> By hiding math's great masterpieces from students' view, we deny them the beauty of the subject.
The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.
As a mathematician, how much time do you spend studying proofs written by other people?
I'm genuinely curious, because I think that if I wanted to be a novelist, I would spend a lot of time reading novels before trying to write one. I feel like we never have children read the great "literature" of mathematics, but then expect them to write their own proofs and are surprised when they hate it.
I literally study the proofs of others every working day, for at least a few hours a day. But I am a young mathematician and still inexperienced with research.
I imagine that once you are a professor you can spend less time studying the details of proofs because you can generate them on your own once you understand the big picture. But even so this depends on your field (combinatorics folks, for example, REALLY spend a lot of time studying proofs).
Teachers should have the place of manager and coach with specialists brought in who are subject matter experts. We had this in art (painting,music), PE and computer programming when I was in public school.
To force or demand that teachers be skilled and have deep interests in everything they teach is not realistic.
This comes from my interactions with colleagues at my own undergraduate institution. So it's biased evidence, but my institution was also one of California's largest math teaching credential programs, so I think it's a reasonable sample.
I can't say how many times my 'teaching concentration' colleagues would look at a basic linear algebra problem (which requires a proof) and whine saying things like, "I don't care about this stuff, it's so stupid and pointless, I just want to get my degree so I can go teach high school math."
All irony aside, these are the students who actually enjoyed the structure of high school math, and were rudely awakened when they were asked to apply their minds to reason. As a result, 'teaching concentrations' in undergraduate and graduate school usually provide mathematically watered down courses and automatic passing grades, and the teachers-in-training don't learn anything new. They certainly don't learn, for example, any mathematics that has been done since 1900, or any open problems that aren't Millenium Prize Problems. And god forbid they ask and try to answer an original question (as they expect their students to do when they're confused).
This is what they mean by the "blind leading the blind."
The problem is that math's great masterpieces are problems and proofs. High school math educators are largely the folks who hated doing proofs during their education, and they're just as uninformed about what the interesting open problems are as their students.