Margaret Brown asks if there is any truth in the perception that maths standards are declining. Mathematics rarely excites the media, yet The Guardian's recent front page headline stating that school maths "is in crisis", and its article reporting that university mathematicians believe there has been a decline in the standards of students entering undergraduate courses, drew a response that was apparently both "voluminous" and "vituperative".
Clearly any attack on standards that is laid at the door of "progressive educationists" is likely to grab media attention. But is this panic justified?
The only hard data have come from a Leicester University biologist who pointed out that his own and colleagues' expectations were confounded when his 1993 and 1994 intakes both scored higher on an algebra test than the 1967 intake, even controlling for other factors.
The Department for Education and the Office for Standards in Education agree there are issues that need to be addressed but say that overall "the situation appears to be healthy". They point to a slowly rising percentage of the cohort passing A-level mathematics and to a rapid improvement in the grades achieved; between 1989 and 1993 the proportion of A grades in mathematics rose from 17 per cent to 24 per cent, and A-C grades from 46 per cent to 57 per cent.
Mathematicians have countered that rising grades are no proof of rising standards since it is now easier to get a high grade. Again the evidence is unclear; although in one study examiners and mathematicians judged that A-level papers have been getting steadily harder since 1950, the data did not include candidate performance. The School Curriculum and Assessment Agency agrees there has been some grade drift, especially for modular syllabuses, but that this is not necessarily to be deplored, as cross-subject comparisons show that mathematics is now only .75 of a grade harder than the all-subject average, the margin having been reduced from 1.5 grades.
Combining the grade drift with the improved grade distribution, it seems likely that overall standards of mathematical competence at A level have not changed greatly. Nor can our 18-year-old maths specialists be accused of being behind comparable students elsewhere; the most recent international comparison put England third out of 15 countries, narrowly behind Hong Kong and Japan.
Certainly for many students the A-level syllabus and style of questions have not changed significantly since the 1960s; most textbooks and teaching methods would be instantly recognised by earlier generations. It is true that in the past year or two a growing number of pupils have been taking newer syllabuses such as SMP 16-19 and Nuffield, with greater information technology and statistical emphasis, which aim to make maths more attractive, more relevant and more intellectually challenging. Nevertheless even these must contain the A-level core of traditional material, and universities will as yet have received few such students.
So are the mathematicians wrong in their perceptions? Probably not, for several reasons.
First, there has been a dramatic reduction of more than a quarter in the size of the 17-year-old cohort between 1980 and 1994. For most subjects this drop has been masked by a doubling in the proportion passing one or more A levels, resulting in more pressure for university places. However, the comparatively small rise in the proportion of the cohort gaining a pass in A-level mathematics, from 6 per cent in 1980 to 7.7 per cent in 1993, has not compensated for the reduction in the cohort size, and has led to a small reduction in the overall numbers.
The drop in numbers taking a second mathematics A level is more marked. After a rise from 6,600 to 8,900 between 1980 and 1985, the figure fell to about 4,500. Physics, too, has seen a 20 per cent reduction in numbers. The data on subject combinations show the percentage of A-level candidates entering only science and mathematics subjects has fallen from 30 per cent in 1980 to 17 per cent in 1993.
Thus, although there has only been a small drop in the numbers of those passing A-level mathematics, the reduction in numbers of students who have the traditional preferred subject combination for entry to a mathematics (or physics or engineering) degree of "maths/further maths/physics", or even two of these three subjects, will be much more drastic.
Taken together with the dearth of mature students in these areas, it is not surprising that evidence suggests that in 1994 there were fewer qualified students applying for places in mathematics than there were places available. The lack of ability to be selective about subject combinations is likely to have led to more students being accepted who had studied less mathematics and to low grade-point averages for less popular courses.
It seems probable therefore that even if overall standards in A-level mathematics have remained steady, more of the best mathematicians are moving out of mathematics and science, after A level if not after GCSE, favouring subjects such as law, computing, economics, business and medicine, which, the DFE report notes, now have a generally more competitive entry and lead to higher salaries.
The closure of one mathematics degree course this year and reductions in others owing to a lack of well-qualified applicants is not likely to excite much national concern. There is no apparent shortage of mathematicians; the number of research students remains comparatively high and employment rates for those attaining first degrees in mathematics are no higher than those in the arts and social sciences. And the trend to study a broader mixture of A levels is generally welcomed. For national competitiveness it may be more important that all students continue studying mathematics to the highest level they can cope with than that the number of specialist mathematics graduates is maintained.
There is considerable evidence that those who give up mathematics do so because they perceive it as "hard" or as "boring", that is, having little significance for them. Any recipe for raising standards must not make the subject even "harder" and more "boring" or even fewer students will continue with it, and hence even lower standards will pertain.
Where mathematicians and educationists do agree is that there is scope for increasing standards at all levels of age and attainment. But ignorant diagnoses, such as that the national curriculum omits multiplication tables, algebraic manipulation and the process of proof, or that teachers have been brainwashed by trendy educationists into ignoring basic skills while spending their time on sloppy-minded investigations, are wide of the mark. Ofsted reports support academic research in showing that number skills are heavily emphasised in primary schools, but that continuing rehearsal of uncomprehended routines can be counterproductive. Few schools teach mathematics through investigations and problem-solving, and those that do generally produce good examination results. From pre-school to PhD level, understanding, technique, process and application support one another and need to grow together, accompanied by developing knowledge about the nature of mathematics and its place in global culture. For optimal progress, learners and teachers need the best possible understanding of what significant mathematics each learner has mastered already, what is partially achieved, what is in sight and where it is all leading.
We will accomplish these aims not by argument, but by working together to research and develop better methods of teaching and assessing as well as a curriculum relevant to the 21st century. Perhaps more importantly, we need to create the motivation and to release the necessary funds to achieve the most professional teaching force.
Margaret Brown is professor of mathematics education, King's College London, and chair of the Joint Mathematical Council.