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Archive for August, 2010

Daniel Foster’s piece in the National Review Online plays pretty fast and loose with measures of central tendency:

The public/private disparity is especially stark when one focuses on public-safety compensation in places such as Oakland; police and firemen have accounted for about 75 percent of expenditures from the city’s general fund over the last five years. Average total compensation for an officer in Oakland — a city in which the median family earns $47,000 — is $162,000 per year.

Someone break it down for us in the comments.

I’ll be talking with Ihor Charischak at 6:30 PM Pacific / 9:30 PM Eastern tonight in the Math 2.0 webinar. The timing catches me in the middle of a messy reconception of this WCYDWT thing. It’s keeping me up at night but it’s made for an exciting summer and I’d like share some of that exhilaration with y’all, benefiting at the same time from your comments, criticism, and spark.

Why I Love Teaching

Me:

How many pounds of extra dirt is Stanley going to dig at the end of a full year?

Mr. K:

How many cubic feet of extra dirt is Stanley going to dig at the end of a full year?

Claire Thompson, scooping both of us:

If we had a container whose base was the area of our classroom, how high would all the extra dirt pile up that Stanley shoveled?

Any job that rewards this kind of fine-toothed, detail-oriented nit-picking is fine by me.

WCYDWT: Dirt

Click through to view embedded content.

This clip is from the movie Holes, which is inexplicably billed as a movie for kids. (Sue Van Hattum kindly brought it to my attention.) The grim premise is a penal colony of children, each digging one hole per day in the desert for the duration of their sentences. On our hero Stanley Yelnats’ first day, he accidentally takes another kid’s shovel which is slightly shorter than the rest. Drama!

[High-Quality Download]

Step One:

  • Play the video.
  • Ask the students for questions that perplex them.

Aaron: “does the shorter shovel really matter?” +4 others
Peter: “how long does it take to dig a standard hole?” +2 others
Other questions. +5

This data is invaluable to my curriculum development. Invaluable. Consider last week’s responses to the boat in the river video.

Steve: “How long will it take Dan to go up the down escalator?” +15 others
Other questions. +2

It’s obvious to me which problem has the stronger current. Maybe I can do something about that; maybe I can’t. Regardless, I had to make a more authoritative call on the problem than I prefer. I said, “Okay, let’s talk about the first question. ‘Does it really matter that X-Ray’s shovel is a couple inches shorter?’

“How many pounds of extra dirt is Stanley going to dig at the end of a full year?”

Step Two:

  • Ask the students to guess at the answer to our question.
  • Ask the students to set an upper bound on an acceptable answer.
  • Ask the students to set a lower bound on an acceptable answer.

Our median [lower bound, guess, upper bound] was [100 pounds, 1,000 pounds, 10,000 pounds]

Step Three:

  • Ask the students to define the information they’ll need to solve our question.

VoijaRisa: Weight of dirt and extra length of shovel
Frank: how many days/ week?
Mr_K: Density of dirt
Steve G.: Density of dirt, dimensions
Aaron: mass of dirt, “shorter” shovel length

The movie doesn’t define the shorter shovel’s length, which leads to an awesome moment where the students and the teacher can basically make something up, some number that has no material effect whatsoever on the mathematics they’re practicing but which gives everyone the sense that “this is our problem.” Big win.

Director Andrew Davis didn’t think it fit the narrative of his film to mention the weight of a cubic foot of desert dirt so we faced a similar dilemma w/r/t density.

Step Four:

  • Give them a pile of information to use as they see fit.

Step Five:

  • Give them time to work.

I put twenty minutes on the clock and asked everyone to email me either a scan or a camera photo of their work when they finished.

Example #1
Example #2
Example #3
Example #4

Step Six:

  • After they compute their final answer, ask them to compare it to their error bounds from step two.
  • Play the answer video.
  • Compare the answer to our guesses from step two. Determine who guessed closest.
  • Discuss sources of error.
  • Discuss follow-up questions.

Stacy: I love this problem, but we still don’t have a good way to check our answer.

So this problem receives a certain demerit for not allowing us to observe the answer. I accept that demerit.

These problems require some kind of plan for challenging students who finish early. The attendees offered two approaches I want to highlight here:

Aaron: change the size of the original shovel.
Justin & Anna: how much shorter would the next shovel have to be for the difference to be the same?

Aaron has changed the input quantities and asked his students to find another output. His students will use the same operations on different numbers. From my experience, this leaves the teacher vulnerable to charges of assigning busy work.

Justin and Anna, by contrast, have made the old output quantity the new input. Before, their students were solving for the total quantity of dirt. Now, Justin and Anna have given their students the quantity of dirt and asked them to work backwards to new inputs. This is a great, versatile way to quickly create a new problem for students who finish early.

Open Questions

  • ft3 or pounds? In a workshop recently, we ended this problem by calling a local composting company and asking them how many cubic feet of compost they brought in on each dumptruck. The units on our final answer, then, were “dumptrucks.” There are different, subtle ways to frame the same question. Do you ask for mass, weight, volume, or dumptrucks? Mr. K and I went back and forth over the difference between these two questions. “How many extra pounds of dirt will Stanley dig after a year?” vs. “How many extra cubic feet of dirt will Stanley dig after a year?” What are their advantages and disadvantages? I’m still going over this in my head but my sense is that our students’ early estimates will be more accurate in ft3, but the answer will be more tangible to them in pounds. (See also: How Big Really?.)
  • I’m curious, if you were in that session, what does it do for your engagement with the problem to see your classmates’ guesses? (I welcome any other comments from the participants, of course.)
  • What app will let people email files to a public folder? I planned to use Dropbox but, last minute, I realized it didn’t support that function so I had participants email me their files, which I uploaded to my domain after updating an HTML file, and the whole thing was ridiculous. How do I cut out the middleman (me) here?

Miscellaneous:

  • Here is the session transcript.
  • I had DimDim’s whiteboard up at the start of the workshop, which turned out to be accidentally awesome. Participants started doodling as they waited for the session to start. One participant drew a map of the US and asked everyone to identify their location with a star.
  • I need to type questions after I speak them so the responses in the transcript make a little more sense to me afterwards.
  • I’m crazy enough to look up the shooting locations for Holes. It remains to be seen if I’m crazy enough to drive down to the Mojave Desert with a scale and weigh a cubic foot of dirt, which is clearly what needs to happen here.

I found our last live session fun and extremely profitable. I’m grateful to everyone who participated live in the DimDim room and later in the comments.

I’d like to pull in twenty more volunteers to play around with a math problem tomorrow afternoon and test out some of my recent modifications to the instructional design. If you can commit 45 minutes, please drop a comment in the box using an email address where I can reach you tomorrow. We pulled a lot of people off the waiting list last time, so consider adding a comment even if twenty people have already signed up.

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