I do agree that people should be able to do basic calculations and make certain estimations in their head, but on the other hand, one can cut to the chase more quickly as one advances into high school - assuming a decent elementary school education - and will benefit in particular if they are required to program their own software.
The experience of programming is a pretty good way to demonstrate the limits of software.
I will say that when my youngest son - the possible nuclear engineer (he still has time to change his mind) - was eight and I taught him binary notation, the little brat refused to learn how to do it on Excel. He sat there with a pencil and paper and wrote out all the powers of two up to around 8192 if I remember correctly. I was pretty impressed with that. I assume it was a good exercise to do it that way.
But I did buy him access to Mathematica when he was in high school, and challenged him to write software for Peng Robinson calculations, which he never did, but subsequently he was required to learn to do lots of programming for his Materials Science Engineering degree and was able to cover a lot of material that would have been impossible with hand calculations.
He still writes code - I guess all the kids and many grown ups use Python these days - and using it in his classes. He's finishing up a one year 30 credit Masters in materials science on a scholarship. Recently for one of his classes, he wrote some kind of program that he was running on his laptop when we visited him. He said he expected it to take 24 hours to complete the computational run. His older brother, the artist, who never took a programming course in his life but somehow has developed considerable knowledge autodidactically, after some discussions, made all kinds of suggestions that were way over my head. They might as well have been speaking Estonian. Sure enough, the program my younger son was running crashed, and he took his big brother's advice for streamlining the code and ran the whole thing about 5 hours.
These are the tools for the times in which these kids live. They made, in my son's case, him far more powerful than I could have even imagined being at his age.
It is the structure of math, I think, looking for the elegant solution for the least cost, that is valuable, because computer time is precious, and we don't want to waste it and we certainly don't want it spitting out the wrong results.
In the text of the paper I discussed in the OP, this was reported as the issue for various numerical schemes. Of course, to be capable of making decisions about what to program, one has to know the mathematical structures to include, which is the point.
I'm OK with young people using technology - it is essential to do so in their era - but only if they really understand how the technology works. That's the issue, and I feel for educators, such as yourself, who need to get that across. I suppose it isn't easy.
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