Here's the thing:
Up here in Scotland quite a few schools are getting excited about "early presentation", which means getting kids to sit some exams (say) a year early in order to allow more time for the "big" exams (here we're meaning the Highers, which are typically sat in fifth year of secondary education, ie the penultimate year assuming you leave school around 18).
I confess I'm not a big fan of this myself, as I worry that we try to cram too much in too soon, and there seem to me to be developmental issues regarding the maturity of pupils for sitting such exams in the first place. BUT! I am keen to see more able pupils being challenged with appropriate work, so here's the question: what work? In particular - say, from a university perspective - what sort of ideas, concepts, methods etc would you say are pretty darn fundamental to being able to cope with further mathematics?
In a way I guess I am asking for people to come forth with their pet hates, ie. where undergraduates tend to go wrong in big ways, and that is part of it. But I'm happy for the net to be cast wider. (I have asked a colleague who works in Electrical and Electronic Engineering and he offered work on logarithmic and exponential functions as being worthy of early coverage, and I see his point.)
Without giving away too many details, this coming session I'm going to have a class in third year (aged 14 ish?) who will be working towards an exam in two years' time. They are a bright bunch and I'm confident I have a lot of time to get through what they need to know AND to cover a good deal more besides, but I need to decide what exactly this bonus DVD material should be. For example, I'm pretty sure I want to cover work on vectors (even though, scandalously IMHO, they don't need to know it for the exam) - but what about, say, matrices?
(BTW, if you're interested in knowing more about the content of maths curricula in Scotland, I'd suggest you look at the SQA website where content can be viewed: here, for example, is the arrangements document for Higher Mathematics.)
Contributions most welcome: you may now start grumping. (And yes, I will be covering fractions!!!)