This is an interesting question, but, understandably, confounds at least two different things. E.g., is it really the case that to "know" a true mathematical fact is to be able to produce its proof on command? I think not. Another diagnostic question: must we understand thermodynamics and the Carnot cycle to drive a car usefully? Must we be able to prove the stability of the proton before setting our coffee cup on the table? Yes, of course, I'm exaggerating... but my exaggeration is in the direction I think is relevant.
Namely, awareness is the key point (and assimilation of the facts into one's world-view... to the extent that they might have some impact and affect one's own decisions).
My opinion on this is in the same vein as my objection to people being told to do every exercise before moving forward: not only are many of those exercises either make-work or pranks, but many are also incomprehensible without understanding what happens in the sequel... which one will not see until after? A bit perverse. Sure, some such pranks are "fun" in Math Olympiads and Putnam and such, but...
The problem that I see is that undergrads are too often conditioned to be paranoid that there's some unfathomable flaw in what they've written... that can only be adjudicated by the oracular professor. One of the worst corollaries of this is that kids are very inhibited about broadening their scope, because they're already worried about defending themselves with regard to a tiny, trivial "plot of land", and are taught to give no credence to their own critical faculties.
So, yes, I think this question raises some good issues, but is literally a bit mis-aimed in its assumptions.