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Hibernating + Open Thread

A reader via email:

Blog monster hungry. Feed me.

I’m not big on the retroactively “sorry I haven’t been blogging” posts. I’d rather proactively explain why I’m not going to be around here for the next several months.

It isn’t for lack of interest in math education or for lack of interesting things happening in math education. For instance:

But my dissertation hearing is scheduled for mid May. I’m in the middle of data collection with lots of writing and analysis ahead and I’m sure I need to become a bit more ruthless in managing my time and writing.

So I’ll see you on Twitter (can’t quit that obviously), at NCTM and other conferences. But I won’t see you around here for a few months.

Please use this as an open thread to talk about whatever while I’m off dissertating. Also here’s all the great classroom action I haven’t written about over the last twelve months. Plenty of food for the blog monster there.

The Frozen Code

Here are the locations of your chain of gelato shops:

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Here is the temperature in the United States today (Fahrenheit):

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So basically business is bad. No one wants your frozen treat.

So what do you do? You lower prices. An across the board cash discount? Maybe. But if you’re Gelato Fiasco, you institute The Frozen Code:

On each day that the temperature falls below freezing, we automatically use The Frozen Code to calculate a discount on your order of gelato dishes. [..] You save one percent for each degree below freezing outside at the time of purchase.

But how do you write this code using the language of variables that your pricing system understands? (Click through for Gelato Fiasco’s answer.)

How would you set this up as a mathematical learning experience for your students?

[h/t reader Nate Garnett]

This is a series about “developing the question” in math class.

2014 Jan 8. Updated to add this important exchange with Gelato Fiasco on Twitter.

Featured Comment

John:

With the clear correlation between temperature and number of chain gelato shops, CLEARLY the temperature causes more gelato shops to be built…or does the number of gelato shops cause the temperature to decrease…am i right?

WTF Math Problems

As I mentioned on Twitter earlier this week, I find a particular kind of math problem extremely exciting now. Here are five of them. I want to know what to call them. I want to know what are their essential features. I want more of them and I want to read more about them.

Here is one of the five, taken from Scott Farrand’s presentation at CMC North.

Here are some points in the plane:

(4, 1), (17, 27), (1, -5), (8, 9), (13, 19), (-2, -11)
(20, 33), (7,7), (-5, -17), (10, 13)

Choose any two of these points. Check with your neighbor to be sure that you didn’t both choose the same pair of points. Now find the rate of change between the first and the second point. Write it on the board. What do you notice?

From Henri Picciotto’s review of Farrand’s session:

Students are stunned to learn that everyone in the class gets the same slope. This sets the stage for proving that the slope between any two points on a given line is always the same, no matter what points you pick.

In an email conversation with Farrand, he proposed the term “WTF Problems” because they all, ideally, involve a moment where the student exclaims “WTF”:

Set up a surprise, such that resolution of that becomes the lesson that you intended. Anything that makes students ask the question that you plan to answer in the lesson is good, because answering questions that haven’t been asked is inherently uninteresting.

These seem like essential features:

  • These problems are all brief. They slot easily into an opener.
  • They look forward and backward. They fit right in the gap between an old concept and the new. They review the old (slope in this case) while setting up the new (collinearity).
  • Students encounter an unexpected result. The world is either more orderly (the slope example above) or less orderly (see problem #2) than they thought.

And the weirdest feature:

  • They require the teacher to be cunning, actively concealing the upcoming WTF, assuring students that, yes, this problem is as trivial as you think it is, knowing all the while that it isn’t.

When did they teach you that in your teacher training?

It’s striking to me that the history of mathematics is driven by the explanations following these WTF moments:

  • We knew how to divide numbers. We didn’t know how to divide by zero. Enter Newton & Leibniz explanation of calculus.
  • We knew how to find the square roots of positive numbers, but not negative. Enter Euler’s explanation of imaginary numbers.
  • We knew what Eucld’s geometry looked like, but what if parallel lines could meet. Enter the explanation of hyperbolic, spherical, and other non-Euclidean geometries.
  • There are lots of WTF moments that haven’t yet been explained.

In school mathematics, though, we simply give the explanations, without paying even the briefest homage to the WTFs that provoked them.

What Farrand and you and I are trying to do here is restore some of that WTF to our math curriculum, without forcing students to re-create thousands of years of intellectual struggle.

So help me out:

  • Have you seen other problems like these?
  • Who else has written about these problems? I believe we’re talking about disequilibrium here, which is Piaget’s territory, but I’m looking for writing local to mathematics.

Featured Comments

David Wees cautions us that the effect of these problems depends on a student’s background knowledge. If you don’t know how to calculate slope, the problem above won’t surprise, just confound. I agree, but the same is true of textbooks and nearly every other resource.

Michael Pershan worries that the “twist” in these problems will become overused, that students will become bored or expectant. (Clara Maxcy echoes.) I demur.

Dan Anderson offers other examples. As do Mike Lawler, Federico Chialvo, Kyle Pearce, Jeff Morrison, and Michael Serra.

Franklin Mason critiques my math history without (I think) critiquing my main point about math history.

Scott Farrand, whose presentation at CMC-North inspired this post, elaborates.

Ben Orlin summarizes the design of these problems in four useful steps.

Terri Gilbert summarizes this post in a t-shirt.

Featured Tweet

December Remainders

Happy New Year. This ThinkUp outfit told me which of my tweets were “biggest” in each of the months of 2014. Twelve “big” tweets, in other words.

Here were my new blog subscriptions in December 2014, some of which might interest you.

What did you fill your head with in December?

Michael Cieply for The New York Times:

The Interview generated roughly $15 million in online sales and rentals during its first four days of availability, Sony Pictures said on Sunday. Sony did not say how much of that total represented $6 digital rentals versus $15 sales. The studio said there were about two million transactions over all.

We gotcha covered, Mike!

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Featured Comment

Stas, with a zinger for the ages:

Probably they took advice from this guy.

Angela Ensminger:

Maybe an opening question to this problem would be:. Do you think Sony had more rentals or more sales? This could lead to some interesting discussions before actually solving the problem.

[h/t Math Curmudgeon]

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