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In most high schools (in America), I think it is safe to say that the highest math subject offered is calculus.
But why is it calculus rather than number theory or some other branch of mathematics?
While calculus has applications, I'm sure other branches of math have applications just as well.
David Bressoud wrote about this last year on his blog in a multipart series called Calculus in Crisis. This post deals specifically with "the rush to Calculus". If you trace his arguments across all the posts in that series, the line of reasoning that emerges is this:
There are no such forces pushing for the adoption in high school of number theory or abstract algebra or whatever, so that's mainly why you don't see them -- where's the demand?
I do think that statistics is making serious inroads into high school mathematics -- I don't know the numbers of how many AP Statistics courses are being offered right now or what their enrollments are, but anecdotally it seems like a lot more of my students are coming in with AP Stats credit than they used to.
And personally I would like to see high school math look a little more like computation + data science + discrete math and linear algebra, some combination that is accessible to a larger group of people and more broadly useful.
I think calculus is just seen as the most relevant, and not for bad reason. The majority of students who are going to take calculus are engineering students. These students need ideas from calculus more than they need to know about something like axiomatic set theory (although linear algebra would be a good class to offer in addition to calculus).
Basically, engineers need to calculate the occasional integral. Although they can do this with a computer nowadays, they should still know what an integral actually is. They're never going to need to know how to solve a Diophantine equation on the spot
"While calculus has applications, I'm sure other branches of math have applications just as well."
Educating yourself on the applications might answer your question. And make your teaching more informed.
I have been a chemist, engineer, military officer, and finance professional and dabbled in applied psychology and economics. Calculus is needed for that stuff. Number theory is not.
Math courses are arranged (in high school and first couple years of college) to service the mass of students who ARE NOT math majors. Learning about their courses and what math is involved in their HW problems, derivations, etc. would help to inform you on why certain classes are part of a standard curriculum and others are not.
P.s. Richard Feynman (pretty good math guy, won the Putnam with no study as last minute team member at MIT) started out as a math major. He ditched it because 'the only use of the things you learn are to become a math professor and teach more math majors'. He went to EE instead and then to physics.
Calculus is the greatest piece of human technology in history, and it is taught because it continues to be not only the most efficient technology but also the most broadly applicable. A highly "applications" based perspective on calculus would say that as a technology, a symbolic technology, more problems have been solved with the tools of Calculus throughout history than any other part of mathematics. It is important to note that the problems solved by calculus that I hint at are not at all limited to mathematical problems: calculus has solved the largest number of human problems in science and society than any other part of mathematics.
It happens to be the tool of our times, and perhaps in the future there will be some other such symbolism which can earn even higher respect for is use as a technology, but so far we are lucky that Calculus continues to be the most efficient tool in human history.
