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Caro(a)s colegas

O 22th Congress for the Psychology of Mathematics Education (PME22)
realiza-se no proximo ano na Africa do Sul. O respectivo primeiro anuncio
ira em breve ser enviado aos membros do grupo PME. Os colegas que quiserem
saber informacoes acerca do grupo e do congresso poderao consultar os
enderecos seguintes:
http://www.sun.ac.za/local/academic/education/pme22/pme22.htm
http://www.unifr.ch/psycho/pme/pme.html

 Caso queiram receber o primeiro anuncio do PME22 pelo correio deverao
enviar-me uma mensagem com o vosso endereco postal.
Cumprimentos,
Joao Filipe Matos
PME Regional Contact
Secretary of the International Committee

****************************************************
Prof Doutor Joao Filipe Matos
Dep. de Educacao
Faculdade de Ciencias da Universidade de Lisboa
Campo Grande, C1 - 2
1700 Lisboa - Portugal

Ph. direct: +351 1 7500118
            +351-1-7573141 EXT. 2223 or 1101
Fax: +351-1-7500082
Ph. +351-1-8491557 (home)
      +351-1-9291926
    mobile: +351-936407607
E_mail: joao.matos@fc.ul.pt
http://correio.cc.fc.ul.pt/~jflm (1st version)
****************************************************

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X-Comment:  Educacao em Matematica
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Joao filipe,

Nao preciso que me envies pelo correio o 1=BA anuncio do PME 22, mas
agradecia se mo deixasses no meu cacifo.

Um abra=E7o

Margarida

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mcesar@fc.ul.pt wrote:
>=20
> Joao filipe,
>=20
> Nao preciso que me envies pelo correio o 1=3DBA anuncio do PME 22, mas
> agradecia se mo deixasses no meu cacifo.
>=20
> Um abra=3DE7o
>=20
> Margarida
Salve Margarida
Como vai ?
Parece que voc=EA enviou a mensagem para o lugar errado.
Um abra=E7o do amigo Bigode

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A conference in honor of the 65th birthday of Ubi D'Ambrosio and celebrating
his role as the originator of ethnomathematics and his influence in
mathematics education and in the history of mathematics, will take place
on Tuesday, January 6, 1998, beginning at 9:00 am, in the Omni Inner Harbor
Hotel in Baltimore, MD, the day before the beginning of the Joint Mathematics
Meetings.  Confirmed speakers include Marcia Ascher, Paulus Gerdes, John
Fauvel, Dirk Struik, Reuben Hersh, and Jeremy Kilpatrick.  The conference is
sponsored jointly by the International Study Group on the Relations Between
History and Pedagogy, Americas Section (HPM), and the International Study
Group on Ethnomathematics (ISGEm).  To register for the conference, please
send a check for $50 (US), made out to HPM, along with your name, addresses,
and phone numbers, to Karen Michalowicz, 5855 Glen Forest Dr., Falls Church,
VA 22041.  The fee is chiefly to cover the cost of a festive birthday dinner.
(If it is difficult for you to send a US dollar check due to currency
conversionproblems, please send a note to that effect and we will collect
the fee at
the conference itself.)  Please direct any questions to Victor Katz at
vkatz@maa.org (but please do this off-list).

                                                Victor J. Katz

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Lamentavelmente n=E3o foi poss=EDvel ler o material enviado, provavelment=
e
devido =E0 incompatibilidade de Mac ou W95 com meu sistema bem mais
simples. Mande um texto mais curto que terei o maior empenho em divulgar
por essas paragens.
um abra=E7o
Bigode

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Mathematics Realism and Its Discontents
WHAT IS MATHEMATICS, REALLY? By Ruben Hersh . Oxford University Press:
344 pp., $35
By MARTIN GARDNER

     A physicist at M.I.T.
     Constructed a new T.O.E. [Theory of Everything]
     He was fit to be tied
     When he found it implied
     That seven plus four equals three.


     Reviewing Reuben Hersh's "What Is Mathematics, Really?" was an
agonizing task because I have such high respect for him as a
mathematician and such low respect for his philosophy of mathematics.
Now retired, Hersh belongs to a very small group of modern
mathematicians who strongly deny that mathematical objects and theorems
have any reality apart from human minds. In his words: Mathematics is a
"human activity, a social phenomenon, part of human culture,
historically evolved, and intelligible only in a social context. I call
this viewpoint 'humanist.' "
     Later he writes: "[M]athematics is like money, war, or
religion--not physical, not mental, but social." Again: "Social historic
is all it [mathematics] needs to be. Forget foundations, forget
immaterial, inhuman 'reality.' "
     No one denies that mathematics is part of human culture. Everything
people do is what people do. The statement would be utterly vacuous
except that Hersh means much more than that. He denies that mathematics
has any kind of reality independent of human minds. Astronomy is part of
human culture, but stars are not. The deeper question is whether there
is a sense in which mathematical objects can be said, like stars, to be
independent of human minds.
     Hersh grants that there may be aliens on other planets who do
mathematics, but their math could be entirely different from ours. The
"universality" of mathematics is a "myth." "If little green critters
from Quasar X9 showed us their textbooks," Hersh thinks it doubtful that
those books would contain the theorem that a circle's area is pi times
the square of its radius. Mathematicians from Sirius might have no
concept of infinity because this concept is entirely inside our skulls.
It is as absurd, Hersh writes, to talk of extraterrestrial mathematics
as it is to talk about extraterrestrial art or literature.
     With few exceptions, mathematicians find these remarks incredible.
If there are sentient beings in Andromeda who have eyes, how can they
look up at the stars without thinking of infinity? How could they count
stars, or pebbles, or themselves without realizing that two plus two
equals four? How could they study a circle without discovering, if they
had brains for it, that its area is pi times the radius squared.
     Why does mathematics, obviously the work of human minds, have such
astonishing applications to the physical world, even in theories remote
from human experience as relativity and quantum mechanics? The
simplest answer is that the world out there, the world not made by us,
is not an undifferentiated fog. It contains supremely intricate and
beautiful mathematical patterns from the structure of fields and their
particles to the spiral shapes of galaxies. It takes enormous hubris to
insist that these patterns have no mathematical properties until humans
invent mathematics and apply it to the outside world.
     Consider 2^1398269minus one. Not until 1996 was this giant integer
of 420,921 digits proved to be prime (an integer with no factors other
than itself and one). A realist does not hesitate to say that this
number was prime before humans were around to call it prime, and it will
continue to be prime if human culture vanishes. It would be found prime
by any extraterrestrial culture with sufficiently powerful computers.
     Social constructivists prefer a different language. Primality has
no meaning apart from minds. Not until humans invented counting numbers,
based on how units in the external world behave, was it possible for
them to assert that all integers are either prime or composite (not>prime).
In a sense, therefore, a computer did discover that 2^1398269minus one is
prime, even though it is a number that wasn't "real" until it was socially
constructed. All this is true, of course, but how much simpler to say it in
the language of realism!
     No realist thinks that abstract mathematical objects and theorems
are floating around somewhere in space. Theists such as physicist Paul
Dirac and astronomer James Jeans liked to anchor mathematics in the mind
of a transcendent Great Mathematician, but one doesn't have to believe
in God to assume, as almost all mathematicians do, that perfect circles
and cubes have a strange kind of objective reality. They are more that
just what Hersh calls part of the "shared consensus" of mathematicians.
     To his credit, Hersh admits he is a maverick engaged in a
"subversive attack" on mainstream math. He even provides an abundance of
quotations from famous mathematicians--G.H. Hardy, Kurt Godel, Rene
Thom, Roger Penrose and others--on how mathematical truths are
discovered in much the same way that explorers discover rivers and
mountains. He even quotes from my review, many years ago, of "The
Mathematical Experience," of which he was a co-author with Philip J.
Davis and Elena A. Marchisotto. I insisted then that two dinosaurs
meeting two other dinosaurs made four of the beasts even though they
didn't know it and no person was around to observe it.
     A little girl makes a paper Moebius strip and tries to cut it in
half. To her amazement, the result is one large band. What a bizarre use
of language to say that she experimented on a structure existing only in
the brains and writings of topologists! The paper model is clearly
outside the girl's mind, as Hersh would of course agree. Why insist that
its topological properties cannot also be "out there," inherent in what
Aristotle would have called the "form" of the paper model? If a
Hottentot made and cut a Moebius band, he would find the same
timeless>property. And so would an alien in a distant galaxy.
     The fact that the cosmos is so exquisitely structured
mathematically is strong evidence for a sense in which mathematical
properties predate humanity. Our minds create mathematical objects and
theorems because we evolved in such a world, and the ability to create
and do mathematics had obvious survival value.
     If mathematics is entirely a social construct, like traffic
regulations and music, then Hersh argues that it is folly to speak of
theorems as true in any timeless sense. For this reason, he places great
importance on the uncertainty of mathematics, but not in the sense that
mathematicians often make mistakes. The fact that you can blunder when
you balance a checkbook doesn't falsify the laws of arithmetic. Hersh
means that no proof in mathematics can be absolutely certain. That two
plus two equals four, he writes, is "doubtable" because "its negation is
conceivable." No proof, no matter how rigorous, or how true the premises
of the system in which it is proved, "yields absolutely certain
conclusions." Such proofs, he adds, are "no more objective than
aesthetic judgments in art and music."
     I find it astonishing that a good mathematician would so
misunderstand the nature of proof. Benjamin Peirce, the father of
philosopher Charles Peirce, defined mathematics as "the science which
draws necessary conclusions," a statement his son was fond of quoting.
Only in mathematics (and formal logic) are proofs absolutely certain. To
say that two plus two equals four is like saying there are 12 eggs in a
dozen. Changing four to any other integer would introduce a
contradiction that would collapse the formal system of arithmetic.
     Of course, two drops of water added to two drops make one drop, but
that's only because the laws of arithmetic don't apply to drops. Two
plus two is always four precisely because it is empty of empirical
content. It applies to cows only if you add a correspondence rule that each
cow is to be identified with one. The Pythagorean theorem is
timelessly true in all possible worlds because it follows with certainty
from the symbols and rules of formal plane geometry.
     In his worst attack on the absolute eternal validity of arithmetic,
Hersh uses the analogy of a building with no 13th floor. If you go up
eight floors in an elevator, then five more floors, you step out of the
elevator on Floor 14. Hersh seems to think this makes eight plus five
equals 14 an expression that casts doubt on the validity of arithmetic
addition. I might just as well cast doubt on two plus two equals four by
replacing the numeral four with the numeral five.
     Hersh imagines that because the concept of "number" has been
steadily generalized over the centuries, first to negative numbers, then
to imaginary and complex numbers, quaternions, matrices, transfinite
numbers and so on, this somehow makes two plus two equals four
debatable. It is not debatable because it applies only to positive
integers. "Dropping the insistence on certainty and indubitability,"
Hersh tells us, "is like moving off the [number] line into the complex
plane." This is baloney. Complex numbers are different entities. Their
rules have no effect on the addition of integers. Moreover, laws
governing the manipulation of complex numbers are just as certain as the
laws of arithmetic.
     Within the formal system of Euclidean geometry, as made precise by
the great German mathematician David Hilbert and others, the interior
angles of a triangle add to 180 degrees. As Hersh reminds us, this was
Spinoza's favorite example of an indubitable assertion. I was
dumbfounded to come upon pages on which Hersh brands this theorem
uncertain because in non-Euclidean geometries the angles of a triangle add
to more or less than a straight angle 180 degrees.
     Non-Euclidean geometries have nothing to do with Euclidean
geometry. They are entirely different formal systems. Euclidean geometry
says nothing about whether space time is Euclidean or non-Euclidian.
Hersh's claim of triangular uncertainty is like saying that a circle's
radii are not necessarily equal because they are unequal on an ellipse.
     Hersh devotes two chapters to great thinkers he believes were
"humanists" (social constructivists) in their philosophy of mathematics.
It is a curious list. Aristotle is there because he pulled numbers
angeometrical objects down from Plato's transcendent realm to make them
properties of things, but to suppose he thought those forms existed only
in human minds is to misread him completely. Euclid is also deemed a
humanist without the slightest basis. (The person most deserving to be
on Hersh's list of maverick anti-realists is, of course, the
mathematician Raymond Wilder. He and his anthropologist friend Leslie
White were leading boosters of the notion that mathematical objects have
no reality outside human culture. Hersh calls White's essay "The Locus
of Mathematical Reality," a "beautiful statement" of social
constructivism.
     John Locke is on the list because he recognized the fact that
mathematical objects are inside our brains. But Locke also
believed--Hersh even quotes this!--that "the knowledge we have of
mathematical truths is not only certain but real knowledge; and not the
bare empty vision of vain, insignificant chimeras of the brain." The
angles, in other words, of a mental triangle add to 180 degrees. This is
also true, Locke adds, "of a triangle wherever it really exists." A
devout theist, Locke would have been as mystified as Aristotle by the
notion that mathematics has no reality outside human minds.
     The inclusion of Peirce as a social constructivist is even harder
to defend. "I am myself a Scholastic realist of a somewhat extreme
stripe," Peirce wrote in Vol. 5 of his "Collected Papers". (All of
Hersh's arguments against realism, by the way, were thrashed out
by>medieval opponents of realism.) In Vol. 4, Peirce speaks of "the
Platonic world of pure forms with which mathematics is always dealing."
In Vol. 1, we find this passage:
     "If you enjoy the good fortune of talking with a number of
mathematicians of a high order, you will find the typical pure
mathematician is a sort of Platonist. . . . The eternal is for him a
world, a cosmos, in which the universe of actual existence is nothing
but an arbitrary locus. The end pure mathematics is pursuing is to
discover the real potential world."
     Hersh devotes many excellent chapters to summarizing the history of
mathematics, and he ends his book with crisp, expertly worded accounts
of famous mathematical proofs. An odd thing about this final chapter,
though, is that Hersh writes as if he were a realist. This is hardly
surprising, because the language of realism is by far the simplest,
least confusing way to talk about mathematics.
     Over and over again Hersh speaks of "discovering" mathematical
objects that "exist." For example, the square root of two doesn't exist
as a rational fraction, but it does exist as an irrational number that
measures the length of a diagonal of a square on one side.
Mathematicians "find" complex numbers "already there" on the complex
plane. After saying that Sir William Rowan Hamilton "found" quaternions
while he was crossing a bridge, Hersh reminds us that quaternions did
not exist until Hamilton "discovered them." Of course, he means that
until Hamilton "constructed them" on the basis of a social consensus of
ideas, they didn't exit, but his wording shows how easily he lapses into
the language of realism.
     We must constantly keep in mind that although Hersh talks like a
realist, his words have different meaning than they have for a realist.
Once humans have invented a formal system like plane geometry or
topology, the system can imply theorems that had previously been
difficult to "discover." Their discovery is, therefore, of theorems that
can be said to "exist" outside any individual mind, but have no reality
beyond the collective minds of mathematicians.
     Hersh closes this chapter with a beautiful new proof by George
Boolos of Godel's famous theorem that formal systems of sufficient
complexity contain true statements that can't be shown true within the
system. Boolos' proof is flawless, a splendid example of mathematical
certainty, as are all the other proofs in this admirable chapter.
     Let the great British mathematician G.H. Hardy have the final say:
"I believe that mathematical reality lies outside us, that our function
is to discover or observe it, and that the theorems which we prove, and
which we describe grandiloquently as our 'creations,' are simply our
notes of our observations. This view has been held, in one form or
another, by many philosophers of high reputation from Plato onward, and
I shall use the language which is natural to a man who holds it. A
reader who does not like the philosophy can alter the language: it will
make very little difference to my conclusions."


Joao Pedro da Ponte

-------------------

Departamento de Educacao da Faculdade de Ciencias
Universidade de Lisboa
Edificio C1, Campo Grande
P-1700 LISBOA, PORTUGAL

tel. 351-1-7500049 (office)
tel. 351-1-3630861 (home)
fax. 351-1-7500082
http://correio.cc.fc.ul.pt/~jponte

------------------

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X-Comment:  Educacao em Matematica
Status: RO

At 22:25 14-10-1997 +0100, you wrote:
>Lamentavelmente n=3DE3o foi poss=3DEDvel ler o material enviado,=
 provavelment=3D
>e
>devido =3DE0 incompatibilidade de Mac ou W95 com meu sistema bem mais
>simples. Mande um texto mais curto que terei o maior empenho em divulgar
>por essas paragens.
>um abra=3DE7o
>Bigode
>
Ola!
Aqui vai uma nova tentativa de nevio do programa:

IV Ciclo de Palestras em Hist=F3ria da Matem=E1tica=20
Universidade Lus=EDada do Porto

A Matem=E1tica Na China Antiga
Maria Jos=E9 Costa - Esc. Sec. Augusto Gomes
22 de Outubro - 17h30

Matem=E1tica Na Mesopot=E2mia
Fernanda Estrada -Univ. do Minho
29 de Outubro - 17h30

O Conte=FAdo Matem=E1tico Da Simetria Na Cultura Isl=E2mica
Ant=F3nio Costa - UNED
7 de Novembro - 17h30 (data a confirmar)

Desenvolvimentos Em S=E9ries de Pot=EAncias=20
Segundo o Matem=E1tico Indiano Nilakantha (s=E9c. XVI)
Jaime Carvalho Silva -Univ. de Coimbra
19 de Novembro - 17h30

Ser=E1 Poss=EDvel Dividir Uma Circunfer=EAncia Em 7 Partes Iguais? E em=
 65537?
Teresa Viegas - Univ. Porto
26 de Novembro - 17h30

Os Precursores do C=E1lculo,
Tangentes e Quadraturas na Primeira Metade do S=E9c. XVII
Manuel Ferrari - Univ. do Porto
3 de Dezembro - 17h30

Ainda ocorrer=E3o mais duas palestras, dias 17 de Dezembro e 14 de Janeiro,
cujo conte=FAdo ser=E1 brevemente anunciado

Nota: Todas as Palestras ocorrem na sala 18 de Pavilh=E3o C da Universidade
Lus=EDada do Porto e ter=E3o a dura=E7=E3o aproximada de 2 horas.

No fim de cada palestra, justifica=E7=F5es de presen=E7a ser=E3o fornecidas=
 a quem
as solicitar

Um abraco
eduarda

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Eduarda:

Nao sei como chegaram as tuas mensagens as outras pessoas. A mim chegaram
totalmente ilegiveis.

Um abraco do

Paulo

------
Paulo Abrantes
Dep. Educ., Fac. Ciencias
Universidade de Lisboa
Campo Grande, 1700 Lisboa, Portugal
Tel. 351-1-7573141 ext. 1101
FAX. 351-1-7500082

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Mime-Version: 1.0

Paulo Abrantes wrote:
>=20
> Eduarda:
>=20
> Nao sei como chegaram as tuas mensagens as outras pessoas. A mim chegar=
am
> totalmente ilegiveis.
>=20
> Um abraco do
>=20
> Paulo
>=20
> ------
> Paulo Abrantes
> Dep. Educ., Fac. Ciencias
> Universidade de Lisboa
> Campo Grande, 1700 Lisboa, Portugal
> Tel. 351-1-7573141 ext. 1101
> FAX. 351-1-7500082

Linhas cruzadas Paulo, estou recebendo mensagens que deveriam ir para
outros destinat=E1rios.

Um abra=E7o do Bigode

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X-Comment:  Educacao em Matematica

Ola a todos!
Aqui vai mais uma tentativa hoje sem acentos!

IV Ciclo de Palestras em Historia da Matematica=20
Universidade Lusiada do Porto

A Matematica Na China Antiga
Maria Jose Costa - Esc. Sec. Augusto Gomes
22 de Outubro - 17h30

Matematica Na Mesopotamia
Fernanda Estrada -Univ. do Minho
29 de Outubro - 17h30

=B7 O Conteudo Matematico Da Simetria Na Cultura Islamica
=B7 Ant=F3nio Costa - UNED
=B7 7 de Novembro - 17h30 (data a confirmar)

Desenvolvimentos Em Series de Potencias=20
Segundo o Matematico Indiano Nilakantha (sec. XVI)
Jaime Carvalho Silva -Univ. de Coimbra
19 de Novembro - 17h30

Sera Possivel Dividir Uma Circunferencia Em 7 Partes Iguais? E em 65537?
Teresa Viegas - Univ. Porto
26 de Novembro - 17h30

Os Precursores do Calculo,
Tangentes e Quadraturas na Primeira Metade do Sec. XVII
Manuel Ferrari - Univ. do Porto
3 de Dezembro - 17h30

Ainda ocorrerao mais duas palestras, dias 17 de Dezembro e 14 de Janeiro,
cujo conteudo sera brevemente anunciado

Nota: Todas as Palestras ocorrem na sala 18 de Pavilhao C da Universidade
Lus=EDada do Porto e terao a duracao aproximada de 2 horas.

No fim de cada palestra, justifica=E7=F5es de presenca serao fornecidas a=
 quem
as solicitar

eduarda moura

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X-Comment:  Educacao em Matematica


Integrado no plano de accoes elaborado pela Comissao de
Acompanhamento dos Programas de Matematica do Ensino
Secundario, foi criado um sitio WWW na Internet, no
Terravista, no seguinte endereco:

http://www.terravista.pt/IlhadoMel/1129/

Ai' se encontram informacoes e documentos de apoio 'as
actividades de Acompanhamento, tais como:

-Ajustamento dos programas do ensino secundario de
Matematica - Opcoes fundamentais;
-Documentos entregues nas
sessoes de sensibilizacao promovidas entre Abril e Junho de
1997 com representantes de todas as Escolas Secundarias;
-Laboratorios de Matematica no Ensino Secundario - Proposta da
Comissao de Acompanhamento do Programa de Matematica do
Ensino Secundario aprovada na sua reuniao de Julho de 1997;
-Oficinas de Formacao.

Mais documentos serao adicionados a este sitio 'a medida que
forem sendo produzidos.

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X-Comment:  Educacao em Matematica


No ambito dos trabalhos de Acompanhamento dos Programas de
Matematica do Ensino Secundario, irao ainda ser promovidas
sessoes de discussao da aplicacao do Ajustamento do Program
de Matematica ao 10o ano, todas as 4as feiras das 11h 30m as
12h 30m atraves do IRC. Nessa altura estara aberto o canal

#matematica_10

na rede RCTS, a que todas as escolas estao ligadas. Basta
usar um programa qualquer de IRC para se ligar a um servidor
do tipo

irc.pop-uc.rcts.pt

Agradece-se que todas as pessoas se liguem com "nickname"
identificavel.

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_______________________________________________________________
 Seminario Nacional de Historia da Matematica
(Sociedade Portuguesa de Matematica)
 _______________________________________________________________
 Dia 16 de Novembro de 1997
 _______________________________________________________________

 10.30-11.00 - Recepcao e Cafe'

 11.00-11.40 - Ubiratan D'Ambrosio (Univ. Campinas)

 11.40-12.20 - Eleanor Robson (University of Oxford) - "A survey of Old
Babylonian mathematics"

 14.30-15.00 - Luis Saraiva (Univ. Lisboa)

 15.00-15.30 - Ana Isabel Rosendo (Univ. Coimbra) - "A vida e obra
 matematica de Inacio Monteiro"

 15.30-16.00 - Conceicao Coelho (Porto): "Teorema Fundamental do Calculo"

 16.00-16.30 - Cafe'

 16.30-18.30 - Mesa redonda: Que caminhos para a cooperacao
 luso-brasileira em Historia da Matematica?
 com a participacao de Ubiratan D'Ambrosio, Antonio Leal Duarte, e Luis Sara=
iva.
 Moderador: Jaime Carvalho e Silva
 _______________________________________________________________
 Local: Sala Jose' Anastacio da Cunha
 Departamento de Matematica da Universidade de Coimbra
 _______________________________________________________________
 Apoios: Centro de Matematica da Universidade de Coimbra,
 Programa Praxis XXI 2/2.1/MAT/458/94,
 Associacao de Professores de Matematica
 _______________________________________________________________

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