21. CDTLink: Teaching Mathematics And Training Mathematicians Bourbaki, N. is a group of mostly french mathematicians, which was formed in the1930s with the aim of writing a thorough unified account of all mathematics http://www.cdtl.nus.edu.sg/link/nov2003/tm2.htm

Triannual newsletter produced by the C entre for D evelopment of ... earning INSIDE THIS ISSUE TEACHING METHODS November 2003 Vol. No. Print-Ready COVER STORY A Vision for Effective Teaching TEACHING METHODS The Contract Game Teaching Mathematics and Training Mathematicians OUTCOME-BASED EDUCATION Outcome-based Education (OBE): A New Paradigm for Learning LEARNING ISSUES Motivating Students in a Writing Class Writing Educational (Learning) Objectives to Facilitate Student Learning Collaborative Learning Online: Setting the Stage CDTL NEWS CDTL Monograph Series TLHE 2004 Welcome to CDTL/Goodbye FROM THE FACULTIES March 2004 November 2003 July 2003 March 2003 ... January 1997 Teaching Mathematics and Training Mathematicians Professor S L Lee Head, Department of Mathematics Introduction Teaching mathematics and training mathematicians are two fundamental responsibilities of a Mathematics Department. The problem of determining what to teach, how to teach and how much mathematics to teach to students is not a mathematical problem with a unique solution. It is a controversial issue, in which many mathematicians, scientists and engineers do not agree with one another. Different people hold different views and different expectations of mathematics. Although the aim of teaching mathematics in general may be different from that of training mathematicians, the two activities have an overlapping set of objectives. I will:

22. 1939A.D. Nicolas bourbaki, a Greek name, was actually a pseudonym, or nom de plum,that a group of french mathematicians used to publish under. http://faculty.oxy.edu/jquinn/home/Math490/Timeline/1939AD.html

Bourbaki (a man of mystery?), Mathematicians of the year or Alan Baker Nicoloas Bourbaki The work of the french mathematician Nicoloas Bourbaki influenced the change in thinking about the structure of mathematics from 1940 on. One of the Bourbaki publications, Part 1 of the Fundamental Structures of Analysis , was directly used to develop later curriculum. It influenced the curriculums in the areas of set theory, algebra, general topology, fuctions of a real variable, topological vector space, and integration. Nicolas bourbaki, a Greek name, was actually a pseudonym, or nom de plum , that a group of French mathematicians used to publish under. The members of the group writing under this name, did not stay consistent, but in general, most of them were from the University at Nancy, and many of them had appointments at American universities. The number of mathematicians publishing under this name usually was around 12 at a time; the most ever in the group at any one time was 20. Four well-known members of the group were C. Chevalley, J. Delsarte, J. Dieudonne, and A. Weil. The only rule of the group was that they most retire from the group at 50 years of age. The work of Bourbaki influenced a change in thinking about math to the degree that a "new math" curriculum was developed to try to address the issues that Bourbaki brought to the surface of mathematical education. His publishings began in 1939. The influential publishings were a general surbey of math. They were trying to develop all of math from a few broad axioms, giving complete proofs for all of mathematics. Set theory was being used to axiomatisize, in a system of first order logic, building on the Axiom of Global Choice. He, or they, developed properties of a lot of "key math structures," like topological spaces and groups. One easy way to understand Bourbaki's work is to see that "the Bourbaki system is a ‘big theory' rather than a mosaic of ‘little theories.'"

23. 1798 A.D. be remembered because there is a crater on the moon named after him, and he is recognizedon the Eiffel Tower in Paris with other great french mathematicians. http://faculty.oxy.edu/jquinn/home/Math490/Timeline/1798AD.html

The Fourier Series states that any function can be represented by a trigonometric series in the interval (-pi, pi) by the equation a /2 + the summation from n=1 to infinity of (a n cosx + b n sinx), where a and b are real numbers. Poisson's advances on this Fourier series provided the basis for the work of Direchlet, as he generalized the series on the bases of the convergence, and for the work of Riemann. The Poisson distribution first appeared in 1837 in "Recherches sur la probabilite des jugements ". This distribution describes the problem that a random event will occur in a time or space interval under conditions that the probability of that event occurring is very small but the number of trials is very large so that the event actually occurs a few times. As if his mathematical contributions were not enough, Poisson would be remembered because there is a crater on the moon named after him, and he is recognized on the Eiffel Tower in Paris with other great French mathematicians. Author : Maggie Cooper References: Eves, Howard, "

24. Israel21c Search branch of discrete mathematics as he had been led to believe; but worse, exceptfor a professor who became his supporter, french mathematicians remained aloof http://math.haifa.ac.il/toufik/personal/Focus03.html

See Focus (Haifa University, Autumn 2003) for the original version Israel's First Druze Math Lecturer Feels at Home Here M y grandfather was a human calculator; he could do all kinds of calculations in his head. My father only went to the fourth grade, but he could do four-figure multiplication in his head. I guess I inherited their genes." Toufik Mansour, 35, was explaining how he became the first member of the Druze community in Israel to become a university lecturer in mathematics. "I have loved math since the age of 0," continued the University of Haifa’s newest math teacher and researcher. His parents were more concerned that he wasn’t studying to become a doctor, as their community at the time considered a physician to be the top professional. The University played a defining role in the young Mansour’s choice of career direction. His economic situation had forced him to drop out of the Master’s program at the Technion, where he had earned his Bachelor’s in Math. After three years of working as a teacher in different schools, from elementary to high school, he returned to his higher education, but now transferred to the University. The University of Haifa, he acknowledged, opened doors for him. But perhaps even more important was that “I felt at home here,” as he expressed it. “I was well treated, both mathematically and economically.”

25. Lebesgue, Henri Leon Building on the work of others, including that of the french mathematicians EmileBorel and Camille Jordan, Lebesgue developed (1901) his theory of measure. http://euler.ciens.ucv.ve/English/mathematics/lebesgue.html

Lebesgue, Henri Leon Henri Leon Lebesgue, b. June 28, 1875, d. July 26, 1941, was a French mathematician who revolutionized the field of INTEGRAL CALCULUS by his generalization of the Riemann integral. Up to the end of the 19th century, mathematical analysis was limited to continuous functions, based largely on the Riemann method of integration. Building on the work of others, including that of the French mathematicians Emile Borel and Camille Jordan, Lebesgue developed (1901) his theory of measure. A year later, Lebesgue extended the usefulness of the definite integral by defining the Lebesgue integral: a method of extending the concept of area below a curve to include many discontinuous functions. Lebesgue served on the faculty of several French universities. He made major contributions in other areas of mathematics, including topology, potential theory, and Fourier analysis. Author: Howard Frisinger Homepage e-mail: webmaster@euler.ciens.ucv.ve

26. Can Mathematical Meaning Allow Cultural Analysis? An Illustration He was a catalyst in the process of the ‘conversion’ of french mathematiciansto the calculus, although he did not contribute to it in any way. http://christophe.heintz.free.fr/papers/FramingMeaningOfInfinitesim.htm

Chistophe Heintz http://christophe.heintz.free.fr Can Mathematical Meaning Allow Cultural Analysis? An Illustration Travelling Concepts II: Frame, Meaning and Metaphor , Amsterdam: ASCA Press . Although our present understanding of concepts leads us to a cultural and historical analysis of their meaning, such an analysis has rarely been explored in the domain of mathematics. It is because the discipline is believed to derive from reason only, that it is assumed that cultural analysis can have no bearing on mathematical concepts. Against this commonly held notion, I shall argue that the cultural analysis of mathematical concepts and their meaning is both possible and fruitful. Concepts are socially constructed and their meaning is the result of social interactions combined with cultural and historical phenomena. It is because scientific concepts, and in particular mathematical concepts, are no exceptions that cultural analysis can indeed take place in this purportedly most rational of domains. th -century France, is susceptible to such an approach. I will explain the social, cultural and historical context from which the concept of infinitesimals arose and analyse the social processes through which it acquired its meaning. This will also involve a comparison of cultural analysis and the traditional history of mathematics, showing that the former has much more explanatory power than the latter. Finally, I hope to demonstrate how an appropriate concept of mathematical meaning renders the cultural study of mathematics possible. In doing so I hope to show that the concept of infinitesimals, like other mathematicals, has properties that allow for a cultural analysis of their meaning.

27. Indo-French Cooperation In Mathematics Each year a support from the french Minstry of Foreign Affairs is provided, whichenables two french mathematicians to visit Pondicherry University and give http://iml.univ-mrs.fr/infrcoop/agreement.html

Pondicherry Poitiers Paris VI agreement A tripartite cooperation agreement has been signed between the Universities of Pondicherry Poitiers and Paris VI in november 1993. Each year a support from the french Minstry of Foreign Affairs is provided, which enables two french mathematicians to visit Pondicherry University and give lectures there, and two or three indian mathematicians to visit France for one month each. A report on this programm has been written by Professor P. Jothilingam Under this programm Mrs Gayatri came to France to prepare a thesis under the supervision of In the following visits took place: a one month visit from S. A. Katre (Pune University) to Gilles Lachaud a one month visit from H. P. Patil ( Pondicherry University ) to Y. Manoussakis (LRI, Orsay ). a one month visit from Sitaramaiah Varanasi Pondicherry University ) to J.-L. Nicolas (Institut Desargues, Université de Lyon I In the other direction

28. Fractals help of the computer in the 1960s, Mandelbrot returned to earlier research questionsfirst posed between 1915 and 1930 by french mathematicians Gaston Julia http://curvebank.calstatela.edu/fractal/fractal.htm

The Mandelbrot Set weds the graphing of complex numbers to the recursive power of modern computers. MandelZoom takes approximately 15 seconds to load. Be patient. MandelZoom (C) Louis P. Santillan 2001-2002 Instructions: Click to zoom IN. O to zoom OUT. R to reset to the original screen. C to CHANGE COLORS. For source code, email Louis here. Back to . . . Curvebank Home Page The points of a Mandelbrot Set are bounded as follows: x: -2 x y i i x i Size: radius or distance from (0,0) The full Mandelbrot Set is plotted within the inscribed circle of radius . Other views showing the fractal edge are displayed by zooming in on only a portion of the bounded area. Sample calculation: Mathematicians in the early 20th century investigated curves that had highly intricate and detailed shapes. Moreover, they realized that while a region might be bounded and thus the area finite, the perimeter or border might seem to be infinite. These curves - the Koch Snowflake for example - with finite area and infinite perimeter, were given the name of "pathological." This particular area of research in mathematics has generated colorful names: Cantor's dust, Polya's sweeps, Peano's dragons, Sierpinski's carpet and others. When the edge of a curve under many iterations is broken, repeated, scaled down, and then scaled down again as the iterations progress, the curve has now become known as a fractal. This relatively new word in mathematics was first coined by Benoit B. Mandelbrot and introduced to mathematicians and computer scientists in

29. The Scientist - People He is currently researching the history of mathematics and helping edit the worksof two great french mathematicians, Jacques Bernoulli and Pierre de Fermat. http://www.the-scientist.com/yr1994/august/people_940822.html

The Scientist 8[16]:23, Aug. 22, 1994 News People By Neeraja Sankaran Chemist And Mathematician Are Named Winners Of Two 1994 Kyoto Prizes Paul C. Lauterbur , a chemist and director of the Biomedical Magnetic Resonance Laboratory at the University of Illinois College of Medicine, Urbana-Champaign, and , a French mathematician who is currently an emeritus professor at the Institute for Advanced Study, Princeton, N.J., have been named winners of the 1994 Kyoto Prizes in the advanced technologies and basic science categories, respectively. The two researchers will each be honored with a commemorative gold medal and cash award of about $430,000 during award ceremonies to be held in Kyoto, Japan, November 9-12. The Kyoto Prizes, considered Japan's highest award for lifetime achievement, are presented by the Inamori Foundation, a nonprofit organization in Kyoto whose mission is to recognize individuals and groups whose work has had a significant beneficial impact. Lauterbur, 65, was the first scientist to make an image using nuclear magnetic resonance (NMR). He predicted the technique's potential when he first described it (P.C. Lauterbur, Nature, 242:190-1, 1973). This work was pivotal in the development of the magnetic resonance imaging (MRI) scanner, which is now widely used in medical diagnostic imaging, providing a noninvasive method to look at the brain, spinal cord, pelvic organs, heart, and joints without surgery or X-rays.

30. André Weil--Life And Work In the 1930s Weil was a founder of Bourbaki, a group of french mathematicians whowrote a highly influential multivolume series of treatises that organized http://www.ams.org/new-in-math/cover/weil-obit.html

In the 1930s Weil was a founder of Bourbaki, a group of French mathematicians who wrote a highly influential multi-volume series of treatises that organized and unified mathematical knowledge. The work, Elements de Mathematique, offered, for the first time, a survey of the leading work in practically every field of mathematics. In 1994 Professor Weil received the Kyoto Prize in Basic Science from the Inamori Foundation of Kyoto, Japan, an award that is frequently referred to as Japan's Nobel Prize. The award citation noted that Weil, who was recognized for his lifetime achievement in mathematics, "altered the very course of 20th century thought in mathematics. His so-called Weil Conjectures have provided the guiding principles for algebraic geometry, which, in turn, have given rise to the accurate and efficient transmission of information through coding theory. Today, Dr. Weil's work continues to play extremely important roles in fields ranging from elementary particle physics to encryption and computer security." During the war, Weil left France for Finland to avoid the draft, feeling that "as a soldier I would be entirely useless, but as a mathematician I could be of some use." The Finns turned him over to the French authorities, who imprisoned him for six months. While in prison Weil created his theorem on the Riemann hypothesis, described as "a jewel of modern number theory" and one of his greatest mathematical proofs. He was released in exchange for agreeing to join the French army. After the War Weil came to the United States, where he held academic positions at Haverford College and the University of Chicago, in addition to spending two years in Brazil at the University of Sao Paulo.

31. Some Contemporaries Of Descartes, Fermat, Pascal And Huygens born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his businessto be acquainted and correspond with the french mathematicians of that http://www.maths.tcd.ie/pub/HistMath/People/17thCentury/RouseBall/RB_Math17C.htm

Some Contemporaries of Descartes, Fermat, Pascal and Huygens From `A Short Account of the History of Mathematics' (4th edition, 1908) by W. W. Rouse Ball. Bachet Mersenne Roberval Van Schooten ... Rolle Bachet was born at Bourg in 1581, and died in 1638. He wrote the , of which the first edition was issued in 1612, a second and enlarged edition was brought out in 1624; this contains an interesting collection of arithmetical tricks and questions, many of which are quoted in my Mathematical Recreations and Essays . He also wrote , which exists in manuscript; and a translation of the Arithmetic of Diophantus. Bachet was the earliest writer who discussed the solution of indeterminate equations by means of continued fractions. Mersenne Marin Mersenne , born in 1588 and died at Paris in 1648, was a Franciscan friar, who made it his business to be acquainted and correspond with the French mathematicians of that date and many of their foreign contemporaries. In 1634 he published a translation of Galileo's mechanics; in 1644 he issued his Cogita Physico-Mathematica , by which he is best known, containing an account of some experiments in physics; he also wrote a synopsis of mathematics, which was printed in 1664.

32. About Hte IUM students of the University have the opportunity to spend one month at ENS (inParis) attending seminars headed by leading french mathematicians; and the http://ium.mccme.ru/english/general.html

INDEPENDENT UNIVERSITY OF MOSCOW Back to the main page About the Independent University of Moscow The Independent University of Moscow was founded in 1991 on the initiative of a group of well-known mathematicians who now comprise its Academic Council. This group includes the following members of the Russian Academy of Sciences: V.I.Arnold (chairman of the Council), S.P.Novikov, Ya.G.Sinai, L.D.Faddeev, V.A.Vassiliev adn the following professors: A.A.Beilinson, the late R.L.Dobrushin, B.A.Dubrovin, A.A.Kirillov, A.N.Rudakov, V.M.Tikhomirov, A.G.Khovansky, M.A.Shubin. Professors P.Deligne and R.MacPherson of Princeton and MIT also played crucial roles in the founding of the University, as did the well-known instructor and organizer of mathematical olympiads, N.N.Konstantinov. In December of 1996, the first seven graduates of the University received their diplomas. The University is a private institution of higher learning for the training of professional mathematicians. Its founding organization is the Moscow Center for Continuous Mathematical Education. The curriculum of IUM generally requires 5 years to complete (students can sometimes shorten or lengthen this term, depending on their individual needs and interests). In order to successfully graduate from the University and receive a diploma, a student must pass exams in all required courses and in some elective courses, and then must write and defend a thesis. In their first and second years students study the following subjects:

33. Elsevier Author Gateway Published from 1836 by the leading french mathematicians, the Journal des MathématiquesPures et Appliquées is the second oldest international mathematical http://authors.elsevier.com/JournalDetail.html?PubID=600731&Precis=DESC

34. Read This: A Mathematician Grappling With His Century during World War II. Schwartz found his way to ClermontFerrand, wheremany french mathematicians had relocated. Life as a Jewish http://www.maa.org/reviews/grappling.html

Read This! The MAA Online book review column A Mathematician Grappling with His Century by Laurent Schwartz Reviewed by Robert Dobrow "Many people nowadays seem to consider scientists, mathematicians and others, like people uninterested in moral questions, locked away in their ivory towers and indifferent to the outside world," writes Laurent Schwartz, winner of the Fields Medal in 1950, in his autobiography A Mathematician Grappling with His Century What Schwartz grappled with was the social and political issues that wracked France, Europe and the world in the mid-20th century. Chapter titles of this remarkable story include Trotskyist, The War against the Jews, Algerian Involvement, For an Independent Viet Nam, The Distant War in Afghanistan. The reader will find few theorems in this book. Except for a brief discussion on Schwartz's most important mathematical contribution, the concept of distributions, which generalized the notion of function and allowed for a considerable broadening of calculus, most of the math in this book is written from a historical perspective and largely accessible to non-mathematical readers. As Schwartz writes, mathematics "concerns only about 15% of the volume." What concerns the rest is Schwartz's great passion besides mathematics: his devotion to the struggle for oppressed people and for human rights.

35. Article - Algebra - Presented By ©NewsFinder.Org - All Rights Reserved Important contributions to algebra study were made by the french mathematiciansGalois and Augustin Cauchy, the British mathematician Arthur Cayley, and the http://www.newsfinder.org/comments.php?id=465_0_1_0_C

36. George Boole (1815 - 1864) from the local Mechanic s Institute, Boole struggled with Isaac Newton s Principiaand the works of 18th and 19th century french mathematicians PierreSimon http://www.home.gil.com.au/~bredshaw/boole.htm

George Boole (1815 - 1864) T he original `working class boy made good', Boole was born in the wrong time, in the wrong place, and definitely in the wrong class - he didn't have a hope of growing up to be a mathematical genius, but he did it anyway. Born in the English industrial town of Lincoln, Boole was lucky enough to have a father who passed along his own love of math. Young George took to learning like a politician to a pay-rise and, by the age of eight, had outgrown his father's self-taught limits. A family friend stepped in to teach the boy basic Latin, and was exhausted within a few years. Boole was translating Latin poetry by the age of twelve. By the time he hit puberty, the adolescent George was fluent in German, Italian and French as well. At 16 he became an assistant teacher, at 20 he opened his own school. Over the next few years, depending mainly on mathematical journals borrowed from the local Mechanic's Institute, Boole struggled with Isaac Newton's Principia and the works of 18th and 19th century French mathematicians Pierre-Simon Laplace and Joseph-Louis Lagrange. He had soon mastered the most intricate mathematical principles of his day. It was time to move on.

37. Zaremba he published his results in French mathematical journals meant that his work becamewell known and highly respected by leading french mathematicians such as http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Zaremba.html

38. Images Of Mathematicians On Postage Stamps Images of mathematicians on Postage Stamps Issued by french Polynesia in 1993 to commemmorate the 15th Congress of Australian mathematicians or the 15th http://jeff560.tripod.com/

Images of Mathematicians on Postage Stamps RECENT CHANGES: On March 20, Francisco gomes_teixeira1.jpg and gomes_teixeira2.jpg were added. On March 17, lagrange.jpg, monge.jpg, and keldysh.jpg were replaced with better quality images. monge.jpg now has the block of four stamps in the set. On March 15, caratheodory.jpg and thales.jpg were replaced with higher quality images (actually both files were replaced with the same image, showing both stamps unseparated). On March 15, democ2.jpg, democ3.jpg, zu_chongzhi.jpg, zhang_heng.jpg, and nunes4.jpg were added. On March 13, avicenna7.jpg was added. On March 7, gazeta1.jpg and gazeta2.jpg were replaced with higher quality images. Thanks to Bert Jagers for these images. On Feb. 26, goldensection.jpg, moebius2.jpg, moebius3.jpg, impossible4.jpg, keldysh3.jpg, metric25.jpg, petrovic.jpg, schmidt2.jpg, calculate1.jpg, icm90.jpg, and bougainville.jpg were added. Thanks to Magnus Waller for these images. ABEL, Niels Henrik. Issued by Norway on April 6, 1929, upon the death centenary abel1.jpg

39. MacTutor History Of Mathematics D'Alembert Biography of this noted french thinker by J.J. O'Connor and E.F. Robinson. Includes links to related thinkers in the mathematical tradition. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/D'Alembert.html

40. French Vocabulary For Mathematicians french Vocabulary Part I. AE. Write down the corresponding English and submit to check your answers. It will count how many matches you have. Then, try again. After some practice, go to the answer page and click the entries to see the french/English pairs and additional comments http://chanoir.math.siu.edu/frvoc.html

French Vocabulary: Part I A-E Write down the corresponding English and submit to check your answers. It will count how many matches you have. Then, try again. After some practice, go to the answer page and click the entries to see the French/English pairs and additional comments. anneau application base biunivoque champ composante continu convergence corps courbure crochet croissant droite endomorphisme ensemble entier exposant Part II Part III Answers

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