vanesch said:
I entirely agree with your analysis - I didn't mean to say that "in the old days it was much better". What has happened, however, is the following: this "self construction of your own knowledge" (one of its favorite promotors is our good own Charpak) has its limits. While it can be fun to illustrate something, or to get started or so, you cannot expect from your average - good student to rediscover, on his own, even guided, about 2500 years of development from the smartest people in history. The trial and error method is simply too slow and inefficient. But because this was seen as "progressive", instead of considering as an eventual method to ACHIEVE educational goals (which could be tested scientifically, by comparing different methods on representative samples of students, and see how good or bad the methods are in reality), they became, for pseudo-political reasons, the GOAL. And because these methods work very well for learning how to color cubes, but not on how to fit together a correct abstract proof in Euclidean geometry (how do you do that "constructively", except if your name is Euclid or the like), they CHANGED the objectives: the method became the goal.
Absolutely correct, vanesch - I think it takes an incredibly skilled and dedicated teacher to properly pull off the constructivist teaching method because, as you point out, it has to be done much more efficiently than the 'trial and error method': the teacher (they call them 'facilitators' now; 'teacher' is politically incorrect - I still use the word 'teacher'

) has to really know their stuff and, as you say further down...
vanesch said:
You cannot require from your average math teacher to have the level of a hotshot university professor: otherwise he'd not be teaching in high school, for the salary he gets and the working conditions he has. But even that would not be sufficient because usually a university professor has not the RIGHT pedagogical skills to deal with adolescents (he usually takes a certain maturity in the student for granted).
So, we're stuck with pretty 'average quality' teachers and, even if we got real discipline experts into high schools, they wouldn't know the crucial pedagogic principles (and are generally not terribly good communicators in any case

)
vanesch said:
Well, I would think that part of the interest of teaching mathematics in high school is to make kids used to ABSTRACT reasoning, exactly because they should get their thinking INDEPENDENT of practical contexts. The main goal is NOT that they have an intuitive understanding of geometrical problems as "pieces of cardboard", but that they can think about it without making this association ; and then, once they master the abstract way of thinking, that they can apply it to cardboard situations (but not that they need the cardboard picture to help them do the reasoning). So I think - especially in mathematics - that people trying to do that missed ENTIRELY the whole educational interest of mathematics teaching ! The aim is NOT to be able to solve practical problems - the aim is to learn to think abstractly. The by-product is that you can also solve practical problems.
Again, I agree with you. It boils down to how best to teach abstract reasoning. I need to do a lot more research on this, but intuitively (and yes, intuition is not good enough on such an important issue) I would have thought that referring backwards and forwards from the abstract to the concrete (making constant links throughout the teaching) may be an effective method. The currently accepted 'best practice' teaching principles in primary school mathematics is to start off with the concrete and gradually introduce abstraction - but while this makes sense at the primary school level, I can see (now that I'm thinking about it more

) that maybe it wouldn't work at the higher levels. But on the other hand, studying Calculus at university level - the textbooks cover the abstract theoretical concepts, but then also include sections of practical applications from discipline areas like physics and biology...
vanesch said:
Again, I agree that we should not return to the 50-ies teaching per se. There were problems, the main one being that the "good" student was the perfect parrot. But AT LEAST he learned something, and those that DID understand, got a very good education. Now, the good average to brightest are simply WASTED (except if their parents can give them extra education, themselves, or by paying extra courses). The dummies have fun, and are mislead in thinking they are good at maths, which makes them make choices where they get completely blocked, 3 or 4 years later.
Hmm, yes - true. But this does not hold only for maths. At least in this country, it seems to hold in every discipline area. I teach people who have actually (in some cases) completed all their years of formal schooling and 'passed', and yet they still cannot write an essay (or even a basic sentence - or even spell my name correctly

) And they have absolutely
no knowledge of history or current events
vanesch said:
I think the goals and the methods are 2 different things: the goals should be: a good capacity of abstract reasoning (correct proof vs. flawed proof etc...), and good problem solving skills. For the methods, all options are good, but you could test their efficiency scientifically on representative samples.
When I studied maths in high school (in another country), we studied geometry and the formal geometrical proofs intensively as part of the mathematics curriculum. In Australia, however, students do not study the formal geometrical proofs any more. I find this absolutely incomprehensible (primary school children are also not taught how to do 'long division' any more - crazy!)
vanesch said:
Maybe that's the mistake: that they think they have to take practical courses instead of fundamental ones. It is a matter of INFORMING people correctly. University professors, instead of being in their ivory tower, should try to CONVINCE their business partners who cofinance them, that a certain amount of fundamental and theoretical teaching is absolutely essential in the education of good co-workers. I know that this tendency has already started, where businesses take care of the "practical" education, and ask universities to ensure the more theoretical aspects.
No, it is not. It is the fault of the pure mathematicians that fail to convince the value of their work to society. Of course, there needs to be public funding of these matters, but I think there is also the error of a certain ivory-tower arrogance from the part of the universities.
Fundamental materials are to universities what are long-term investments for businesses.
And at this point, vanesch, is where our differences begin - my understanding of capitalism (derived from observations of what is happening in the higher education sector) is that it is not interested in investments that do not yield a return in the
short-term. If you can 'mass produce' enough engineers who can basically 'do the job' (however superficial their knowledge is), and you can do this more quickly and with less money by making them all do applied maths courses and cutting out the theoretical, abstract, 'useless rubbish' (note, I do not believe it is 'useless rubbish'!), then that's 'good enough'. Personally, I think the whole system will implode on them (the 'free market' economic rationalists) precisely because, as you write, "theoretical teaching is absolutely essential in the education of good co-workers" - and by the time they realize it (business types need to actually see the proof; they are hopeless at extrapolating!) it'll be too late: the university system will be in tatters by then. Personally, I don't think pure mathematicians have either the political nous or the persuasive skills to convince business and political leaders of anything - you cannot convince those in power to see what it does not suit them to see when it comes to their short-term profit margins.