Now Michael Gove's intention is to scrap the present modular A and AS levels, and replace them with old-style two-year A levels, with assessment largely based on examinations at the end of the second year. The Minister's aim, supposedly, is to "revive the art of deep thought". It's hard to quarrel with the Minister's stated aim here, since the inculcation of better thinking skills in the population can surely only be welcomed. So why are Cambridge and ACME objecting to these reforms to the examination system, and are they right?

Several arguments have been advanced. Let me comment on the major ones.

(1) Harder examinations and the move away from modular courses might discourage more students from taking mathematics at school after age 16, and this in turn would reduce the pool of people from which future mathematics students are drawn. In the worst case, this could even lead to the possible closure of some university mathematics departments.

(2) Only a minority of students taking A level mathematics go on to do maths at university, but many others might well use their maths foundation in a wide variety of other degree programmes. Hence at the A level stage, it is more important to get lots of folk to do the course so that the general level of maths education is higher than it would otherwise be.

(3) In any event, for those who do actually wish to do mathematics degrees at university, A level maths has always been a necessary but not sufficient qualification for doing so. Rather, potential mathematics students have always been advised to take additional courses and/or examinations, such as Further Mathematics, AEA papers in mathematics (Advanced Extension Awards), or STEP (sixth term entry papers).

(4) Nowadays most university degrees are modularised, so surely it makes sense for the school exams like A levels to be modular too.

What should one make of these arguments? Point (4), I think, can be dismissed out of hand. It's a silly point, partly because it seems to imply that all stages of education should be done in the same way, and I'm unaware of any good arguments for that; and in any event, it wouldn't surprise me if the current university fashion for modular degrees eventually went into reverse, as I'm not sure there are terribly good educational arguments for such degrees either.

*The numbers game*

Points (1) and (2) are really about 'bums on seats', encouraging as many people as possible to take mathematics as far as A level, and if need be making sure the course is not perceived as 'too difficult' in order to achieve that result. I understand the argument, but must confess to finding it profoundly depressing as it seems to imply that school pupils in the UK are such a feeble bunch, and have such limited ambitions, that they cannot (or are unwilling to) cope with a course that is perceived by them, their parents, or their teachers as 'difficult'. This is a great shame, if true.

Why might we not wish to accept this line of argument? Well, I can think of lots of reasons. First, UK school pupils do not come out enormously well in objective international assessments of mathematical attainments - see, for instance, the OECD's PISA surveys. Yet our governments constantly remind us of the importance of international competition for the UK economy, and of the need to raise our achievement levels in science and mathematics, and rightly so. Thus simply accepting fairly weak performance as the best we can do is surely not really very satisfactory. Somehow, we need to shift the whole culture of learning in the UK so it is much more widely understood and accepted that young people need to work hard from an early age in order to learn and succeed. It sounds terribly boring, perhaps, but what's the alternative? And in maths, we might also need a new generation of superbly well qualified, enthusiastic and inspiring teachers.

Second, comparisons that I am able to make personally with other countries don't leave us looking good. Thus I have looked at syllabuses and textbooks from Russia, Kazakhstan, Hungary and to a more limited extent from Singapore, and the level they expect young people to reach is significantly better than what we expect in the UK. True, there is more rote learning than we seem to be comfortable with in the UK nowadays, and probably less 'problem solving', but in terms of basic knowledge and techniques we are well behind.

Third, from talking to my own students over many years, it is almost invariably the case that UK students struggle with the maths they need for economics, while our overseas students almost always cope far better. In fact a few foreign students have even commented to me how shocked they were to discover how little maths many UK students actually knew and could use. This is not a very comforting situation for the UK.

Overall, then, it seems to me that a case can be made for raising maths standards at all levels in our schools, and in that wider context the Minister's proposed reform is perhaps an important first step. The Cambridge and ACME arguments are not, in the end, wholly convincing.

*University entrance*

Point (3) is about doing maths at university, and the extras that good students need to secure university entry, notably to English universities where degree courses normally last three years. The situation is a bit different here in Scotland as most degree courses are still four years' long - though that might change soon if the forthcoming budget cuts really bite hard.

I'm familiar with these 'extras' partly because I taught STEP mathematics to my son to prepare him for Cambridge entrance, and he was successful; and partly because for the past three years I have taught some advanced maths to an out-of-school extension class at my son's former secondary school. The school itself, and I suspect many schools, is just not resourced sufficiently well to be able to cope with this level of maths teaching. Yet although the STEP syllabus is largely the same as most A level courses, or the Advanced Higher in Scotland, the type of question one encounters is a world away from the typical A level question. For the latter normally splits each question into a series of parts, each following logically from the one before. This is fine in terms of showing that the students have grasped a particular mathematical technnique, but is not so good in terms of creativity and problem solving.

In contrast, a typical STEP question basically says, 'here is an interesting question, solve it', without giving any clues about the area(s) of mathematics that need to be deployed, or the sequence of steps needed for a solution. This is a lot more challenging, especially when the questions have to be done under the usual examination-type time pressure. But this kind of maths is both fascinating and fun, and it would be great if more people could be exposed to it.

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