Re: Euclidean Geometry in Schools

In article <BEE35E7F.766A%jean-claude.arbaut@xxxxxxxxxxx>,
Jean-Claude Arbaut <jean-claude.arbaut@xxxxxxxxxxx> wrote:

>On 25/06/2005 18:32, Herman Rubin wrote:

>>>> The Bourbaki approach to measure theory was from the functional
>>>> analysis standpoint. To me, this is quite restrictive.

>>> My knowledge in algebraic topology (and friends...) is insufficient for me
>>> to have a definite opinion. I tend to see algebraic topology as a
>>> "follow-up".

>> Do you know "general topology"?

>Some, from university courses and from Kuratowski's book.
>Well, I don't know *everything* in that book. ;-)

Kuratowski's book is obsolete. Try Kelley.

>> What one can do with spaces
>> with weak separation axioms? In some work of mine on the
>> topology of random variables on topological spaces, I found
>> it convenient to use finite non-Hausdorff spaces, and the
>> results of assignment programming.

>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.

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. 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.

>>>> This does not mean that I do not think the abstract ideas should
>>>> not be used to understand the special cases and what makes them
>>>> special; it does not work the other way, as the special part
>>>> needs to be unlearned.

>>> Unlearn functionnal analysis ? Rather extreme :-) Even unlearning general
>>> topology, by the way.

>> No, one does not have to unlearn functional analysis. But one
>> has to unlearn the idea that the restrictions of functional
>> analysis are to be followed.

>> The teachers seem unable to unlearn the idea that numbers are
>> strings of decimal digits, to be manipulated. They cannot
>> understand that the representation by decimal digits is a
>> derived representation and a convenience, not at all basic.

>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.

>>> I'm still not convinced it's a good idea... Well, I don't have evidence it
>>> was a mistake either. It's rather an obscure feeling that changing
>>> everything may be dangerous :-) But when you see the condition of education
>>> today, it's not much better. Teachers know how to teach I think, but they
>>> have very little control on what happens.

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".

>> But it is necessary to "change everything". This HAS happened
>> before, and will have to happen again.

>If you are sure... But I doubt anybody knows the exact consequences of such
>changes. One knows what one expects, not what will happen.

>>> And that's certainly a good thing. But they are not prepared to teach, and I
>>> know that for having been one of them ;-)

>> The degree they get in their fields is abysmally low, and efforts
>> to raise it do not get far. The real problem is that they can no
>> longer think abstractly.

>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.

>>> Good news. Maybe there is still something to learn from this failure.

>> Yes; get those who cannot understand out of a position to
>> keep others from understanding.

>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.

>> 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.

>>> :-) I suspect it's almost the same here, since they cannot have a degree in
>>> every field.

>> I do not care if an elementary school teacher has calculus.
>> But I do care if they can understand algebraic notation,

>Algebraic notation for pupils who are merely learning to count ?

What does algebraic notation have to do with counting, or
any branch of mathematics?

>> which should be taught with beginning reading, as it is
>> language and not mathematics at all, and that they can
>> learn the structure of the integers and rational numbers.

>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.

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.

>>> Difficult question, but worth asking. The problem is, ability is not easy to
>>> measure, especially for young children.

>> It is not easy to measure, and except for urging children to do
>> more, it is not necessary to be that great in measuring it.



>>> Not that I'm not religious, but I think religion is not entitled to have its
>>> say in that matter. And by the way, *which* religion ? If some religious
>>> group imposes its opinion, it's more fundamentalism than education.

>> You are definitely correct.


>>> I find difficult to take such decisions when the whole country is involved.
>>> All are convinced to be right, but *at most* one is, and they look like
>>> sorcerer's apprentice. Yet, this task must be done.

>> The only way I can see to get anything done is to remove the
>> idea that the government should have anything to say about it.

>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.

>> This holds for many other things, including research. Pure
>> research is very weak in the US now, because of the effective
>> government control.

>I'm not sure whether private companies would pay for research in
>mathematics.


>> It did well before WWII, with no government
>> support whatever. The US has a long history of private schools,
>> weakened by the government.

>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.

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


--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

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