What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then ohhhhhhhh.
Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for ohhhhhhh.
But there’s a common problem: too often, kids can beat the puzzles without feeling any of those things.
I find this, for example, with lines in the coordinate plane.
The suite of coordinate games should fill weeks, or months. But there’s a cheat code: y = mx + b. In my class, lots of students arrive already knowing this formula, and it means they can solve would-be-tricky problems (“See this line? Give equations for five more that never touch it!”) with boredom and ease (“Uh…. y = 2x + 1, y = 2x + 2, y = 2x + 3… do you want me to keep going?”).
Do they understand what the cheat code means, or why it works? Rarely. But that doesn’t bring my vanquished puzzles back to life.
So I need new puzzles. Ones that work for kids who have never heard of “y = mx + b,” but which also challenge students who have.
I’m still growing my portfolio, but here’s some of it:
How much of high school math would be easier if students understood that graphs express relationships between variables?
Answer: Basically all of it.
My hope is that questions like this can build the groundwork early.
It’s natural to see this and think, “Wait, a hyperbola? That’s way too hard. You’ve got to walk before you run. And you’ve got to run a mile before you run a marathon over hot coals.”
But I see it a different way: Before you make an earnest study of lines, you’ve got to study some non-lines.
If I were trying to teach you about animals, I might start with cats and dogs. They’re simple, furry, familiar, and lots of people have them lying around the house. But I’d have to show you some other animals first. Otherwise, the first time you meet an alligator, you’re gonna be like, “That wet green dog is so ugly I want to hate it.”
Equations aren’t etched in stone. They’re flexible and mutable as dry-erase marker. All those fun manipulations we use to solve equations can be used to reconfigure and redecorate them, too. And this quick task opens up two fun conversations:
Fun conversation #1: “Hey, check out how the same x-y pairs work in all these equations!”
Fun conversation #2: “Which of these equations is best for generating points on the line?”
Time has smiled on this little puzzle. Not only does it build intuitions about slope, but its room for freedom and ambition (“I’m going to do a point where the x-coordinate is a million!”) brings out interesting ideas and puts student conceptions on display.
Sometimes, I want puzzles with simple numbers. They’re more forgiving, inviting varied approaches and intuitions.
But sometimes, I want puzzles with hard numbers. These ones are rigid as brick. They demand greater technical competence, and usher students towards strategies that are more abstract and general – strategies they can carry forward.