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Archive for September, 2012

[3ACTS] Taco Cart

This task is one possible response to this week's check for understanding. It was a pile of fun to produce.

Release Notes

Real to me. My wife and I were on a beach recently and found ourselves in this math problem. This happens to every math teacher, I'm sure. We use our own product. We employ mathematical reasoning in our own lives in obvious and subtle ways. I've tried to discipline myself not to miss those moments, to instead write them down, photograph them, and turn them into a task where students experience the same dilemma my wife and I did.

Google Maps. The game here is to screenshot a bunch of tiles from Google Maps, align and stitch them together in Photoshop, and then fly around that large image in AfterEffects.

Use appropriate tools strategically. The sequels aren't optional here. One sequel suggests that the cart will start moving towards you and asks "at what location will both paths take the same time?" The other asks for an even faster path than either of the two originally posed.

In both cases, I enjoyed setting up and solving the algebraic models.

But as I contemplated solving one equation and finding the minimum of another, symbolic manipulation never occurred to me. Without any teacherly presence hovering over me, nagging me to rationalize my roots, the most obvious, practical solution was Wolfram Alpha — no contest.

A teacher at a workshop pulled off a similar move this week and felt embarrassed. He said he had "cheated." Tools like WolframAlpha require us to come up with a more modern definition of "cheating." (And of "math" for that matter.)

The ladder of abstraction.

Referring back to the check for understanding, here are ways the original task had already been abstracted:

  • the dog and the ball are represented by points; their dogness and ballness have been abstracted away,
  • very little of the illustration looks like the scene it describes, for that matter; the water and sand are the same color; the image of a dog swimming after a ball has been turned into the remark "1 m/s in water,"
  • points have already been named and labeled,
  • important information has already been identified and given,
  • auxiliary line segments have already been drawn; the segments AB and BC and DC don't actually exist when the dog is running to fetch the ball; they have been abstracted from the context later.

My version of the task starts lower on the ladder. You see the sand and the sidewalk. You see what it looks like to walk in each. They aren't abstracted into numerical speeds until the second act of the problem, after your class has discussed the matter. I do draw a triangle on the video, which is a kind of abstraction. I didn't see any way around it, though.

BTW. Andrew Stadel also has a nice task involving the Pythagorean Theorem and rates.

[LOA] Check For Understanding

Adapted from the May 2012 issue of Mathematics Teacher:

A dog is running to fetch a ball thrown in the water. Point A is the dog's starting point, point B is the location of the ball in the water, and point D can vary. Given that the dog's rate of swimming is 1 meter per second and its rate of running is 4 meters per second, determine where point D should be located to minimize the time spent fetching the ball.

Some questions to consider here:

  1. In what ways has this context already been abstracted?
  2. Can you de-abstract (recontextualize? concretize?) the context? Describe a task that would allow students to learn about the process of abstraction rather than just encounter its result.

[LOA] Lies We Tell Ourselves

  • The number of party guests increases according to the function g(t) = 2t + 4, where t is the number of hours after the party started and g is the number of guests.
  • The number of iPads sold increases according to the function s(t) = 2t + 4, where t is the number of weeks after the iPad went on sale and s is the number of iPads sold in millions.
  • The number of points the team scores increases according to the function p(t) = 2t + 4, where t is the number of minutes after halftime and p is the number of points scored.

Party guests. iPads. Points.

They're all the same to the student who doesn't understand abstraction, the process by which we turn those contexts into words and symbols. The idea that any one of those contexts will engage that student any more than another is a fiction.

BTW. Clarifying: the issue at hand isn't that these three problems are simplistic or false abstractions of a context. It's that they start at a high-level of abstraction. (This isn't a revisitation of pseudocontext, in other words.)

2012 Sep 27. Nathan Kraft points us to some research that says, "This kind of superficial personalization, indeed, increases engagement and achievement." So I may have overstated my case considerably. The point of this ladder of abstraction series, though, is that investments in making abstraction more explicit are way more worth our while, not that other investments aren't important also.

101questions Updates

[cross-posted to the 101questions blog]

The big changes:

  • You can upload files now. No more pasting links to external content. You no longer have to upload your image to Dropbox or ImageShack or anywhere else (an incredibly cumbersome step for a lot of people) just to get material onto 101questions. We're no longer restricted to YouTube's hardline interpretation of Fair Use either.
  • You can download files now. Click "Actions" on any uploaded first act and then click "Download." It's awesome. It downloads whatever file the user uploaded (it won't pull down content uploaded to YouTube or Vimeo) along with a text file with all the submitted questions.
  • You can get more responses more quickly by sending your link around. It bummed people out that they'd link to a first act and other people couldn't add a question unless they saw it randomly come up on the homepage. "You should be able to add a question to the page itself," they said. I resisted but I was wrong and now you can.

The small changes:

  • A pile of corrections to aspects of the UI that annoyed me, Amazon S3 integration, automatic comment subscription, a lot ground laid for the winter update.

#5: Kids care less about context — "real world" problems — than they do about problems that start at the bottom of the ladder. "Real world" is a risky bet.

Real World

Here is a "real world" problem:

The caterers Ms. Smith wants for her wedding will cost $12 an adult for dinner and $8 a child. Ms. Smith's dad would like to keep the dinner budget under $2,000. Ms. Smith would like to invite at least 150 guests to her wedding. How many children and adults can Ms. Smith invite to her wedding while staying within budget?

There is nothing to predict. Nothing to compare. The important information has already been abstracted. The question has been fully defined. The problem, as a whole, has been stretched tight and nailed to a board. The student's only task is to represent the important information symbolically and then apply some operations to that representation.

And so hands go up around the room. The students attached to those hands say, "I don't know where to start." The task has hoisted them up to a middle rung on the ladder of abstraction and left their feet dangling in the air. Students are frustrated and disengaged in spite of the "realness" of the task.

Fake World

Meanwhile here is a "fake world" problem:

Here are questions you can ask at the bottom of that task's ladder:

  1. What are the new percents? Write down a guess.
  2. Which quantities change?
  3. Which quantities stay the same?
  4. What names could we give to the quantities that are changing?

These questions include students in the process of abstraction. Each student guesses the new percents and is consequently a little more interested in an answer. Students aren't just asked to accept someone else's arbitrary abstraction [pdf] of the context. They get to make their own arbitrary abstraction of the context. (Why ABCD? Why not WXYZ?) All of these tasks prepare them to work at higher levels of abstraction later.


My preference is a combination of the two — a context that is real to students and a task that lets them participate in the abstraction of that context.

But I can't tell you how many conversations I've had with teachers (veteran and new) and publishers (big and small) who tell me the fix for material that students don't like is to drape some kind of context around the same tasks. Rather than expanding and enriching their tasks to include the entire ladder of abstraction, they insert iPads or basketballs or Justin Bieber or whatever they perceive interests students.

Real-world math is a risky bet. Bet on the bottom of the ladder. Here are some of those bets:

  1. With the wedding task above, the teacher can ask students to pick any combination of children and adults they think will work. Any combination. 100 kids and 50 adults? Fine. Now tell me how much it costs. We're all invested for a moment in a problem of our own choosing. Then we assemble student work side-by-side and notice that we're all doing the same kind of calculations. Then we say, "All your work looks the same. What's happening every time?" The students are participating in the symbolic abstraction.
  2. Louise Wilson is using the images and videos on 101questions to give students practice just asking questions about a context. Asking questions is the assignment. Getting answers isn't.
  3. Andrew Stadel is giving his students daily practice with estimation, another task at the bottom of the ladder.

We ask our students to work most often at the top of the ladder and the result is a pervasive impression that a successful math student is a student who can memorize formulas and implement them quickly and correctly. Those are, of course, great and useful skills, but mathematicians also prize the ability to ask good questions, make good estimations, and create strong abstractions. These are skills where other students may excel. There is unrewarded excellence in our math classrooms because we have defined excellence narrowly as being good at abstract skills. You can only find (and then reward) that excellence by betting on the bottom of the ladder of abstraction.

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