I spent several formative years growing up in Italy. Not only was I impressed by the importance of Latin -- which I didn't study formally, except through other Romance Languages -- but I discovered the importance of Roman Civilization, building a lot on the Greek.
There's nothing like history to help one understand why an "ideal circle" should be one we'll never come across, as no real circle is really ideal. Or why we belabor the "number versus numeral" distinction (somewhat bogus). Or why we insist points have no dimension, but if you line up an infinite number of them, suddenly you've got a line.
Ah philosophy (they tell you it's math, but at bottom, it turns into something a lot more squishy).
Regarding Algebra, in my neck of the woods we're doing more to converge exposure to variables with simple computer programming. Defining little functions, such as to give the nth triangular number, is a great way to learn why we say "nth" i.e. we don't want to commit ourselves to just one number, we want to use the very same definition to return any triangular number, including the 2091092th.
That's *not* the whole of what Algebra is about of course (we mention it got started in Baghdad by the way -- that belongs in the standards), but it's a good way to start phasing in a lot of named variables in place of numbers, getting kids used to *templates* and/or *recipes*. Plus we don't just work with number types anymore. Symbolic processing is *so much more* than mere numerical methods.
Re those Triangular numbers:
IDLE 1.1 >>> def tri(n): return sum([i for i in range(1, n+1)])
>>> tri(2091092) 2186333921778L
>>> ((2091092+1)*2091092)/2 2186333921778L
So why are those two answers the same? (And what's that silly "L" doing there?). This takes us immediately to the child Gauss, and his game to add the first n consecutive integers. Kids like this story because the premise is Gauss was misbehaving (we say the whole class was) and the assignment to add 1 + 2 + 3 + ... + 100 was for punishment. The punch line is Gauss discovered this short cut and *out smarted* the teacher!
Why do I share this story? Because I think it touches on an important aspect of education. A younger generation likes to feel it outstrips, outsmarts, out-thinks the generations before it. At least that's a hallmark of western civ, of the USA. We're not really into ancestor worship, even though we respect specific ancestors (like some of those movie actors in those old b/w classics, and like Gauss). We like to feel we're "making progress" which means we leave older adults behind, eating our dust (not trying to be mean -- those adults had their chance), as we race towards new horizons.
What's stultifying are those adult systems which act like their job is to "bring kids up to speed" in the sense of making carbon copies of the teachers. After years of work, you'll be no further along than your fuddy dud mentors. That'd be a horrible fate, by many internal criteria. To be just like Wayne or Dom or Kirby or Pam in terms of one's understanding of mathematics -- what a failure that'd be! Math is more like music. Each generation pioneers its own.
So if you really want to be an effective teacher, you'll not get in the way of students surpassing you, and understanding in ways you simply do not. Let them also teach *you*. Make it a two way street, from the get go.
In any case a superior education system allows and encourages this. We're not even *trying* to reassure parents that their kids are just recapitulating the math *they* had in school. No way Jose. These kids are onto something new. It's not your childhood all over again. It's theirs and it's different, sometimes extraordinarily so. Get used to it.