Eddie* came after school because he wasn't happy with his 6/10 on graphing log functions. I asked him to, without a calculator, sketch a graph of y = 2^x and y = log_2(x), then state the domain and range of each.
He did an admirable job sketching the graphs. He made a table of x vs 2^x, inverted them to make a table for the log function, and plotted points appropriately, connecting them with a smooth curve. An all around serviceable graphing effort.
Then for Domain and Range for both, he wrote "All real numbers." (Dun Dun Duuuuuun.)
Eddie, I say, your graphs are beautiful, but tell me how you decided on the domain and range. Oh, well, he says, I didn't really know. I just guessed, because sometimes it's all real numbers. OK Eddie, Domain refers to possible values of x. Look at your graph of the log function. What x's are not on the graph? Easy, he says, the negative ones.
Why should that be? Why can't you take the log of a negative number? Right now I'll give you an 8/10. You can have a 10/10 when you can tell me, in your own words, why the log of a negative is undefined. Invite its inverse function to the party. I hear he's a big help. This is not that difficult, you will just need to focus on the question for a little while. Maybe 10 minutes of concentrated effort. You are capable of figuring this out.
Eddie takes off down the hall and returns, breathless, 5 minutes later, bursting into my room Kramer-style, "Miss Nowak! You can't take the log of a negative because it's impossible! You just can't! Because it's not in the domain." Did he talk to someone in Precalc? Another teacher? I wonder. "I'm aware of that Eddie. I want you to tell me why it's not in the domain."
Eddie comes back a few minutes later, iphone in hand, and starts reading wikipedia at me. It was a drive-by Walesing. "EDDIE! No points for reading crap that you don't understand verbatim off wikipedia! Here's a secret, Ed: I ALREADY KNOW THE ANSWER! I'M NOT LOOKING FOR ENLIGHTENMENT HERE! I WANT TO KNOW THAT YOU KNOW! You have just spent 20 minutes searching for an easy way out, when in 20 minutes you could have figured it out for yourself! You can do better!"
I really hope that I go into school tomorrow and can write a happy ending to this story. Everyone send calming, logical brain waves Eddie's way.
Confrontations like this make me wonder if the "They can just look it up on their iphones" crew really know what they are saying. Knowing the domain of log(x) is not tops on anyone's list of crucial life skills...that's not my point. But insisting Eddie understand the why will make him re-investigate the behavior of the exponential function that is its inverse. And exponential functions are slapping you in the face every time you turn around, whether you realize it or not. Open your credit card statement? Slap. Open up a 401k? Slap. Nuclear waste will never totally go away! Slap. Your repairables are depreciating? Slap. The effects of one-child policy in China? Modeling the adoption of disruptive technologies? (Yeah, Disrupting Class includes graphs on a log scale.) The Richter Scale - Slap. The point is we spend our lives (incoming: barely coherent analogy) in an intellectual ocean - facts float in and float out - sometimes we can grab onto a mechanized vehicle that will move us around faster, sometimes we are provided with a guide, more often, not. We can increasingly get our hands on any discrete bit of information we want. But to do anything with all this information, we kind of need either direct experience with the ocean floor or very good maps. I'm in the underwater expedition and map-making business.
*Eddie is not his real name, of course.