Re: Euclidean Geometry in Schools

In article <BEE23E2A.73D0%jean-claude.arbaut@xxxxxxxxxxx>,
Jean-Claude Arbaut <jean-claude.arbaut@xxxxxxxxxxx> wrote:



>On 24/06/2005 21:26, Herman Rubin wrote:

>> In article <BEE1633E.7226%jean-claude.arbaut@xxxxxxxxxxx>,
>> Jean-Claude Arbaut <jean-claude.arbaut@xxxxxxxxxxx> wrote:



>>> On 24/06/2005 03:57, Herman Rubin wrote:

[Lots of attribution deleted.]

.....................

>>>>>>> This PDF discusses a french reform of the teaching of mathematics during
>>>>>>> the
>>>>>>> seventies, which was a complete fiasco. If you can read french, I think
>>>>>>> it's
>>>>>>> worth reading that :-)

>>>> I can read French, but I do not feel like going through 63
>>>> pages of philosophy.

>>> You are forgiven :-)

>>>> Frankly, I do not think that
>>>> Dieudonne quite got the conceptual idea of abstract
>>>> mathematics, as there was much of Bourbaki which is, from
>>>> the standpoint of someone used to abstract topology and
>>>> measure theory, rather concrete.

>>> In France they are considered rather abstract :-) But I don't know what a
>>> french topology expert would think.

>> There are quite a few branches of topology. Most of the work
>> done in topology is of the "combinatorial" topology, geometric
>> topology, and similar branches.

>Maybe the *new* work, and I'm still not sure it's true.

>> 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"? 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.

>> 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.

>>>> This does not mean that
>>>> special cases are not useful, but the real understanding is
>>>> not there.
>>>>>> Even after quickly skimming that chapter, I envy the French for
>>>>>> being willing to admit faults of the "New Maths". In New Zealand the
>>>>>> reformers of around 1970 never admitted their mistakes, so our school
>>>>>> curriculum has had to evolve slowly and painfully from that stage, and
>>>>>> IMHO needs to evolve a lot more.

>>>> In the US, the program was HIGHLY tested, and succeeded
>>>> quite well when those who could understand the concepts
>>>> taught it. I doubt that any innovation has been that
>>>> well tested.

>>> I was not born, but from what I read or was told, I agree it was not very
>>> well tested.

>> The program was tested on tens of thousands of students,
>> mainly in private schools, before it was made generally
>> available. There is no way the public schools would have
>> considered adopting something which did not originate from
>> them unless massive testing had been done. I was not
>> involved with the project, but I was there at the time,
>> and I know many who were. It worked when those who taught
>> it could understand logic and the concept of cardinal numbers.

>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.

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

>>>> It was a real surprise that there was a
>>>> problem with the teachers understanding it, and this has
>>>> even gotten worse. We will never get good teaching of
>>>> mathematics done if those teaching it cannot understand
>>>> anything other than memorization of facts and grinding
>>>> out of answers. They may think they are teaching such
>>>> concepts as commutativity by giving the word and a
>>>> formal definition, but unless one is a candidate for
>>>> a PhD, understanding is not likely to occur if that is
>>>> all that is done with it.

>>>>> Sad. But it's what happens when we let mathematicians take education
>>>>> decisions. That's not their job!

>>>> It might not be their job, but those who are now making
>>>> the decisions have NO understanding of mathematics. In
>>>> fact, they have no understanding of any structured
>>>> subject; grammar is not taught, either, and history has
>>>> become the Marxist idea that one needs to know only what
>>>> the condition of the peasants was at each time.

>>> Well... Here teachers have at least a BSc and often a master's degree,


>> I am very aware of the contents of those degree programs; at
>> this time, high school teachers usually need a degree in their
>> fields.

>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.

>> Someone who went through the original new math program
>> before it was dumbed down in an attempt to get the teachers to
>> be able to handle it knew almost as much real mathematics as
>> these high school teachers are supposed to know.

>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. 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.

>> Those who
>> became elementary school teachers take a course in "mathematics
>> for teachers" which is at an abysmally low level.

>:-) 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,
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.

>>> and "teachers teachers" are almost always teachers as well :-)

>> They may have been, but they are a special group of college
>> teachers, with their own specialty, who push their philosophy
>> and do not know what understanding means.

>Harsh but... I cannot deny that completely :-) But one the other hand, those
>knowing what undertanding means, don't always know what *teaching* (or
>learning) is.

>> Part of their
>> philosophy is that children should be with their "peers", by
>> which they mean their age group, regardless of ability.

>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.

>>> But I don't know who *really* take decisions, that's rather obscure...
>>> I think usually, groups of teachers and scientists give their advice, then a
>>> minister take a decision.

>> In the US, it is almost entirely done by organizations of
>> teachers, sometimes with some political input,

>> mainly by
>> religious groups,

>:-(

>> or those fighting the impositions of the
>> religious groups.

>:-)

>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.

>> When there is action by a state or
>> federal office, the people involved are typically from the
>> schools of education, and these do not understand subject
>> matter, but claim to be able to tell how to teach it.


>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.
This holds for many other things, including research. Pure
research is very weak in the US now, because of the effective
government control. It did well before WWII, with no government
support whatever. The US has a long history of private schools,
weakened by the government.

--
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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