Let's just call them "theories of engagement" for now. Every teacher has them, these generalized ideas about what engages students in challenging mathematics. Here's the theory of engagement I'm trying to pick on in this series:
This theory says, "For math to be engaging, it needs to be real. The fake stuff isn't engaging. The real stuff is." This theory argues that the engagingness of the task is directly related to its realness.
This is a limited, incomplete theory of engagement. There are loads of "real" tasks that students find boring. (You can find them in your textbook under the heading "Applications.") There are loads of "fake" tasks that students enjoy. For instance:
No context whatsoever in any of them. Perhaps the relationship actually looks more like this:
I'm being a little glib here but not a lot. Seriously, none of those tasks are "real-world" in the sense that we commonly use the term and yet they captivate people of all ages all around the world. Why? According to this theory of engagement, that shouldn't happen.
Here are fake-world math tasks that students enjoy:
- Ihor Charischak's Jinx Puzzle.
- David Masunaga's Magic Octagon.
- Malcolm Swan's Area v. Perimeter.
- NRICH's Factors and Multiples Puzzle. (Megan Schmidt: "… I have one student in particular who is not particularly motivated by much … when I bust out a puzzle, he’s all in.")
- Matt Vaudrey's Magical Triangle Theorem.
- Andrew Stadel's Weekly Puzzle.
- The meaning of the sequence 3, 3, 5, 4, 4, 3, … , which drove kids bananas the day I wrote it on the board at the end of a test.
- The proof that 2 = 1.
- 2013 Dec 3. The four fours puzzle.
- 2013 Dec 5. Timon Piccini's Broken Calculator.
My point is that your theory of engagement might be limiting you. It might be leading you towards boring real-world tasks and away from engaging fake-world tasks.
We need a stronger theory of engagement than "real = fun / fake = boring."
- Write about a fake-world math task you personally enjoy. What makes it enjoyable for you? What can we learn from it?
- Write about an element that seems common to those enjoyable fake-world tasks above.