I almost feel that math ought to be learned through physics. At least, I remember my physics instructor explaining math concepts in a mich more terse and understandable way than my math professors.
I have a PhD in physics, and this is exactly how I learned the material. I mean, really learned it. I did fine in plain math classes, but never quite connected with the material.
Calculus and linear algebra both really only 'clicked' when applying them in my physics classes.
Physics is like the "word problems" we'd have in elementary sho, but for advanced math.
With no offense intended to mathematicians, a lot of the higher level material in my pure math courses tends to be of the "How many angels can dance on the head of a pin?" type. Debating the finer points of details that are mostly irrelevant minutia, yet which are given just as much text space as serious results that actually do things.
Physics' real-world origin prevents a lot of that sort of thing, and places emphasis on a practicality I find deeply appealing. Cauchy-Riemann equations are important, here's where they come from, here's what they do, bam done.
Although, there have been many cases where results from Pure Mathematics have found to elegantly describe phenomena in Physics, often years after the initial result which might have been regarded as the equivalent of "angels dancing on a pinhead" by physicists of the time.
Let's forget about the pure/applied divide. That's not what I am getting at and I want to make my language very clear. Let's instead talk about "maths for its own sake" and "maths for x" (physics, computer science, what have you).
Calculus is a nasty kind of maths to study for its own sake. It's not intuitive, and it's shaped by motivations that aren't at all clear until you've learned its applications.
Stuff like abstract algebra or graph theory on the other hand are - to me - "quite nice". They're certainly much more intuitive than anything in calculus world.
So, basically I agree with you - in high school, maths (by which we really mean geometry and calculus) should be hand in hand with physics. I mean that is its reason for being in the curriculum, to prepare future engineers. People who want to study "real maths" can learn something easier to grasp, and maybe a bit more profound.
