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!

Student self-teaching: results

The practicum: see a few posts back, but it's basically "group quiz with a twist", namely, after the class does the group quiz together, one person selected at random has to get up in front of the class and explain how to do the quiz. The prize is no homework for the whole class.

This went surprisingly well, but not exactly as planned (surprise!). By treating the first part of class like a real quiz, with everyone working on a what looked exactly like a quiz sitting right in front of them, I think we really got everyone's attention -- in fact, despite my explanation I'm sure some students took a while to realize it wasn't a real quiz. The promise of no homework helped as always, which I still find shocking considering how few students turn in the homework anyway.

Nevertheless, I think that ten minutes of confronting a problem you have no idea how to do while the class is silent all around you can have an illuminating effect on the limits of your knowledge. One benefit of a practice quiz during class time is that you experience the sinking feeling you get from taking an exam you weren't prepared for, but before you've actually done any damage to your grade. Thus, sinking feeling of despair transforms into motivation, hopefully. A glimpse into the fiery depths, if you will, but with a chance for salvation immediately afterward.

Anyway, the first not-as-planned part came during what was supposed to be the self-teaching section. As it turned out, it ended up being more of one-on-one teacher-teaching, with little helpful student-student interaction. We rolled with this, though, and just devoted more time to helping everyone get their questions answered. In retrospect, I think I should have more actively encouraged the students I could see getting it to help their classmates.

Once George and I explained to everyone how to do the problems (without actually giving answers), the class voted on the answers to each problem. There were some disagreements on some problems, but the majority of the class that voted was usually right, and both 2nd and 4th hour were able to democratically arrive at all 5 correct answers.

There was also a lot of shyness in the second half of the practicum, where one person picked at random was supposed to come up and do the quiz on the overhead now that we'd gone over everything together and voted on the answers. I had a student pick a name from a hat at random in each class, and in one class that random person flat out refused to participate. It wasn't a belligerent thing, either, just an eyes-downcast, mumbling, embarrassed cop-out from a student who usually won't shut up! This was a good reminder of an important point. These kids try to act all tough and invincible and too-cool-for-anything, and it's all just as much of an act now as it was when I was their age -- with all that hard exterior they are just marshmallows inside!

I ended up taking volunteers in that class, one person for each of the five questions. Several students were really itching to get up there and show that they knew how to do the problem, and it's hard to refuse that kind of enthusiasm.

So, all in all it wasn't a disaster in terms of people bouncing off the walls or hanging from the ceiling. But did it work? Was it worth doing this versus the usual pre-quiz review of similar problems on the board?

By the numbers, correcting for students who missed the Thursday practicum and students who did not take the quiz at all, class averages were up slightly (about 0.5-1 points out of 15 total) compared to the most recent quiz. Now, is that meaningful? Perhaps the subject matter was just easier that week than it had been the week before. This would be a good problem for the stats classes in 1st and 3rd hour, but I think the answer would come down to insufficient data to draw a statistically significant conclusion.

Anecdotally, yeah, it was seemed like it was working. George and I talk a lot about what I should try to bring to the classroom, whether I'm presenting, just sitting in and helping out, or planning an activity and teaching like this. We've come to a conclusion that one of the most important things I can do is shake up the routine that the students are used to. If they're used to coming in and tuning out because they know what's coming and they don't like it, then anything I can do to rattle their cages and get them engaged in something new is worth it. And, judging by the increased attention span during the practicum relative to their usual quiz review, I'd say that's about right.