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Archive for November, 2008

This is why I only use ten percent of my textbook's printed pages:

The text has already imposed a rigid, powerful framework around an interesting drawing of a ski-lift. It has labeled the points, scaled the axes, and written the questions. The textbook has told my students how to care. The student can interpret this drawing only as the textbook intends.

To a certain extent, I have no problem with this. I want my students to interpret this drawing in a particular way. I want to use it to learn slope. But by applying this powerful framework in advance, the textbook has told my students exactly how they should be curious, which isn't any kind of curiosity at all. It doesn't train my students to draw these strong, interesting connections on their own and it presumes their engagement with the problem.

For example, if a textbook were to repurpose my last What Can You Do With This? prompt, it would run like this:

Just a guess.

The textbook would apply the most powerful framework to the problem, imposing a definite line of inquiry on the student before she even gets around to asking herself, "why does the tennis ball blur like that?"

By contrast, an application of the Rule of Least Power to the problem looks like this:

I put this picture up, just a picture, totally absent any mathematical framework, the least possible power I can apply here, and I ask, "What do you guys notice about this photo?"

The moment any student mentions the blur I drive the conversation her direction. The student has given me permission to apply more power to the situation. I ask, "Does anyone know why cameras do that?"

Several students take photography as an elective and mention shutter speed. I have the students take out their cell phone cameras and take a picture. I ask them to explain the camera's pause1.

Having been given permission now to talk about shutter speed, I apply more power:

We talk about "1/25″ and what it means to photographers. I might draw another blurred tennis ball on the board, one with a longer blur, and ask them to describe the differences. (A: a longer blur would mean it was dropped from a greater height.)

Finally, after this careful, deliberate application of power, I ask, "Can anyone tell me how high up off the ground this tennis ball was dropped?" No one can, not without measurements, and once someone mentions that, I project the last picture.

And we take on the problem. We have voluntarily committed ourselves to a mathematical framework. That commitment wasn't forced upon us by an external agent. (Again: the involuntary commitment.)

The Rule of Least Power, as I have applied it to my classroom, means:

  1. Tell no student to care.
  2. Tell no student how to care.
  3. Apply increasingly powerful frameworks to mathematical objects only as the class cares about them.

Please don't confuse this with hardcore, Waldorfian constructivism. I have an agenda, a standard to meet, a lesson objective. But I don't fence my students onto a narrow path to my objective. I instead pave the ground beneath them so that the path to my objective is the easiest and the most satisfying to walk.

  1. Perhaps we digress with these images.

Lately I am a man obsessed. As others are obsessed by numerology, the year 2012, or the birth certificate of President-elect Obama, I am obsessed by the Rule of Least Power and how succinctly it explains why I have never found the right place for a textbook — any textbook — in my math classroom.

Whenever my mind starts to spin down for sleep, it wanders to this computer programming axiom and everything becomes hypnotizing and clear. In this waking dream, I see a spider's web connecting disparate artifacts:

  1. my textbook;
  2. The Wire, Friday Night Lights, The Shield, and 24;
  3. What Can You Do With This?
  4. the Muji Chronotebook;
  5. and the Rule of Least Power, most crucially:

Use the least powerful language suitable for expressing information, constraints or programs on the World Wide Web. — W3, The Rule of Least Power.

And then I'm inches from some grand unification theory of curriculum design. It's close. It's killing me. If I could find seven contiguous hours, I might fully articulate the network and I'd finally have an operational theory, an operational aesthetic, really, putting only a few miles between me and dy/dan: algebra, volume one.

For the last month, I have had this single image banging around in my head, hogging valuable CPU cycles. I couldn't find it anywhere else so I shot it myself. Click the photo for high quality. See the pilot for instructions.

BTW: The comments feature no fewer than two dozen lesson inspirations, at which point the questions become (I think) which lesson inspiration a) will sustain the most interesting math the longest? and b) which prompt can be summarized the most succinctly, the most viscerally? I think those are two of the most important metrics for evaluating these ideas.

Under that light, you have Ben Wildeboer: "Calculate time before impact with the ground." It's visceral. The student wonders first how she'll find that information in a static image. It seems impossible. The result is a page of physics function work.

Also: "How high was the ball when it was dropped?"

Or: "How long has it been in the air?"

My work for both of those questions:

The first answer is off by nearly a meter. That's just under 100% error.

BTW: A reader writes to let me know I blew the math here.

It seems that the work shown is using different reference points. At the top of the diagram the top part of the ball is used, and at the bottom of the diagram, the bottom part of the ball is used. I think the top of ball should be used for both or the bottom of ball; either of which would require knowing the diameter of the ball.

He's absolutely right, which would explain the 100% error.


This desk makes me question my convictions.

I have been convicted for some time that, to be a good teacher, you need not have experienced a bright light on the road, a deep voice summoning you to the job. To succeed here (at least in the short term) you need some combination of self-reflection, intelligence, and good humor. The rest can be taught.

But that desk testifies to certain attributes of good teaching that cannot be taught. That desk tells the story of a student who was so bored by her teacher's instruction that she spent a not-insignificant fraction of her school year tunneling through an inch of wood. More importantly, it tells the story of a teacher whose tedious instruction was her lesser fault.

Her greater fault was oblivion. She had no idea what any of her students were doing at any given moment of class. She kept sacred that invisible curtain between student and teacher. She knew none of her students and knew nothing of what they did during the hours she thought they were paying attention to her.

I don't know if anyone can untrain that kind of oblivion, to say nothing of training the kind of hyperattunement common to all good teachers, the kind of "court sense" that let Magic Johnson connect no-look passes, which manifests in the classroom as a certain omniscience, as "eyes in the back of your head," as constant awareness of who is working, who needs refocusing, who is scheming, cheating, and plotting, at all times.

If that kind of oblivion can't be cured (without great expense, anyway) we must direct ourselves, then, to identifying its precursors in our applicant teachers.


Kilian splices together his own highlight reel, going a long way to explain how someone managed such sui generis blogging from a generic Blogger template.

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