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?
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.
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.
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.
— Nacho Santa-María (@nacheteer) January 11, 2015