They start to see the need to invent these kinds of functions. The idea that ln(x) is a human invention necessary for the solution of a peculiar little problem gives them some ownership where there was once only mysticism. I like that.
The journey through calculus has taken us through the basics of integration. We last left off with numerical integration as a technique for handling definite integrals that are analytically challenging. We now move into the nether region of traditional calculus. We begin studying all of those sundry things that are for some reason a part of the standard calculus 1 curriculum. The first, and arguably most important hurdle, is the concept and application of the magic number e. An almost ludicrous amount has already been written about e, so I’m going to launch right into its calculus applications. Namely, the Natural Log and Natural Exponential functions. No two other functions strike as much fear into the hearts of the young’ns.
Cornally runs through his logic gauntlet, “Ok, I have to teach about this because it is in my curriculum. I have to teach about logarithms because they give kids conniptions, and they’re an integral part of the mathematical structure of many scientific disciplines.” However, this real-worldiness sometimes gets to me. I’m not addicted to it, as any addiction will ultimately end up breeding ridiculousness. The idea here is to motivate with whatever it takes. I don’t want to give my style an edu-jargon name, because as soon as that happens it begins to die the slow painful death that is canning. What I’m trying to say is that lessons should be driven by genuine human emotion. e.g.: Do we need this technique to solve an interesting problem? (The Dan Meyer Method) Or, does this math show something beautiful about mathematics and human thought? (a la Blum-Smith) This sometimes requires real-worldiness, but sometimes things are just interesting (astronomy), or beautiful (fractals).
This is the case with the Natural Logarithm: Our simple algorithm for finding antiderivatives fails in one case. This was terribly over-looked when I took calculus in college, but the more I thought about it, the more I realized I had a great hook on my hands. This is totally obvious to anyone that plays calculus for fun, but hey, kids by definition don’t. By reverse-power rule:
As with most of my HITC:AC (do you understand why we can’t have nice acronyms!?) you either totally see through what I just wrote or you just skipped that math entirely. Here’s the skinny: The inverse of x cannot be integrated in the usual way because it results in an nonsensical answer (1/0). This befuddles the average calculon. Most students just hope they never get presented with this problem on a test. This is a sentiment you must fight. This hope, that by some magic they won’t have to deal with some strange math idea is a hold over from their summative-obsessed past. (obligatory SBG nod) They don’t see math as a series of genuine mysteries; they see it as a sad funeral march of things they either have memorized or don’t. I can practically hear the dirge now — the violas are out of tune — and that pisses me off.
The integral above is a problem I present to them for the sake of math. Lockhart is living in me somewhere. How can we solve this, I patter. What’s even the issue? The kids vocalize, and in a beautiful chorus I get a deep understanding of what they know in a way that a standard problem could never provide. Some don’t even realize that this is an issue. Some are truly bothered the way I want them to be. This function surely has area beneath it, but yet no antiderivative to help us find it? These kids help scaffold the former type in a way I never could.
I order them to try anything. They start drawing. They start making slope-function type connections. 1/x is the derivative of some unknown function, but who?
This function is reporting slope values. Who, if anyone, starts of with hugely positive slopes and then increases forever, but yet slows. This function is also undefined at zero, because its derivative show infinite slope in that region. They start drawing, and a surprisingly large number come up with something along the lines of natural log!
This is where the inquiry train has really come to its natural end. I’m not harboring any delusions that kids will follow Napier to this thing’s natural end, and I also don’t really have that kind of time. The traditional lesson on natural log and its connection to 1/x begins. However, now that the students have thought about it from this perspective, it doesn’t take nearly as much song and dance at the ol’ whiteboard from me. They start to see the need to invent these kinds of functions. The idea that ln(x) is a human invention necessary for the solution of a peculiar little problem gives them some ownership where there was once only mysticism. I like that.
A Green Analogy For Teaching Logs/Exponents:
We haven’t even mentioned e, yet! The way I present e is one of the only little tricks I’ve come up with on my own (I think). I like to talk about the family tree of operators. Each has its partner, which both complements and undoes its significant other. Plus has minus, division has multiplication, and exponentiation has the logarithm. I use a silly little analogy of “logging.” If something is logged (lnx), our conservationist friends instruct us to plant a tree. That is, exponentiation (up in a tree, yea, I know you’re laughing). The same reverse logic applies. If you want to retrieve something from a tree say this here x:
You must use the correct saw (or logging operation, I mean you can’t use the same machines to fell soft pine as you can old-growth oak, that would be right silly!) to cut down that tree and retrieve your stranded feline x, you must use the logging operation appropriate for that tree! Take a think about it, you’ll find the analogy richer than you think.
Ok, enough of the silly analogy, but hey, silly analogies are the secret indulgence of most high school math teachers, and I’m no exception.
There’s way — W A Y — more to natural log and e forthcoming.
Spoiler Alert: IGOTAHIGHSPEEDVIDEOCAMERA! (omg)