I remember becoming suddenly enlightened around the age of 16 when I started doing A level maths. Until then, I passed tests but I never really understood what I was doing.
It was only when I started on my A levels that I realised, or should I say *understood* that the '-5' in , say, 8-5 was a term itself, and not "8 and 5 with a minus operation". I only really understood *why* I was solving equations for x^2+x+1 = 0 at this point, before that I never really understood where the zero came from.
I never really understood the rules behind algebraic manipulation until this point, either: I'd try to multiply both sides of an eqation by 2, but end up only multiplying one of the terms on either side (Id change 1 = x + 3 into 2 = 2x + 3, say) but then I would understand why sometimes I could do this and other times I couldn't (for instance 1 = x*3 going to 2 = x*6 is valid.) Looking back, I was pretty dumb when I was a kid, actually.
Its strange how little it is possible to understand about maths while still being able to get A grades in your tests.