I was gonna say, don't waste good acid!Couple of weeks.
No acid, but I did read a bit of it while otherwise intoxicated - dunno if it make any more sense, but I don't think it made any less.
But didn't the operations come first historically? The (rigorous) philosophy of it came later when people realised that it was a handy tool and the explanations as they stood weren't adequate. So in a sense people are learning it in an order that is somewhat justfiable."But they do not get to grips with the basics of calculus; they learn how to perform certain operations, and so the knowledge they acquire is for the most part purely procedural. A major problem with undergraduate maths courses is that the lack of conceptual knowledge doesn't prepare them well for the transition to HE programmes that rely on maths. There is quite a literature on this."
Well, when we started doing differentiation we did derive it in a way that made sense. It's just that at university we did it in a different way as we had a more subtle understanding of various types of limits and things. But my understanding is that that derivation came later so I don't have a problem with seeing them that way round. What I'm saying is that I certainly don't feel as though the methodology of calculus was just plucked out of the blue when I was doing my a-levels - although we had a very good teacher, maybe that wasn't the usual experience."I'm not arguing for Philosophy, just conceptual understanding. It's a question of connecting one procedure to the others, so that you can understand it as a concept -- a mathematical, not philosophical concept -- as well as a procedure. But there are time constraints imposed because everyone needs to know enough procedures to pass the exam."
Well, it's interesting, I did maths and further maths at a-level and in the further maths group there were only about six or eight of us who were all quite bright (if I can say that) and everyone was obviously interested enough to do maths twice; on the other hand the normal maths group was very mixed ability and about thirty people and it was in this that we first studied differentation obviously. I think the justification was most likely presented along the lines of "you may not stricly need to know this for the exam but here it is". It was certainly a far cry from "You don't need to know this so shut the fuck up"."That's cool. Obviously, it varies. I guess at A level it's the prior knowledge of the class versus the constraint of the exam. A really good, homogeneous class can probably do a lot."
Yes, I've seen this quite often with my students - I'll say something like "Oh come on, I know you can do this, it's just a fraction!" and they respond "Oh, I didn't know this was a fractions question...". As if maths (or any subject) is naturally divided up into nice discrete, non-overlapping topics. Which is, probably unavoidably, how it's taught of course.I'm not arguing for Philosophy, just conceptual understanding. It's a question of connecting one procedure to the others, so that you can understand it as a concept -- a mathematical, not philosophical concept -- as well as a procedure. But there are time constraints imposed because everyone needs to know enough procedures to pass the exam.
But you don't understand, this is the only place I can spazz out and be a dork. My friends have all read theory and stuff but they don't really care about it--beyond a few extremely cracked out/impressionistic remarks here and there-- because they are artists, which means they mostly care about a) being hot/looking good b) fucking other scenesters and c) maintaining a base-level of abject poverty that doesn't quite match their earning potential.I suppose Advanced Dungeons & Dragons tournaments do eat into one's time somewhat.