Hi, I remember teaching interest theory during
my first year at Drake University. And while
I was explaining the force of interest, my mind
suddenly went blank. Fortunately, one of my students,
Terrence, understood exactly where I was headed
and he did help me out. It was a little bit humbling
to ask. Now, one of the most difficult sentences
to say in mathematics is also one of the most
important. I do not understand. It is easy to
treat those words like an admission of failure,
because it is easy to worry that they will make
us look unprepared or maybe incapable. But in
my experience, the opposite is true. The people
who learn the most are often the ones who are
willing to admit when they're confused. Actually,
if you think you already understand something,
you may stop asking questions. But the moment
you say, I do not understand this yet, you've
identified exactly where learning can begin.
Mathematics has a way of exposing false confidence.
You can read a proof and think it makes perfect
sense. And then you can try to reproduce it on
your own and realize you only recognize the explanation.
You did not truly understand it. I find that
I truly understand an idea when I can apply it
to a new setting or maybe build it from first
principles. So here are some standard questions
that I ask. Why is this assumption necessary?
Where did this equation come from? Or my favorite,
can I program this idea or build a spreadsheet
that models the concept? Questions like these
lead to a much deeper understanding than simply
asking what is the next step. I've also discovered
that one of the best ways to test my own understanding
is to teach the idea to someone else. The moment
I struggle to explain it clearly, I know I've
found a gap in my own thinking. And there's another
benefit to admitting that you do not understand.
you give someone else permission to do the same.
Often you are not the only person who is confused.
You are simply the first person with the humility
to say it. So do not measure your progress by
how rarely you are confused. Measure it by what
you do when you are confused. Do you pretend
to understand or do you become curious? In the
end, humility chooses curiosity.