PROFESSOR: Two goals really-- one
is to introduce students to an area
of mathematics that really isn't covered much in the high school
curriculum.
I think there's a lot of fascinating problems
in combinatorics and probability that kids
don't see much when they're in their earlier studies,
and they're extremely relevant.
I guess there's sort of three main reasons I give students
at the first day of the course as to why combinatorics
is an interesting subject.
The first is something that I guess has always been true,
but we've only really become aware of it
in the last 50 years-- is the universe
is sort of fundamentally discrete and digital,
whether you're talking about quarks inside atoms,
or electrons in orbits around atoms,
or the way molecules are built, or the DNA in our cells.
All of these things are made up of discrete structures,
and to understand how they work and how they interact,
we need to be able to understand how these pieces fit together.
So whether you're a physicist trying
to understand subatomic particle collisions
or a chemist trying to understand the patterns
in the periodic table, combinatorics and combinations
of objects are an important part of that process.
Building on that, and one of the major themes of the course,
is probability theory, and sort of any question
about probability is really two questions about counting-- sort
of how many things could possibly happen,
and how many of those things correspond to what
you're interested in?
And if I were to pick out one piece of mathematics
that I think everybody should know,
whether they're interested in math or not,
it would be sort of a basic understanding of probability
theory.
I mean, we spent a lot of time making
kids memorize the quadratic formula, and learn trig ratios,
and various other miscellaneous material that they are probably
not that likely to use in their everyday life
if they don't end up going into mathematics.
But probability theory is pretty fundamental to the decisions we
make everyday in our lives, especially big decisions
like how we invest our money, or decisions about our health,
or other decisions that have a big impact on how we live.
And a lot of people either never learn probability
or are very confused by it, and I
think that's an important skill for people to learn.
And then sort of the third thing,
for the people who are interested in mathematics
and combinatorics-- which tends to be the people who come
to MIT to take these courses--
is to expose them to some interesting and challenging
problems that, I guess, open up some areas
of the mathematical universe that they may not have known
existed.
A lot of the students that I see are, you know,
kids who were at the top of their class
or who were bored in their math classes.
They know everything that's in the textbook,
and I had much the same experience
when I was in high school.
And it's pretty easy to go through--
there's sort of a linear progression of math courses.
You know, take algebra, geometry, trigonometry,
calculus-- and view it as sort of a linear sequence,
and think that eventually you'll get to the end of that sequence
and know what there is to know.
But in fact, it's a much broader subject,
and there's a huge amount of material out there
that is very accessible to high school students
but is never incorporated in the curriculum.
I didn't really learn any serious combinatorics
until I was a sophomore or even a junior at MIT,
but I found the subject fascinating.
And I think it's very accessible and useful to younger students.