As an AoPS Academy campus director, a big part of my job is meeting with parents of prospective students. One of the most common complaints I hear is that their children never show any work. Parents are surprised when I push back, gently, on the underlying assumptions. In fact, showing work is sometimes a bad idea, and forcing students to show work unnecessarily can be harmful.
Doing all math work mentally is, of course, not a good habit. Adults understand this because they have been through the entire course of schooling, and they know what lies ahead for their children. They know that Algebra, Geometry, and other higher level classes will require students to lay out their thoughts on paper in order to have any hope of solving the problems.
But many students who come to us can get everything right in their school math classes without showing any work at all. Why should they? The directive to write down lots of intermediate steps often feels like something that is done for the teacher’s benefit, not for the student’s.