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**Educación matemática. De Félix Klein a Hyman Bass**

Diego Pareja Heredia. Universidad del Quindío La constancia escrita de esta charla aparece en la página web. Incluir hipervínculo con botón derecho El texto y las diapositivas de esta charla se encuentran en:

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Félix Klein ( ) Matemáticas. El programa de Erlangen. Botella de Klein. Educación matemática. 58 estudiantes y descendientes Énfasis en la “unidad del conocimiento”. La inseparable alianza: ciencias y humanidades. En matemáticas, la interrelación: matemáticas puras, matemáticas aplicadas. Asumir seriamente la formación de los docentes de educación media. Matematicas Elementales desde un punto de vista avanzada. Los nombres de Klein y de Bass son hitos, para delimitar nuestra exposición en el ámbito histórico. La educación matématica se inicia con las escuelas filosóficas griegas. Precisamente el nombre “matemáticas”, tal cual, (no, matemática, que es un nombre espúrio), tiene su origen en los pitagóricos con el significado de “aquello que se puede aprender o entender”, o conocimiento adquirido, o más ampliamente, conocimiento adquirible por medio del aprendizaje. La contracción del significado de la palabra matemáticas; de conocimiento general, a las matemáticas propiamente dichas, parece que está ya, presente en los trabajos de Aristóteles ( AC), pero no aun, en la obra de Platón ( AC). De acuerdo a las investigaciones hechas por el matemático Salomon Bochner, entre los primeros comentadores de las obras de Euclides, se desprende que, los primeros pitagóricos, tenían una especie de escuela de graduados para adultos, a la cual asistían dos tipos de personas: los asistentes regulares y los participantes ocasionales. A los últimos los llamaban auditores (oyentes) o akoustimaticoi, mientras que, a los estudiantes regulares, los denominaban mathematikoi (matemáticos). 22 Santos en España y un solo matemático. Julio Rey Pastor.

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Hyman Bass Profesor de la Universidad de Michigan, matemático en la línea de ascendencia intelectual de Gotinga (Kaplansky, Mac Lane, Bernays, Landau) Fue presidente de la American Mathematical Society. Educador (25 estudiantes y 89 descendientes). Presidente hasta 2006, de la International Commission on Mathematical Instruction (ICMI). El primer presidente fue Felix Klein(1908). This page is archived - For information only Hyman Bass is the Roger Lyndon Collegiate Professor of Mathematics and Mathematics Education at the University of Michigan. Prior to 1999, he was Adrain Professor in the Mathematics Department at Columbia University, which he once chaired. His mathematical research interests include algebraic K-theory, commutative algebra and algebraic geometry, algebraic groups, and geometric methods in group theory. He has held visiting appointments at universities and research institutes in Princeton, Paris, Bombay, Madurai, Cambridge, Berkeley, Rome, Rio, Mexico City, Stockholm, Trieste, and Salt Lake City. He is a member of the US National Academy of Sciences, the American Academy of Arts and Sciences, and the Third World Academy of Sciences. Bass is a past president of the American Mathematical Society. He formerly chaired the Mathematical Sciences Education Board at the National Research Council, and the Committee on Education of the American Mathematical Society. He is President of the International Commission on Mathematical Instruction. Since 1996 he has been collaborating with Deborah Loewenberg Ball and her research group at the University of Michigan on the mathematical knowledge and resources entailed in the teaching of mathematics at the elementary level. He has also worked to build bridges between diverse professional communities and stakeholders involved in mathematics education. Edmund Landau attended the French Lycée in Berlin, graduating at the age of 16 which is two years earlier than was normal. He then studied mathematics at the University of Berlin. His doctoral work there was supervised by Frobenius, and Landau received his doctorate in 1899 for a thesis on number theory. Landau was always interested in mathematical puzzles and even before he received his doctorate he had published two books on mathematical problems in chess. On 9 June 1900 he wrote a letter from Paris, where he was studying, to Hilbert giving an outline of his ideas for proving the prime ideal theorem for algebraic number fields. Landau's main work was in analytic number theory and the distribution of primes. He gave a proof of the prime number theorem in 1903 which was considerably simpler that the ones given in 1896 by Vallée Poussin and Hadamard. One consequence of his simpler proof was that it enabled him to obtain results concerning the distribution of prime ideals in algebraic number fields. His masterpiece of 1909 was a treatise Handbuch der Lehre von der Verteilung der Primzahlen a two volume work giving the first systematic presentation of analytic number theory. He also wrote important works on the theory of analytic functions of a single variable. Darstellung und Begründung einiger meuerer Ergebnisse der Funktiontheorie [1]:- ... contains a collection of interesting and elegant theorems of the theory of analytic functions of a single variable. Landau himself discovered some of the theorems and demonstrated others in a new and simpler fashion. Schoeneberg writes [1]:- Written with the greatest care, Landau's books are characterised by argumentation which is complete, and as simple as possible. The necessary prerequisite knowledge is provided, and the reader is led securely, step by step, to the goal. Landau wrote over 250 papers on number theory which had a major influence on the development of the subject.

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**Antecedentes Históricos**

La cultura griega. Matemáticas y filosofía. Leibniz, el gran universalista. La revolución educativa en Alemania. Federico Guillermo III. Emperador de Prusia. Ministro de Educación: Guillermo Von Humboldt (egresado de Gotinga y hermano de Alejandro). Las Universidades de Berlin (hoy Universidad von Humbold) y Bonn Breve historia del término Ph. D. German Humboldt-Universität zu Berlin , byname University of Berlin , formerly (1810–1949) Friedrich Wilhelm University coeducational state-supported institution of higher learning in Berlin. The university was founded in 1809–10 by the linguist, philosopher, and educational reformer Wilhelm von Humboldt, then Prussian minister of education. Under Humboldt's guidance the university, originally named after Frederick William III of Prussia, developed into the largest in Germany. It enrolle The establishment of La Institución Libre de Enseñanza in 1876 had an enormous impact on the reinstatement of science in Spain. Founded by a group of professors who refused to adjust their methods to the official religious, political or moral dogmas, the aim of the institution was to defend the freedom of education. This idealism obliged them to pursue their goal on the fringes of intellectual society and thus create a private, education establishment . Julio Rey Pastor entered a world of accelerating scientific activity, particularly in Spain. The establishment of La Institución Libre de Enseñanza in 1876 had an enormous impact on the reinstatement of science in Spain. Founded by a group of professors who refused to adjust their methods to the official religious, political or moral dogmas, the aim of the institution was to defend the freedom of education. This idealism obliged them to pursue their goal on the fringes of intellectual society and thus create a private, education establishment. Additionally, the German defeat of France in the Franco-German war of 1870, marked the end of French hegemony in continental Europe and a significant shift from a French-orientated Spanish culture to one of German orientation. This was of great importance since it would transcend to Latin America through mathematicians such as Rey Pastor, Esteban Terradas and many others. When Santiago Ramon y Cajal became the first Spaniard to win the Nobel Prize, together with Camillo Golgi in 1901 for their work on the structure of the nervous system, it seemed that the Spanish scientific community had finally awoken from a period of prolonged stagnation. Following this in 1907 was the creation of La Junta Para Ampliación de Estudios, which arose as a result of Spain's cultural self-reflection after losing her final colonies. The key objective of the organisation was: Pedagogic renovation through the provision of grants in order to elevate Spanish culture and bring it to the level of the most advanced countries in Europe, such as Germany. This was to have a direct effect on Julio Rey Pastor. Educated at home until the age of twelve, Rey Pastor began studying at his local secondary school, El Instituto Sagasta, in He had broad interests in his youth and before he became completely occupied with mathematics, he wrote poetry. Having failed the mathematics section of the entrance examination for the Military Academy in Zaragoza, he began studying science at Zaragoza University in It was here that his true vocation for mathematics was awoken and he published his first paper in 1905, entitled Sobre los nœmeros consecutivos cuyo suma es a la vez cuadrado y cubo perfecto (Consecutive numbers whose sum is both the square and the perfect cube). He graduated with a PhD in algebraic geometry from Madrid University in 1910. Between 1908 and 1910, Rey Pastor founded the Real Sociedad Matemática Espanola with the support of José Echegaray and General Benitez. In 1911, Pastor was appointed secretary of the society and, in the same year, he became Professor of Mathematical Analysis at the University of Oviedo. The university, situated in Oviedo in northern Spain, was an ancient one founded in It was here that he wrote the controversial inaugural address for the academic year , in which he candidly discussed the lamentable state of Spanish science since the sixteenth century. As a consequence, Rey Pastor was accused of being unpatriotic and his reputation suffered significantly. An example of his frank discourse may be seen his 1915 Inaugural Discourse, where Rey Pastor discusses the development of science in Spain. You can see a translation of part of this address. Between 1911 and 1916, La Junta Para Ampliación de Estudios funded Rey Pastor to carry out a series of visits to Germany. This resulted in two major publications on geometry in 1912 and The 1916 monograph was on synthetic geometry in n-dimensions and introduced [1]:- ... concepts of great generality (for example the definition of the curve) and developing them in all their consequences. In 1915 Rey Pastor moved to a chair at Madrid University, where he published the highly acclaimed, Fundamentos de la geometría proyectiva superior. However he was not one to remain fixed in one place for a long time and went to Barcelona in 1915 to give a series of lectures at the Institut d'Estudia. His lectures there on n-dimensional geometry and conformal mappings, developing the work of Schwarz, was written up by Esteban Terrades who attended the lectures, and the course was published in Catalan. Rey Pastor was invited by the Institución Cultura Española to lecture at the University of Buenos Aires in Although he was still a young man, only 29 years old, he was asked to help promote mathematics in Argentina and a way was found to enable him to do this. A contract was proposed which enabled him to spend six months each year in Argentina and six months in Spain. Rey Pastor was pleased to sign the contract:- ... to direct the advanced study of the exact sciences in Argentina. It has been asserted that Don Rey Pastor was responsible for the creation of a distinctive Argentinean School of mathematical research and the reconstruction of science in Argentina. When he took up a 6-year contract at Buenos Aires University in 1921, the Science Faculty had only one PhD programme, which had barely progressed since Between then and Rey Pastor's arrival, there had been some small improvement in the engineering courses, but all the other further mathematics courses were in a severe state of stagnation. Rey Pastor needed to persuade the engineering professors of the importance of mathematics beyond the elementary textbooks, which they had previously been using. He had been invited by the Institución Cultura Espanola, to give a lecture course at the University of Buenos Aires in This first course, given as a visiting lecturer, was an introduction to Klein's Erlangen Programme. In this course, Rey Pastor presented his students with the concept of geometry based on group theory, using methods of establishing invariants of each group, with topological methods being the most general. His second course, given in 1921, was a specialised one for engineering students and included the following topics: functions of a complex variable, conformal mapping, advanced geometry (non-euclidean), mathematical analysis and mathematical methodology. Many of these topics, although commonplace in Europe, were entirely new and revolutionary to Argentinean mathematicians. As a consequence, Rey Pastor succeeded in gaining popularity amongst the students, whilst receiving stern criticism from his contemporary old school professors, who regarded him a foreign usurper. Rey Pastor focused mainly on teaching engineering, but he recruited many pure mathematics students to his courses since he regarded the engineering courses on the techniques of calculation good preparation for pure mathematicians. He believed it was important to maintain an understanding of both areas of mathematics and it was this characteristic, panoramic vision of his, which allowed his students to appreciate the profundity of the new concepts he was teaching them. By separating mathematics from its purely technical aspect, he attracted a wide audience on both sides of the Atlantic Ocean. Rey Pastor founded the Sociedad Matemática Argentino in In 1927, he was given a permanent appointment at the University of Buenos Aires and held two chairs: one of Mathematical Analysis and the other of Higher Geometry. This was to have a profound and transcending impact on Argentinean mathematics. In 1928, he founded an influential mathematical seminar El semenario Matemático Argentina (similar to the one he had set up in Madrid). The seminar published a bulletin, which contained the first modern research in Argentinean mathematics. He also brought important foreign mathematicians to the university to give short courses, including: Frederigo Enriques (1925), Francesco Severi (1930), Tullio Levi-Civita (1937), Emile Borel (1928) and Jacques Hadamard (1930). In the 1940's, Pastor's best students began gaining international recognition. Amongst them were: Alberto González Domínguez, who became an important quantum physicist; Alberto Calderón, who would become the chairman of the Department of Mathematics at Chicago University; and many more who considerably enhanced the collective teaching capacity of the mathematical community. In 1931, Rey Pastor published one of his most elegant works on analysis in the Rendiconti del Circolo Matematico di Palermo, an Italian mathematical journal. It dealt with the study of the method of summation of series. This article by Rey Pastor is framed by a long series of works, begun at the beginning of the twentieth century, on problems of summing series, convergence algorithms, singular integrals and comparative studies of series and integrals. He first presented his work in this area in 1926 in his course on "Series and Integrals" which he gave at Buenos Aires University. The same course, slightly amplified, was repeated in Madrid in The same year Rey Pastor presented a summary of his ideas in his paper given to the International Congress of Mathematicians at Bologna, which he attended with a large group of his Argentinean students. He continued working on problems related to the theory of summation of divergent series throughout the 1930's and published much of his work in international journals. The themes which Rey Pastor dealt with in this period had considerable influence on the development of Argentinean mathematics. In 1952 he was expelled from Argentina despite his efforts to remain apolitical. However, he retired with the satisfaction of having initiated the transformation of Argentinean mathematics. He contributed exceptionally to the development of investigation into pure mathematics and trained a new school of Argentinean engineers with a modern outlook. Furthermore, his teaching opened the doors to the study of the history of science and laid the foundations for generations of secondary school teachers with his influential textbooks. In 1954 he returned to Argentina and joined La Academía de Lengua. The history of mathematics had always interested Rey Pastor and late in his career his interests in historical topics extended to cartography. Of course Spain has a reputation for remarkable cartography so his monograph (written jointly with E Garcia Camarero in 1960) on the history of Spanish cartography was a particularly useful addition to knowledge of the topic. Examining his textbooks gives us insight into Rey Pastor's ideas concerning teaching of mathematics. In the introduction to Algabraic Analysis, Rey Pastor comments that rather than follow the general tendency of elevating elemental problems to the point of abstraction, it is his goal to simplify complicated questions whilst maintaining a rigorous approach. He adds that all abstract thought requires a pre-existing knowledge base, which many students lack when they arrive at university and, which they expect to gain over the course of their degree. According to Rey Pastor however, it is a didactic mistake and historically absurd to attempt to approach the concepts of analysis in this contrary manner. He died at his home in Buenos Aires.

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Consecuencias Reformas. Sistema nacional de educación. Seis horas por semana de matemáticas. Escuelas normales. Universidades. Berlín y Bonn. El apogeo de las matemáticas en Alemania. Los primeros frutos: Jacobi, Dirichlet, Grassmann, Kummer, Weierstrass. Heine, Kronnecker, Riemann, Dedekind, Cantor. Frege, Klein, Lindemann, Hilbert, Haussdorff, Zermelo, y otros más, también importantes. Se instituyó un sistema nacional de educación primaria, secundaria y escuelas normales. Se destaca el hecho sin precedentes que, en el currículo de los gimnasios aparecían 6 horas semanales de matemáticas a lo largo de los 10 años que duraba tal ciclo educativo. Por esta época se crearon las universidades de Berlín (1.809) y de Bonn (1.818). Carl Jacobi, Lejeune Dirichlet, Hermann Grassmann, Ernst Kummer, Karl Weierstrass, fueron los primeros frutos de la nueva era. Discípulos, ó bajo la influencia de los anteriores surgieron Heine, Kronecker, Riemann, Dedekind, Hankel, Cantor. La nómina viene a completarse con Klein, Frege, Lindemann, Hilbert, Hausdorff y Zermelo, entre los más conocidos.

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**Matemáticas y educación matemática**

Educación matemática en Alemania. Klein y su compromiso con la educación media.Trascendencia de pi y de e. Inicio de la separación entre lo que se enseña y el frente de las matemáticas. El estatismo en la educación. Principio del momento lineal de Euler . Expuesto en la Academia de Berlin. Ver prólogo en la edición deL Algebra de Euler

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Las grandes preguntas ¿Cuáles han sido las causas que han propiciado la separación entre lo que enseñamos, y lo que ahora es noticia mundial en matemáticas? ¿Qué hacer para cerrar la brecha entre lo que el profesor enseña y aquello que actualmente es motivo de investigación en las matemáticas? ¿Es este un problema local, o es un problema de alcance universal?

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El gran vacío Lo que enseñamos ahora y lo que se enseñaba en el siglo XIX. El álgebra de Euler y el álgebra de Chrystal frente al álgebra de Baldor. La degradación de contenidos. La Conjetura de Poincaré y la Clasificación de las álgebras de Lie. Los problemas del milenio. La Hipótesis de Riemann, P vs. NP. Las Ecuaciones de Euler-Navier-Stokes. Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equations P vs NP Poincaré Conjecture Riemann Hypothesis Yang-Mills Theory The Euler and Navier–Stokes equations describe the motion of a fluid in Rn (n = 2 or 3). These equations are to be solved for an unknown velocity vector u(x, t) = (ui(x, t))1in 2 Rn and pressure p(x, t) 2 R, defined for position x 2 Rn and time t 0. We restrict attention here to incompressible fluids filling all of Rn. The Navier–Stokes equations are then given by @ @t ui + n Xj=1 uj @ui @xj = ui − @p @xi + fi(x, t) (x 2 Rn, t (1) 0), div u = n Xi=1 (2) = 0 (x 2 Rn, t 0)

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**La gran eclosión matemática**

Teoría de conjuntos. Análisis y Topología. Teoría de medida. Lógica simbólica. Álgebra lineal y álgebra moderna. Espacios abstractos. Teoría analítica y algebraica de Números Probabilidad y Estadística.

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**Buscando una solución. La Escuela Bourbaki**

Objetivo central. Acortar la distancia entre lo que se enseña y lo que se investiga. Influencia y Consecuencias. El caso de hispanoamérica. “Rigor” vs. “compresión”.

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**Solución “Made in USA” Bourbaki y la “Nueva Matemática” El grupo SMSG**

La reacción de los grupos ilustrados Morris Kline y su Juanito “Por qué el profesor no sabe enseñar”. El regreso a las bases. La contrarreación.

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El Caso de Colombia La historia de la educación matemática aquí, ya ha sido bien contada. ¿Con qué conocimiento matemático queremos enriquecer nuestra cultura? ¿Qué es lo que debemos enseñar para llenar el gran vacío? Gardner also notes another example of the Bellman's rule: Carroll's constantly reiterated reply "I don't know", when asked to explain what he had in mind with the Snark. The Bellman's rule-of-three Another rule that has given rise to widespread speculation is the Bellman's rule-of-three: What I tell you three times is true. It runs as an underground current through the whole poem, breaking the surface only sporadically, as in Fit 1, Stanza 2, or Fit 5, Stanza 9. Gardner mentions, among other examples of conjecture, Chaos, Co-ordinated, a science fiction story by John MacDougal, and cites Norbert Wiener as saying in his book Cybernetics that the human brain, just like a computing machine, probably works on a variant of the famous principle expounded by Lewis Carroll. Gardner also notes another example of the Bellman's rule: Carroll's constantly reiterated reply "I don't know", when asked to explain what he had in mind with the Snark. [edit] Hidden meanings? As already stated, the Hunting of the Snark is unusual among Lewis Carroll's poems for its length and its dark nature. This also fits with an attempt to find a hidden personal message within its pages. Many believe that this hidden message should be in the repeating stanza They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap. But no convincing theory yet explains it. Lewis Carroll once wrote: "Periodically I have received courteous letters from strangers begging to know whether The Hunting of the Snark is an allegory, or contains some hidden moral, or is a political satire: and for all such questions I have but one answer, I don't know!" According to Gardner, there are more than three such denials on record. By the Bellman's rule-of-three, we therefore must conclude that if the Reverend Charles Lutwidge Dodgson said he didn't know what the unimaginable[9] something is, which is sought with symbols and with faith, hope, and care, then he really didn't know.

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**Educación y matemáticas**

¿Se puede cambiar la educación matemática al margen de la educación en general? ¿A donde debemos apuntar para iniciar la transformación del sistema educativo? ¿Cómo crear masa crítica que genere una revolución educativa?

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CODA (I) Mayor impulso a la educación avanzada – doctorados y postdoctorados – Reingeniería para las facultades de educación – educación universalista y humanista – ciencias y filosofía. Mayor compromiso de la universidad para elevar su nivel académico buscando llenar vacantes con los más capaces. Creación de institutos de alto nivel, donde se formen los profesores universitarios del futuro.

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CODA(II) Enseñar Aritmética en el sentido clásico – Teoría de números – como una parte del Algebra Abstracta – “Comprimir” el cálculo infinitesimal dentro del análisis matemático. Geometría euclidiana y no euclidianas. El enfoque Thurston-Bass-Ball.

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“Para descubrir algo en matemáticas, hay que superar, las inhibiciones y la tradición. No podemos vencer barreras, sin ser subversivos” Laurent Schwartz.

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