It would only have that sense if you added. Recently one of my friends told me that there is distinct difference between 'know of something' and 'know about something' expressions. It's like when saying did you know strawberries aren't berries, but bananas are?, i didn't know if.
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In my understanding, ' as we know it ' usually follows a noun phrase and means like the building as we know it = the version/condition of. Why do you think that he doesn't know him from his schooldays means that he does know him? Can it be used in formal conversation/writing?
What is the correct usage of phrase you don't know what you don't know?
Because you've just told them!