eSOANEM, I'd certainly get behind a class like that... it's just that it sounded to me that you were suggesting that Noether's theorem is as something important, while I believe that, in that context, it's a mere curiosity, and likely to do more harm than good if the teacher isn't vey competent and fully grasps the concept which is being talked about.
eSOANEM wrote:Teachers should not be afraid I think to introduce concepts which cannot fully be understood at their students level. I am reminded of something I read a while ago about a school in Germany which teaches complex numbers shortly after it teaches negative numbers. I doubt very much those children can grasp the full majesty of the complex world, but their strong grounding in complex arithmetic will undoubtedly help them when they need to do proper maths with complex numbers.
This really goes straight to the point I am trying to make: mentioning extra material to stir curiosity and encourage further research is great, but I believe that teachers must, first, clearly define what the students must be expected to learn, and what is flavor knowledge. And this core of what students should learn has to, in my opinion, be fully understood - or at least have all the essential assumptions laid out clearly.
I agree entirely. Having had the good fortune of being in sets of high-performing students for sciences my entire career, I often find myself accidentally assuming that the majority of an arbitrary class "get" the basics. My suggestion should therefore be thought of more as a means of stretching the bright students or a response to the question "why is energy/momentum etc. conserved" (which will usually indicate that at least one student wants to be stretched) and, if necessary should be done as a more one-on-one discussion whilst the rest of the class does some problems to become fluent with the basics.
moiraemachy wrote:I feel really divided on the example you gave about complex numbers: it'll probably help the students with their algebraic manipulations later, but is is also dangerous - it may enforce the view of mathematics as the manipulation of strange symbols with arbitrary rules. The fact that it took some time for the mathematical community to fully accept complexes counts as evidence against teaching them: it suggests that a student that accepts that complexes work may be only submitting to his teacher's authority. In my experience, when introduced to complexes, it's the best students who squirm, and try to game the new system to make inconsistencies arise. After all... if you don't provide a good explanation for why complexes are ok, they'll begin to wonder why it's not valid to just define problems away.
True, although the same argument could be easily applied to teaching negative numbers.
Anyway, as I say, I read the article a long time ago, it's quite possible the school no longer exists let alone has the same syllabus. Personally, I think that the benefits outweigh the potential cost but it's certainly not a clear cut thing.
As a more "horizontal" example of teaching different syllabi: my school has a lot of links to a couple of schools in Ethiopia (one of the ex-teachers founded a charity providing aid including a lot of school-related stuff) and so quite a few of the teachers at my school have paid visits over there and seen some of their lessons even if they haven't taken any themselves. Now that we've started doing a tiny bit of group theory in our final year before uni, the teacher teaching it to us told us that, in Ethiopia, some really basic group theory is taught along with basic arithmetic with the idea being that, without an understanding of groups, you can't really understand how arithmetic works. That seems a bridge too far to me, but I'm not sure if that's because it really is, or just because I wasn't taught that way.
On the subject of trying to game the system, I remember that I didn't really have any problem with complex numbers when they were introduced, I just accepted them, and played around seeing where it led; likewise with taking the derivative and the whole "are you dividing by 0" debate we must have had a hundred times in our lessons. Anyway, I didn't try to find contradictions in complex numbers, instead, I took the idea of "take unsolvable calculation and define answer as "blah" and run with it" a bit too far.
I tried to come up with a consistent real system of dividing by 0 (the thread's probably still floating around the maths forum somewhere). It didn't work very well although, by defining it on the Riemann sphere instead of the real field, I'm pretty sure it would yield something a bit like the surreal numbers but complexified.