Now you’re using schooling in a sense that doesn’t involve a school.
tr.v. schooled, school·ing, schools
1. To educate in or as if in a school.
2. To train or discipline:
You do not need a school to educate, train or discipline someone. Ask my mama.
Correct. Schooling may not occur in a school. The only schooling done in Public schools is indoctrination. The word is both a noun and a verb.