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→Note to Ed Poor about Pedagogy: new section
Regarding the use of "guises" in the Instruction section, this is totally legit. When students begin using the Axiom of Choice, they are rarely told that it is the Axiom of Choice, and rarer still are they told the bizarre consequences and debate over the axiom. I myself was taught Zorn's Lemma first, and freely used it before I took some set theory and learned about the Axiom of Choice. Calling it a guise reflects the atmosphere of concealing the true nature of Zorn's Lemma in order to get students to use it and think it's not so bad. That wears down their suspicions for when they learn about the truly strange implications of the AC. It's not a bad analogy to compare it to drug dealers who get you hooked, and then you won't know how you could have done mathematics without it. To water this section down by simply stating that Zorn's Lemma is equivalent, is to ignore the pedagogical deception that is occuring (intentionally or unintentionally). -[[User:Foxtrot|Foxtrot]] 05:06, 5 December 2008 (EST)
Regarding the use of "guises" in the Instruction section, this is totally legit. When students begin using the Axiom of Choice, they are rarely told that it is the Axiom of Choice, and rarer still are they told the bizarre consequences and debate over the axiom. I myself was taught Zorn's Lemma first, and freely used it before I took some set theory and learned about the Axiom of Choice. Calling it a guise reflects the atmosphere of concealing the true nature of Zorn's Lemma in order to get students to use it and think it's not so bad. That wears down their suspicions for when they learn about the truly strange implications of the AC. It's not a bad analogy to compare it to drug dealers who get you hooked, and then you won't know how you could have done mathematics without it. To water this section down by simply stating that Zorn's Lemma is equivalent, is to ignore the pedagogical deception that is occuring (intentionally or unintentionally). -[[User:Foxtrot|Foxtrot]] 05:06, 5 December 2008 (EST)
== Note to Ed Poor about Pedagogy ==
Ed:
A couple of things before we begin:
#I come in peace. We have had bitter arguments in the past. I come in peace.
#The bulk of this message was written before the recent flap about deleting the axiom of choice article, though it does relate to that article. It was written just after your "grad students, stop showing off" comment. You need to understand this context. But it has suddenly become extremely timely. Please be patient.
#The article someone wrote giving the pronounciations of the numbers from 1 to 10 was (sorry about personal remarks!) utter garbage. That's not the way to go. I hope to explain. But I think things that you have written may have contributed to people thinking that's what you wanted. I hope to convince you that it isn't what you should want.
#I'm sure you are aware that the people to whom you issued the "three weeks" ultimatum are dead and gone.
There's a whole lot that I'd like to say on the subject of math pedagogy. (And that awful game of "wff 'n proof" :-)
For background, you might want to look at some things I wrote a few days ago [[Talk:Boolean_algebra]] (you might not have seen it yet) and my draft of an article about paradoxes including some material incidentally relating to AC, in my sandbox [[User:SamHB/Mathematical_paradoxes]].
I see that you are complaining about "grad students showing off" when they should be working on more fundamental issues of logic and proof theory. While I don't agree that they were showing off (they were simply trying to fix a disaster), I do agree that the page is a disaster. Among other things, it and its talk page have been block magnets from day one.
But we do need a page on that topic. You might want to look at my paradoxes draft to get an idea of the level that I consider appropriate. Note: It was '''not''' intended as a draft of AC. It just happens to talk about AC in another context.
Now, about proof theory. I already expressed in [[Talk:Boolean_algebra]] the view that mathematical logic should be considered a topic that is simply out of reach for CP. But, from what you have written above, you are interested in proof theory on a much more basic level. What you have written at [[Talk:Addition]] also shows that you want CP's math and science articles to reach all the way down to the fundamentals.
I'd like to argue that reaching too high and reaching too low are '''both''' pitfalls to be avoided. I've already discussed the issue of reaching too high, and that we have to be careful that, wherever we do reach, the path is solidly filled in. But, when we reach too low, we turn CP into a "textbook of common-sense notions". We are an '''encyclopedia'''. Something you look things up in when you already have basic knowledge of the subject, either to go on to the next step or to fill in gaps. We are not an '''introductory textbook'''. People do write such books (remember Dick and Jane?) and they are used in primary school. But I doubt that we have the expertise or time to write such things. (I speak only for myself. If you can do it, by all means do so, but I question the wisdom of following this path.)
Take the case of proof theory, which you raised both here and on the Boolean algebra talk page. The real "professional grade" topic is too high. (Except maybe for people who liked [[WFF_'N_PROOF]] :-) But you raised the question "how to prove a theorem of geometry". I submit that that's too low. The question in a student's mind that you would be trying to address would be something like "What does it mean for a proof of Pythagoras' theorem to be correct?" What does the proof mean? What does the statement mean? I submit that these are common-sense notions. 7th graders (or whatever) just need to be shown the statement and the proof, and they'll figure it out. Of course, they need common sense knowledge of how to interpret the sentences, but that isn't something we can teach them. As they go on in their mathematics education, they will become more adept at these issues. And, in graduate school, they may learn about formal proof theory. By the way, the existing [[Pythagorean_Theorem]] article is excellent. Among the best math pages we have. It needs no additional supporting articles from below.
[[User:SamHB|SamHB]] 11:27, 5 December 2008 (EST)