In 1989 there were 85,000 A-level mathematics candidates. This year, there were just 55,000. Such is the extent of our failure to nurture young mathematicians. Government policy for able pupils is largely ineffective and offloads what should be central responsibilities on to schools.
Schools are required to make provision for able pupils, yet the Qualifications and Curriculum Authority and the key stage 3 strategy have generated no framework within which such provision might be developed.
Not surprisingly, Ofsted repeatedly highlights the lack of provision for able pupils in ordinary classes. We urgently need a curriculum and assessment framework which supports such provision in a natural way.
The official requirement to "identify giftedness" is misguided. Able children are not medical cases, with "ability" as something one diagnoses and then treats. Ability is latent and develops in the context of the provision made.
Like all human talents, it can be cultivated or destroyed, with many well-intentioned schemes (for instance taking GCSEs early) having unfortunate consequences. Able pupils need stimulating provision which allows them to develop at their own pace.
We therefore need two different target groups. First, a loosely defined large group of about 20 per cent, chosen to follow an enhanced day-to-day curriculum for one or two top sets. Such a group includes those who would benefit from higher expectations and those likely to study some highly numerate subject at university. Its size allows improvement within the ordinary timetable; its flexibility avoids giving parents and pupils the misleading impression that they have been identified as having some "magic gene".
There is also a second, smaller group, roughly 2 to 5 per cent, who may need special provision. These are pupils who stand out in mathematics, often by a long way. They should be expected to complete ordinary classwork and to get it 100 per cent correct. But they may still need something more.
(My books Maths Challenge 1-3, Oxford University Press pound;2.99-pound;6.50, provide a possible parallel course for such pupils in Years 7-10 to use alongside routine classwork.) In many year groups you may think that there is at most one such outstanding pupil. However, it is always worth trying to find a partner and to teach two or more at a level slightly above that of the junior partner.
The most able pupil may then progress more slowly, but will benefit hugely from working with another and from the opportunity to explain (thereby taking some of the strain off the teacher).
To spot hidden potential, set a "puzzle of the week", run a maths club, set the occasional harder problem and enter 35 per cent of each year group for the UK Mathematics Challenges (junior, Years 7-8; intermediate, Years 9-11; senior, Years 11-13) www.mathcomp.leeds.ac.uk These also provide stimulus for a larger group.
Alternatively, dip into Charles W Trigg's Mathematical Quickies (Dover pound;4.98), one of Raymond Smullyan's logic puzzle books such as To Mock a Mocking Bird and Other Logical Puzzles (Oxford University Press pound;7.99), the Australian online mathematics competition at www.amt.canberra.edu.au David Wells' The Penguin Dictionary of Curious and Interesting Numbers (pound;7.99), any of Brian Bolt's puzzle collections (Cambridge University Press), any collection of the Chinese shape puzzles called tangrams, publications from the Mathematical Association (MA) or the Association of Teachers of Mathematics (ATM).
Able pupils need a regular diet of school mathematics with two key features. They need experiences which reveal the power of mathematics, and the satisfaction of using simple ideas to solve non-trivial problems. Only in this way will they develop that respect for mathematics which can persuade them to study at higher levels. Every teacher needs a standard source of such problem material, such as the ones listed above.
But such pupils first need to master basic technique in a more flexibly robust way than their peers. Our neglect of this constitutes the biggest weakness in current practice. At each stage they need to develop total fluency, and regularly to go beyond routine one-step exercises to solve lots of simple multi-step problems.
Pupils arrive in secondary school having practised a certain kind of mental arithmetic in Years 5 and 6. But they have not learned to integrate these "atomic skills" in ways that allow them to solve "molecule-sized problems" reliably. Thus there is scope to review and extend key stage 2 arithmetic via such problems as: "Two cyclists cycle towards each other along a road.
At 8am they are 42km apart. They meet at 11am. One cyclist pedals at 7.5 kmh. What is the speed of the other cyclist?"
Such problems require pupils to extract simple information given in words, and to choose and implement successive appropriate operations (subtraction, multiplication, subtraction and division) to obtain the answer. At present, many students leaving school with good A-levels cannot solve such problems quickly and reliably.
This example might be tackled as a whole-class activity, with the problem presented on the board, and pupils expected to work mentally and to explain their solutions, including the final step "19.5 divided by 3". Once one problem has been understood, other multi-step problems can be tackled. If odd-numbered problems are easier than the even-numbered ones, reviewing the first two or four problems at the end of the lesson ensures that everyone has some success, no one is bored, and weaker pupils see that harder problems can be solved in the same way.
A regular diet of multi-step problems strengthens basic technique and helps pupils, especially able ones, appreciate the "connectedness" and power of mathematics. While teachers should embed such material in their day-to-day teaching, it is time the QCA took the lead in devising curriculum and assessment structures that encourage, rather than obstruct, high quality maths teaching.
Cambridge University Press: http:uk.cambridge.org
Oxford University Press: www.oup.co.uk
Tangram puzzles: www.ex.ac.ukcimtpuzzlestangramstangint.htm
Tony Gardiner is a reader in mathematics and mathematics education at Birmingham University