[tsimionescu]

If that were how people learned the multiplication table, you would see that people take longer to come up with the result of 9*9 than it takes them to compute 3*5. I have never seen anyone work this way, and so I believe it's far more likely people just remember the results in a table. "9*9=81" is simply a fact you have stored, not a computation you perform, even subconsciously.

Edit: I should also note that it's pretty well known people learn arithmetic as symbol manipulation and not some higher order reasoning. The reason this is pretty well established is that historically, the switch from Roman numerals to Arabic numerals led to a huge flurry of arithmetic activity, because it was so much easier to do arithmetic with the new symbols. If people had learned by subconsciously calculating the underlying linear functions and not through symbolic manipulation, the switch would have been entirely irrelevant. Yet for most mathematicians in Europe at the time, doing 27*3 was much easier than doing XXVII*III.

[jltsiren]

People are more than their conscious minds. A neural network can compute a linear function with a small domain without having to store each case separately.

I never memorized the multiplication table, because I found it boring and unnecessary. 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.

Some other things I had to memorize. Those were usually lists of arbitrary names with no apparent logic behind them. And if I didn't need them often enough, they never became more than lists of random facts. For example, I often can't tell the difference between sine and cosine without recalling the memorized definitions.

Or, to give another example, Finnish language has separate words for intercardinal directions (such as northeast). Usually when I need one of them, I have to iterate over the memorized list, until I find the name for the direction I had in mind. Similarly, I had to iterate over the six locative cases in Finnish grammar whenever I needed a name for one of them.

[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".