Meh, there are many things though that are important that a child won't realize so they have to be forced to do it. One such thing being school itself (regardless of the homework). Most children are not self motivated to learn anything without immediate gratification so without forcing them they will end up pretty misinformed.
Unfortunately all of that neuroplasticity is wasted on the young. :)
> Most children are not self motivated to learn anything without immediate gratification
That’s definitely not true. I’ve spent enough time hanging out with small children of various backgrounds to observe that they are in general far more curious and self motivated to learn and explore than adults are. (Of course, some adults maintain this same imagination and curiosity.)
What they often aren’t motivated to do is memorize lists of irrelevant-seeming facts or work through endless identical arithmetic problems.
They are only curious about things that provide immediate gratification. i.e. they won't spend the effort to learn trigonometry to quantify the length of a shadow they see from a flagpole. Once they learn the general intuition behind the angle of the sun and the shadow, they will lose interest and move on.
>What they often aren’t motivated to do is memorize lists of irrelevant-seeming facts
This is basically the issue. They don't understand that many things are important to learn to build the necessary base to move beyond a superficial understanding of the physical world.
If you are complaining that teachers and textbooks do a crap job explaining why someone should bother learning lots of the school curriculum, I agree 200%.
Trigonometry per se, as it is currently taught, is a boring and not very important subject, not worth spending more than about 2–3 weeks on sometime in maybe 10th grade (for a typical student), maybe 10 hours of total class time. Its form is a historical anachronism, dating from a time when there were no pocket calculators or general-purpose computers, and the only way to do science or engineering involved doing lots of hand arithmetic and consultation of big lookup tables printed in books, so it was important to be able to convert many types of problem into a form compatible with the available tables, and thus it was essential to memorize a large number of abstract formulas as shortcuts to tricky reasoning. Nowadays, you can easily solve a broad range of such problems using a computer algebra system or other type of computation, and learning to understand the reasoning is more practically important than memorizing the formulas, but the standard trigonometry course hasn’t caught up.
Moreover, the trigonometry taught in high schools now is based on the understanding from of several centuries ago, and hasn’t changed to incorporate any more recent advances in related fields. The current trigonometry course could be skipped altogether and replaced with a basic course in geometric algebra, from which the current content of a typical high school trigonometry course is a completely trivial set of corollaries. https://en.wikipedia.org/wiki/Geometric_algebra (Obviously a high school course would not be at the same level of abstraction as this Wikipedia article.) This would have the advantage of radically simplifying and clarifying nearly every college mathematics, science, and engineering course.
If you really like trigonometry and you wanted to make an interesting course on the subject, or on triangle geometry more generally, one could certainly be devised. Such a deeper and more interesting course would then deserve more work and more respect. Students would probably even find solving the problems to be fun. There are many surprising and beautiful theorems related to triangle geometry which students should be exposed to as part of their cultural heritage but currently are not.
I've said this before, but I learned far more about trigonometry from my industrial arts classes than I did from my math classes. Mechanical drafting, wood shop, metal shop, and the experience of helping my dad and the neighbors do rough carpentry was a far better use of my time. It's hard to build risers for a set of stairs, or figure out your cuts in a set of rafters without trig.
The worst part was having to fight to fit those useful electives in around the college-prep track bullshit I had to take. There was some assumption that nobody would ever want to take both shop and AP courses, so they had a nasty habit of trying to schedule them in the same blocks.
Though, to be honest, the only part of high school that is memorable is the second semester of my senior year, after I'd locked up the valedictorian spot and gotten accepted to college. I cut a lot of school to go fishing and cut firewood that spring, after basketball season finished up.
Unfortunately all of that neuroplasticity is wasted on the young. :)