It's been a fair few years now since I finished doing maths at school. Now, I will freely admit that I am not necessarily as up to date with how maths is taught as I used to be, but I have seen some large snippets here and there and I have looked at the occasional past paper.
In the UK grades have been rising for years in Maths, but there has always been the question, are they rising because students genuinely are getting better, or are standards for a given grade falling. Now, I don't have much direct contact with school students so I'm not sure I can comment on this, but over the years I have been involved in training a number of engineering graduates and I can tell you that something has changed. I don't think it is necessarily the knowledge that they have, most good engineering graduates have a good array of basic maths such as integration and partial differential equations (the Classics). But, what is missing is the skills to apply the tools that they know to applied problems with an undefined solution.
This is a serious problem, engineering only advances by people solving problems that have never been solved before. Doing this is a basic skill, a skill that has to be taught, and a skill that has to be practiced. You do need some basic knowledge to do this, you do need some tools to solve the problem, but most problems can be solved many different ways and so you don't need to know every piece of knowledge, just a good array of basic tools that you can apply. But without the skill of applying what you know to a new problem you can not progress into new territories.
Today this discussion hit the front page of the BBC new website:
Two reports have been authored recently with slightly differing points of view of British maths education. One report puts the British maths education in the top 10 in the world, the other believes that the current education is too superficial, rushing pupils through topics. This is the crux of the problem, the tools that a mathematician has is a form of knowledge, something that can be learnt fairly quickly by many pupils. But the ability to apply these tools is not knowledge but a skill, something that needs to practiced over and over again in order to learn it. There are no shortcuts, it is all about quality time spent practicing.
To me, this all boils down to one thing, circular motion. When I was at school doing A-levels I was in one of the last years to study circular motion in applied maths. This was important because in this one subject you got taught a few basic equations and skills. These tools were then applied, along with tools you had already learnt at other stages of the course, to problems of circular motion with undefined solutions. Like building with Lego bricks you had to solve the problem on your own terms. The problems were relatively simple and there were often clues on how to solve them in the question, but fundamentally you were left on your own. Learning something by wrote would not have helped you solve the problem, the only way to do it was to just get stuck in and have a go. Maybe your first attempt wouldn't work but you should learn something from it and try again.
I can understand why this might have been removed from the curriculum. There was always a slight element of luck in how long it would take you to answer the question. Measuring a students ability in the application of a skill like this is far more difficult than measuring if they know a snippet of knowledge. But come the real world, and having to deal with real engineering problems, much of the knowledge you acquire as a student counts for little. You can always read a book at bedtime to learn something, but learning a new skill takes effort and good quality time. You have these as a student, but with the pressure of real world deadlines in trying to solve real world problems, very few people have the chance to properly learn a new skill.