Monthly Archives: January 2012
The Space Between the Numbers commented on my last post that some people call launch problems like the infection one I did for concavity “Anchor Problems,” the idea being (and I quote from her) “you can keep referring back to the problem to help students latch onto the new learning by remembering this solid, relate-able context.” I think that this describes what I was trying to do perfectly, and I love having new language to talk about my teaching. So, thanks for that!
I wanted to share one more Anchor Problem that I am using in my AP Class tomorrow to introduce Differentials (which I used to what I thought great success in my non-AP class last year). I am relating the idea of making a prediction with differentials/a tangent line to making stock predictions. With charts like this:
… I’m going to have them predict the stocks price in the near and far future using the graph in any way they can. Most students last year figured out to draw a tangent line, use the slope of that tangent line to see how fast it is changing currently and then multiply that by the number of months to get the change in price, and then add that to the original price. It’s an intuitive idea that sounds way more complicated when you try to describe it. I added the “% confidence column” this year to try to get at the idea of a prediction being less and less accurate the further you are from known data.
Eventually, I want us to codify our process into a rough equation like this:
Which we can then use to look at the “equation” for using differentials:
(I think my face is crooked… I always write slopey)
This worked well last year because even when we were solving abstract problems that had nothing to do with stocks, I would ask questions like “well, how fast is your stock changing right now?” and “how many months in the future or past are we predicting?“ which served to connect the abstract to the intuitive situation. I also think it gave students a mental picture of what they are doing.
Hope it works this year too…
One of the problems I have with math instruction that goes CONCEPT-PRACTICE-APPLICATION is that you miss out on some great opportunities to teach difficult concepts using application. When I introduced concavity this year, I did a great 10 minute activity that paid huge dividends when discussing what can become the somewhat tricky conceptual math of the second derivative and inflection points.
We got back from break this past Sunday, so I started just by asking what everyone had done for break. I had a secret agenda for this though, because from their answers I chose the person who went to the most exotic or most random place and chose them as Patient Zero for an infectious disease that was going to infect our classroom. I assigned everyone a number (I used a deck of cards, but I have done this before with the random number generator on the calculator too). Then, while Patient Zero stood on one side of the classroom everyone got as far away from him or her by standing on the other side of the classroom, so as not to catch their exotic disease. Then Patient Zero picked a card and whoever’s card got picked got infected with the disease and came over to the sick side of the room. I stood at the board and recorded data for how many total people had been infected up to each round. Then, all cards were replaced and all sick people picked someone to inflect. Then, again. Then again and again until the whole classroom was infected.
The disease spread exponentially at first, but once people started picking others that were already sick and the supply of healthy people dwindled, the spread of the disease slowed down, a nice beautiful logistic curve. Here is the data we collected:
After we hand-graphed the points together, I had them then write bullet point stories for why the graph looked like it did. Most students were quickly able to notice that the infection spread quicker and quicker at first and then started slowing down for the reasons I noted above. Many even picked out the inflection point (but not by name) by saying that this was the point the infection was spreading fastest. (The data weren’t as pretty in my other class, but that’s okay! The main point still worked, and it’s nice to see that models aren’t perfect).
After that, we were ready for Calculus. This was a perfect thing to do after a 3 week break, because I then had them tell me everything they could in terms of Calculus to fit with the stories they wrote. We had a very lively review of everything I had wanted to review grounded in this conversation about infection, and the conversation really primed us to talk about inflection points in a deep and meaningful way. The rest of the lesson was like cutting through sponge cake (is that a saying people say?) The idea of an inflection point possibly occurring when the second derivative is zero made far more sense than if I had tried to state this fact and then show why, or make more abstract graphical arguments.
My goal this year has been to motivate well all the math occurring in the classroom, both with application and pure mathematical ideas, and I think this is a good example of success! More to come on other applications of concavity.
Happy New Year! I’m not going to apologize or feel guilty for not blogging in a while, but I will warn that this is completely non math teaching related.
And the winner of “Favorite Student Interaction of 2011″ goes to…
Mitch was a student in my non-AP Calculus class this fall and was struggling. His overall grade was floating in the 30s and it was looking pretty bleak. One day, I popped out of my apartment to grab something from the dorm kitchen. I passed through the dorm and noticed Mitch sitting in the common room, which was strange because he did not live in my dorm. I greeted him and then continued on my business, going into the narrow alleyway that leads to the tiny, little enclosed kitchen.
I looked behind me and Mitch had followed me in there. Grinning, he says “I’m sorry Mr. Rami, but I’m going to have to kill you now” (NB: Rami is my Arabic name).
My mind: What!? Is he serious? I mean, he could be, he’s not doing well in the class. But no, that’s crazy. Things like that don’t happen in real life. I’m not Samuel L. Jackson in one of those turn-around-the-violent-inner-city-school stories.
My mouth: [makes a nervous chuckle]
At this point, he pulls something out of the bundle he was carrying… a bejeweled Bedouin dagger in its sheath.
My mind: Ho-o-okay, that’s officially the first time someone has pulled a weapon on me. I think he is serious. I’m trapped in this narrow kitchen and I’m officially freaked out.
My eyes: [widen in panic]
I think he could tell I was a bit nervous, so in an attempt to soothe me he pulls the knife out of its sheath. He says, “Don’t worry, it’s not sharp.” Then, he puts the knife up to his face and demonstrates that it’s not sharp by running it along each of his cheeks.
My mind: Okay, I feel like saw Hannibal Lector do that in a movie and then slice the crap out of someone to eat their brains. This image in my mind is slowly morphing into Mitch doing the same to me. Not very helpful.
My face: [has a look of panic that has spread from eyes to other facial features]
Finally, seemingly confused that his actions are having eliciting these reactions from me, Mitch shows me that the face of the blade has “Rami” engraved in Arabic (رامي) on it. It was a gift, and this was the wonderful way that he decided to present it to me. He handed me the engraved dagger and a Jordanian keffiyeh, a completely unprompted, middle of the semester gift, which was perhaps perplexing, but really nice. He told me he wanted to give it to me because I enjoyed “Bedouin things” (not sure how he came to that conclusion). I asked him where he got it and he said he wasn’t sure because his driver had picked it up for him. Well… okay. Sweet gesture though.
I honestly don’t think he was giving it to me so that I would bump his grade, or get in my favor in any way. I think he actually just wanted to express his appreciation to me… for something, I’m not sure what. It reminded me of what I found to be one of the most surprising things I found out about teaching. I was always a really good student and teachers liked me so I think I kind of assumed that teachers only really liked the good students. I quickly found out that this is patently false – some of my favorite students over the past three years have been some of the ones who have struggled most. Conversely, I have less-than-enjoyed a handful of the A students I have taught.
So thank you Mitch for reminding me how wonderful it is to be able to see potential in everyone and for making me wet my pants for the first time since 1st grade.*