Checking Math Work

19 Dec, 2024

When I take a math test (in Calc/Stats/Physics), I rarely check my work. Which is unfortunate -- despite me basically optimizing myself to get everything right on the first try¹, I inevitably make mistakes, which costs points and makes me feel a little sad. This is all despite the fact that I usually have the time to check my work.

(Meta-cognition warning) For me, I've come up with several reasons why I probably don't check my work. The first and most obvious is that, for me at least, checking work is not entirely necessary -- I've already done the math, I'm reasonably confident in my answer (I almost never guess blindly), and my past results have proven that (overall) not checking my work is a safe bet I can put my money on. Another good reason to not check my work is that checking math (and to some extent, programming) work is hard. This whole post is dedicated to explaining why.

Theorem: Checking one's math work is hard.
Proof: Consider a problem that you've solved. How do you check its work? Most people (or at least the ones that I know/imagine) would simply retrace their line of reasoning and their steps, make sure they did all their multiplication and subtraction and parentheses correctly, then simply move on as they don't find glaring issues in their work. But this is wrong.
The real way to check your own work is to do the problem again. From scratch. Without your previous answer.
This sounds absurd. But by doing the problem again (when hopefully you have forgotten the answer and how you did it the first time), you essentially get a second opinion on the answer -- if the two opinions concur, you can have confidence that since two roads both lead to Rome, all roads lead to Rome and that your answer is correct. Likewise, if your answers differ, you can then do an in-depth comparison of both methods and catch a bug hiding in one of your answers, and correct it.

So why is checking your own work so hard? The answer to the second is the concept of familiarity. In the opening example, if I simply skim over my steps, no critical thinking is involved. The only issues that are ever picked up and solved are those of the trivial/glaring variety -- the missing plus sign, the bad addition, the missing case. Really, all skimming is (from my point of view) is your brain's way of saying "look what I have done" and "look how grand and glorious and correct it looks". Rather than trying to find flaws in your argument and criticize your logic, the point of checking your work, your brain simply tries to reinforce its own potentially incorrect beliefs². Adding on to this problem is the problem of familiarity -- you're familiar with your solution as soon as you read it, and it lulls you into a false sense of correctness.
However, when you redo a problem from scratch (and actually invest time and effort into doing so), you avoid both the problem of familiarity (after all, you shouldn't remember the answer) and of ego -- there's no answer to check, no easy way out, and nothing is familiar if you start from scratch.

The only issue with redoing a problem is that it is costly on both time and mental energy. Which is why I don't like checking my work -- I always think it'll cost me too much time and energy and it will result in minimal benefit. And so, I don't check my work. I've still gotten by alright³.

Checking work is hard. Math tests are hard. I still don't know how we as a society acts like it's no big deal and that it's a run-of-the-mill event.

Next time you're checking your test, check harder. Force yourself to think harder and do harder -- don't just idly sit scanning your paper, do something! Criticize your own arguments! Verify all calculations! That is really, checking your own work. It's called checking your work, not validating the easy steps in your work or assuming your work is correct.