I'd say one of the valuable things an SAT tutor could teach is an attitude: to take initiative, and reject resignment to failure.
The biggest difference I noticed in how I would take a test versus other people I tried to coach is that I viewed the test as a fun game like a challenging video game level. And those who struggled on the test viewed it as dreadful judgement being rendered on them.
It's like when you can tell someone is extremely self-conscious while dancing: Beyond teaching them any actual dance moves, you have to turn off the part of their brain which is blocking their natural mental resources for problem solving, and that's often the fear they are inadequate to the task, will disappoint their supporters, and that it will hurt their future prospects.
The biggest difference I noticed in how I would take a test versus other people I tried to coach is that I viewed the test as a fun game like a challenging video game level. And those who struggled on the test viewed it as dreadful judgement being rendered on them.
It's like when you can tell someone is extremely self-conscious while dancing: Beyond teaching them any actual dance moves, you have to turn off the part of their brain which is blocking their natural mental resources for problem solving, and that's often the fear they are inadequate to the task, will disappoint their supporters, and that it will hurt their future prospects.