I still think it’s better to give them tests and show them that the only way to pass the test is by practicing before hand.
I actually lost interest in the subject around that time, even after scoring high enough in regional competition previously to get a free choice of any high school.
So watch out, you may kill someone's enthusiasm by making them practice simple things the way you think is best to satisfy some silly rules. I ended up more happy not getting full marks on something I could do well than showing the work.
As would plenty of other adults, myself included. Whether that's evidence that children don't have much different self-motivation than adults, or that my self-motivation skills are still childish - that I'm not sure.
Each shortcut or omitted step is more mental load and for these beginners they will slip. Just missing a negative sign or adding a constant would cost the points on the test. Worse, if the tests are multiple choice questions (which are stupid imo) they get 0.
So to build their mechanical ability I don't even allow them to show how smart they are. I stop them when they're being too smart. I only reward complete steps.
It sounds rough and it kind of is but after using these methods for years they still come to me and ask me to tutor them, rather than our parents or a private tutor. They know that my diligent checking of their work cuts down on their study time and I'm still tutoring them into college now.
I wouldn't read too much into it... there's often times other factors involved. Just because you are a sibling and not a parent might be reason enough, at some age, to prefer you; and by college, it may already be a habit.
(not sure about the "private tutor' part, have they actually tried that, and was the tutor ok? because normally that's what a kid would prefer, since he/she doesn't feel the need to challenge the tutor's authority. Maybe a really poor experience/ personality mismatch with one private tutor? ).
The private tutors they've tried are actually math teachers. I think teaching a class is a very different skillset than tutoring.
According to my brother he came to the realization that an hour studying with me makes up for multiple hours with the tutors. I'm stubborn and authoritative when I coach, I challenge them and it takes me a lot of effort to butt heads with a teen, but I think they see the results.
Math is tough in that you don't get to enjoy it until mechanical concerns become an afterthought. So when my youngest sister, who initially didn't enjoy math, said "wow that felt good" about solving a problem I knew I was going somewhere.
I ran into a situation that isn’t described: even if you’re right, they want you to prove that you didn’t cheat. Thanks to the absurdity of physics in imperial units, you can almost always show some bland dimensional conversions, but generally they love seeing an explicit change of basis for say straightforward calculus problems.
tl;dr: embrace your inner windbag, it won’t matter as long as they’re right (and one day they may appreciate it when they’re slow like us!)
This is a good idea, and having tried it, I know it sorta-kinda works a little. When I’ve tried it with my kids, what happens most often is one of two things. Sometimes if it gets too intimidating they give up and won’t try without hints. And sometimes it gets them to write some intermediate steps, but they still skip over the easy pieces, try to do 2 or 3 simplification steps at once and make mistakes. That should, you’d think, be convincing about the importance of writing down more granular steps, but for whatever reason they just hate making it mechanical and they keep resisting the idea of writing incremental steps. I’ve tried too many times to point out how much it help avoid mistakes, but they just think I’m a windbag and asking them to do boring things.