Note that when I talk about what schools actually do, I'm referring to the English educational system: YMMV.
So, on with the show.
To a first approximation, I think that school mathematics should provide three things.
Basic skillsFirst off, everybody should leave school numerate. They should be arithmetically competent, and be able to cope with percentages, ratios, proportions and the like. They should also have a reasonably well-developed number sense, and be able to make plausible estimates and approximations.
For example, the total cost (say) of a shopping basket shouldn't come as a complete surprise, nor should the cost per head of population of something that costs the country a billion pounds, nor the fraction of one's own disposable income that that amounts to. And how to work out whether a 500g box of cereal or a 750g box is the better deal should not be a mystery (or rely on the shelf label helpfully giving the price per 100g).
This is the kind of stuff that the mythical general public probably think of as mathematics. But it's not the only stuff that you comes in handy in 'real life'.
We also all need some basic statistical understanding, at least we do if we aren't going to have the wool pulled over our eyes all the time. By this I mean understanding data and its presentation, what can be reliably inferred from it, and the nature of uncertainty in what is reported (or predicted!).
This is, I think, a bare minimum of the mathematics that you need to cope with everyday life (without being taken advantage of) and take meaningful part in a democracy (so that your decisions have a fighting chance of being well-informed).
I leave you to decide for yourself whether this utilitarian need is met. (And if it's as much of a Bad Thing as I think it is for it not to be met.)
Intellectual developmentOutside that, mathematics is there to provide you with intellectual development.
The purpose of this material is not to provide every pupil with a bag of mathematical techniques which they will employ in day-to-day life or in their jobs. Attempt to persuade them of a utilitarian value are doomed to failure, because they all know adults. The purpose of this material is the same as exposing them to great literature or art; it's to expand their mental horizons.
This shouldn't consist (only) of methods and algorithms to be learned, but try to provide an appreciation of the structure and process of mathematics. Mathematics is, after all, one of the subjects where you don't just have to take somebody else's word for it. You can give a satisfying argument about why something is the case. There's a huge amount here that can be explored with no more than numbers and basic algebra.
For example, nobody should leave school knowing that 'a minus times a minus is a plus' without also knowing that this isn't just an arbitrary rule, that it follows from some more fundamental choices, and that those choices themselves are made for a good reason.
Currently, there is a lot of material in the compulsory maths curriculum for school pupils up to the age of 16. There's a lot of learning to do, and a lot of skill to be acquired. And there's a lot of teachers working extremely hard to help their pupils acquire the knowledge and develop the skills.
I think that the curriculum is drastically overloaded, and encourages the wrong (I did say I have opinions) kind of learning: a procedural, or instrumental, skill based learning which is quicker to achieve, but doesn't provide the deeper understanding that an appreciation of the mathematics requires.
I'd much rather see a much smaller curriculum, with time for a deeper understanding of fewer topics to be developed.
You can even show how if gives an enriched understanding of the world around us.
"How much shorter is a short cut corner to corner compared to going along the pavement round the edge of a rectangular lawn?" is an interesting question, and nobody is pretending that a builder or a plumber needs to use Pythaogoras in everyday life to work out how long a pipe or a rafter has to be.
Note that this isn't going to rob anybody of job opportunities. The requirement to have some arbitrary number of GCSE's (including English and mathematics) at some arbitrary level of achievement isn't selecting the people with the required level of mathematics to be able to do a job—though there may be some hope that it selects out those who can't read, write or count.
Preparation for further studyAnd of course, we also have to prepare those who might go on to study maths at higher level. What we should be providing here is a basis on which undergraduate study can be built. A good foundation for the more advanced and abstract material covered in a degree course.
The current preparation expected (and required) by Universities is A-level, which includes a large chunk of calculus. I feel in a better position to comment on this, because every year I see a lot of students who have been moderately to very successful at A-level mathematics. As the years have passed, I've become more convinced that this does not provide a particularly good foundation for what we do at university.
Again, the problem seems to be one of an overstuffed curriculum. I see a lot of students who can sort of do a quite a lot of stuff. But a large proportion of them seem to have, again, an entirely instrumental learning: when I see this, I do that.
In fact, some of them by this stage have a fairly well-developed conviction that that is what mathematics is: it's a bag of procedures and algorithms which you learn to apply to the problems that you're been taught how to solve using them. These get downright panicky, and in some cases even resentful, at the idea of having to explain why some mathematical fact is true, or why some procedure works.
Unfortunately for them, there tends to be much more emphasis on why things are true or work in undergraduate maths than they are used to.
So, back to my previous point. They should be arriving with a basis on which further undergraduate study can be developed.
I don't think this is well-served by having a lot of stuff which so many students have learned to do without knowing why it works.
I think it would be better served by having a reduced curriculum, but with a deeper understanding of the material. I'd much rather see students arrive with a strong understanding of algebra, and a decent idea of what constitutes a proof, than see them arrive with an extensive bag of tricks which they don't really understand. OK, it would be even better if they had the deeper understanding of all the stuff they currently meet. But it takes time and effort to develop that understanding. You can't have both.
Yes, that would mean that we have to start calculus from scratch at university. But it's pretty standard to do that anyway (if quite fast) to try to give the deeper understanding that we want; I'm not sure that it would make a big difference to what we do in the first year.
And even if it did, that might not be a terribly Bad Thing.
There's a whole other discussion to be had about what the content of a mathematics degree ought to be, but one point on which we could probably all agree is that very little of the content of a mathematics degree is of great relevance to one's life after graduation. For those who do use mathematics subsequently, whether commercially or academically, the most important thing is likely to be the ability to learn new mathematics efficiently, and to understand what's going on.
And I'd be happy to maintain that this is better done by developing understanding, if necessary at the expense of some extent of coverage.
- I think I'm pretty safe in these opinions. I might be entirely wrong, but nobody will ever do the experiment (i.e. completely overhaul mathematics education) to find out.
For a nice summary of what I called instrumental and deeper learning, there is Richard Skemp's article
Relational Understanding and Instrumental Understanding, Mathematics Teaching 77 20–26, (1976).