What is Problem Solving?

by Richard
Rusczyk*

I was invited to the
Math Olympiad Summer Program (MOP) in the 10th grade. I went
to MOP certain that I must really be good at math… In my five
weeks at MOP, I encountered over sixty problems on various
tests. I didn’t solve a single one. That’s right – I was
0-for-60+. I came away no longer confident that I was good at
math. I assumed that most of the other kids did better at MOP
because they knew more tricks than I did. My formula sheets were
pretty thorough, but perhaps they were missing something. By the
end of MOP, I had learned a somewhat unsettling truth. The
others knew fewer tricks than I did, not more. They didn’t even
have formula sheets!

At another contest later that summer, a younger student, Alex,
from another school asked me for my formula sheets. In my local
and state circles, students’ formula sheets were the source of
knowledge, the source of power that fueled the top students and
the top schools. They were studied, memorized, revered. But most
of all, they were not shared. But when Alex asked for my formula
sheets I remembered my experience at MOP and I realized that
*formula
sheets are not really math*. Memorizing
formulas is no more mathematics than memorizing dates is history
or memorizing spelling words is literature. I gave him the
formula sheets. (Alex must later have learned also that the
formula sheets were fool’s gold – he became a Rhodes scholar.)

The difference between MOP and many of these state and local
contests I participated in was the difference between problem
solving and what many people call mathematics. For these people,
math is a series of tricks to use on a series of specific
problems. Trick A is for Problem A, Trick B for Problem B, and
so on. In this vein, school can become a routine of ‘learn
tricks for a week – use tricks on a test – forget most tricks
quickly.’ The tricks get forgotten quickly primarily because
there are so many of them, and also because the students don’t
see how these ‘tricks’ are just extensions of a few basic
principles.

I had painfully learned at MOP that *
***
true mathematics is
not a process of memorizing formulas and applying them to
problems** tailor-made for those formulas.
Instead, the successful mathematician possesses fewer tools, but
knows how to apply them to a much broader range of problems. We
use the term “problem solving” to distinguish this approach to
mathematics from the ‘memorize-use-forget’ approach.

After MOP I relearned math throughout high school. I was unaware
that I was learning much more. When I got to Princeton I
enrolled in organic chemistry. There were over 200 students in
the course, and we quickly separated into two groups. One group
understood that all we would be taught could largely be derived
from a very small number of basic principles. We loved the class
– it was a year long exploration of where these fundamental
concepts could take us. The other, much larger, group saw each
new destination not as the result of a path from the building
blocks, but as yet another place whose coordinates had to be
memorized if ever they were to visit again. Almost to a student,
the difference between those in the happy group and those in the
struggling group was how they learned mathematics. The class
seemingly involved no math at all, but those who took a
memorization approach to math were doomed to do it again in
chemistry. The skills the problem solvers developed in math
transferred, and these students flourished.

We use math to teach problem solving because it is the most
fundamental logical discipline. Not only is it the foundation
upon which sciences are built, it is the clearest way to learn
and understand how to develop a rigorous logical argument. There
are no loopholes, there are no half-truths. The language of
mathematics is precise, as is ‘right’ and ‘wrong’ (or ‘proven’
and ‘unproven’). Success and failure are immediate and
indisputable; there isn’t room for subjectivity. This is not to
say that those who cannot do math cannot solve problems. There
are many paths to strong problem solving skills.
**Mathematics
is the shortest**.

Problem solving is crucial in mathematics education because it
transcends mathematics. By developing problem solving skills, we
learn not only how to tackle math problems, but also how to
logically work our way through any problems we may face. The
memorizer can only solve problems he has encountered already,
but the problem solver can solve problems she’s never seen
before. The problem solver is flexible; she can diversify. Above
all, she can
**create**.

**
***Source: Art of Problem
Solving** *

** Richard Rusczyk was the winner of 1989's
USA Mathematical Olympiads
(USAMO) and the
founder of
Art of Problem Solving (AoPS). *