Or, Math Class is Too Full of Spoilers
In grad school, my wife took a class that assigned no homework. The topic was an advanced, hyper-specific area of research—the only plausible problems to give for homework had literally never been solved. Any answer to such a question would have constituted novel research, advancing the field and meriting a publication in a professional journal. The professor assigned no homework for the simple reason that there was no practical homework to assign.
This tickled me. I’d never thought of good questions like a fossil fuel. A nonrenewable resource. Built up over eons and consumed in minutes.
But the thought kept popping back up: Good questions are a resource. And in this new light, something started to make sense, an uncomfortable little fact that had nagged at me since my first year teaching.
I lurched sideways into the profession, like a bowling ball that hops into the next lane and knocks over the wrong pins. I knew nothing about pedagogy, other than whatever naïve intuitions I’d gathered as a student. So that year, whenever we encountered new types of problems, I simply told my geometry students exactly how to answer them.
It felt natural; it felt like my job. Need to prove these triangles are congruent? Do this. Need to prove that they’re similar? Do that. Need to prove X? Do Y and Z. I laid it all out for them, as clean and foolproof as a recipe book. With practice, they slowly learned to answer every sort of standard question that the textbook had to offer.
Months passed this way. But something wasn’t clicking. I kept seeing flashes and glimpses of severe misunderstandings—in their nonsensical phrasings, in their explanations (or lack thereof), in their bizarre one-time mistakes. Despite my best intentions, something was definitely wrong. But I didn’t know what.
And, more worryingly, I didn’t know how to find out.
I’d already coached them how to answer every question in the book. How, then, could I diagnose what was missing? How could I check for understanding? For every challenge I might give them, every task that might demand actual thinking, I’d equipped them with a shortcut, a mnemonic, a workaround. The questions were like bombs defused: instead of blasting my students’ thoughts open, they now fizzled harmlessly.
Good questions are a resource, and I’d squandered mine.
I couldn’t articulate it then, but I began seeing questions differently. They were not just obstacles to overcome, or boxes to check. Instead, they were my fuel. My kindling. My only real chance to ignite reflection, curiosity, and the impulse to seek out deeper truths.
Questions were not just things to answer; they were things to think about. Things to learn from. Giving the answer too quickly cut short the thinking and undermined the learning.
Good questions, in short, are a resource.
Solving a math problem means unfolding a mystery, enjoying the pleasure of discovery. But in every geometry lesson that year, I blundered along and blurted out the secret. With a few sentences, I’d manage to ruin the puzzle, ending the feast before it began, as definitively as if I’d spat in my students’ soup.
Math is a story, and I was giving my kids spoilers.
Now, I don’t believe in withholding the truth from kids indefinitely, or forcing them to discover everything on their own. But I’m aware now of a danger: if I tell them too early how the film will end, they may turn it off halfway through, and I’ll have a hard time convincing them of everything that they’re missing.
Questions aren’t merely targets that need to be hit. They’re also our arrows for hitting bigger, more elusive targets: concepts, connections, ways of thinking.
Questions aren’t the enemy. They’re the ammo.
As I move through my career, I find myself increasingly overcome with the opposite difficulty: too many good questions. Too much cool stuff. A sprawling wealth of worthwhile problems, such that my students and I couldn’t possibly get around to half of them.
But sometimes, I still slip up. I think my students are learning deep concepts and strategies, but they’re only parroting phrases or mimicking procedures. Too late, I realize what’s happening… but by then the suitable questions are used up. They can answer without understanding. And it’s back to the drawing board for me.