One more thing. You speak glibly of “problem solving acumen” as if that’s what students acquire by not being taught algorithms. Well, if that were the case then I would expect, of the thousands of students we mathematicians see graduating from public schools and coming through our university classes, a few of those displaying an ignorance of basic procedures would compensate for this with vastly superior problem solving skills.
In university math, problem solving is the ultimate determiner. More than anything you can throw at them, we face students with open-ended problem solving that relies fundamentally on them having a solid grounding in the subject. Nobody cares more about “problem solving acumen” in mathematics than professional mathematicians.
Alas … the hard reality is that one can reliably predict failure … miserable failure, in practically every student in our large first-year courses who demonstrate lack of knowledge of basic procedures on that first midterm.
You’d think every now and then we’d pull out a paper and pass it around to the other professors marking the midterm and say “Hey, look at this guy … doesn’t know long division, but … look at how clever he is! He just works around it so elegantly, and does it much more efficiently!”
But … that never happens. *Never!*