I have to disagree with this a little bit. The skills you use for mental arithmetic are very similar to the skills for pen and paper. I would view pen and paper as training wheels.
It might be argued that mental arithmetic should also at least be taught, but the only way I know of to do that which will tend to generalize to a majority of students is to teach them to use an abacus (seasoned abacus users eventually stop needing to actually have the physical device). And yes, we should almost definitely be teaching children how to use an abacus.
It's my impression that doing basic mental arithmetic effectively tends to depend on understanding what happens to the magnitudes of numbers after various operations and making good approximations, which has very little resemblance to the brute force long multiplication/addition/division that is taught, but is teachable; yet approximations are almost never acceptable in school, despite being very useful in real life. Meanwhile, more advanced stuff requires random numerical tricks like these:
The first thing you described is what is called "Fermi estimation". Basic mental arithmetic is being able to do addition, subtraction, multiplication and division in your head. That kind of arithmetic is strictly a prerequisite for Fermi estimation. The main purpose of Fermi estimation is not to make the calculation easier (although it can be used for that too), but to get answers while not having very good data.
For example, the other day I was trying to figure out, supposing I built a small 4-foot deep swimming pool in an apartment (not on the base floor), if I could expect it to weigh less than what the builders would assume the floor needed to be able to tolerate over that area. I didn't know the mass of a cubic foot of water off the top of my head, so I went off of guessing at how many 2 liter bottles would fit in the volume, and how much a crowd of people would weigh if they all clumped together tightly. Based on these estimations, I knew my answers would be within an order of magnitude of reality. Since the first answer was several orders of magnitude larger than the second answer, I knew that the final answer to my question was "no".
I generally do this kind of thing several times per day, because my brain is filled with crazy ideas like building swimming pools in apartments.
The answers Feynman came to could be done with ordinary mental arithmetic, assuming you have good tools (better than I have) for moving numbers between working memory and short term memory, and the tricks he used to get there faster are shortcuts you simply learn with time and practice.
But basic mental arithmetic is very similar to pen-and-paper arithmetic. The main differences for me are small mechanical adjustments to make use of results that I have cached. For example, if I'm doing division, it might be easier to multiply the number by four first, and then divide by four at the end. And of course if I'm getting a rough estimate, I'll do some rounding here and there to make things more convenient.
The other major difference is that I usually reduce a fraction as far as possible before doing long division, since that makes the multiplication steps (necessary for getting the remainder) simple table look-ups instead of long-multiplication.
Metric would have made your indoor pool example way easier.
4 foot is around 1.2m,so you have around 1.2 metric tons of water per square meter. Most buildings are designed for at most a few 100kg per square meter.
> The skills you use for mental arithmetic are very similar to the skills for pen and paper.
Speak for yourself. Pen-and-paper arithmetic, to me, is a completely mechanical process, pure symbol manipulation, whereas mental arithmetic is very, very heuristic-heavy and involves a much richer mental model that often borrows from geometry.
It might be argued that mental arithmetic should also at least be taught, but the only way I know of to do that which will tend to generalize to a majority of students is to teach them to use an abacus (seasoned abacus users eventually stop needing to actually have the physical device). And yes, we should almost definitely be teaching children how to use an abacus.