Kyle Peterson is a Professor of Mathematics at DePaul University.
Let $n$ be an integer greater than one. Since $n$ and its successor,
$n+1$, are relatively prime, their product, call it $n_2 = n(n+1)$,
has at least two distinct prime factors. By similar reasoning, the
product of $n_2$ and its successor, say $n_3 = n_2(n_2+1)$, has at
least three distinct prime factors. Continuing in this way, we can
construct, for any positive integer $i$, a number with at least $i$
distinct prime factors. Hence the set of primes is not finite.
Not a bad little argument, eh? It's my favorite proof of the
infinitude of the primes, and I first read it on the homework turned
in by one of my students, whom I'll call Sara. If you are a regular
reader of American Mathematical Monthly, you may remember this
argument from a note by Filip Saidak in the December 2006 issue, two
months after Sara handed in her homework.
Sara was an average student, certainly not the ``best" of the class. I
had expected her, like many of her classmates, to construct an
argument similar to Euclid's classic contradiction argument, since
that's what my carefully chosen sequence of problems pointed to. (Or
so I thought.) I remember being so floored by what I read in Sara's
paper that I initially thought there must be some error. It was so
different from what I was expecting to see! But no, it was correct,
and I immediately ran across the hall to share it with a friend.
This is the potential of an IBL class. Given only the necessary
preliminaries, along with some time and space to think, Sara had come
up with something truly novel. It was just a month or so into my first
semester of IBL teaching, and, like the hack golfer who hits a
hole-in-one, I was hooked. I had never had a student so thoroughly
surprise me, and I wanted to experience more of those surprises.
I've been teaching IBL for about five years now, and while a gem of
that magnitude is rare, I find smaller surprises happen on a regular
basis in my IBL courses. For one thing, just because a student comes
up with a proof, the way they arrive at the proof often takes its own,
fascinating path. Other sorts of surprises include the time a student
who doesn't seem to be paying attention pipes up to point out a flaw
the rest of the class missed. Or when a student who would rather die
than go to the board in the first few weeks leaps out of her chair to
go to the board in the last few weeks. People laughing and smiling...
in a math class! Rather than: What do I have to talk about today?, I
walk into the room thinking: What will the kids show me today?
My favorite overheard conversation last year:
Student 1: ``So I think we have a bijection. Do we have a bijection?"
Student 2: ``I don't know...(mumble, mumble)... Wait! Yes!"
Student 1: ``We do?"
Student 2: ``Yes!"
Student 1: ``Yes!"
-Slap!- (the students give each other a high five)
Not every IBL course will necessarily produce a revelation like Sara's
proof, but they all produce everyday miracles that will delight and
surprise you. Using a lecture-only format pretty much guarantees that
your students won't surprise you, since they will be doing
their best to mimic what you do. To paraphrase something I've often
heard Ed Parker say: Why should we limit our students to what we know?