Musings on the Chain Rule (Sorry, Newton)

Very, very rarely do I teach something and think, "Wow. That went really well!" In fact, I think it's happened exactly once. And it happened with the chain rule, which for some probably brings back horrible memories of first-year college calculus. For whatever reason, it couldn't have been more different in my calculus classes and I wanted to share my strategy in case someone else might find it helpful.

I was inspired by Think Thank Thunk's use of gears to teach the chain rule and by Sam J. Shah's box method. I started out with a simple gear example. We worked as a whole class because I didn't have enough gears set up for all the kids to play around, but they had actually just finished discussing gears and gear ratios in their engineering classes (yeah, my school's that awesome) so it worked fine.

The moral of the story is that when you compose gears, speeds multiply. They got that. I then invoked a little poetic license to use "gears" as a metaphor for functions and "speeds" as a metaphor for derivatives. They were confused. I don't blame them. But THEN we put it all together and the confusion turned to glee! (Okay, maybe not quite glee, but you get the point...) After asking them what kinds of functions we don't yet know how to differentiate, we came to the conclusion that even though we know how to take the derivative of 2 to the power of x, we don't know how to take the derivative of 2 to some function of x.

So we connected those functions back with when we did function composition at the beginning of the year, and I reminded them that to make a complicated function like f(x) = 2^(6x+3) we had to take a trip to the function factory, where we immediately went to the assembly line where they made f(x)'s. Yes, I actually drew the following picture on the board, conveyor belt and all (I find that when it comes to cheese, go all the way or go home):

The conversation goes something like this: "What's the first thing that happens to x?" "It gets multiplied by 6 and added to 3." "Right, so it goes into the 6 blah plus 3 machine. What does it come out as?" "6x+3" "Then what happens to it?" "It gets raised by a power of 2" "Right, so it goes into the 2 to the blah machine. What does it come out as?" "2 to the 6x+3." "Are we done?" "Yes."


When we go to take derivatives, we'll say stuff like: "The derivative of '2 to the blah' is 'ln of 2 times 2 to the blah'." Although I'm pretty sure this "blah" nonsense is standard language when it comes to the chain rule, it's still kind of funny to see my kids at the board saying things like "2 to the blah." When they parrot those funny things back at me, I have that strange realization that sometimes they're actually listening to what I say, and if they are then it goes down into their notebooks as Calculus with a capital "C". And what would Newton think of "2 to the blah"? Sometimes I worry about these things.

Anyway, we loved the chain rule. By the end of the first class, they were begging me to put a really crazy one on the board. At the beginning of the period, I had made them repeat after me: "I will not be afraid of the chain rule." At the end of the class, they asked why I had made it seem like it would be so scary - it was the easiest thing they had learned all year! I must admit that part of me wished they had to suffer through the chain rule just a *tiny* bit more by learning about u-substitution or whatever awful way I first learned it, but of course as teachers we must resist the urge to do things a certain way simply because that's how we learned them.

How do I foster inquiry?

One of my classes has spent the last month or so on basic trigonometry: right triangle trig, the unit circle, and graphs of trig functions. We have a school-wide spring exhibition coming up in exactly one month, and since trig seems to be one of the few high school math topics that has actual real-world applications that are both authentic and accessible to teenagers (a balance that's difficult to strike) I've decided to do a  project surrounding trigonometry and sound (specifically music).

I've spent the last couple of days just fiddling around with Audacity, a great free audio editor that you can download here. I've experimented with importing entire songs and just with playing single notes or chords on the piano. You can also record yourself, which gets really fun for obvious reasons. In any case, any sound will generate a neat image which, if you zoom in enough, is just a complicated sound wave: 

If you keep zooming in it looks more like a combination of the sine waves that we all know and love, and if you go to Analyze --> Frequency Analysis, you get a nifty frequency distribution of all the different notes in your clip. Even niftier for those of us who dig this sort of thing is that Audacity is really performing a Fourier transform on the sound wave! Gosh I wish someone had showed me this when I took Linear Algebra. Orthonormal, schmorthonormal.

What's cool is that even if you play a single note, like an "A", you don't just get a single peak at 440 Hz (a perfect A). Instead, you get a bunch of overtones that contribute to what you hear on different instruments.Once I had the basics (and I mean, the very basics. I know literally nothing about music) of Audacity down, it was just so much fun to play around. You can generate tones, superpose waves on each other, and create all sorts of nifty fade and filter effects. I wonder about having my kids try to re-create sound waves in GeoGebra by superposing simple waves until they get close to the original complex waveform. What about looking at what happens to a sound wave when it has to travel through a glass of water? Or, recording their voice and a friend's voice and trying to harmonize them by changing the pitch - a little experiment in intervals.

A colleague and I spent a good hour this afternoon recording ourselves saying simple words or phrases, and then playing them back in reverse and trying to repeat the reversed word and then play that back in reverse to see if it sounded anything like the original word - which was a jolly good time. It turns out that the word "potato" is incredibly difficult to do this with because of the little "p" at the beginning which is just a puff of air that is somehow really difficult to capture in reverse. But I digress.

... or maybe I don't. Maybe this is actually the point. I still can't think of a "project". But what I really, truly want is just for my kids to play around like I did, generate some interesting questions or "what if?"s or ideas, and try to tackle them. From experience, my fear is that when I ask my kids to simply play around and write down some observations, they freak out a little. They want you to ask a specific question so they can come up with a specific answer. It strikes me as a little odd that research - for some, that coveted culmination of one's academic career - is almost exclusively about asking good questions, yet K-12 education (and even college, in many cases) is all about getting the right answer to questions that are given to you. 

So my conundrum: I want my kids to experience the "wow" factor of playing around with some neat software and exploring the connection between the mathematics of sound and why we hear things the way we do. I don't want to give them a laundry list of questions to answer because that immediately decreases the coolness by a factor of at least 100. But at the same time, I want them to have enough direction that they don't immediately flounder. Any words of wisdom out there?

Counting my Blessings

Our school is currently going through its WASC accreditation, which means we've had a committee of several outside teachers/administrators strolling our halls, meeting with teachers and students, and poring over a  long "self study" document that our staff has been putting together all year. During one of our WASC meetings today we were praised on what we do, which was really nice to hear - apparently, our students are happy and successful, our teachers work hard to make sure that they learn, and all in all there's lots of great stuff going on at our school. Unfortunately, our scores on the state tests aren't necessarily our best attribute (the reasons for this are a whole post unto themselves, and probably don't even need to be explicitly stated for most of the people reading this). As a result, our AYP, API, etc. are lower than we'd like -- which means that this became a major topic of discussion at the aforementioned meeting because these scores are what the state uses as evaluative metrics.

While the WASC committee was incredibly helpful and generally positive about our school, they really wanted to impress upon us how much we want to raise those AYP/API scores to avoid becoming an "NCLB Program Improvement (PI) School." I was incredibly curious about what this meant, so I came home and looked up the requirements for PI schools here. Both committee members at this meeting were able to speak to some of the consequences of becoming a PI school - having state-appointed administrators come in to implement programs, being forced to read lesson plans that have been scripted down to the minute, and even - yes, I could not believe it - BANNING NOVELS IN ENGLISH CLASSES. Folks, I was floored by this. True, no one's "banning novels" explicitly. But apparently PI schools are required to cover so many writing and reading samples in English classes that it's virtually impossible for teachers to teach entire novels, and those who do receive threatening memos from their administrators. I realize that struggling schools need change, and that change isn't always easy, but - for goodness sake - does anyone out there really, truly think that preventing our kids from reading novels is going to make them more educated??? Just like a bad teacher can "get in the way" of a beautiful subject, it seems that the government have managed to "get in the way" of education. I'm 100% for accountability at all levels (school, teacher, student) but I also believe that this accountability happens locally. And certainly not by banning novels. There are very few things that shake me out of complacency in my old age (ha) but this really did it. What can I do to change the status quo?

I left the meeting a little shaky (which I believe was the intention) but also feeling incredibly lucky. Lucky to work at a school with people who are so passionate about education and about constantly refining their practice. Lucky to have the freedom to experiment with new ideas and to make changes in my classroom as I see fit. I always knew I was lucky to begin my teaching career at such an amazing school, but today the point was really hammered home.