At a time when we see so much ill- informed and unjustified criticism of schools, teachers and education in general, it was very refreshing to find, in the latest Bulletin of the Institute of Mathematics and Its Applications (August 1997), an article which comprehensively and authoritatively deals with many of these criticisms and shows the positive progress being made by students and teachers. The article is by John Cooper, head of mathematics at King Henry VIII School in Coventry.
His main concern is to report on the success achieved by one scheme, the MEI Mathematics Scheme, with its modular examinations at A level in mathematics and further mathematics. In doing this he effectively deals with many other issues as well. His starting point is the fact that between 1984 and 1995 there was a 41 percent drop in the number of students taking A level mathematics, and between 1984 and 1994 a drop of 61 percent taking further mathematics, after a period of 20 years in which numbers had been roughly the same.
A constant criticism is that 'standards are falling'. Such critics are shown to have confused two uses of the word 'standards'. The system of awarding grades used in times past was that of 'norm referencing' which means that each year the same proportion of students is awarded a particular grade irrespective of the quality of entry, so grade A is a measure of exclusivity. This method of grading measures relative, not absolute, performance.
In contrast, the present system of 'criterion referencing', means that 'thresholds are designed into examination papers' to ensure that anyone who receives a grade A is of the same calibre as previous years. New syllabuses have been introduced which 'of necessity' bring new criteria for assessment. John Cooper, in criticising people who say that standards must have fallen, draws an analogy with the suggestion that because more athletes have broken the four minute barrier the mile must now be shorter!
Many of those who claim standards have fallen are critical of 'modular' courses. These are courses where a student is tested and marked on sections of work as they are completed. The claim is that this is easier than the old fashioned 'linear' course where a big exam is held at the end of a long course. Cooper defends modular structures in several ways. First, direct comparison cannot be made with linear examinations because early failure to cope with a modular course may lead to a change of course, and the connection between hard work and academic achievement is clear to the students from the beginning of the course. He also notes that success in mathematics depends on an appreciation of the connection between ideas. The idea that having taken a module it can be forgotten is simply untenable.
Also the uniform scale used in MEI makes it more difficult to achieve high grades. Under the old scheme of examination papers being taken at the end of the course, it was possible for a student to perform very well on one paper and relatively poorly on the second and still get a top grade (for example marks of 100 percent and 45 percent averaging to 72.5 percent would probably lead to an A grade). In the new system this is not possible: 'no allowance is made for this kind of regression', '60 marks on the uniform scale is needed for a module grade A, ie a total of 6 x 60 for an A level grade A'. This again ensures that grades accurately reflect mastery over the whole of the course content.
The examination papers are designed so as to test work requiring different levels of ability. Typically candidates are expected to achieve three quarters of the available marks (ie A 75 percent, C 56 percent and E 38 percent) and there is no question choice. 'Shorter examinations examining more manageable chunks are less dependent on memory and allow more difficult concepts to be tested at each stage.'
In his general comments John Cooper notes the widespread concern about the state of mathematics in this country. Far less time is spent on mathematics than in other comparable Western democracies, he argues, and the national curriculum has cut the time devoted to mathematics in junior and secondary school.
He regrets that we allow students to give up mathematics at 16, and that far too few have the chance of getting good grades at A level. He sees the examination at 16 as a hindrance and that at 14 as a retrograde step. On the positive side, he notes that the six students in the British team came fifth in the mathematics Olympiad an international competition for senior school students held annually which brings together the very best school mathematicians from around the world. It is better suited to other countries than to Britain because it concentrates on pure mathematics, whereas in this country there is a much greater emphasis on applied mathematics. The team beat teams from China, Taiwan, Singapore and Germany, and two British students from the same school won silver medals with the same score (one following a traditional course the other a modular course). Although this example represents a small group of the most able, it does indicate that British mathematics teaching is achieving world class standards.
He concludes that much of the media criticism is not just ignorant, but mischievous, undermining the hard work of students and teachers. He is critical of league tables and calls for demands for a single examination board to be resisted. This he believes would lead to 'a sterile curriculum impervious to innovation'. He applauds the success of MEI modular mathematics as a 'tribute to ten years of hard work and research by mathematicians free from government interference... Politicians must not be allowed to ruin it.'