Asking the Right Question

I’ve been poking around on math reddit lately, and in particular /r/askmath and /r/learnmath. And one thing I’ve really noticed is that many of the posters are really bad at asking questions.

Learning to ask good questions is really important, for a couple reasons. First and most obviously, it’s easier to get help with things and learn things if you can ask better questions. Second, and maybe more importantly, framing questions well is a lot of what makes you a good mathematician.

The most boring way to ask a bad question is just to not include enough information. Sometimes this is just laziness (“I don’t understand how to do this problem, please help”). And I’ve definitely seen questions asked that are thin disguises over “I don’t want to do my homework; can someone do it for me?”

But more often, badly-phrased questions result from deep confusion on the part of the asker. If they understood the material well enough to ask their question clearly and correctly, they wouldn’t need to ask it in the first place.

A lot of math is less about answering questions than about figuring out exactly what question you should be asking, and how to make it precise. We tend to sweep this under the rug a bit when teaching, in a way that I suspect probably leads to a certain amount of confusion.

When we teach, we often ask a question, and then demonstrate a tool to answer it, without necessarily stopping to explain why that question is a good one, or how people settled on asking exactly that question. And this often leaves our students with the sense that what they’re doing doesn’t really mean anything.

This is a major reason students fall back on figuring out what “type of problem” they’re working on, and then following “the steps” to get the answer. They see math as a sort of opaque box, and a question asks them to perform the correct magical ritual to get the answer. Because if the words you’re using—and your questions—don’t have a meaning, that’s all you can really do.

And that’s how you get questions like “how do I find solutions to $f(x) = \sin(x)+ \tfrac{1}{2}$.” I can tell what the original question probably was. But because the student doesn’t really understand what a “function” is, they ask a question that is, read literally, completely nonsensical.

There’s a different type of bad question that comes up: questions of the form “prove to me that this definition is correct”. Or, relatedly: “what is the true meaning of this ambiguous statement”.

I saw one question asking whether $-3$ was a prime number. And that’s actually a tricky question, but for a basically dumb reason. In elementary number theory, we usually define prime numbers to be positive integers. That’s part of the definition, so $3$ is prime and $-3$ is not. Why? Because we said so.

And then when you start doing algebraic number theory, you define primes as a factorization property in a ring, and you see that $-3$ is in fact a prime in the ring of integers. So is $-3$ a prime? Yes, by the definition of prime elements in a ring. Apparently the same question, two different answers—determined by the definition we happened to pick at the time.

Another great example of this problem is questions about order of operations. What’s the value of $6/2 \cdot 3$? It could be 1; it could be 9. I can’t tell you what the original author meant. I can only tell you that they’re a sloppy writer.

This happens in a lot of cases where we’re just discussing a notational convention. There are no underlying facts that make notation right or wrong; screwing up order of operations isn’t really the same type of error as adding numbers incorrectly. It’s just an aid to communication¹.

My favorite type of badly written question does this exactly right. Sometimes, you see a question that’s basically “this one thing feels kind of like this other thing, but I can’t tell you how. Can you tell me?”

These people are doing good math. They’re noticing a pattern, and trying to put it into words. They’re maybe not quite there, and sometimes it’s hard to answer the question clearly. But it shows great instincts.

And this is how math tends to actually get done! When we teach, we tend to define terms, then state theorems about them, and then prove the theorems. But this is exactly backwards from how math is often actually done. First we understand what’s going on; then we figure out what the rule is and write it down; and finally understand what conditions are important and give those conditions names. That is, we formulate a proof, then state the theorem, and then define the terms.

These questions are working on step 1. I want to encourage them.

Have any thoughts about questions students ask? Any good techniques for helping your students write better questions? I’d love to hear about them in the comments below!