How I do Proofs

On how to read proofs, my hope was that I wouldn't be presenting any proofs that are hard to read. And given their level of skill, I didn't want them trying such shortcuts either. That said, the trick is to recognize weasel words where the author's laziness makes the reader do work, and then do that work.

When someone says "clearly" what they mean is "the proof is routine, and I don't want to clutter my line of reasoning digressing into a proof of it." If you don't find it clear, then you have to produce that proof (or at least an outline of it). Filling in these details is often one of the hardest parts of reading someone else's proofs.

"Without loss of generality" means, "I'm about to set up a bunch of stuff that looks like it could be a special case, but it really isn't." As a reader it is then your job to figure out why it isn't a special case. Once you are convinced that it really isn't a special case, then you can accept the specific setup that just got made.

And "similarly" means "the proof for this piece is pretty much the same as what you just saw, and I don't feel inclined to write it all out again." In that case you need to convince yourself that this is true.

Furthermore they were staring at a theorem that was complicated and abstract enough that none of them had any idea why it was true. They had just been given a method of attack, but had no particular reason to have any confidence in it.

I expected and wanted this surprise. I wanted them to take this handout seriously. I thought that the experience of seeing how easy an apparently impossible problem really was would make them pay attention.

When I taught 11th grade English, I started the year by asking students to write a two-page composition (and gave them a concrete, well-defined topic), due the following Monday. They panicked, hardcore: "How do you expect us to do this? That's 2 whole pages! How can we ever get that muc done?"

Now, even when I substitute teach, I have to spend a decent amount of time hand-holding.

Particularly in classes, it feels like there are not many teachers who take the "Let's try this out and see what happens. It's okay if it doesn't work this time, because we're learning a new skill. It might take a few tries," approach. It's a necessary approach, I think, when learning higher-level thinking, particularly systematized thinking.

"The advanced reader who skips parts that appear too elementary may miss more than the reader who skips parts that appear too complex" – G. Polya

However I think that this is useful, even for students with decent problem solving skills.

The biggest issue that I was trying to solve is clearing up misconceptions about what a proof actually is. A beginning student knows that proofs have to be utterly convincing. They know that even many basic and obvious things require proof. So they are uncertain what techniques are OK to use. Worse yet, all of their liberal arts education pushes them towards believing that the way to make something convincing is to add verbiage, examples, supporting references, and so on. All of which is the exact opposite of what you actually want to do in a mathematical proof.

A second issue that I was trying to address is the art of reading problems and figuring out what it actually says. This is why I included specific advice on how some common phrases actually cover two separate claims.

A third issue is that students arrive at a first course where they have to do proofs having seen (and usually ignored) lots of examples of worked proofs, but never having done one. Even if you use divide and conquer elsewhere, there can be a barrier to realizing that this is the technique you need to use here.

And that assumes that the students are competent at divide and conquer. But there are many paths through academia where you can arrive at the first course where you have to do mathematical proofs without having explicitly learning to divide and conquer. You can complain about that situation, or do something about it. I chose to do something about it.