Yes, на can mean both division and multiplication depending of context. But if there is no any context, I would interpret it as multiplication by default.

There are a lot of defaults in math, for example operation priorities with standard functions:
sin 2 x = sin(2*x)
sin x cos x = (sin(x)) * (cos(x))
You are not only saying in in a same way (without any operation), but you are also writing it in a same way - without brackets.

Yes, на can mean both division and multiplication depending of context. But if there is no any context, I would interpret it as multiplication by default.

There are a lot of defaults in math, for example operation priorities with standard functions:
sin 2 x = sin(2*x)
sin x cos x = (sin(x)) * (cos(x))
You are not only saying in in a same way (without any operation), but you are also writing it in a same way - without brackets.

which was about the time when children start to know that на is a common abbreviation in speech for умножить на (or разделить на)

I think they know as soon as they start to multiply numbers, at least numbers more than ten. As for smaller numbers we usually use "дважды два", "пятью три" instead of "два на два", "пять на три".