Hi, when I studied to become an actuary, I did
not have access to all of the educational tools
that we have today. There were no online videos
or manuals or practice problems or discussion
boards. There definitely was nothing that adapted
to my level of understanding. Usually what I
had was a textbook, a set of problems, and maybe
a solutions manual. When I was stuck, I usually
remained stuck for a while. I would reread the
problem. Maybe I would search through earlier
chapters. I would try an approach, and then I
would realize it did not work, and then I would
start over. Sometimes I might spend hours trying
to understand a single idea. Now, at that time,
I'm not sure I considered all of this a gift.
But in some ways, I think it was. Because an
explanation was not immediately available, I
had to learn how to ask my own questions. What
do I know? What am I trying to find exactly?
Which assumption am I making? Where exactly does
my understanding break down? Here's a specific
example. How does the force of interest relate
to the force of mortality? These questions taught
me more than the formulas did. So, 20 years later,
I helped create tools that made it easier to
understand actuarial concepts. I believe good
explanations are valuable. The right hint or
the right illustration can open a door that might
otherwise remain closed. But I have also come
to recognize a tension. The tension is if help
arrives too quickly, it can take something away.
We may learn to follow someone else's reasoning.
without developing our own. We may know which
formula to use without understanding why it works.
We may become so conditioned to immediate answers
that we interpret confusion as a sign that something
has gone wrong. But confusion is not always a
problem to eliminate. Sometimes confusion is
the beginning of learning. Not knowing creates
space for curiosity. It forces us to slow down,
to notice what we have overlooked, and to form
questions that we would never ask if the answer
was already there. Now, this does not mean we
should try to withhold help indefinitely or make
learning unnecessarily frustrating. It just means
that good help should not merely rescue us from
the struggle. It should help us to participate
more fully in it. There is a difference between
being abandoned in confusion and being given
room to think. Mathematical perseverance grows
in that space. So the next time you become stuck,
Perhaps you do not need to escape the feeling
right away. Simply pause. Try to describe what
you understand and where the mystery begins.
What I have learned is that sometimes the most
valuable thing a teacher can give is not an answer.
It is enough time and enough confidence for you
to continue the search.