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:
- Tell no student to care.
- Tell no student how to care.
- 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.
- Perhaps we digress with these images.