Adrian
Posts:
197
From:
Houston, TX
Registered:
1/1/07
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Re: If you really want math reform....
Posted:
Jan 8, 2007 9:55 AM
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Yes, I don't think Goldenberg was following that thread too closely. My position is that pedagogy isn't really the issue. You can memorize and recite postulates from the Elements ("old school") or do nothing but problems. I do think there is something to the problems approach, incidentally. I think there ought to be a way to make a book entirely of problems grouped into sections that one moves from one section to the next so that when they are done, they know Algebra, say. Nevertheless, if all those problems are "solve for x" types of problems with no "show that" problems, then you simply aren't doing math -- at least that's what I claim. You may well be doing something else that is very important -- physics or engineering, perhaps -- but not mathematics which is the proving of theorems.
And, yes, "proving theorems" is precisely the use of deductive reasoning to justify a proposition (the statement of a theorem) based on axioms or other theorems. I guess I wasn't that explicit about that, but the fact that it is this strict deductive approach is part of what makes it formal. So, that is what I am talking about when I am saying "formal justification" -- as opposed to informal heuristic rationales. Empirical methods are actually fallacious when applied to mathematical theorems. They are an okay basis for a student to suspect that an assertion might be true, but not at all okay to roll out with as justification for what is a purely a priori proposition. In fact, both of these -- empirical and heuristsic justification for formal a priori propositions -- are really conditioning a student to reason fallaciously which is why I would like to avoid that as much as possible with my own children. (Maybe this is how sometimes trailer trash rednecks end up smarter and with better judgment than university educated folks.)
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