I'm going to begin by stating some standard-issue frustrations, some non-standard-issue frustrations, then I'm going to reflect on three ways I could teach exponential functions and logs.
First, the personal baggage: This is my first year teaching, I just got out of college, I've never taken a class in education, and I'm teaching at a yeshiva high school in NYC with 3 preps.
What this means is that I'm feverishly thinking about the entire three-year math sequence in New York. I'm really committed to teaching this stuff in a way that reveals its true value and combats my students grumpiness about math. But because I'm teaching at a yeshiva that means that I also have way less time to teach the same curriculum as everyone else does. The average public school has 180 days for teaching math--I have 124.
That means that I'm going to do a worse job in the classroom, and there's just nothing that I can do about that. What's lost in those 56 days is so much of the context and the meaning behind math. (Or, to put this another way, Sam Shah spends 13 days on exponential functions?!)
What does that mean for me? It means that I can't spend three days on exponential functions, including continuous growth. NY State gives me one day, more or less.
Which brings me to the decision: how do I teach it? I have three opposing inclinations.
1) Teach them science. Teach them population growth, or--even better--simple differential equations. Teach them dynamics! Get them to understand how these things are actually used in the world everyday by people--that is, by scientists.
PRO: It's true. They'll believe it and appreciate it.
CON: It'll take too long, both for me to prepare and also for them to understand. It isn't a standards-efficient activity.
2a) Teach them finance. True, you don't need to understand exponential functions to operate in the real world. But it'll help you understand credit cards and APR . I could start with exponential growth in general and then move into continuous growth.
PRO: The kids will like that it has to do with the "real world." "Oh, this stuff is actually useful!" Also, the amazing blogo-verse has already provided me with worksheets ready to go. In addition, it's a bit more standards-efficient then what I would cook up.
CON: When kids say "This stuff is actually useful" they're talking about today's lesson, not Algebra II. And they mean "Unlike everything else that we've learned." We have to be careful to distinguish "real-world" and "everyday-life." An example is "real-world" if people use it to understand the world. By this test, almost everything in Alg2/Trig passes. Almost. (I'm looking at you absolute value inequalities...) But very few of the "real-world" applications are "everyday-life" applications.
2b) Teach them this stuff: http://www.mathalicious.com/?cat=98. I forgot about mathalicious.
3) Teach them problems. Rules. Methods. Algorithms. This is what they're used to, but they find it boring and I find it SUPER boring.
PRO: I won't fall farther behind the pace.
And this is my choice every time I sit down to figure out what to do in the classroom. It's a fight between science, the everyday, and, *sigh*.