> So you really feel like most 8th graders graduating from your group > theory class could, say, prove that every group of prime order is > cyclic or that a cyclic group is abelian? Of course, I don't mean > give an example of an abeliean cyclic group or something like > that. Can they put together a coherent and sufficiently formal > argument that proves these assertions or an assertion like it? If > that is indeed what you are doing, then perhaps there is more to it > than you at first let on. If not, then I think we are back to > teaching them some particular fact without teaching them the real > justification for that fact. Instead what you are doing is giving > them a rationale for why someone might think the fact is true, not > the actual justification for the fact that *proves* that it is > true. Something like that is little more than a mnemonic device to > remember the fact by.
Don't let Paul hear you disparaging anything to do with memorization! > > It's really much deeper even than math education -- a fact without > justification is not knowledge.
Really? Gosh, we mathematics educators would NEVER advocate actual knowledge. We're all about rote memorization and regurgitation, don't you know?
> What do your students *know* when they are done? Have they > acquired any knowledge? Or, do they just have vague rationales for > facts that they believe without justification?
Hmm. You mean that at some point, we have to look at what students can DO with what they've learned, and mere recitation isn't enough? How radical. Watch out, Adrian: someone is going to shoot you for being too fuzzy.