## Algebra II: What’s The Question?

Dan Goldner, whose recent blogging has been fantastic:

I need an overarching theme, question, or mission for Algebra 2 that transcends and motivates the required skills. The content goals of Algebra 2 are to invert, transform, solve, and apply graphical, tabular, and analytic representations of linear, quadratic, exponential, sinusoidal, and rational equations. That’s R. What’s S?

I count myself fortunate to have never taught Algebra II, the current configuration of which in California probably violates some of the conventions and treaties the US has signed w/r/t human rights abuses. [NB. The R and S thing will make more sense if you click through.]

## Walking The Problem Solving Talk

Allison Cuttler, reminding me how satisfying it is to read math teachers' descriptions of their own problem solving processes:

My boyfriend and I spent the afternoon at my favorite neighborhood coffee shop, basking in the San Diego sunshine. I was working on my end-of-year comments when I suddenly remembered the dartboard problem. I asked him to tell me how he had solved it. He looked at me somewhat disappointedly. "Really? But then you'll never keep thinking about it your way." At that point I still didn't have a "way" but a small fire lit inside of me – How could I not have a "way"? Some idea, some line of reasoning? What would I say to a kid who asked me for a help with a problem and didn't have anything of his own to show?

## Grand Forks, ND

I added an #anyqs component to the workshop I facilitated in Grand Forks, ND, last week. This was new for me. At the end of the first day, I assigned homework:

Give yourself one photo or one minute of video to tell a mathematical story so perplexing that all of your students will want to know the ending, without you saying a word or lifting a finger.

I received e-mails all the way through the night and into the morning before the second day's session. I loaded each entry into a slidedeck and reeled them off to the group over 25 minutes.

At the same time, I had the participants working in a Google Form. For each entry, they'd write up a) the name of its designer, b) the question it provoked, c) the perplexity of that question (ie. how bad they wanted to know the answer).

The yield on that investment of 25 minutes was incredible. We spent the next hour mining and interpreting data and drawing conclusions about effective curriculum design with digital media. It was some of the productive professional development I've ever been a part of.

I posed several rounds of questions for table discussion. The first round began after they had just submitted all of their questions:

1. Was the exercise fun? Was the exercise useful?
2. What will be the most common question for your entry?
3. How curious will people be about it?
4. Which ones were you most curious about?

Then I sent out a link to the Google Form results and posed the second round:

1. How effective was your entry at provoking a common question?
2. How could you have made your entry more perplexing?
3. Whose entry was the "best"?
4. What do we mean by "best," anyway?

While they batted those questions around, I dug into the spreadsheet and found the median response for each entry's perplexity rating. Six of them tied for the highest median rating:

We reviewed those six, briefly, and then I posed a third round of questions:

1. What's special about these entries?
2. If you had a favorite that didn't make the cut, what isn't captured by this measurement (the median perplexity rating)?
3. Dan and Nancy both feature leaky faucets filling up a container. In what ways are those two entries different?

Selected Answers To Those Questions

• The unanimous consensus was that the exercise was fun. Fun isn't necessarily a prerequisite for an effective PD exercise, but man does it ever help.
• One participant said the #anyqs exercise was useful, mainly, for "training my eyes." She elaborated that after just one pass at #anyqs the day before, she was already more alert to the applications of math in her life outside the math classroom.
• The issue of subjectivity has been one of the most fascinating conversations about #anyqs online, and so it was in Grand Forks also. Are the questions of math teachers about this image a useful proxy for the questions of students? Will a student from North Dakota have a different question about a video of a wheat thresher than a student from California? One participant noted that the high school and middle school teachers in the workshop asked questions that were linked closely to their content areas? (Strong data mining, right?) The sum of my thinking to date? Yes, the process is subjective. In spite of its subjectivity, field testing my curriculum with teachers has improved it immensely for students. Perplexity can transcend our demographic differences.
• I forgot to mention this in Grand Forks but the easiest, best way to make your video-based problems more perplexing is to use a tripod. Or to simply put your camera down on something sturdy. The reason being is that it's so much harder to gauge so many different measurements (speed, for instance, or height) when the camera is wobbling back and forth and up and down, throwing your subject around in the frame.

General Remarks

• I have designed a lot of different constructs to explain to myself (and others) perplexing curriculum design. None has been as effective as mathematical storytelling. I'm particularly chagrined to think back on all the times I've broken a problem down into the four tasks of "verbalization, visualization, abstraction, and decomposition." That construct resonates with my grad school peers, but it's terrible vocabulary for teacher professional development. (ie. "Okay, so where do you find the decomposition of this task. How would you help the student abstract the problem space?" etc. Gross.) I've never heard table groups reference "decomposition" in one of my design activities. The language of storytelling, in contrast, was a constant feature of their conversations.
• One participant: "We need a website for sharing these." Yes.
• Another participant: "Kids should bring in their own photos and videos." Maybe.
• This is the dy/dan drinking game: every time I put my readers to work to make me smarter or more effective in my studies or at my job, drink. I was legally unsafe to drive after receiving hundreds of pages of student work for my Michael Benson experiment. I was black-out drunk after using the work of @salmathguy, @reimerpaul, @eduz8, @techsavvyed, @fnoschese, and @wpeacock202, for fodder in my #anyqs workshop. Y'all should be so lucky to have readers like y'all.
• There was a horrible moment in the early morning of the last day when I planned to hand each participant 28 strips of paper (one for each person's #anyqs entry) on which she'd write her question and the name of the designer. Once we finished, I figured we'd trade the slips back to each other, a process that would probably take forty-five minutes on its own, right? So take a shot for Google Forms as long as you have the bottle out.

Your Homework

• How are the two leaky faucet videos [Dan, Nancy] different? Which one is better? Define "better."
• One participant submitted this video of Carl Lewis' 1984 Olympics long jump. He was perplexed by the parabolic motion of Lewis' jump. Instead, nearly all of his colleagues (and yours truly) wondered how fast Lewis was running at lift-off. Given infinite resources, tools, the ability to travel anywhere in time and space, how would you capture Lewis' long jump in a way that highlights the perplexity of his parabolic motion?

And Now A Word From Our Sponsor

If any of this seems interesting to you, let me recommend my Perplexity Session, which I'll be hosting in Mountain View on 9/10/11.

Here's John Scammell with a celebrity endorsement:

As someone who was fortunate enough to see one of the early incarnations of this workshop, I can tell you that it is incredibly valuable professional learning. Dan is a skilled facilitator, and more importantly a great teacher. I highly recommend it.

## The Perplexity Session — 9/10/11 — Mountain View, CA

I've been facilitating the same workshop around the world for the last year, tightening it down, and refining it to the point where I'm very excited to offer it to a public crowd of math teachers. I'd like to bring 75 of you into a room and leave six hours later with a stronger understanding of perplexing curriculum design and mathematical storytelling. We'll determine where digital photos and videos fit into that design and we'll develop the technological skills for designing curricula with any kind of media.

If you'd like to learn more, check out the session website.

If you'd like to sign up, head over to the Eventbrite page. I'm limiting registration to 75 people.

What: The Perplexity Session
Where: The Mountain View Hilton
When: 8:30 – 3:30. September 10, 2011.
Cost: \$75

Bagels, coffee, and lunch are included, as is a huge pile of digital curricula and resources I haven't found time to post here. If you'll need a room for the night, I have a block room rate from the Hilton. Call them up and use the code "DYI."

The other extremely important thing about this session is that you'll have to sign this video release. I have contracted a local video production company to film our work together. This whole workshop is operating at net zero profit so I can get it down on film for wider distribution.

Let's do this! I'm excited to work with you guys.

2011 June 14. John Scammell in the comments:

As someone who was fortunate enough to see one of the early incarnations of this workshop, I can tell you that it is incredibly valuable professional learning. Dan is a skilled facilitator, and more importantly a great teacher. I highly recommend it. I love that it is being filmed for posterity.

## [Help Wanted] Does The Medium Matter?

2011 June 13: No more help needed on this one. Thanks for piloting this study for me, team.

If you're still in school, if you still have students around, I'd be thrilled if you'd help me answer a question that's nagging me.

If you pose this question to a student:

It took Michael Benson six minutes to run one mile in a timed trial at his school — his fastest time ever. How long would it take him to run a 26-mile marathon?

Does the student multiply six by 26 getting 156 minutes? Or, more sensibly, does the student multiply six by 26 getting 156 minutes and then add some time to compensate for the fact that Benson isn't going to be able to maintain his fastest time for one mile over 25 more of them?

My speculation is that math class pretty effectively conditions that sensibility out of a student by the fourth or fifth grade. It's very difficult to succeed at math class if you don't train yourself to ignore entropy, gravity, friction, and a million other factors mitigating these tidy word problems.

But does the medium matter? If you present those two sentences in black and white on paper, are the results worse than if you present them in a medium that's more befitting the context? Like a video? Or a newspaper clipping?

How You Can Help Me

1. Print out equal numbers of these four pages. They're four different versions of the same question.
2. Shuffle the four versions and pass them out randomly, in equal numbers, to your students. (Doesn't matter to me if this is calculus or sixth-grade math. The more classes the better.)
3. Tell them you can't answer any questions about the problem. They should write those questions down. Let's leave calculators off the table. Ask them to do their best and to be sure to write down their reasoning.
4. Give your students as much time as they need to do this one problem.
5. Collect the papers.
6. Get the papers back to me:
• via email: dan@mrmeyer.com
• via fax: 831.325.0095
• via mail: PO Box 429, Mountain View, CA 94040

I'll need to know the letter grade of each student, too. And the course they're taking. Can I get any volunteers to raise a hand in the comments? I'm not going to pretend this isn't a huge favor. You can consider me in your debt.

2011 June 7. To clarify, would you be sure to send the forms back without the student names. Feel free to just attach numbers to the top. ("Student #1," "Student #2," etc.) Or anonymize them in some other way. I only need some way to link the student's response to her overall class performance.