Hi. What does it mean to fail? Well, in school,
failure seems fairly easy to define. We take
a test, and then we receive a grade. And if our
score falls below a certain number, well, we
fail. My career is in actuarial science, and
the actuarial exams makes this distinction especially
clear. Each exam is created on a pass or fail
basis. At the end, you receive one of two results.
You either pass or failed. Now, it is true that
this distinction serves a practical purpose.
Schools and professional organizations need ways
to evaluate progress. But when we carry that
definition into mathematics itself, it just seems
like we are losing something critical. Because
math is so much more than arriving at the correct
answer. For starters, doing math involves asking
questions, looking for patterns, trying examples,
making conjectures, and then following ideas
whose destinations we cannot yet see. Sometimes
we try an approach and discover it does not work.
Sometimes we make a mistake. Sometimes we reach
an answer but still do not understand why it's
true. And sometimes we spend hours on a problem
without finding an answer at all. So, did we
fail? Perhaps we need to separate failing in
an exam from failing at mathematics. A person
may know key concepts and yet fail in an exam
because of calculation errors. And just as bad
is if a person can pass an exam by memorizing
the right formulas without developing much understanding.
Test scores can give useful feedback, but it
likely does not tell us everything. In fact,
often when mathematicians work on questions,
those questions do not have known answers yet.
Most of their attempts do not work. They test
an idea, they discover a problem, then they adjust
their assumptions, and then they begin again.
This is not an interruption of mathematics. This
is mathematics. The unsuccessful attempt reveals
something. It closes one path, exposes a hidden
assumption, or maybe suggests a better question.
even when it does not provide the answer and
may deepen our understanding of the problem.
I have experienced this process many times in
my career. Yet it is easy for us to view a wrong
answer differently. A wrong answer often communicates
we were not good at math. So we begin to protect
ourselves by maybe not raising our hands or avoiding
difficult problems. It is too easy to wait for
someone to show us the correct method so we would
not risk trying to the wrong one. Eventually,
success in math came to mean avoiding failure.
But what if it means almost the opposite? What
if growing in mathematics requires choosing problems
difficult enough that our first attempts probably
will fail. So this week, we're going to explore
failure and perseverance, not simply as obstacles
that we must endure, but as essential parts of
discovery. We will look at mathematicians who
struggled for years and questions that remained
unsolved for centuries. But we will also ask
what those stories might teach us about our own
lives. Because an unsuccessful attempt is not
the same as an unsuccessful person. I believe
that failure is not getting a wrong answer. I
believe the deeper failure is allowing our fear
of being wrong to keep us from exploring math
at all.