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Archive for February, 2009

February 7, 2008:

I bought my car new on January 15, 2006. Today, February 7, 2008, it has 37,846 miles on it. On what date will I need my 120,000-mile tune up?

February 20, 2009:

On what date will my car need its 120,000-mile tune up?

In 2008, my students proceeded admirably through a challenging problem, successfully navigating proportional reasoning, but let's not pretend I did anything for their ability to see the world through a mathematical framework.

In 2009, my students had to ask themselves, "what do I need to know in order to answer this question?" a line of inquiry thoroughly absent in 2008, a line of inquiry thoroughly absent in my textbook, which supplies only relevant information and, in some cases, "helpfully" suggests a route to the solution.

As my (patient) readership has no doubt realized, the impotency of our textbooks to do anything but teach procedure has recently whacked me over the head. Part of this, I realize, is fundamental to the print medium, which doesn't permit a layered application of mathematical structures, but part of this is the inexcusable lack of imagination of publishing houses, whose bundled supplements are both costly and unhelpful, who don't understand that they need to help students less:

Dan bought his car new on January 15, 2006. It's a four-door sedan with 16-inch wheels. Today, February 7, 2008, the car has 37,846 miles on it. He lives 24 miles from his job and drives, on average, 48 miles per hour. The weather in his hometown ranged from 23° to 107°. On what date will he need his 120,000-mile tune up?

I felt like talking to my homeroom class about the education excerpt of President Obama's Joint Sessions speech so I ripped it and then excerpted it a little further. [entire education address; smaller excerpt]

I realize this is Pacific Standard Time and some of you are just finishing lunch right now on the East coast but how cool is it that our President delivered an enormous policy address in our nation's capital last night and I can have the (middling-quality) video in front of my students for discussion the next morning. Obviously, everything is going to be just fine.

The excerpt we'll be addressing, for good and bad:

In a global economy, where the most valuable skill you can sell is your knowledge, a good education is no longer just a pathway to opportunity. It is a prerequisite.

Right now, three-quarters of the fastest-growing occupations require more than a high school diploma, and yet just over half of our citizens have that level of education. We have one of the highest high school dropout rates of any industrialized nation, and half of the students who begin college never finish.

This is a prescription for economic decline, because we know the countries that out-teach us today will out-compete us tomorrow.

So tonight I ask every American to commit to at least one year or more of higher education or career training. This can be a community college or a four-year school, vocational training or an apprenticeship. But whatever the training may be, every American will need to get more than a high school diploma.

And dropping out of high school is no longer an option. It's not just quitting on yourself; it's quitting on your country. And this country needs and values the talents of every American.

There was the struggle between classroom management and engaging instruction. I invested myself equally in both until the depressing day I realized that my investment in engaging instruction also paid off certain dividends in easier classroom management. I spent that day re-evaluating my assumptions about teaching and re-balancing my investment portfolio.

Then there is the more current struggle between teaching skills (multiplying two exponentials) and teaching concepts (proportional reasoning). I figured, until recently, that in a 120-minute classroom, any time we spent on goofy conceptual digressions was time away from skill instruction we'd have to make up later.

So it's strange, then, that after a semester of frequent digression, my classes are still on pace with every other Algebra 1 class and my kids set the curve for the semester final exam.

We spent thirty minutes on Friday, for example, investigating Jessica Hagy's infographic work. I cherrypicked some interesting relationships, covering up Hagy's graph in each, and asked the students to draw the relationship, also labeling each "direct" or "indirect" variation because, why not.

By the end, we were disputing Hagy's graphs on technicalities, altering the intercepts ever so slightly to reflect the fact that (eg.) plastic surgeries could, at first, make someone less frightening to children.

After digressing for fully 25% of the period, we got down to the new business of adding and subtracting polynomials. And it struck me as I put an example up on a slide and asked them "what can you do with this?" how little time I spend "teaching" anymore, how these goofy conceptual digressions have trained my kids to look for connections, not just between "plastic surgery" and "frightened children," but between "old skills" and "new skills."

I realized, Friday, why we lost dozens of hours in the first semester to goofy conceptual digressions but still outpaced the school.

We didn't need those hours anymore.

Which of the following five images would best drive a rigorous, analytical, mathematical discussion between a teacher and students, leaving the students best prepared to interpret the world outside their school? Defend your choice.

A. [larger]

B. [larger]

C. [larger]

D. [larger]

E. [larger]

[If you have never found a standard-issue measuring cup utterly mesmerizing, it's possible you haven't satisfied some of the prerequisites for this course.]

When we teach math we are helping our students establish a framework for interpreting the world. One of the worst ways I know to help them establish that framework is to print an illustration of a real-world scene in a textbook, write in only the relevant measurements, and tell the students in the text of the problem which formula or strategy to apply. This leaves a student helpless and unprepared (in the mathematical, analytical sense) should she ever encounter the world that exists outside the pages of her textbook.

So we instead bring digital media from the world into the classroom, simulations of the world as students experience it, artifacts which students can discuss and to which they can apply frameworks of their choice. In order to leave students capable and prepared for their encounters with the world, this media must be captured and presented very intentionally.


We must capture this media — an audio clip, a photo, or a video, for example — so that it most closely approximates the student perspective, so that the media appears as nearly as possible to be the world as the students experience it, rather than as some audio, photo, or video that has already been interpreted for them by a photographer or a cinematographer or an editor. It must be captured to allow for the most possible classroom uses, the most possible interpretations, and the most possible framework applications — allowing even for the application of useless frameworks.

  • Frame wide on the scene rather than narrow. Capture the entire scene inside a static frame and add 5% to the margins. This forces a useful question on the student, "Where do I focus my attention?"
  • Compose the shot parallel to the plane of action. Extravagant camera framing makes mathematical photogrammetry difficult but also makes the camera operator an interloper on the scene. An extreme low angle, for example, begs the question, "Who shot this?" when you'd rather the student forget about everything but the scene itself.
  • Record ambient audio only, exactly as the student would hear at that scene.
  • Use location lighting only, exactly as the student would experience at that scene.
  • Maintain a fixed camera position. Use a tripod. A handheld camera allows the camera operator to impose her point-of-view on a student's interpretation of the scene. We want the student to decide for herself what parts of the scene are relevant, important, or useful, not the camera operator.
  • Record several alternate takes, changing variables, illustrating different iterations of the scene.
  • Take a photo of a ruler or meter stick within the plane of action for reference.
  • Do not edit the video. Edits invalidate timecodes, and, again, they impose an outside party's interpretation of a scene on the student when we'd rather the student interpret the scene for herself. There are, of course, two unavoidable edits — where you start the clip and where you end it.
  • Allow at least five seconds on either end of the scene, though more time is preferred. This will force students to decide when something relevant has begun and ended.
  • No narration. This allows the teacher and students to determine the dialogue.
  • Shoot the highest-definition video possible.
  • Shoot at the fastest possible frame rate given the location lighting.


The goal with classroom presentation is to eliminate the presence of interlopers, to eliminate everyone from the scene but the student, including the teacher, to whatever extent possible. The teacher exists here to scaffold and curate the artifacts, not to suggest, explain, or gesture.

  • The artifact must beg an obvious, compelling, seemingly unsolvable question, a question which begs for more questions and for more information. If the teacher has to suggest, explain, or gesture in order to persuade the student of an artifact's interest, then it isn't compelling or obvious enough for classroom use. (This is the most challenging criterion in this framework. This is the criterion that begs most loudly for open sourcing. The technical aspects of this framework are fairly intuitive but the creative aspects of this framework are extremely challenging and demand collaboration.)
  • Declare nothing and ask only two questions: "What is the next question?" and "What measurements do you need to answer that question?" Again, our ideal digital media shouldn't require any introduction or explanation.
  • Impose a mathematical framework on the scene only as students request it. This is where textbooks fail, imposing a grid or labeling points or establishing measurements before the student has even begun to process the scene. Once the overarching question has been introduced (eg. "will the water balloon land on the target?") the students will see the need for measurements. ("We don't know how high it was dropped.") The teacher then plays the exact same clip with a measurement grid superimposed on the footage. The ideal digital media artifact consists, then, of multiple remixes of the same video or picture or audio files.
  • It is essential to add a timecode, if for no other reason but for student reference ("Can you scan back to five seconds in? I thought I saw something.") but also because the timecode will allow for time-based calculation. The timecode needs to take the format Minutes:Seconds:HundredthsOfASecond not Minutes:Seconds:Frames.
  • Invite the students to estimate the answer to the chosen question. Take five student estimates and post them on the wall. This offers the students a low-stress opportunity to consider a correct range for the answer. It also invests them in the problem.
  • Offer the students something tangible to manipulate or measure. A hard copy of a video still, for example, or a digital copy of a video still they can import into a dynamic Geometry system like Geogebra.
  • The final element in one of these digital media artifacts must provide the payoff for the mathematical work. It needs to confirm, for example, that the water balloon did or did not land on its target, or it needs to flashback to the moment the water balloon was dropped, revealing the height of the drop. The classroom discussion can then turn to possible sources of error or calculations of percent error.

This isn't gospel. Please edit the wiki if you feel so led. This simply makes the most sense to me of my last three years teaching.

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