Re: Euclidean Geometry in Schools




On 25/06/2005 23:34, Herman Rubin wrote:

> Kuratowski's book is obsolete.

Old, but still worth reading I think.

> Try Kelley.

I will !


>> Usually topology taught in BSc amounts to metric spaces, Hilbert spaces, an
>> usual theorems, for application in integration and complex analysis.



>> There is at least an application of non-metric spaces with distributions.

This is one year later, in fact

> Those getting the American degrees which "qualify" teachers
> of high school mathematics usually do not even know finite
> dimensional metric spaces, or what a derivative or integral
> are, except computationally. It is not just these; we get
> many candidates for PhD's in mathematics whose undergraduate
> degrees do not cover this.

Incredible!

> It is the result of the dumbing
> down which comes from the pressure to teach to the level of
> those who are in the classroom, and also down to the level
> which they are willing to accept.

There should be limits to dumbing down!


>
>> But real numbers *are* (infinite) strings of digits!
>
> NO! They can be represented by infinite strings of digits,
> but this will not give any insight whatever. It is vital
> to give meaningful characterizations, not those which may
> be cute to compute with, but provide no understanding.

Ok, I think I understood with your comment above :-) Usually, we learn
properties such as "every majorized subset of R has a sup", and we learn
analysis from that, after high school. Infinite strings of digits are only a
consequence.

What I find surprising is that basic formulas for differentiation and
integration are learnt in high school (up to integration by parts, but not
change of variable). As a reference point, high school is ended by a
national exam at age 18 (usually 18, but can be slightly different).

And in first year of university, we learn rigorously analysis from
definition of R, and Riemann's integral. Lebesgue's integral is learnt in
third year (= B.Sc I think, called "licence"), along with topology, and
basic group theory (Sylow theorems, decomposition of finite abelian groups),
and other subjects (can be number theory, projective geometry, etc.).
After third year, students can continue in IUFM (preparation for future
teachers), which is ended by a national competitive exam. Otherwise, they
continue in 4th year (= M.Sc ? Was called "maîtrise", now "mastère"), then
5th (Was DEA or DESS, now 2nd year of "mastère"), and then PhD in three
years at least. I'm quite confused with studies equivalences.


> They do not know how to teach concepts, largely because they
> think in terms of memorization and routine. This was the
> problem which teachers had with the "new math", and still do.
> The only "classical" course, below advanced courses, which
> did much about it was "Euclid".
>
>> I disagree here. But maybe we are not talking about the same degree.
>> Or maybe not *all* students are concerned.
>
> We can teach abstract thinking in kindergarten. It is
> much harder the later one waits.

I'm glad I was not taught too abstract thinking ;-) But you are right, it's
not good to wait too long.


>> And where will you throw them ? To trash ? You seem too elitist.
>
> Put them to teaching literature, and possibly art. They
> cannot teach mathematics, because they do not know any.

Ok, I didn't understand very well. Seems like the situation is much worse
than I thought.


>>> This did happen for reading;
>>> for a 20-year period, the use of phonics was forbidden in
>>> most schools, and words were memorized as whole strings, with
>>> the particular letters essentially arbitrary as far as the
>>> children were taught.
>
>> And that was a bad idea... We tried that with "global reading".
>> Another fiasco, but it still has not ended.
>
> And the fiasco of removing mathematical concepts, grammar,
> structure in all fields, is rampant and has not ended.

Funny. It has begun in France some years ago in high school... Maybe we are
just blind followers. Well, I eggagerate a bit, the educational standard is
still quite good, but it's a trend.


>> Algebraic notation for pupils who are merely learning to count ?
>
> What does algebraic notation have to do with counting, or
> any branch of mathematics?

Well, not much, but I think the concept is more difficult, and maybe more
easy to understand when pupils know how to count. Maybe I'm wrong.

>> Structure, structure... Sounds like set theory, and it was *not* a good
>> idea.
>
> Set theory is a unifying structure in mathematics, but the
> new math never got beyond set algebra, which is quite weak.
> The approach to arithmetic from the Peano Postulates does
> not require learning set theory, or even set algebra, first.

I agree, for Peano Postulates.

> It is a good idea, in all branches of learning. I believe
> that music teachers can get away with it. But it is now
> almost prohibited in language, history, geography, as well
> as mathematics and science.
>


>> Who decides, then ?
>
> Nobody and everybody. There is no reason that two children should
> go to the same school just because they live in the same area.

That can help social relationship.

>
>> But I don't like the idea of private school for the rich, and
>> the left-overs for the others.
>
> Private schools are not just for the rich. The public schools
> CANNOT be substantially improved at this time, but if there
> were an adequate number of affordable private schools when the
> educationists started their activities, the miserable performance
> of their schools would have been apparent in a few years. At no
> time do we need an educational monopoly, and at this time, the
> monopoly makes it impossible to do anything with it.

Correct. We are so used to public schools that sometimes I forget other
systems can succeed.

> A public university cannot tell a legislator that the high schools
> in his district turn out totally incompetent graduates.

:-)

.

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