What if you are very, very stupid?
Understanding a proof in a math textbook is one thing; being able to reconstruct it without help is a whole other beast entirely. My classmates and I learned this the hard way at university. Most of our exams were oral exams, and nothing exposes a lack of deep knowledge faster than trying to explain a concept to someone.
Trying to explain an idea that you think you understand, only to watch the explanation fall apart with every word you speak, is an absolutely gut-wrenching feeling.
Unlike written exams, where visual learners may be able to parrot back memorized notes that they barely understand, an oral test demands creative thinking in real time. When presented with a conjecture, students not only need to recall relevant definitions and theorems, but they also need to apply them—sometimes in ways they never anticipated.
That brings us to the obvious question: How does one prepare for such an exam? Or, to put it another way: How can you tell if you’ve studied enough to truly understand a subject? Or, to rephrase yet again, this time in the words of Monty Python’s John Cleese, “If you’re very, very stupid, how can you possibly realize that you’re very, very stupid? You’d have to be relatively intelligent to realize how stupid you are.”
And indeed, as psychologist David Dunning, one of the discoverers of the Dunning-Kruger effect, describes in his book Self-Insight: Roadblocks and Detours on the Path to Knowing Thyself: “If you’re incompetent, you can’t know you’re incompetent [...] The skills you need to produce a right answer are exactly the skills you need to recognize what a right answer is.”