989 may well have stemmed from our failure to foresee, and to pre-empt, the loss of the largely un-sung post-war generation of highly competent and socially motivated mathematics teachers, many of whom retired in the mid-late 1980s.

the crude economic model is all wrong!

The crude consumerist model ignores three crucial factors.

• The first is that mathematics remains mathematics, and human beings remain human. So if university students are to come to grips with mathematics in any meaningful sense, there are things that need to be mastered and understood before the age of 18. Hence the scope for flexibility in simplifying what is expected at school level lies principally in identifying and concentrating on what has to be learned and mastered before the age of 18 (as was done in the Numeracy Strategy for the corresponding age of 11), and then choosing any additional material carefully and selectively.

• The second is that, while one cannot simply “drop the price” to persuade more punters to buy into an unpopular product (for reasons explained in the first bullet point), one can vary the “exchange rate”, by increasing the official “exchange value” of certain mathematical qualifications (as used to be the case – informally – with Further Mathematics), and so remove avoidable disincentives.

• The third is that human motivation is more subtle than naive economics might suggest. Satisfaction is a significant part of the perceived “return” on any investment of effort. (How else can one explain 40 000 people suffering the discomfort of running the London Marathon? Or 600 000 students taking part each year in the national mathematics competitions?) Hence, provided the rewards reflect the effort involved, and provided the subject is shaped and taught in a way that captures the imagination (as mathematics has consistently done for 4000 years), large numbers of moderately able students will take up the challenge irrespective of whether it is more demanding than the norm

good question

This is why we conclude that “the key to reversing the recent decline is more likely to lie in rediscovering what it is about elementary mathematics up to age 16 that appeals most strongly to those with a modest mathematical bent”.

this was Carolyn's point

This observation applies not just when seeking occasional specialists, but also when recruiting for bread-and-butter positions. It should not be necessary to spell out the implications for key developments – in defence technology, in information security, or in nuclear power! Political decisions in such areas are sufficiently delicate that policymakers could do without the additional burden of having to assess whether the education system in its present state will produce sufficient numbers of highly trained engineers for these “security critical” sectors

I hadn't thought about the 'in-between' sector of nuclear power.

Carolyn mentioned that defense companies are required to hire mathematicians with American citizenship. (Carolyn--have I got this right?)

But in fact, you have to have high levels of security clearance to work in industries, such as nuclear power, that could easily be converted to weaponry or terror, too.

This failure to produce an adequate supply of high quality, home-grown graduates in mathematics and in other highly numerate disciplines is a relatively recent phenomenon. It stems in part from our failure in the last 15 or so years to take steps to nurture a sufficiently large pool of mathematical talent at school level. But it also derives from the fact that national and international developments have allowed the resulting vacuum to be filled to some extent by imported talent. Policymakers, who may not appreciate the increasing extent to which our society depends on mathematics, have (perhaps unconsciously) allowed this international mobility to camouflage the urgency of considering what needs to be done to achieve a more stable supply of high quality home-grown mathematics graduates in the future.

this is what worries me

In an era where power and wealth increasingly derives from “intellectual property”, the UK is in danger of becoming totally dependent on imported intellect.

Ed was saying the other day that if nobody learns math, then everybody with smarts & ambition will become a lawyer.

back to Singapore

in the last 15 years or so, much of our mathematics teaching, and most of our assessment at all levels, have become fragmented – with multi-step tasks being routinely reduced to (and assessed as) a collection of unrelated “one-step routines”.

and bring back proofs

Some years ago QCA made the bold decision to try to reverse the trend .... by demanding that GCSE examinations should include simple proofs and the solution of problems that are not broken down into a sequence of one-step routines.

[snip]

“The capacity to reason, justify, explain and prove is central to being successful in mathematics. However, these qualities need to be explicitly developed and nurtured over time in just the same way as calculation skills or techniques for solving equations. Many teachers do not have a sufficiently secure understanding of the progression in these skills from one National Curriculum level to the next.

don't trust the tests.....

Scores on national tests at KS3 have clearly risen; but this appears to be out of line not only with results at KS2 and with subsequent performance at the end of KS4, but also with the evidence from international comparisons. Thus the evidence is consistent with the suggestion that gains at KS3 may only be apparent and due largely to a mathematically and educationally counterproductive tendency to reduce mathematical thinking to “one-step routines” (e.g. by “teaching-to-the-test”).

[snip]

Within a centralised curriculum and assessment structure, political demands for measurable “improvements” in performance have made examiners and moderators nervous about material, which they know candidates often find “hard”. This has led to the increasing neglect of more demanding material in examinations – and hence in the classroom. This will be very hard to reverse if higher grades at GCSE become available in modular form.

declining numbers of high school teachers

....failure since 1996 to monitor the seriousness of the known shortage of mathematically qualified teachers teaching mathematics in our secondary schools. While the figures themselves are almost incredible, the subsequent official inertia is even more breathtaking. Smith notes (Table 2.2) that

(i) the 1988 survey indicated that there were 46,500 mathematically qualified teachers in English secondary schools;
(ii) the 1992 survey produced the worryingly lower figure of 43,900; and
(iii) the 1996 survey gave rise to the positively scary figure of 30,800.

British PhDs

British mathematics postgraduates with a PhD from a British university are now largely unemployable in British universities. The level of research output, which British university departments are required to demonstrate in order to obtain adequate levels of funding from HEFCE, can now only be achieved by sucking in increasing numbers of older and more experienced researchers from overseas. Mathematics Departments have no choice but to appoint the best applicants, and at present British applicants stand little chance of being shortlisted.

we may not have this problem

the need to remove systemic pressures on students to avoid harder subjects

the really bad news about the top kids

In the recent, and highly instructive, international comparison (TIMSS 2003) the mathematical performance of our most able 14 year olds was almost embarrassing. The sampling in this study is robust, the items used are straightforward, and the results fairly reliable. Because of the number of new developing countries taking part, the DfES agreed that it was inappropriate to make comparisons with the “International average”, and agreed to a smaller “Comparison group” of countries (including Hungary, Italy, New Zealand, Singapore, USA and a few others). While the average student score was fixed at 500, a markedly higher level (625) was fixed as the “Advanced benchmark”. In the “Comparison group” of countries, 13% of 14 year olds scored at this higher level – which might be taken as a rough indication of those who are well-positioned to subsequently study mathematics and other highly numerate subjects with some prospect of success post-16, or at university.

Naturally some countries in the “Comparison group” had a larger percentage performing at or above this level, while some fared worse. A mere 7% of the USA sample scored at or above this ”Advanced benchmark” level. And the International average was just 6%. But the results for England should have struck Ministers and officials as far more disturbing: the percentage of English 14 year olds scoring above the “Advanced benchmark” was just 5%!