We already live in such a world. Or rather, there exist critical things that no single mind fully comprehends.That is not the same thing. I am talking about technology which literally nobody understands. That's the premise in a lot of science fiction: a distant future dystopia where technology is slowly decaying and nobody understands it well enough to reproduce it. At best, people are able to scavenge parts from one broken artifact and use them to fix another.
And this is not a far-fetched idea. It has already happened. The fall of the Roman Empire led to a lot of technology being forgotten for centuries. I have also heard that, for example, CRT displays cannot be reproduced (because there are no more factories) and the knowledge to build a manufacturing process for high-quality CRTs would essentially have to be rediscovered.
Don't you think the insistence on learning (and testing to demonstrate something was learned) so much "circus math" (math done for show, because there's vanishingly little other reason to do it by hand in current year) might have something to do with how so few students ever even get a taste of real linear algebra, statistics, calculus, and let alone more advanced topics?
My original reply in this thread was about the use of calculators being bad, particularly in elementary school. I wouldn't exactly call the ability to quickly multiply 5 * 6 in your head "circus math." It's more like a bare minimum basic skill.
It's like learning how to type properly when you want to become a programmer. Yes, it's not strictly necessary to be able to type as a programmer. However, if you are in full "hunt and peck" mode, you are going to be very slow at getting your thoughts onto the screen. It's going to take so much mental effort just to input a line of code that your concentration on the deeper aspects of the problem will be adversely affected.
The same goes for math.
I'm still rather happy my 9th grade teacher taught us row reduction, made us do a few by hand and one big 3x4
If you can't do basic arithmetic, you aren't even going to be able to do one by hand. I would even go so far as to say you can't even understand what's going on in the process or what the bigger picture means (e.g. that matrix-vector multiplication is a linear combination of column vectors).
Are you going to be able to understand the Gram-Schmidt procedure or QR-decomposition or singular value decomposition without being able to do them by hand? I don't know, but I somehow doubt it. All of these processes lean heavily on basic high school algebra. If someone needs a computer to solve a quadratic equation (with integer roots) then they're not going to have much fun learning how to diagonalize a matrix.
Meanwhile a Canadian friend reported to me that his "linear algebra" class in university consisted almost solely of doing row reductions of various dimensions by hand, all semester.
That's really sad. My linear algebra courses taught me about the four fundamental subspaces of a matrix (and their relationships to one another), various matrix factorizations ([orthogonal/unitary] diagonalization, QR decomposition, singular value decomposition), the Gram-Schmidt procedure, the Cayley-Hamilton theorem, and many many more interesting things, as well as how to prove a lot of facts about this stuff. We also worked in abstract vector spaces and polynomial vector spaces and learned about quadratic forms and the normal equation which will give you any polynomial regression you want.
Anyway, the whole point of all of this is not to say that every single person needs to know all of this math. Only that some do. And I believe that if you create a math education system that leans heavily on technology without any manual work, you will not raise any students who are truly competent enough to do it. I have first-hand experience as a math tutor (for elementary and high school students) for the past 4 years. I have seen upper-elementary and high school students, unable to do basic arithmetic without a calculator, absolutely struggle to learn more advanced concepts. The reliance on the tool is a huge barrier to deep thought.