As a neophyte, the only thing I find missing is proof by cases, exhaustive proof (sometimes there are just a few cases that can be trivially proven separately, yet a comprehensive proof is elusive), and (while it's a bit tangential) a couple tips on how to read others' proofs (e.g., what do things like "clearly" and "without loss of generality" precisely mean in this context, and what to do when you encounter them).
I didn't include proof by exhaustion because it wasn't going to come up in the course. But in principle it is the same as any other technique, but the trick is figuring out which cases to break it into. There is an art to that which varies by topic, and I don't feel up to trying to explain it in general. :-)
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.
(As a thought experiment) I find it very interesting that the same flow/dissection can be applied to outside math, as set of tools to solve any problem - even to make those indecisive moments interesting. I just went through the flow to make a 'decision'.
Tangential, but I don't understand this reaction after he tells them in class they're going to go through a proof together:
'You should have seen the shock on their faces! Some started complaining. So I said, "No, seriously. You will all prove this. Just wait and see. It will work."'
Are people really that timid when all they're being asked to do is try?
These students in all likelihood had never, ever, been asked to produce a mathematical proof. They were being told that they were about to write one as a group, after having a general description of how to do it, without seeing any worked out examples first. It is quite possibly the first time that they had ever been expected to do anything for a first time in a math class without seeing a worked example first.
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.
This is (1) probably not useful to anyone with even rudimentary problem-solving skills and (2) more "How to do anything" than "How to prove theorems". But it's for people with scarcely any problem-solving skills. It's kinda sad that a university mathematics course is full of people with scarcely any problem-solving skills, but that's hardly btilly's fault.
I wouldn't expect there to be a lot of HN participants who need it, though.
I agree that it is just divide and conquer, presented in detail, with some specifics thrown in on techniques that are useful in mathematical proofs.
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.
Huh? It's right there: second column, fourth box down. You could quibble that it isn't "one of the techniques", but the effect of that is to invalidate your original complaint that all the chart tells you to do is to apply known techniques.
OK, I looked at the arrows. (1) What's your point? (2) Is there any actual reason why you haven't said already what it is, rather than all this "Look at the flowchart" / "Look at the arrows" nonsense?
There is an arrow from "Do known techniques apply?", labelled "No". It leads to "Try to find a new technique". There's an arrow back from there to "Do known techniques apply?". Obviously, if you succeeded in finding a new technique applicable to the problem, the set of "known techniques" has expanded and you then follow the "Yes" arrow from "Do known techniques apply?".
Is it your opinion that there is something wrong with this? If so, what?
(If you respond with another passive-aggressive "Look at X" reply, I shall ignore it unless looking at X immediately convinces me that you've been right all along and I've been missing something.)
(1) When I see the phrase "known techniques" in the context of math it generally refers to techniques known in the field. Novel techniques are not "known" in this sense. The different, literal, trivial interpretation of "known" makes the word superfluous. That was the motivation for my first comment. In both cases (your point of view being the second) my comment is true; by this flowchart a technique must be known to be applied. The only way I could see someone disagreeing is if they're confused about what "known" means.
(2) It seemed obvious, and moreover too trivial a point to merit more than just pointing it out. I apologize if I've caused any offense.
* To clarify, I don't consider "find a new technique" to be a technique, which is why I said to look at the arrows.
Please don't dismiss the "literal, trivial interpretation". If you read the explanations that came with the flowchart, they try to make it very clear that "known" means "known to you".
Also consider the context. This was a handout to students in a first linear algebra class, that was meant to help them learn to do basic proofs on their homework problems. Nobody expected them to be engaged in original research. Any useful technique they needed to "discover" was very likely to be well-known to lots of people, including me.
Finally it is not clear to me why you think that the word "known" is superfluous. There is a world of difference between the stage where you are running through the techniques you know, trying to find one that fits, and the stage where you're engaged in expanding your list of available techniques. I was trying to get at that difference.
It reminded me of the Feynman method a bit though :)
1. Write down the problem.
2. Think real hard.
3. Write down the solution.
More seriously, the classic book by Gy. Polya "How to solve it" is also very useful:
http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d... (non-affiliate)
http://www.amazon.com/How-Solve-Aspect-Mathematical-Method/d... (affiliate)