A student raises his hand in my lecture room and the question forms before he can stop it. I'm afraid I don't
understand. Four words carrying more
weight than their sound suggests because
what he means is not that this
particular equation has eluded him or
that he has lost the thread of the last
10 minutes. What he means is that he is
standing on ground. He cannot feel that
the calculus or the quantum mechanics or
the differential equations have passed
through his mind without building
anything solid. and uh tomorrow there
will be an exam that measures whether
the solid thing exists.
What he means is
make it stop moving. What he means is
tell me when it will feel like the other
things I know, the things I trust, the
things that stay known and the answer he
wants, the answer he thinks he deserves
after attending the lectures and reading
the textbook and working the problems is
that understanding will arrive. That
this sensation of skating over frozen
water he cannot see through is
temporary. That one more explanation,
one more worked example, one clearer
analogy will be the one that makes it
click. And after that click, the
knowledge will be his in the same way
his native language is his, automatic
and immovable and requiring no vigilance
to maintain. The night before an exam is
when this request becomes impossible to
honor. There is no time remaining for
understanding to arrive. The exam is in
8 hours. In 8 hours he will sit in a
room with a paper that asks him to
demonstrate possession of something he
does not possess. And the institutional
apparatus does not grade effort or good
faith or the sincerity of his confusion.
It grades answers. So the question
sharpens not when will I understand but
what do I do with the fact that I don't.
There is a story people tell about me
repeated in mathematics departments for
70 years. Sometimes as a joke, sometimes
as a coan, sometimes as a piece of
advice so brutal it sounds like
pedagogical malpractice. A young
physicist came to me the night before a
deadline that carried the weight of an
exam. He said, "I'm afraid I don't
understand the method of
characteristics." And I said, "Young
man, in mathematics, you don't
understand things. You just get used to
them. The line sounds like dismissal. It
sounds like a genius telling a
non-genius that comprehension is for lesser subjects, that mathematics is inherently beyond reach, that some people have access and others don't. It sounds like the kind of thing you say when you've never experienced confusion yourself when understanding arrives for you as effortlessly as breathing and you cannot fathom why others struggle. But that is not what I meant. And the context matters.
The physicist did not leave that conversation unable to work He left it able to work. The advice functioned, which means it was not dismissal. It was something else. What it was, a description of what knowledge actually is in mathematics and everywhere else, delivered by someone who had looked carefully enough to see it.
