I collect stories about first encounters with mathematics told me by my fellow professional mathematicians, and this is one of them, about multiplication as repeated addition of the same quantity:
%% When I was 7 or 8 year old, ny uncle decided to teach me to count up to a million. I do not remember up to what number I could count at that time. Teaching was done in an abbreviated form: ... after a thousand goes two thousand, then three thousand, ..., ten thousand, eleven thousand, twenty thousand, thirty thousand, ..., one hundred thousand, two hundred thousand, three hundred thousand, ..., one million. I accepted that as a literal truth, and came to conclusion that one thousand multiplied by 37 gives one million. That is, it was assumed that one thousand is followed by one thousand and one, but I somehow missed that point, since it was not told to me explicitly. But when I told to my big brother about my discovery that one thousand multiplied by 37 makes a million he looked at me as if I was crazy, I was embarrassed, started to think and everything somehow fell into its place (although I do not remember how long it took). %%
And here is another story, about division (and scaling):
%% I could not understand the ``invert and multiply'' rule for dividing fractions. I could obey the rule, but *why* was multiplying by 4/3 the same as dividing by 3/4?
My teachers could not explain, but I was used to that. I couldn't work it out for myself either, which was less usual.
Finally I asked my father, who was an accountant. He said: if you divide everything into halves, you have twice as many things. Suddenly not just fractions but the whole of algebra made sense for the first time. %%