Maybe your son will starting showing his work on his own when given math problems that require multiple steps and which he cannot do in his head. I would tell the teacher she should not be assigning detention for your son's "offence" and talk to the principal if necessary. OTOH, she would be within her rights in giving a student lower marks for not showing the work.
Homeschooling and private schooling are not cheap, but if a smart boy is coming to think of his teachers as adversaries, everything should be considered.
