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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

by gjm11 16y ago

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.

lkozma 16y ago
From the related book I mentioned in the other comment: "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
btilly 16y ago
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.

by gjm11 16y ago

lkozma 16y ago

From the related book I mentioned in the other comment: "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

btilly 16y ago

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.