[tsimionescu]

Whether it happens consciously or unconsciously, computation takes time. So, if your theory that the brain computes the results instead of remembering them were true, it should take measurably longer to compute 9*9 than it takes to compute 2*3. I am certain that doesn't happen for me, but it could be measured for others as well.

> When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.

This is prefectly explained by some results becoming memorized as you see them more and more, and makes no sense if your unconscious mind were computing things. If your brain was computing these results unconsciously because it had learned the function to apply, it should have come up with results automatically for any (small) multiplication. That it didn't, and you had to consciously do the computation for some numbers, is pretty clear proof that you slowly memorized the same multiplication table, but only filled it in gradually.

Overall I'm not advocating for the importance of cramming the multiplication table. I'm just saying that people who want to do mental arithmetic, or even pen-and-paper arithmetic, can only realistically do it if and when they learn the multiplication table by heart. And, that the reason the multiplication table is taught to children is strictly to have them memorize it so that they can do arithmetic without a calculator at realistic speeds.

[jltsiren]

From my point of view, what happened with the multiplication table was practice without memorization, while the word lists were memorization without practice. Two different approaches to learning with two different outcomes.

[tsimionescu]

Oh, absolutely, you learned it through practice instead of pure memorization, which overall tends to be a better strategy. But just like chess masters tend to mostly learn game positions (through practice, note trying to do rote memorization) and not some advanced logic, I am pretty sure you learned the specific results and did not learn some subconscious algorithm deeper than "lookup 9*5 in the stored table".