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.

"*Eddie is not his real name, of course."

ReplyDeleteOf course. His name is Edward.

On a more serious note, I have to say that your blog is one of my favourite things to read nowadays. I very much enjoy reading your posts, every time you tell a story like this one or anything providing a small window into your world (such as when you explained what's your favourite theorem and why). You're sharing this bits of reality I can connect to and learn from and relate to my own ideas, and I appreciate that.

I'm just a college student with a healthy interest in the education process, and hope to teach someday. Your blog showed up one day in my Google Reader recommendations, and frankly it's been the best recommendation I've received from it.

Keep up the good work, and thanks!

*twitch* The third paragraph from the end needs a second pair of quotes. Sorry for inflicting my OCD on you.

ReplyDelete"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." Nailed it, Kate! Your experience described here needs to be shared with so many more. Yes, we have information at our finger tips, but not knowing how to use it is another story. For example, I can read about a great lesson or idea on your blog, but to put it to use (and know when/where to appropriately use it) is another story. It's the classic climb of Bloom's taxonomy. I look forward to a hopefully happy ending to this story.

ReplyDeleteOops! Forgot to add that I really appreciate the way you helped Eddie think through his understanding of the problem...from positive feedback on the graph to further prompts for true understanding of the domain/range.

ReplyDeleteA follow-up question: did you require this same level of understanding of the class as a whole on the initial assessment? If so, was it a mere, "explain how you know" type of prompt or some other method? -Matt

@lb - Thanks for reading. Lately I feel like I'm just kind of ranting. I'm glad you find it worthwhile.

ReplyDelete@Oscar - Thank you.

@Matt - On the assessment in class, there was a question like "State the domain of log(2x-5)" so they had to apply this knowledge. I frame my after school make-up grades as: in order for me to change their grade, they have to prove to me they know what is up. So it does end up being more rigorous than I can practically demand from 120 students all the time. I'm ok with that. If they want to avoid the ordeal, they could just study more and get full credit in the first place.

Also fun to note: TI calculators biff horribly the graph of the logarithm, so students have to genuinely understand what's going on with the asymptote if they're relying on one.

ReplyDelete>Confrontations like this make me wonder if the "They can just look it up on their iphones" crew really know what they are saying.

ReplyDeleteOh God thank you thank you thank you.

I just got that excuse from a colleague as we were discussing how to set up the final for our district's algebra 1 class.

Reading that (on the district discussion board) gave me a feeling akin to heartburn.

I am NOT an old fogey for sticking to the idea that you should be able to do simple integer operations sans calculator. (I'm only 27!) If you don't know how, the answer (dun dun DUUUN) is to learn it. I can help.

I have allies! Allies in this fight!...Just none teaching algebra at my school ;-/ Oh well.

Too cool!! I love this! WHAT A TEACHING MOMENT! (applause applause)

ReplyDelete"Here's a secret, Ed: I ALREADY KNOW THE ANSWER! I'M NOT LOOKING FOR ENLIGHTENMENT HERE!"

ReplyDeleteFantastic.