Hi, Paul Lockhart compares doing mathematics
to climbing a mountain. Boy, do I love that analogy.
Yes, climbing a mountain is difficult. The legs
are tired. The path disappears. We choose the
wrong direction and we retrace our steps. We
may even wonder, why did we start climbing in
the first place? Mathematics can feel the same
way. We might begin with a question, but we do
not know exactly where it will lead. We may try
one path and discover it goes nowhere. We return
to where we start and notice something we might
have missed. And we try again. Eventually, perhaps,
we reach the summit. We find the answer. And
there is joy, not just in the answer, but because
the journey was difficult. But reaching the summit
is not the only benefit of climbing. Along the
way, we become stronger. We learn how to navigate.
We begin to recognize which paths are promising
and which ones may lead us in circles. And when
we finally reach that higher ground, yes, we
can see things that were invisible from below.
Ideas that once seemed unrelated suddenly appear
connected. We go from memorizing, say, the law
of logarithms to understanding why the law of
logarithms works. The landscape has not changed.
We have. And this is why making mathematics too
easy can sometimes take away something valuable.
Of course, we should explain ideas clearly. Of
course, we should remove unnecessary confusion
and make the path accessible. We should not assume
that all difficulty is bad. Think about it. If
someone carries us to the top of a mountain,
yes, we could see the view, but we missed the
climb. We would not develop the strength or the
judgment or the confidence that comes from finding
our own way. There is something similar about
being given a mathematical answer before we had
the opportunity to really explore the question.
Reaching the destination. without experiencing
the journey, is shallow. And perhaps this is
also true beyond mathematics. What I have found
is that some of my most important understanding
comes slowly, through questions, through mistakes,
uncertainty, and even perseverance. I would rarely
choose the difficult path simply because it was
difficult. But when the path leads somewhere
worth going, I have found that the struggle can
be part of the gift. So if you're working on
something difficult today, perhaps the question
is not only how quickly can I reach the answer,
perhaps another question matters just as much.
What is this climb allowing me to see? And more
important, who am I becoming along the way?