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Archive for April, 2009

I love the iPhone game Flight Control for all the reasons I love a good lesson plan.

  1. It builds from a simple, visceral premise. "Land the planes. Don't let any collide." ¶ Which packs the same clear punch as "what is the combination?"
  2. Harder, differentiated challenges arise naturally from that premise. Which is to say, as you get better at the game, it doesn't just double the speed of the planes or throw up concrete clouds or reverse the controls. It introduces different planes into the airspace, planes which move slightly faster. ¶ In the same way, a good lesson plan doesn't adapt itself to faster learners by doubling the length of the same problem set or imposing artificial constraints like, "what if one of the buttons was broken?" It tells the learner, "okay, we dusted the lock for prints and found out that these four numbers get pressed a lot. What can you do with this?"
  3. Those new challenges necessitate new skills. In its early stages, Flight Control accommodates a player's sloppiness but when you have three 757s approaching the landing strip and three helicopters holding in a pattern you have to keep your approaches extremely tight. ¶ The combination lock forces the need for permutations.
  4. Those new skills are assessed simply and clearly. A lesser game would assign separate point values for larger planes or include bonus multipliers. Flight Control assesses your skill along one simple metric: "How many planes have you landed?" ¶ After all the calculations in "Will it hit the can?" the assessment was simply "Were you right?"

Not every game or lesson can accommodate this aesthetic. Nor do I expect them to. But these are my favorite. These are my students' favorite. And they are too few and far between. We need more.

Jason Dyer, responding to my NLOS cannon post:

Since I’ve done this over the summer with real life bottle rockets, a launcher that could be set at any angle, and a vertical target, I’m not finding the computerized version nearly as interesting. I’ve also run a simpler version of this in my classroom with wads of paper. Why must everything be digital? [emph. added]

Hopefully I've made clear by now my preference for pedagogy over technology. If digital media makes for inferior learning, then, by all means, let's stuff it in a burlap sack and toss it in the river. My preference is also for the real thing over a digital simulation of the real thing. That said, there are three circumstances where digital media is preferable to the real thing:

  1. The real thing is too expensive. I'd rather let every kid hold a photo of a measuring cup than spend $100 for a class set of measuring cups. It's too expensive to take a class trip to the Yucatan Peninsula so perhaps we can forgive ourselves for showing photos of the Mayan pyramids instead. I'd much rather copy and paste Google's satellite imagery into a Keynote presentation than charter a plane to take my kids up in groups.
  2. The real thing is too mathematically noisy for classroom use. Jason prefers a real demonstration of projectile motion using bottle rockets to my use of online simulators but that introduces acceleration and wind resistance— mathematical noise — into the system. Let's not romanticize the real or the digital. They are both deficient. They both require a cost-benefit analysis.
  3. The real thing can't be iterated precisely enough. I wanted to show my students several misses with "Will it hit the can?" — long, short, and to the side — and at least one success. If my students were live with me, on the scene, they would see many, many, many misses, most of which would be mathematically unhelpful. My students can also measure and manipulate digital media (by modeling a parabola in Geogebra, tracking motion in Logger or Tracker, etc.), something they can't do with live events.

Michael Caratenuto:

Personally, I think that this particular image lacks opportunities for inquiry. Perhaps if it was presented with other kinds of door locks leading students to come up with and answer the question, “which is the most secure lock?” [emph. added]

This is exactly right. The latest WCYDWT? installment has provoked the usual litany of Really Interesting Bite-Sized Questions, the sort of prompts that will play great in the Applications & Extensions & Assorted Mindblowers section of your lesson plan but which, on their own, aren't a lesson plan. Those questions don't provoke the kind of iterated, increasingly difficult practice that students need for skill development.

Again, this image on its own is insufficient. With some creative modifications, however, it will carry you through permutations. Here is that lesson plan in its broadest strokes.

Start with the image.

Tell them the code is 1 digit long. Tell them the code is 2 digits long. Tell them it's as long you want it to be. I respected the rule of least power here, which meant that when I took this photo I tried to stay out of the way of your lesson planning. Have them write down all the possible codes for n=1, n=2, n=3, etc. The increasing obnoxiousness of the task will motivate a formula for the general case. That's arrangements.

Tell them the lock is a 4-digit lock. Now turn on the blue light.

Ask them to list the possible codes. You can iterate this a bunch of times until they have discovered on their own this tool that mathematicians call a factorial.

Remind them it's a 4-digit lock. Then put up this image. It will be confusing, but only for a second. Ask them to list every possible code.

Iterate this with two and three buttons until they have generalized permutations. Then maybe you iterate the entire thing with another keypad lock.

Then maybe you dip into the comments of the original WCYDWT? post and help yourself to some very-interesting follow-up questions. I recommend Alex's.

Let me close by saying how shocked I am at how little all of this costs.

[Update: Bruce Schneier has a good follow-up on information leakage. Two photos.]

[Update II: due to the peculiarities of many car door locks punching in "123456" tests both "12345" and "23456." Consequently, there is a number string 3129 digits long that will test every five-number comination.]

[Update III: more information leakage.]

[Update IV: more information leakage.]

I'm mixed. On the one hand, YouCube is a pretty interesting way to compare remixes of a thing (ie. David After Dentist) to the thing itself.

On the other hand, this strikes me as just another one of those tool that depends entirely on a teacher's pre-existing digital storytelling skills but which also distracts her from developing those skills. (ie. Why learn how to make one video really well when you can put six average videos on a cube!)

Download high quality here. Here's the pilot but I need to modify the prompt somewhat. Every math teacher reading this likely sees the mathematical potential in this image. Most could come up with a question right now like, "If this is a four-number combination lock, then how many combinations will you have to try to break in?"

Lately in these threads I get lists of those questions, which is great, but questions don't constitute a lesson plan. So consider this the new prompt: what is the lesson plan? what will the students do? what is the best plan to provoke sustained, rigorous inquiry?

Let's push this forward.

BTW: My lesson plan.

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