Surgeon Atul Gawande has a fine article in The New Yorker about his experience hiring a retired master surgeon to coach him on his scalpel technique the way top tennis and golf stars have swing coaches. Famous opera singers often employ vocal coaches, but, it's highly unusual for surgeons to seek outside criticism. I once attended a lecture by Tom Wolfe, who mentioned that surgeons and fighter pilots are the two most arrogant professions he had encountered.
Gawande is a moneyball surgeon who compares his annual performance (e.g., percentage of operation X that have complication Y) against national averages. He noticed that when he hit his mid-40s he had stopped improving. So he sought out an old teacher to watch a few of his operations and give him notes, which turned out to be quite useful.
Gawande goes on to visit a public school to watch ex-teachers coach teachers, and makes a big deal about how we should have — revolutionary idea! — coaching for teachers. Of course, we already have a lot of teacher teachers, maybe too many.
The economics of coaching works like this. Consider tennis. Lots of young people love the game and play for free in the hopes that someday they'll get good enough to get paid. But even those who do become tournament pros are typically physically washed up by 25 or 30. So, there is a huge supply of potential coaches relative to current tournament pros. Coaching offers them a chance to stay in the game in some fashion, even at reduced pay.
In contrast, in teaching, it's not clear when the average teacher gets too old for the classroom, but it's considerably older than when the average tennis pro gets too old for Wimbledon. What is clear, however, is that a lot of teachers get sick of teaching other people's children and would like to transition into a nice, child-free education job dealing mostly with other grown-ups, especially because the pay isn't less, it's the same or even higher. That's one reason for the huge expansion over the years in the number of staffers and consultants in school districts, most of whom are ex-teachers. Promoting your best teachers (to the extent that any school district knows who their best teachers are) to teach teachers might well hurt students more than help teachers.
Also, note that most relationships between individual sports athletes and their coaches are mutually voluntary. In contrast, school districts assigning ex-teachers to coach current teachers is less likely to be a good fit.
The problem with teaching is that it's a lot like being a doctor in the 18th Century. Erasmus Darwin, the grandfather of Charles Darwin and Francis Galton, was the most celebrated doctor in England. King George III asked him to cure him of his madness. Erasmus, no fool, turned him down.
You see, 18th Century doctors had relatively few ways to actually cure anybody of anything, so their reputations for healing the sick mostly depended upon their skill at "prognosis." Erasmus Darwin was the best at figuring out which potential patients would likely improve on their own and which wouldn't. He avoided the latter like the plague, even if they were the King of England. Similarly, nobody knows (or much cares) how good, say, Harvard is at teaching undergraduates. But Harvard is outstanding at prognosis of high school seniors. And that's what counts at present in education.