I recently mused about my daughter's soroban (abacus) lessons. Yesterday we had a nice moment over dinner. She and a boy from her class (both second graders) were sitting at the table in a restaurant, and I was challenging them with doubling addition: "What's 1+1? 2+2? 4+4?" and on up. They got as far as 4096+4096, but couldn't do 8192+8192 in their heads.
We got to a certain point (1024+1024, maybe?) and my daughter thought for a moment, then produced the right answer. When I praised her, she said, "I just imagined a soroban in my head and used that." Yes!!! That's the way it's supposed to work! She's actually learning math.
(I once TAed in a gifted program for junior high schoolers, and one of them had memorized a log table and could do not just large multiplications but even exponentiations in his head. That seems a little extreme...)
(Back In The Day when I was doing a lot of hexadecimal debugging on hardware, if I got stuck on a train without something to read, I would run through the hex multiplication tables in my head. I got good enough at it to be useful for work, but I haven't used it much in a long time, so it's mostly gone now...)