With all due respect, science professors at the highest levels are required to excel at a specific type of pedagogy.
We are pretty bad (usually) at teaching mediocre to poor students. We excel, however, at teaching the high end of the curve. That is, essentially, what our success depends on. I’ve never met a successful scientists that did not excel at teaching. We teach (apprentice) brilliant and informed students to become truly exceptional, and use that training to answer important and significant questions that serve the common good.
This is pretty much worthless for dealing with the complexity of a classroom, which includes the full range of aptitudes. So in that sense, do not come to use for advice. However, it is very closely relevant to dealing with gifted children. That is something we do have some understanding of.
The “show your work” piece is really important. Give him some perspevctive. Math at the most interesting levels is not about established answers. To make progress you always have to show your work, to convince people you are right when they do not see it clearly. Showing your work is just as important as getting the right answer.
Having him tutor other kids is a great way to reinforce that too. It’s not just about getting the “right” answer. We also have to be able to demonstrate its the right answer. (Once again, we are getting into soft skills).
@Christy, I think your son is lucky to have you. Kids like him can do really well in homeschool, if they get the right support and direction.