The route to rigorous reasoning

21st March 1997 at 00:00
In come notions of proof, out goes the mathematics of uncertainty. Victoria Neumark reports on the new core at A-level which should give high fliers the chance to shine

In September A-level maths students will embark on new syllabuses, based on the core of knowledge agreed by the School Curriculum and Assessment Authority and issued by Secretary of State Gillian Shephard last month. Despite many teachers feeling that the changes had been rushed through with too little time for consultation, it looks as if the core may be a fruitful development on the higher rungs of the education ladder.

Over the past couple of years, academics running maths and maths-related courses at universities have campaigned for more rigour in school mathematics teaching, particularly at A-level. Now, with the pure maths content of the core increased from 40 per cent to 50 per cent and the obligatory applications (chosen from statistics, mechanics or discrete or decision maths) raised from 10 to 25 per cent, it looks as if they have got their wish. At the same time, those who are keen on making sixth-form maths more user-friendly are pleased that there is a steady drift away from courses assessed purely on final examination. At present there are 32 linear A-level courses, of which 17 also have a modular option (taken by 20 per cent of students). By the year 2000 all courses will be available in modular form, and will very likely be more popular with students, even though only one re-sit is allowed for any module.

The core content now contains, for the first time, work on mathematical notions of proof, in line with the demand for more rigorous reasoning in last year's Standards Over Time report from SCAA and OFSTED. In, too, come vectors, ditched in 1993 from the 1983 version of the common core; out goes the mathematics of uncertainty. In line with the more precise approach is a list of formulae which have to be remembered by students and the requirement that there be one calculator-free paper which carries 25 per cent of the marks. Though at this level, banning calculators may be a sop to a politicians' bugbear, it is none the less useful, thinks Ann Kitchen of the Association of Teachers of Mathematics, to approve arithmetical accomplishment, even if to a lesser proportion of the marks.

Likewise, in the controversy over graded or "stepped" problems, many teachers may feel that, post-Cockcroft, examiners went too far in laying out problems in easy stages. One of the new core assessment objectives is the demand for "construction of extended arguments for handling substantial problems presented in unstructured form" - effectively a prescription for at least some questions to be couched as mysteries to be solved, rather than step-by-step guided analysis.

Although little in the new core has not previously been encountered in the elective syllabuses, these changes should offer a chance for candidates at the top end to demonstrate their ability - and that should please the universities, says Richard Brown, professional officer for maths at SCAA.

But Professor Margaret Brown, of King's College London, says this is not surprising because the consultative body was mostly composed of academics and examiners, with little teacher involvement. She is concerned that the new AS-level, to be completed over a year instead of the former two, is too difficult for those intending to take it as their only sixth-form maths course. For example, she says, the demand for laws of logarithms rather than some logarithmic functions is the kind of constraint that may deter students who need maths but are not mathematicians.

Roger Porkess, of the Centre for Teaching Mathematics at the University of Plymouth, agrees. He believes that schools, with the aid of new project syllabuses such as his Mathematics in Education and Industry A-level, have only just begun to woo back A-level pupils. Changing the core again, with the proposed limits on modular re-sits and a top-heavy AS course, will dampen the enthusiasm creeping back into 16-19 maths education.

Mrs Kitchen, shadow chairwoman of the ATM and research fellow at the centre for mathematics education at the University of Manchester, is also concerned that the AS core may be too heavy for a year (effectively, with exam times, only two terms) and estimates the real proportion of teaching time at AS level to be 55 to 60 per cent.

Nevertheless, Mrs Kitchen is optimistic about the new A-levels and urges teachers and students to see their advantages. In providing the universities with students who have a more homogeneous body of knowledge, the core will, she hopes, also enable students to be better equipped to face their degree. Not just the 4,000 or so of the 67,000 A-level maths students who go on to do maths degrees, but the others who do maths-related degrees and who at present may not even have taken maths A-level. If they take an AS level with an agreed core, they may fit much better into their undergraduate courses.

School students beginning A-level will find the pre-requisite statements helpful. For the first time, it is clearly set out what students coming from the intermediate tier with a C or above at GCSE will need to master to continue on to A or AS-level maths. While they may not entirely obliterate the infamous GCSEA-level gap, it should narrow it for middle tier students. Top tier (especially A* students) at GCSE should easily be able to move on to do A and AS-level within a two-year tertiary education.

Before studying A-level maths, students should know:

* sine and co-sine rules * quadratic equations * the difference between rational and irrational numbers * the laws of indices for positive integer exponents * changing the subject of a simple formula or equation * the volume of a cone and a sphere * the properties of a circle.

While these may well have been stock bits of knowledge that any maths teacher worth their salt assumed students should know, they have never before been set out as a stated baseline for A-level, so that students from the middle tier will know what they have to catch up on before they start A-level.

Standards Over Time suggested that fewer syllabuses should be offered. SCAA has ruled that each board can offer one A-level maths syllabus, with another "project" syllabus. Eight have so far registered: one each from the Associated Examining Board, the Northern Ireland and Welsh boards, the University of London Examinations and Assessment Council and the Oxford and Cambridge Schools Examination Board, OCSEB also offering two project syllabuses. This should, says Mrs Kitchen, who also works for NEAB, offer choice for students who want to emphasise or avoid coursework, and diversity for novel ideas like MEI or NEAB 16-19 to be offered as project syllabuses.

Where does this leave the 5,000 students who do further or double maths, traditionally the route to maths degrees? Professor Brown sees the new possibility of "one-and-a-half maths" as largely replacing this opportunity for bright students who need challenging. She sees the reduced demand in comprehensives and the call of the broader syllabus for many able students (maths with English, art or history) as having contributed to an evolution of tertiary maths.

Mrs Kitchen is not so sanguine. She believes that further maths could be resuscitated if colleges and schools were properly funded for running it. At present, it only attracts a third or a quarter of the funding of a "proper" subject, so only the richer institutions can afford to run it. Yet it is the copper-bottomed route to maths prowess.

Within the new maths core, then, lie many chances for improving performance, not least its contribution to the debate about post-16 maths and its role in the intellectual future of the country. SCAA has emphasised the values of higher maths; others are still wondering about the popularising of this crucial ingredient of the modern world. After all, as Mr Porkess points out, 67, 000 students took A-level last year: a drop of about 30,000 from 10 years ago.

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