Working Backwards

I teach a lot of students who are still learning the basics of proof-writing. My calculus students are seeing their first college math, and often my number theory class is the first really proof-heavy class that a lot of the students take. So I spend a lot of time helping students figure out how to write good proofs. The single best piece advice I’ve come up with is to get comfortable working backwards.

Math students are generally comfortable working forwards. At the beginning of our education this is the only way of working we have: we’re given a problem, like “add these two numbers together”, and we use some algorithm to work out the answer. I like the way Jordan Ellenberg describes this:

[U]ntil algebra shows up, you’re doing numerical computations in a straightforwardly algorithmic way. You dump some numbers into the addition box, or the multiplication box, or even, in traditionally minded schools, the long-division box, you turn the crank, and you report what comes out the other side. Algebra is different. It’s computation backward.

But even once we get to high-school algebra, we don’t really tend to feel like we’re working backwards. In practice we develop an algorithm for solving equations and turn the crank; instead of the addition box or the long-division box we now have the quadratic-equation box. This effect is strong enough that when I give my calculus students problems where I explicitly tell them to guess and check¹, they feel very confused and want me to tell them what steps to follow to finish the problem.

But this wanting to know “the steps” is a trap—it leads to treating problems like a black-box request for an algorithm, rather than thinking about what’s actually going on. And as soon as problems require any sort of creative engagement, the search for steps fails.

This first hits students hard when they learn to do integrals. When I taught Calculus 2, I spent a couple lab periods having my students do integral worksheets while I answered questions and gave advice. Fairly frequently, they would spend five or six minutes staring at an integral, before giving up and asking me how to start; I sometimes pointed out that we only had four or five things to try, and if they’d just tried all of them they’d have found one that worked already. But they were uncomfortable with the idea that there wasn’t one correct thing to try.

But all of this becomes especially important when you start doing proofs, because in proofs there is no straightforward algorithm—and often you can’t even really work forward.

Working forward in proofs can be quite useful, to be fair. Given a set of hypotheses, you can start listing off things that the hypotheses obviously imply. Especially in the context of a class, you can often see “okay, if I know these three things, it looks like I should try applying that theorem and see what I get.” But this has really serious limits. You can’t plausibly write down all the implications of your hypotheses.

Instead, you need some idea of where to go. So you should start by looking at the goal, the thing you want to prove. Figure out what you could know that would be enough to finish the proof.

If you can see how to do that, then great, you win. If not, now go look at the hypotheses, and figure out what you can reasonably and easily conclude from them. Can you see how to get from there to the things you wanted?

You develop a sort of push-pull dynamic: work backwards from the goal, then forwards from the premises, then backwards from the goal again. And hopefully, eventually things will meet in the middle.

This seems pretty mundane. But a lot of students are comfortable working forward, and deeply uncomfortable working backwards. They can draw conclusions, but are bad at staying focused on the ultimate purpose of whatever they’re doing. So just prompting them to think about their goals in the middle of proofs can be really helpful.

I do this a lot while lecturing. In the middle of a proof, stop and ask the class to remind me what I’m actually trying to do. They find this surprisingly hard. Sometimes they even struggle to remember what the actual theorem we’re trying to prove is, despite the fact that it’s still written on the board; it’s easy to get lost in the weeds of whatever you’re doing right now and forget the broader context. But without that context, everything you’re doing is kind of pointless, and it’s really difficult to decide what to do next.

And that idea of the importance of context, of focusing on your actual goals, is just as true outside of math class as it is inside. I see this a lot when I give feedback on people’s writing and speaking: they will keep saying things, and the things will be correct, but they won’t do a good job of saying things that are relevant to their goals and message, and of telling us why the things are relevant.

But you can also see this in a lot of bad planning and management. If you lose track of why you’re doing what you’re doing, you’re much less likely to actually achieve your goals. You’ve forgotten what they are, so if you meet them it’s essentially by luck!

This is one of the dangers, for instance, of getting too reliant on management-by-metric. Originally you create a metric to measure how well you’re achieving some goal. But over time, people forget about the goal and remember the metric—and do things that improve the metric, but in ways that don’t advance the original goal.

Staying focused on your actual goal, and working backwards, is really helpful for learning to write proofs. So if you’re teaching people how to write proofs, this is worth explaining explicitly, and then actively training. Keep asking people why they’re doing what they’re doing, and how it gets them closer to the conclusion they want to prove!

But it’s also a really transferable skill, one that can help you in almost all aspects of life. Another example of general thinking skills that studying math helps develop, but are helpful everywhere.

Do you have any suggestions for how to help students get comfortable working backwards? Any other tips for teaching students to write proofs more fluently? Please share them in the comments!