The Law of the Broken Futon

By Ben Orlin 

Try asking random adults about their math education. They refer to it like some sort of NCAA tournament. Everybody gets eliminated, and it's only a question of how long you can stay in the game. "I couldn't handle algebra” signifies a first-round knockout. “I stopped at multivariable calculus” means “Hey, I didn’t win, but I’m proud of making it to the final four.”

It’s as if each of us has a mathematical ceiling, a cognitive breaking point, beyond which we can never advance.

But there’s a new orthodoxy among teachers, an accepted wisdom that just about anyone can learn just about anything. It takes grit, effort, and good instruction. But eventually, you can bust through any ceiling.

I love that optimism, that populism. But if there’s no such thing as ceilings, then what do students keep thudding their heads against?

Is there any way to bridge this canyon-wide gap in views?

I believe there is: the Law of the Broken Futon.

In college, my roommates and I bought a lightly used futon. Carrying it up the stairs, we heard a crack. A little metallic bar had snapped off. The futon seemed fine—we couldn’t even tell where the piece had come from—so we simply shrugged it off.

After a week, the futon had begun to sag. “Did it always look like this?” we wondered.

A month later, it was embarrassingly droopy. Its curvature dumped all sitters into one central pig-pile.

And by the end of the semester, it had collapsed in a heap on the dorm room floor.

Now, Ikea furniture is the fruit fly of the living room: notoriously short-lived. There was undoubtedly a ceiling on our futon’s lifespan, perhaps three or four years. But this one survived barely eight months.

In hindsight, it’s obvious that the broken piece was crucial. The futon seemed fine without it. But day by day, butt by butt, weight pressed down on structures never meant to bear the load alone. The framework warped. The futon’s internal clock was silently ticking down toward an inevitable failure.

And, sadly, so it is in math class.

Say you’re acing eighth grade. You can graph lines, compute slopes, specify points. But if you’re missing one vital understanding—that these graphs are the x-y pairs satisfying the equation— then you’re a broken futon.You’re missing a piece upon which future learning will crucially depend. Quadratics will haunt you; the sine curve will never make sense; and you’ll bail after calculus, consoling yourself, “Well, at least my ceiling was higher than some.”

Why not wait to add the missing piece later, when it’s actually needed? Because that’s much harder. In the intervening years, you develop shortcuts that do the job, but warp the frame. You’ll need to unlearn these workarounds—bending the futon back into its original shape—before you can proceed.

Once under way, damage is hellishly difficult to undo.

This, I believe, is the ceiling so many students experience in high school and early college. It’s not some inherent limitation of their neurology. It’s something we create. We create it by prizing right answers over deep reasoning. We create it by saying, “Only clever people will get it; everyone else just needs to be able to do it.” We create it by saying, in word or in deed, “It’s OK not to understand. Just follow these steps and check your answer in the back.”

We may succeed in getting the futon up the stairs. But something is lost in the process. Moving forward without key understandings is like marching into battle without replacement ammo. You may fire off a few rounds, but by the time you realize something is missing, it’ll be too late to recover.

A student who can answer questions without understanding them is a student with an expiration date.

Ben Orlin is a teacher in Birmingham, England. His blog is Math with Bad Drawings.
Email: ben.orlin@gmail.com