Summer school right now involves six hours of Geometry instruction followed by three hours of planning for the next day's Geometry instruction, which basically leaves me fully tapped for tweeting, blogging, smiling, anything but sleeping. I'd say something laced with regret here but the fact is I enrolled some truly incredible students who challenge me and crack me up for the better part of those six hours. These kids make for light work.

Their proficiency does cause its own kind of trouble, though, because my strongest and weakest students space themselves out dramatically over six hours, requiring all kinds of differentiation. My favorite recent method, particularly with today's investigation of reflections, is to say, "okay, now do that with just a compass and straightedge."

I had a method in mind but several students each did me one better.

One student made kind of stunning use of SSS congruency. Another dripped sweat all over the page constructing perpendicular bisectors, copying angles, copying sides in an incredible (but functional) mess. Another used the method I chose but did it in three fewer arcs.

I have five more days to enjoy this.

[BTW: I have determined that at least 20% of this is garbage.]

When they want to see more methods, they'll let you know. When they become discontented with their ideas of proof, they'll let you know. And it WILL happen. Because there will always be kids who ask themselves and their peers: "Why does that work? Why does that make sense? How do you know?" And that's all we need to nurture in them: their own natural curiosity, rather than suppress that and replace it with curiosity about only the following: What does the teacher think? What does the teacher want me to say or do? What do I need to do to get an A?

I like this.

My library has already been cut. We will have no bookroom clerk, making novels almost an impossibility and replacement costs much higher than previous years for sure. We will lose one adviser, the person we send students to when they are problematic. We will have a total of fifteen more students each day, meaning that we’ll teach five and a half classes for the same pay as we usually get for teaching just five. The reproduction clerk is gone, meaning that we all will have three-hundred copies per month, end of discussion, and we’ll have to allot time to make those copies ourselves instead of dropping them off and picking them up later. The whisper has it that our athletics director will go away. There’s even talk of moving to two administrators, dropping from our current three. All these things mean that folks will be placed back into classrooms, where the newly christened teachers will be the first on the chopping block. This news comes before what are rumored to be even larger final budget cuts. I can only imagine what further decimation will happen after that.

I put up this slide and asked Mika to pick a point out. I asked her to tell Jason across the room which point she was thinking of. She stumbled and stammered a bit. "It's sort of to the left of the one that's near the center," etc.

And then I added labels.

And it became a little clearer why we label points. Mika relaxed. Everything looked easier.

In 2007, I told my students that we name lines using two letters and I gave several examples. Today, I asked Mike how he would tell Kelsie across the room which of these lines he was looking at. First, it was easy.

Then it was difficult.

The same went for how we name angles.

This math thing is easier to approach if I ask myself, what about this concept is useful, interesting, essential, or satisfying, and then work backward along that vector, rather than working toward it from a disjoint set of scattered skills. There is probably a book I should read somewhere in all of this.

Postscript

Also: I didn't hate our opening exercise in which I gave each student a) a compass, b) a straightedge, and c) a map of the Meyer family's South Pacific archipelago, Meyeronia, and d) five questions. [pdf]

1. How many miles is it from Kenneth to Christy?
2. Which island is farther from David? Barbara or Christy?
3. List all the islands that are three miles from Kenneth.
4. Find a location in the water that is the same distance from Tom & Bob. How many are there?
5. Find a location in the water that is the same distance from Tom & Bob & Kirsten. How many are there?

While maybe not reflective (on its own) of any massive change in my pedagogy since I started teaching, the growing size of each year's lesson folder does reflect my growing tendency toward visual mathematical multimedia.

Sort of.

I never coached tennis. I never sponsored a club. I didn't attend the plays or games or concerts I always felt I should have. I regret that. I never watched a freshman class graduate, never saw them from the start of high school to the finish. I regret that most of all.

But I have made amends with classroom management, time management, and compensation, the challenges which, at various points over five years, had me talking to admissions officers at schools of engineering and medicine. After five years, I am unequivocally a "happy" teacher.

I regard this professional transformation (from miserable to happy, incompetent to competent) with complete stupefication. The arc of a new teacher's development is short and bends in any number of directions. My own was filled from beginning to end with lucky coincidences, chance mentors who appeared and disappeared at the exact moment I most needed them, hobbies from my childhood which came back around to pay off huge dividends in my classroom. I can't explain any of it. I know I could do it all over again and arrive a completely different teacher.

I need to get a fix on some larger issues of teacher development and I can't do that from the ground level here, from the classroom, with blog posts scattered around and squeaked out in the fifteen-minute interval after lesson planning ends and before my wife gets off shift. I am enrolled in in the Ph.D. program at the University of California, Santa Cruz, for the fall, to what end I don't yet know. But I'm ready to spend half a decade or more pursuing the answer to a single, confounding question.

There is math here, certainly, but I have made it a goal this year to stall the math for as long as possible, focusing on a student's intuition before her calculation, applying her internal framework for processing the world before applying the textbook's framework for processing mathematics.

Bad First Question

This one sucks the air right out of the room. We're into the math immediately, having bypassed several easy opportunities to pull in our students who hate math¹.

Jason's First Question

Jason Dyer suggests handing out plastic cups, letting students roll them around, then asking "why do they do that?" I have no problem with this approach. I would like to start from a position of stronger student investment, though.

My First Question

Have them roll some plastic cups around. Then toss up this slide and ask them a question that has a correct answer, yes, but which attaches little stigma to the wrong answers. It's an educated guess and different students will make persuasive cases for all three of these. Ask them to write their guesses down, to put them on the record².

A Lesson Sketch

The conversation can then proceed along some interesting lines where you ask the student to:

1. justify her guess.
2. draw the kind of cup that will roll the largest circle using a fixed amount of plastic. This is fun. Many will draw a really tall cup, which isn't the best use of limited material. A two-inch-tall cup can roll a circle that's a mile wide.
3. make their ideal cup from a page of card stock. The fixed size of the card stock will normalize the results.
4. draw a complete picture of their cup including the auxiliary lines. Can they find the invisible center of the circle it will roll? What's the method?

We do all of this before we start separating triangles, before we write up a proof, before we generalize a formula. We ask for all this risk-free student investment before we lower the mathematical framework down onto the problem.

Degenerate Cases

A cool feature of this formula is how well it handles degenerate cases. For example these two:

1. A cone's roll-radius is the same as its slant height so letting d = 0 should (and does) eliminate D from the formula.
2. A cylinder will roll forever so letting D = d should (and does) return an undefined answer.

Iterate

From there you can pull out of your cupboard (digital or otherwise) any random set of cups and the students should be able to predict the roll-radius within a small margin of error.

And the framework grows stronger.

A Parting Swipe At Textbooks

I didn't dig this out of a textbook³ but this (hypothetical) scan highlights the difference between my math pedagogy and my textbook's.

1. … and, when those students comprise your entire class, good luck with that. [back]
2. It's extremely helpful here that the tallest glass doesn't make the largest circle. [back]
3. h/t Mr. Bishop, Summer School Geometry, Ukiah High School, 1997. [back]

Two things:

1. It isn't "what question can you ask?" but "what can the students do with it?" What is your lesson plan here?
2. If Jason Dyer doesn't come around to tell me I'm doing this wrong, I'll be very surprised.

[high quality: photo, video]

[BTW: I updated the original image because josh g. is exactly right.]

Michelle Baldwin, dissenting from the comments:

In considering Dan Meyer's arguments, I don't really agree with him. At all. It's all about finding the "right" photo to enhance the text.

Is that what presentation is all about? Witty aphorisms and inspiring photos?

You have a thesis. Let's assume there are very real, really real real-world implications to your thesis. Why not cut to that chase? Why make an abstract matter like edutechnology even more abstract with dramatic photography and 140-character pullquotes from your Twitter feed?

1. Show me something real.
2. Give me a space to interact with it.
3. Let me have your thoughts on it.

In this case, if learning really is social, please show me examples of that social learning. Or show me examples of how dangerous it is when that learning is taken out of a social context. If you find it difficult to connect your thesis to video or screenshots or sound clips ("multimedia," basically) then it's possible you are chasing down the wrong thesis or that your thesis doesn't lend itself to a presentation medium¹.

I like that Darren modified the stock photography (adding the "Learning Is Social" placard) to connect it better to his thesis than the average stock photo slide but I wonder if we're approaching the question, "What is presentation?" along two different vectors.

1. I caught David Jakes' Black Coffee presentation on Slideshare last week and was impressed that something like 95% of its 63 slides were screenshots, archival photos, YouTube videos, newspaper clippings, etc., etc. Jakes had done his groundwork. [back]