Hi, the year was 1985. I was a senior in college.
I wanted to join a mathematics fraternity. How
exciting. Anyway, it seemed like a small thing
at the time, but it changed my perspective on
math for the rest of my life. As part of the
application process, I was asked to present an
original mathematics problem. Now, at first,
that did not sound too difficult. After all,
I had solved hundreds of math problems before.
But then I realized I had never created one.
Until that moment, math had mostly been something
that I'd received from other people. Teachers
assigned problems, textbooks provided exercises,
and professors explained concepts. And now, I
had to come up with my own problem. And that
was much more difficult than I expected. So after
days of trial and error, I settled on a sports
problem. Imagine you're kicking a field goal.
If the ball is in the center of the field, moving
closer to the goal posts only increases the angle
between them. But if the ball is placed on a
hash mark off to one side, then the problem becomes
more interesting. First, moving closer improves
the angle. But eventually, the angle begins to
shrink. And that means there must be an optimal
spot. A place where the angle is as large as
possible. An optimal perspective, if you will.
Now, as I worked through the problem, I discovered
that the same mathematics also applies to other
situations. Imagine sitting off to one side while
watching television. There is an ideal place
to sit that maximizes your viewing angle. Different
situation, but the same mathematics. Now, what
mattered most was not the problem, but it was
the perspective shift. For the first time, I
realized that all those formulas All those ideas
and relationships that I had learned were not
isolated facts. They were like tools. Tools I
could use to explore the world around me. It
felt like I had discovered a math tool belt that
I had been carrying all along. And suddenly I
started seeing opportunities to use those tools
everywhere. Math was no longer just about learning
concepts for the next exam. It had become a way
of looking at the world, a way of noticing patterns
in relationships, and even possibilities. Now
perhaps that is one of the most valuable perspective
shifts that I've experienced. Not seeing mathematics
as a collection of answers, but ideas and principles
that are a collection of tools. Tools that I
needed to understand and practice and learn how
to use. And the more familiar I became with those
tools, the more connections I began to see between
them. What once felt like separate topics gradually
became a connected whole. Now there was another
benefit that I did not expect. When I started
viewing mathematics as a tool belt, I put less
pressure on myself to perform perfectly. on any
single test or assignment. Now, of course, I
still wanted to do well, but the goal became
larger than a grade. The goal was to add another
tool to my tool belt to become a little better
prepared for future problems. A difficult exam
no longer defined me. A tricky problem no longer
discouraged me as much because those moments
became part of the process of learning. And looking
back, I think that perspective gave me something
more valuable than a good test score. It actually
gave me a more reasonable hope. Hope that growth
matters more than perfection. And hope that today's
struggles can become tomorrow's tools. And hope
that learning is not about proving what we know.
It is about preparing for what comes next. See
you tomorrow.