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Archive for November, 2011

Pretending Closed Questions Are Open

I was in Avery Pickford's session at CMC-South when he put up this image and polled his participants for questions that interested them.

They asked about the slope of the diagonal. They asked about its length. Avery then constrained their questions. "What things could we count?" he asked.

His participants responded with "the perimeter" and "the number of squares." At that point, Avery just asked the question that interested him:

How many squares does the diagonal pass through?

His session ended on that problem but I'm extremely curious what would have happened had he presented a new image and asked his participants for new questions. I can't be sure but I suspect they would have held out. They'd know from their last experience that Avery had a question in mind and everyone but the apple-polishers would have waited him out.

Open And Closed Questions

If you have a question you'd like your students to answer, ask it. But before you ask it, consider creating a visual — something short and sweet, one photo or one minute of video — that orients your students to the context of your question and makes that question seem like a natural one to ask. Like:

If you'd like your students to pose some questions of their own, ask them what questions they have. But questions about what? Give them something to ask questions about. Consider creating a visual — something short and sweet, one photo or one minute of video — that lends itself to different perplexing questions. Like:

What Happens On Twitter Stays On Twitter

How can you tell in advance if students will be perplexed by your closed question or if they'll have open questions about your photo or video? You pilot it. There's no right way to pilot curricula, only optimizations for different variables that are often in competition with one another. Like:

  1. Are you piloting with students or with some proxy for students?
  2. How easy is it for your participants to give you feedback?
  3. How many participants are giving you feedback?
  4. How helpful is that feedback to your development process?
  5. How far into the development process are you waiting to get feedback?

Here's one optimization: show teachers your photo or video on Twitter and ask them what questions they have about it.

This means (1) you aren't piloting with students, which is unfortunate, though no students are harmed if your idea is a dud, (2) it's easy for your participants to give you feedback, (3) the number of people giving you feedback is proportional to your followers on Twitter, (4) that feedback is often useful — if you plan to ask a closed question, the feedback will let you know if that question is interesting; if you plan to ask for open questions, the feedback will let you know what questions to expect.

Or you might pilot your curriculum on the same day you're teaching it, making modifications for your afternoon class based on feedback from your morning class. You can evaluate the variables for yourself on that one.

Make it work for you. Twitter, #anyqs, your classroom, your faculty lounge, whatever. Make it make you a better teacher. Just understand that when you're using curriculum in the classroom, you're optimizing for an entirely different set of variables than when you pilot that curriculum somewhere else.

The question that bugs me at all hours is "When is video / photo / print valuable?" This video is one minute long and gets me closer to an answer.

The intermediate value theorem says that because you picked purple when the purple slice was big and blue when the purple slice was small and because slices run continuously from small to big, there is a particular slice that makes you go, "Meh," that's exactly in between "I choose purple" and "I choose blue."

I love that students have an intuition about that slice, an informal understanding of probability that we can develop into something formal. We can access that intuition with video by showing that small slice growing continuously into the big. How do you replicate that experience in print, a medium which does a bang-up job with static quantities but has something of a panic attack when those quantities change?

Featured Comment

Avery Pickford:

Know what I’d really love. For every student to be able to click their mouse (or some equivalent) when they would make the switch and to have this data show up on my screen right after the video was done.

2011 Nov 29. Evan Weinberg hacked together something that does what Avery described. The results surprised me.

Last April, fourteen of Palo Alto High School's twenty math teachers petitioned their school board [pdf] against raising graduation requirements to include Algebra II:

We live in an affluent community. Most of our students are fortunate to come from families where education matters and parents have the means and will to support and guide their children in tandem with us, their teachers. Not all of them. [..] We are concerned about the others who, for reasons that are often objective (poor math background, lack of support at home, low retention rate, lack of maturity, etc) can't pass our Algebra II regular lane course. Many of these are [Voluntary Transfer Program] students or under-represented minorities.

Since those students objectively can't pass Algebra II, the next appropriate step is to compile a list of those students and prevent their enrollment in Algebra II in the first place. Otherwise, you're putting them in a position to care about passing a class we can be objectively certain they will fail. If I were a parent of one of those students, this determinism would probably drive me out of my mind.

The signatories are Radu Toma, Suzanne Antink, Kathy Bowers, Judy Choy, Arne Lim, Deanna Chute, Natalie Simison, Misha Stempel, Maria Rao, Charlotte Harris, Scott Friedland, Lisa Kim, Ambika Nangia, and David Baker.

Featured Comment

Jason Buell:

Their hearts I think were in the right places but they whiffed badly. The point isn’t can every kid take Alg 2, but should they.

2012 Jan 16: Coverage from the San Jose Mercury News.

Shoulda Woulda Coulda

Two things I'd do if I were still doing the job instead of just talking about it:

Set Up The Expected Value Spinner

I don't think people who understand expected value understand how hard it is for other people to understand expected value.

Let's say I roll a die. I ask if you want to bet on an even number coming up or a five. You're bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I'd pay you $150 if the even number comes up and $600 for the five. What if I said I'd keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the value of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. ("Tails never fails," betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I'd print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I'd seal my own bet in an envelope.

I'd ask a new student to spin it every day for a month. We'd tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

BTW: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

Host A Steepest / Shallowest Stairs Competition

Tonight's homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you'll post those photos at the front of the classroom. You'll make a big fuss over them. Then you'll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn't have a unit, so you don't need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the tallest steps and the steepest steps. It's possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students' heads around that one.

Be prepared also for students who can't shake the sense that math is here every time they climb up a new set of stairs.

What a cool job the rest of y'all have.

[photo credits: moyogo, vulcho]

2012 Jan 17: Useful description and modifications from James Cleveland.

Sweat The Small Things

"There are five birds and three worms." That's the set-up.

The pay-off is that Tom Hudson found significant improvement in achievement when he asked primary students, "How many birds won't get a worm?" instead of "How many more birds than worms are there?"

Two things that probably go without saying:

  1. Your students with poor math achievement may be achieving poorly at something besides math. Like language.
  2. It's hard not to love a job that rewards this kind of obsessive attention to detail.

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