What is the biggest problem facing humanity this week?
- A. The threat of Grexit
- B. The bittersweet knowledge that someday, when all of this has passed, we’ll have fewer opportunities to use the amazing word “Grexit”
- C. People thinking functions are linear when they’re SO NOT LINEAR
- D. Other (e.g., cat bites)
If you answered C, then congratulations! You are probably a teacher of math students ages 13 to 20, and we all share in your pain.
For everyone else (including you poor cat-bitten D folk), what are we talking about? We’re talking about errors like these (warning—mathematical profanity ahead):
Every rule in secondary school math is actually a statement about objects—shapes, numbers, whatever. But they don’t feel like that to students. They feel like purely symbolic manipulations, rules for how x’s and y’s move around a page, not for how actual things relate to each other. When you see rules that way, it’s easy to fall prey to certain systematic errors.
Why is this particular error so tempting? Perhaps because it closely resembles a real rule students have learned:
This is called the Distributive Law, and it’s actually a deep fact about the numbers, an essential link between addition and multiplication.
You can discover it through numerical examples:
Or you can understand it with arrays:
Or you can use my colleague’s wonderful phrasing and think of an expression in parentheses as a “mathematical bag”:
It’s that students don’t learn the distributive law as a fact about numbers. They learn it as a fact about parentheses.
The notation “f(x)” doesn’t mean “f times x.” It means “what I get when I put x into the function.”
But our faithful symbol-pushers don’t always catch the distinction. They see parentheses, and think, “Oh yeah! That rule I learned should apply here.”
What’s the solution? Here’s my current game plan:
- Teach the distributive law more carefully. Draw pictures. Work examples. Talk about “bags.” Make sure they understand the meaning behind this symbolism.
- Teach function notation much more carefully. Give them the chance to practice it. Think like Dan Meyer and seek activities that create the intellectual need for function notation.
- Keep stamping out the “everything is linear” error when it crops up. Like the common cold, it’ll probably never be entirely eradicated, but good mathematical hygiene should reduce its prevalence.
What’s the only thing giving me pause here, making me doubt my nice, tidy theory of this error? In two words: Jordan Ellenberg.
In How Not to Be Wrong, he catalogs a variety of cases where people falsely assume “all curves are lines.” His examples are never just symbolic. Each time, people are messing up on a conceptual level. They’re not just pushing x’s and y’s around a page, but genuinely believing their own wrong statements.
So is my approach off-base? Are people actually acting on false beliefs about functions, or (as I’ve posited) are they failing to think about them as “functions” at all?
I’d love to hear what you think. And if you’ve got solutions for the Greek economy or cat bites, well, I’m all ears.
UPDATED: edited to correct the fact that I do not know the difference between numbers and letters. Oh well, like I always say, “C strikes and you’re out.”