Victoria Neumark looks at the debate on how to teach mathematics - and reports on an individualised scheme
Mathematics education is the subject of much hot debate. On the one hand there is a call, spearheaded by the London Mathematical Society, for a return to more traditional ways of teaching. This tends to mean more whole-class teaching; more algebra on the syllabus, and earlier; fewer science-style investigations and more teaching to notions of proof.
On the other hand, mathematics educators are convinced that recent reforms could be made to work and suffer only from lack of teacher training. Real-world-based, pupil-centred, activity-oriented, individualised maths schemes, they argue, are the best way to teach a syllabus which is relevant to today.
There are a number of key questions in this ideological struggle: how much practical work needs to be mixed in with the abstract reasoning which is the heart of pure maths, for example? How useful are calculators and at what levels? When and how is it best to set on the basis of ability? And how far is the national curriculum able to produce both numeracy in the population at large and, at the same time, nurture a small group of maths-skilled specialists to be the nation's scientists, technologists and mathematical thinkers?
Margaret Brown, Professor of mathematics education at King's College, London, is a great supporter of the national curriculum, which she helped to develop. In her inaugural lecture in October 1993 she spoke eloquently against the "deficit model" of education, which assumes the learner comes empty to the stream of knowledge poured out by the teacher.
Professor Brown sees those on the "radical right" as ignorant of how students learn and why some of them find concepts difficult. Along with other progressives in education, she sees the developments of skills for use in a changing world as paramount. She characterises the traditionalists as seeking to maintain an old elitist culture which ensured that the majority of students simply could not grasp "high" education like advanced maths.
Few would argue that maths has always attracted a minority, whether by aptitude or from cultural expectation; few would argue with the need to extend numeracy. But many are uneasy with a sharp decline in the numbers taking maths at A level and university - and university dons have expressed dismay at the levels of mathematical proficiency among even the brightest students who study maths at higher levels. For example, a report for the Engineering Council in March 1995 pointed to a perceived decline in mathematical skills among students entering university now compared to 10 years ago. They criticise Professor Brown for not being interested in the lessening of competence at these highest levels and accuse the progressives of "levelling down".
The structure of GCSE maths has been identified as a key problem area. It is possible for above average students to be entered for the Intermediate Tier and to gain a B in that tier without significantly studying algebra.
If they do gain a B, they may go on to study A level maths, perhaps with the idea of a science-related subject at university. Yet they will be starting "real" maths without any knowledge of its grammar - algebra, the foundation of manipulating number. Is it any wonder that 20 per cent of science undergraduates at one university were unable to solve the simple equation: x2 4x = 0?
An even more worrying trend is the effect of league tables in pushing down standards. Where syllabuses at GCSE once ranged widely and, in some cases, included more advanced material, all realistic teachers will now admit that the tendency now is for schools to pick the easier syllabuses and diminish their students' knowledge base in order to raise the school's standing in the league tables. This is the marketplace in action.
Some people fear that other, even more basic values are compromised. Professor Geoffrey Howson of Southampton University and David Crighton at Cambridge, seeing the most able of the country's young mathematicians, encounter worrying numbers who do not understand the fundamental notion of proof: that is, that mathematics rests on induction and deduction and rational process, not on measuring, estimating and individual cases.
Against this, Sue Burns of King's College mounts a stout defence, arguing that new methods have resulted in students who, while they may not display as much mathematical knowledge, have increasing confidence and transferable skills - and more of them are girls, too.
Calculators are another thorny issue. They are of some use, but not if introduced prematurely, agree most maths advisers, such as Graham Last in Barking and Dagenham. The danger comes when calculators are substituted either for routine calculations which ought to rely on mental fluency, or for work involving irrational numbers (like pi) or fractions.
Being decimal, calculators cannot give accurate answers and may, says Tony Gardiner - thorn in the side of progressive methods and reader in mathematics at Birmingham University - foster a sense of inexactness "being ok". Yet, of course, we live in the real world and students must learn to use calculators sensibly; for trigonometry, for example, they are invaluable.
Much of this controversy eddies around individualised learning schemes, condemned by the London Mathematical Society in its 1995 report. At their worst, such schemes can allow a class to wander unchecked through different areas of the curriculum, with the teacher too stretched by 30 different sets of questions and answers to assess individual progress. Topics are rarely taught to the whole class, so whole-group thinking does not develop; learners dictate the pace rather than teachers, so responsive mental maths lags behind; children may develop many misconceptions unnoticed.
At their best, however, individualised learning schemes can stimulate pupils who have not before found much to interest them in maths.