Despite an extra day to try to think things through I remain confused about my own approach to teaching / talking about exponential functions. I’m actually struggling to even understand what the struggle is. After all, running across this fun little connection between and as a college freshman is what convinced me to major in math:
As I mentioned in the first post in this series, my approach to teaching exponentials has not been nearly as formal as my approach to teaching arithmetic. We began by talking about powers where I introduced exponents as essentially a time saver. Probably like just about everyone who has ever talked about exponents, one of the early conversations was about the zeroth power and negative powers. That talk about exponents with my younger son remains one of my favorite math conversations he and I have ever had:
Feels as though you are almost forced to introduce integer exponents early one if you want to talk about place value or different bases or other similar topics, but the path to exponentials from here just isn’t that satisfying to me. You’ll have to introduce fractional powers and then define non-rational powers by some sort of limit process (at least if you want to approach things formally). If you are going to bed at night fearing a Grant Wiggins-like “conceptual understanding of exponential functions” exam, you probably won’t like this path at all.
There are fun topics, though, so I’m not suggesting that integer or rational exponents are a waste of time. Two of my favorite topics here have been finding a formula for the Fibonacci numbers while we were studying quadratic equations:
and talking about Graham’s number, which is one of the most fun math activities that we’ve ever done, and probably as much fun and excitement with integer powers as you are ever going to have. It took me a week to figure out how to put this one together (and I stopped after a week, because there was no way I was going to figure out how to do it anyway!)
With all this background I’m kind of surprised that I can’t really think of a nice, easy transition from exponents to exponential functions. As I was riding home last night I tried to keep a look out for anything I saw that I naturally thought of as being associated with an exponential function – something / anything that kids might see occasionally in their life. I couldn’t find a single thing which made me a little sad. Maybe I’m just not being creative enough.
In yesterday’s post I mentioned a few things from finance and probability where exponential functions appear pretty naturally, but those are well outside the realm of things that kids see or worry about. What I didn’t mention was a different field where exponentials play an incredibly important role – physics. Representing waves in the form is pretty convenient, to say the least, but again is way outside of what might be reasonable examples for kids.
So, I’m lost. A non-formal approach starting with integer exponents does let you talk about some really interesting problems, but doesn’t really seem to set you up too well to move to exponential functions in general. I’m a little frustrated at my inability to find any great (or even reasonable) natural exponential examples to share with kids. And, to top it all off, starting with a formal approach like defining as the limit as n approaches infinity of just seems stupid.
The perplexing thing is that both and ln(x) play such incredibly important roles in math. You’d think that there would at least be a few easy examples you could talk through with kids to introduce / motivate these ideas. I mentioned in yesterday’s post that I gave one formal approach a try. That was in response to a question I saw about logs on twitter when I happened to be talking about them with my older son. That question motivated me to throw together a fun overview of some of the areas in math where logs are part of important results. I wasn’t expecting my son to get much of anything out of it other than to see some really amazing math involving prime numbers. That blog post is here:
Tomorrow, or over the weekend, I’ll try to come out of the fog and write about what I’d like my kids to learn about logs and exponentials. Hopefully I’ll have it all figured out by then. Ha!