If I tell you one thing, kid, it's this: Know your audience!

Well, I've had good days, and I've had flops. This particular incident falls in the flop category, and the lesson is simple: know your audience.

I gave a talk on the Fibonacci sequence, which is the recursive sequence where every number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. There were some cool problems you can look at with this sequence, including some combinatorics puzzles, the old rabbit breeding problem of the Fib-meister himself, and lastly some interesting examples of the sequence showing up in plant patterns in the natural world, called phyllotaxis.

So, to review, I gave a talk about an abstract sequence of numbers and how it connects to flowers. In retrospect, I should perhaps have seen this one coming. It was just about impossible to keep anyone's attention during the talk, possibly because the combinatorics puzzle I led off with was just too hard. Normally a solid opening can land you a good free five minutes of attention span -- instead, I gave them this puzzle:

There are five steps leading up from my apartment door to the main entrance to my apartment building. Since I hurt my knee (true story!), I can only take steps one at a time, leading with my good leg. If I'm in a hurry, I can take two steps at a time, still leading with my good leg. However, sometimes in the morning I don't think of it right away and I'll take one step at a time for the first couple steps, then start taking two at at time. Or, I'll get tired or my leg will hurt so I'll take two at once at first, and then just take one at a time the way to the top. How many different ways could I take one or two steps to the top of the stairs?

I thought this was a pretty darned cool puzzle, since the situation is basically a true story and I was in fact doing exactly this on many morning while my knee was still in really bad shape. The key is to think about how you take the last step to the top -- you either reach it taking a single step, or a double. That means that really the number of ways to climb five steps is really the sum of the different ways to climb three steps or four steps. If we think of a sequence of numbers, where the n-th number denotes how many unique ways there are to climb a set of n steps, we've just proven that the n-th number is the sum of numbers n-1 and n-2. Sound familiar? That's right, it's all tied into the Fibonacci sequence, and the solution to the puzzle above is F(6) = 8 unique ways. (Note: F(1) is the number of ways to climb zero steps -- you just stand there. Just one way to do it. That's why F(6) is the number of ways to climb 5 steps.)

So that went over like a lead balloon. I skipped over the rabbit breeding problem at this point, because that one is ever trickier, and instead I went straight to the plant stuff -- I figured pretty pictures with neat designs might re-capture the lost attention of my students. But, they were pictures of flowers, pineapples and pinecones. If they were pictures of rockets, maybe. Pictures from video games, sure thing. Plants no. So, despite an array of neat images showing patterns in how the Fibonacci sequence describes flower petals, pinecone ridges and many other things, I got no traction at all.

I did fare a little better with an activity where we drew out a simple model of a nautilus shell, which can be modeled by drawing a sequence of squares with sidelengths given by the Fibonacci numbers (see here). I thought this sort of break from math-type stuff and detouring into a more geometric application might appeal to some of my students who I've seen doodling occasionally. I actually think it's a shame that geometry and probability aren't more integrated into the algebra classes like they were in the AlGeoStat sequence of classes in my high school (Plymouth-Canton School District). Frankly, basic algebra is really boring until you learn enough to start looking at more interesting problems, and geometry and intro stats bring a little bit of spice into an otherwise bland offering.

Anyway, the reason the Fibonacci sequence actually spits out something that looks like a nautilus shell is because the limit of the ratio of successive Fibonacci numbers is a constant value (in fact it's the golden ratio phi = 1.61....), so you're really approximating a logarithmic spiral. The upshot of this is that the spiral is self-similar at every scale, so a growing baby nautilus will grow such that it always fits in its shell and doesn't need a new one. While we discussed that effect, going into the math reasoning behind it in class would have been straining credulity at best, and by this point I was belatedly remembering that very important lesson:

Know. Your. Audience!