Hi algebraists,

I am sensing that it may be the time in the semester for me to deliver the following very strange pep talk. At first, it is not going to sound much like a pep talk, but bear with me, because it'll come back around. The pep talk consists of three parts:

Part 1: You suck at proofs

The proofs I have looked at so far on the first few homeworks have generally been pretty bad. There's a lot of assuming-the-conclusion, there's some wacky use of quantifiers, there's the thing where you accidentally assume that the group is abelian, there's accidentally only proving part of the theorem, there's lines that don't follow from the previous sentence, there's extraneous variables, there's unnecessary complications, there's “spitballing” aka long human-words sentences that seek to obscure that you haven't quite nailed down the key idea – I could go on.

“Why are you being so mean to us in that paragraph?”, you sniffle. I promise I'm not: indeed, I hasten to add that everybody has turned in at least one or two expert-level complete correct proofs. What I'm doing with this litany of errors is trying to give you a realistic picture of where you're currently at.

Part 2: It is currently correct for you to suck at proofs

Indeed, if you didn't suck at proofs, (a) I would be very surprised and (b) I would wonder why you were bothering to take this class. It is correct for you to suck at proofs because you're 4-ish weeks into a class about proofs. If you were 4-ish weeks into an underwater basket-weaving class, would you expect to be producing beautiful Longabergers? If you were 4-ish weeks into playing tennis, would you expect to be Serena Williams?

The main reason you suck at proofs is that, besides learning abstract algebra content, you are learning how to write proofs. Even if you have successfully written proofs in other classes, what makes a good abstract algebra proof is often a bit different from what makes a good proof in whatever other class.

As a rule of thumb, at this stage, you should generally expect that most of your proofs have at least something wrong with them. “What percentage of proofs, and how wrong?” you ask, and I gently smile and refuse to give any quantitative answer because that is specifically not my point.

Part 3: The way to get better at proofs is to embrace sucking at proofs

The key feature of human learning is learning from mistakes. This requires making mistakes and receiving feedback about them, either from another person or from the world itself. (You learn how long to cook a pancake by burning a pancake.) I am inviting you to recalibrate your confidence level so that (a) you can more productively engage with feedback, (b) you aren't holding yourself to an unreasonable standard, and, crucially, (c) you can develop the indispensable skill of identifying when you are sucking.

Part 3a: Embracing sucking at proofs helps you hear feedback

If you think that you are awesome at proofs, then whenever someone tells you that your proof is incorrect, your immediate set of emotions will be mostly negative. In particular, it may include such things as anger (“how dare that person impugn my beautiful proof and/or my honor”), self-righteousness (“they just don't understand my beautiful and perfect thoughts”), dismissal (“they clearly don't know what they're talking about”), or even humiliation (“because of this error, my proof is worthless and so am I and I shall thus fling myself into a volcano”). I think if you pause and reflect about these emotions for a hot second, you'll see that each one of them is a direct impediment to learning from mistakes.

(Indeed, even if you think you should be awesome at proofs but you aren't, then you often feel this same set of negative emotions, but this time pointed at yourself. I don't think I need to tell you why that's bad, either for learning or for, like, being.)

If, instead, you walk into the situation with the idea that your proof is likely to be flawed, then suddenly you have the room to engage with emotions like surprise (“oh wow, I can make this work for infinite groups!”), satisfaction (“oh nice, removing that extraneous $k$ makes my proof much easier to read”), and pride (“hell yeah, the part I felt good about was indeed right”).

Part 3b: Embracing sucking at proofs allows you to suck

Or, in other words, it grants you freedom from pretending that you don't suck.

I think a lot of the time, when you leave something blank or say “I don't even know where to start” or similar, it's actually because you're experiencing these incorrect expectations about what your performance should be, and the negative emotions associated with them. It's somehow easier to be like, “well, I'm fine, it's this problem that's wrong” than it is to be like, “I can see how to do 20% of this but then I'm not sure.” Release this self-expectation! Allow yourself to suck! Then you can engage – tentatively, hesitantly, suckily – with a proof that you don't currently see the whole way through.

One of my dumb little maxims is that “anything worth doing is worth doing poorly.” It's worthwhile to suck! It's better, on both your end and my end, to do a partial job that you know is partial than it is to do no job at all. Why, you may reasonably ask? Well, on my end, I can't give helpful feedback on a blank page. If you don't show me the basket you made that we both know sucks, I can't say anything about how (googles hastily) your slews aren't alternating properly between the stakes and bye-stakes. On your end, there's real value in identifying to yourself the parts that you are confident about (proof frame? unpacking definitions?) and the parts where you suck.

Part 3c: Embracing sucking at proofs helps you learn not to suck

The fancy word for the idea at the end of this last paragraph is “metacognition” – thinking about your own thinking. This is another one of those key and inescapable parts of the learning process. Indeed, a bunch of researchers back in the '90s synthesized decades of educational research literature, and one of the points in their executive summary was that effective instruction is that which promotes metacognition.

If you accept that you might suck at proofs, then you can ask yourself a bunch of useful questions: Does this proof suck? Do I know what I'm talking about here? Can I make this more precise? Are these quantifiers in the right order? Are the two things on either side of this equals sign really the same? What do I mean when I say this long string of human words? Am I assuming the conclusion? Am I using all my hypotheses? What mistakes that I commonly make am I maybe making here?

Repeatedly (perpetually!) asking yourself these questions is how you get good at proofs. It is a hallmark of expertise; it is what differentiates a novice prover from an expert. And you can't get there without first sucking, and then embracing the suck.

the bright-side-dark-side bus meme, but both people are captioned "I suck at proofs"

With love and confidence in your proof-writing abilities,

  • Dr Bagley