I suppose there are two things I'd like to see. A bit more care in defining structures and functions considered in school mathematics, and logic being taught as early as possible. Rigorously defining functions like sin, cosine and tangent at GCSE is stupid and unhelpful, but it would be a start to actually define them in some useful sense before asking students to use them as I've seen happen. With regard to logic, you don't have to build up mathematics from scratch, but actually giving students an idea of what logic actually is in terms of mathematical results I think is almost fundamental to encouraging creative thinking in the subject.
Is it helpful for an A-Level student to learn what a limit is before starting calculus? If they're going on to university Mathematics/Physics maybe, but for anything else of course not! But it's a shame that someone coming out of school onto a mathematics course has to effectively relearn almost everything they studied at A-Level.
Do I think school maths is totally symptomatic of 'broken education garrrr!'? No, of course not. There's a lot of good stuff in there, but from what I personally have seen from being in the system myself and tutoring students in the system now, there are certainly improvements that could be made. Most importantly, do I think that mine is the be-all and end-all of opinions? Of course not, but I'd certainly like to engage in discussion about how the system can be improved
.It's great and I wouldnt discourage for a second that teachers make the effort to engage the students with what they need to know, but finding the value of Pi to some arbitrary degree of rational accuracy isnt making you think. It's a clever and interesting way of helping students to remember the relationship between the radius and circumference of a circle.
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