Re: Programming languages for the very young

From: Michele Simionato (michele.simionato_at_poste.it)
Date: 01/25/04 

Date: 25 Jan 2004 06:59:01 -0800

ngo@cartan.de (Nils Gösche) wrote in message news:<87wu7hywa4.fsf@darkstar.cartan.de>...
> Feuer <feuer@his.com> writes:
>
> > What is scalar, vector, or tensor depends on how it transforms.
>
> Yeah, right. When I was a freshman, I had a lot of trouble
> understanding what a tensor is. Teachers in physics classes kept
> telling me about transformation rules and I kept wondering why anybody
> would care about any transformation rules at all.
>
> > When you rotate a complex number, it stays the same.
>
> I think I can rotate it by multiplying it by exp(it) (t real) just
> fine, and it /does/ change, doesn't it? ;-)
>
> > When you rotate a vector, it changes.
>
> Naturally ;-) I guess explaining what a tensor is is one of the
> hardest things in the world. That's because when you get it, all of a
> sudden tensors seem to be an almost trivial concept. You cannot
> understand tensors as long as you still think in terms of coordinates.
> Once you stop thinking in coordinates, you see that tensors are simply
> a certain kind of multilinear maps, the most simple and natural thing
> to define.
>
> I guess the best one can tell a young student is that the real world
> doesn't care about the particular system of coordinates we are using
> to describe it. So, all physical entities and laws ought to be
> independent of our choice of a coordinate system.
>
> Or simply tell him what a manifold and a tangent bundle is; once he
> got that the whole problem simply goes away, but that might not work
> as well for some people...
>
> Regards,

[not posting on comp.lang.logo]

I wonder if there is issue of different teaching of maths in U.S. and
Europe. It seems Nils (who is posting from Germany) got a math
education similar to mine. I was though tensors as multilinear forms
at the first year of college. Only a couple of years later I became
familiar with the idea of "tensor as an object with given transformation
laws". I always thought it was a very ugly way of putting things, BTW.

What surprises me is that apparently American colleges are still
teaching the parabola formula and why you cannot divide by zero
(these things should be studied in junior high school according to
my experience). For comparison, in my first year course of geometry
they started with homomorphisms of vector spaces and only after a month
or so, in the exercises course, they introduced matrices as a way of
representing them, therefore proving the matrix multiplication law
formula and the other properties.

BTW, I did Physics, of course math courses for people doing economics
or architecture were much simpler, but still at higher level than
teaching the parabola.

I think the reason for the difference is that in the U.S. the students
pay a lot of money, so they don't want to fail. In Europe, the students
pay much less, so there are always too many students and the Universities
are not afraid to select between them with really hard courses.
Actually, the failure rate of my geometry course was fearsome. But this
was not considered a problem at all by the University.

Just to raise some new point of discussion,

        Michele Simionato

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