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History of Mathematics in Mathematics Education. Recent Developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang
History of mathematics in mathematics education. Recent developments Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang To cite this version: Kathy Clark, Tinne Kjeldsen, Sebastian Schorcht, Constantinos Tzanakis, Xiaoqin Wang. History of mathematics in mathematics education. Recent developments. History and Pedagogy of Mathematics, Jul 2016, Montpellier, France. hal-01349230 HAL Id: hal-01349230 https://hal.archives-ouvertes.fr/hal-01349230 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. HISTORY OF MATHEMATICS IN MATHEMATICS EDUCATION Recent developments Kathleen CLARK, Tinne Hoff KJELDSEN, Sebastian SCHORCHT, Constantinos TZANAKIS, Xiaoqin WANG School of Teacher Education, Florida State University, Tallahassee, FL 32306-4459, USA [email protected] Department of Mathematical Sciences, University of Copenhagen, Denmark [email protected] Justus Liebig University Giessen, Germany [email protected] Department of Education, University of Crete, Rethymnon 74100, Greece [email protected] Department of Mathematics, East China Normal University, China [email protected] ABSTRACT This is a survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the HPM domain). Section 1 explains the rationale of the study and formulates the key issues. -
Causal Dynamical Triangulations and the Quest for Quantum Gravity?
Appendices: Mathematical Methods for Basic and Foundational Quantum Gravity Unstarred Appendices support Part I’s basic account. Starred Appendices support Parts II and III on interferences between Problem of Time facets. Double starred ones support the Epilogues on global aspects and deeper levels of mathematical structure being contemplated as Background Independent. If an Appendix is starred, the default is that all of its sections are starred likewise; a few are marked with double stars. Appendix A Basic Algebra and Discrete Mathematics A.1 Sets and Relations For the purposes of this book, take a set X to just be a collection of distinguishable objects termed elements. Write x ∈ X if x is an element of X and Y ⊂ X for Y a subset of X, ∩ for intersection, ∪ for union and Yc = X\Y for the complement of Y in X. Subsets Y1 and Y2 are mutually exclusive alias disjoint if Y1 ∩ Y2 =∅: the empty set. In this case, write Y1 ∪ Y2 as Y1 Y2: disjoint union.Apartition of a set X is a splitting of its elements into subsets pP that are mutually exclusive = and collectively exhaustive: P pP X. Finally, the direct alias Cartesian product of sets X and Z, denoted X × Z, is the set of all ordered pairs (x, z) for x ∈ X, z ∈ Z. For sets X and Z,afunction alias map ϕ : X → Z is an assignation to each x ∈ X of a unique image ϕ(x) = z ∈ Z. Such a ϕ is injective alias 1to1if ϕ(x1) = ϕ(x2) ⇒ x1 = x2, surjective alias onto if given z ∈ Z there is an x ∈ X such that ϕ(x) = z, and bijective if it is both injective and surjective. -
Oral Presentation
Oral Presentation A LOOK AT OTTO TOEPLITZ’S (1927) “THE PROBLEM OF UNIVERSITY INFINITESIMAL CALCULUS COURSES AND THEIR DEMARCATION FROM INFINITESIMAL CALCULUS IN HIGH SCHOOLS”1 Michael N. Frieda & Hans Niels Jahnkeb aBen Gurion University of the Negev, bUniversität Duisburg-Essen This paper discusses Otto Toeplitz’s 1927 paper “The problem of university infinitesimal calculus courses and their demarcation from infinitesimal calculus in high schools.” The “genetic approach” presented in Toeplitz’s paper is still of interest to mathematics educators who wish to use the history of mathematics in their teaching, for it suggests a rationale for studying history that does not trivialize history of mathematics and shows how history of mathematics can supply not only content for mathematics teaching but also, as Toeplitz is at pains to emphasize, a guide for examining pedagogical problems. At the same time, as we shall discuss in our paper, an attentive reading of Toeplitz’s paper brings out tensions and assumptions about mathematics, history of mathematics and historiography. TOEPLITZ’S LIFE IN MATHEMATICS, HISTORY OF MATHEMATICS, AND MATHEMATICS TEACHING Before starting our examination of the paper which is our focus in this paper, we ought to have some sense of who its author, Otto Toeplitz, was as an intellectual and educational figure. Toeplitz was born in Breslau, Germany (now, Wrocław, Poland) in 1881 and died in Jerusalem in 1940. His doctoral dissertation, Über Systeme von Formen, deren Funktionaldeterminante identisch verschwindet (On Systems of Forms whose Functional Determinant Vanishes Identically) was written under the direction of Jacob Rosanes and Friedrich Otto Rudolf Sturm at the University of Breslau in 1905. -
Jewish Mathematicians in German-Speaking Academic Culture
Opening of the exhibition Transcending Tradition: Jewish Mathematicians in German-Speaking Academic Culture Tel Aviv, 14 November 2011 Introduction to the Exhibition Moritz Epple Ladies and Gentlemen, Mathematics is a science that strives for universality. Humans have known how to calculate as long as they have known how to write, and mathematical knowledge has crossed boundaries between cultures and periods. Nevertheless, the historical conditions under which mathematics is pursued do change. Our exhibition is devoted to a period of dramatic changes in the culture of mathematics. Let me begin with a look back to summer 1904, when the International Congress of Mathematicians convened for the third time. The first of these congresses had been held in Zürich in 1897, the second in Paris in 1900. Now it was being organized in Germany for the first time, this time in the small city of Heidelberg in Germany’s south-west. 2 The congress was dedicated to the famous 19th-century mathematician Carl Gustav Jacobi, who had lived and worked in Königsberg. A commemorative talk on Jacobi was given by Leo Königsberger, the local organizer of the congress. The Göttingen mathematician Hermann Minkowski spoke about his recent work on the “Geometry of Numbers”. Arthur Schoenflies, who also worked in Göttingen and who a few years later would become the driving force behind Frankfurt’s new mathematical institute, gave a talk about perfect sets, thereby advancing the equally young theory of infinite sets. The Heidelberg scholar Moritz Cantor presented new results on the history of mathematics, and Max Simon, a specialist for mathematics education, discussed the mathematics of the Egyptians. -
Mathematics Meets Physics Relativity Theory, Functional Analysis Or the Application of Probabilistic Methods
The interaction between mathematics and physics was the Studien zur Entwicklung von Mathematik und Physik in ihren Wechselwirkungen topic of a conference held at the Saxon Academy of Science in Leipzig in March 2010. The fourteen talks of the conference ysics have been adapted for this book and give a colourful picture of various aspects of the complex and multifaceted relations. Karl-Heinz Schlote The articles mainly concentrate on the development of this Martina Schneider interrelation in the period from the beginning of the 19th century until the end of WW II, and deal in particular with the fundamental changes that are connected with such processes as the emergence of quantum theory, general Mathematics meets physics relativity theory, functional analysis or the application of probabilistic methods. Some philosophical and epistemo- A contribution to their interaction in the logical questions are also touched upon. The abundance of 19th and the first half of the 20 th century forms of the interaction between mathematics and physics is considered from different perspectives: local develop- ments at some universities, the role of individuals and/or research groups, and the processes of theory building. Mathematics meets ph The conference reader is in line with the bilingual character of the conference, with the introduction and nine articles presented in English, and five in German. K.-H. Schlote M. Schneider ISBN 978-3-8171-1844-1 Verlag Harri Deutsch Mathematics meets physics Verlag Harri Deutsch – Schlote, Schneider: Mathematics meets physics –(978-3-8171-1844-1) Studien zur Entwicklung von Mathematik und Physik in ihren Wechselwirkungen Die Entwicklung von Mathematik und Physik ist durch zahlreiche Verknüpfungen und wechselseitige Beeinflussungen gekennzeichnet. -
Corry L. David Hilbert and the Axiomatization of Physics, 1898-1918
Archimedes Volume 10 Archimedes NEW STUDIES IN THE HISTORY AND PHILOSOPHY OF SCIENCE AND TECHNOLOGY VOLUME 10 EDITOR JED Z. BUCHWALD, Dreyfuss Professor of History, California Institute of Technology, Pasadena, CA, USA. ADVISORY BOARD HENK BOS, University of Utrecht MORDECHAI FEINGOLD, Virginia Polytechnic Institute ALLAN D. FRANKLIN, University of Colorado at Boulder KOSTAS GAVROGLU, National Technical University of Athens ANTHONY GRAFTON, Princeton University FREDERIC L. HOLMES, Yale University PAUL HOYNINGEN-HUENE, University of Hannover EVELYN FOX KELLER, MIT TREVOR LEVERE, University of Toronto JESPER LÜTZEN, Copenhagen University WILLIAM NEWMAN, Harvard University JÜRGEN RENN, Max-Planck-Institut für Wissenschaftsgeschichte ALEX ROLAND, Duke University ALAN SHAPIRO, University of Minnesota NANCY SIRAISI, Hunter College of the City University of New York NOEL SWERDLOW, University of Chicago Archimedes has three fundamental goals; to further the integration of the histories of science and technology with one another: to investigate the technical, social and prac- tical histories of specific developments in science and technology; and finally, where possible and desirable, to bring the histories of science and technology into closer con- tact with the philosophy of science. To these ends, each volume will have its own theme and title and will be planned by one or more members of the Advisory Board in consultation with the editor. Although the volumes have specific themes, the series it- self will not be limited to one or even to a few particular areas. Its subjects include any of the sciences, ranging from biology through physics, all aspects of technology, bro- adly construed, as well as historically-engaged philosophy of science or technology. -
Why Toeplitz–Hankel? Motivations and Panorama
Cambridge University Press 978-1-107-19850-0 — Toeplitz Matrices and Operators Nikolaï Nikolski , Translated by Danièle Gibbons , Greg Gibbons Excerpt More Information 1 Why Toeplitz–Hankel? Motivations and Panorama Topics • Four cornerstones of the theory of Toeplitz operators: the Riemann–Hilbert problem (RHP), the singular integral operators (SIO), the Wiener–Hopf op- erators (WHO), and last (but not least) the Toeplitz matrices and operators (TMO) (strictly speaking, compressions of multiplication operators). • The founding contributions of Bernhard Riemann, David Hilbert, George Birkhoff, Otto Toeplitz, Gábor Szego,˝ Norbert Wiener, and Eberhard Hopf. • The modern and post-modern periods of the theory. Biographies Bernhard Riemann, Vito Volterra, David Hilbert, Henri Poincaré, Otto Toeplitz, Hermann and Marie Hankel. 1.1 Latent Maturation: The RHP and SIOs The most ancient form of a “Toeplitz problem,” which was not identified as such for a hundred years (!), is the Riemann, or Riemann–Hilbert, problem. 1.1.1 Nineteenth Century: Riemann and Volterra Bernhard Riemann submitted his thesis in 1851 (under the direction of Gauss) and presented his inaugural dissertation entitled “Grundlagen für eine allge- meine Theorie der Funktionen einer veränderlich complexen Grösse” (also available in [Riemann, 1876]). Its principal value lay in its pioneering intro- duction of geometrical methods to the theory of functions, and in the objects that we now know under the names of Riemann surfaces, conformal mappings, and variational techniques. Moreover, among the 22 sections of this 43-page text (in today’s format) there was a short Section 19 containing what is known 1 © in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-19850-0 — Toeplitz Matrices and Operators Nikolaï Nikolski , Translated by Danièle Gibbons , Greg Gibbons Excerpt More Information 2 Why Toeplitz–Hankel? Motivations and Panorama (following Hilbert) as the ‘Riemann problem,” one of the cornerstones of the future theory of Toeplitz operators. -
The Calculus: a Genetic Approach / Otto Toeplitz ; with a New Foreword by David M
THE CALCULUS THE CALCULUS A Genetic Approach OTTO TOEPLITZ New Foreword by David Bressoud Published in Association with the Mathematical Association of America The University of Chicago Press Chicago · London The present book is a translation, edited after the author's death by Gottfried Kothe and translated into English by LuiseLange. The German edition, DieEntwicklungder Infinitesimalrechnung,was published by Springer-Verlag. The University of Chicago Press,Chicago 60637 The University of Chicago Press,ltd., London © 1963 by The University of Chicago Foreword © 2007 by The University of Chicago All rights reserved. Published 2007 Printed in the United States of America 16 15 14 13 12 11 10 09 08 07 2 3 4 5 ISBN-13: 978-0-226-80668-6 (paper) ISBN-10: 0-226-80668-5 (paper) Library of Congress Cataloging-in-Publication Data Toeplitz, Otto, 1881-1940. [Entwicklung der Infinitesimalrechnung. English] The calculus: a genetic approach / Otto Toeplitz ; with a new foreword by David M. Bressoud. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-226-80668-6 (pbk. : alk. paper) ISBN-10: 0-226-80668-5 (pbk. : alk. paper) 1. Calculus. 2. Processes, Infinite. I. Title. QA303.T64152007 515-dc22 2006034201 § The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences-Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992. FOREWORDTO THECALCULUS: A GENETICAPPROACHBYOTTO TOEPLITZ September 30, 2006 Otto Toeplitz is best known for his contributions to mathematics, but he was also an avid student of its history. He understood how useful this history could be in in forming and shaping the pedagogy of mathematics. -
Traditions in Collision: the Emergence of Logical Empiricism Between the Riemannian and Helmholtzian Traditions
TRADITIONS IN COLLISION: THE EMERGENCE OF LOGICAL EMPIRICISM BETWEEN THE RIEMANNIAN AND HELMHOLTZIAN TRADITIONS Marco Giovanelli This article attempts to explain the emergence of the logical empiricist philosophy of space and time as a collision of mathematical traditions. The historical development of the “Riemannian” and “Helmholtzian” traditions in nineteenth-century mathematics is investigated. Whereas Helmholtz’s insistence on rigid bodies in geometry was developed group theoretically by Lie and philosophically by Poincaré, Riemann’s Habilitationsvotrag triggered Christoffel’s and Lipschitz’s work on quadratic differential forms, paving the way to Ricci’s absolute differential calculus. The transition from special to general rela- tivity is briefly sketched as a process of escaping from the Helmholtzian tradition and en- tering the Riemannian one. Early logical empiricist conventionalism, it is argued, emerges as the failed attempt to interpret Einstein’sreflections on rods and clocks in general rela- tivity through the conceptual resources of the Helmholtzian tradition. Einstein’sepiste- mology of geometry should, in spite of his rhetorical appeal to Helmholtz and Poincaré, be understood in the wake the Riemannian tradition and of its aftermath in the work of Levi-Civita, Weyl, Eddington, and others. 1. Introduction In an influential paper, John Norton (1999) has suggested that general relativity might be regarded as the result of a “collision of geometries” or, better, of geo- metrical strategies. On the one hand, there is Riemann’s “additive strategy,” which Contact Marco Giovanelli at Universität Tübingen, Forum Scientiarum, Doblerstraße 33, 72074 Tübingen, Germany ([email protected]). Except where noted, translations are by M. Giovanelli for this essay. -
OTTO TOEPLITZ: ALGEBRAIKER DER UNENDLICHEN MATRIZEN 3 Aus Königsberg Und Rademacher Aus Breslau
OTTO TOEPLITZ: ALGEBRAIKER DER UNENDLICHEN MATRIZEN STEFAN MÜLLER-STACH Zusammenfassung. Otto Toeplitz ist ein Mathematiker, dessen Schicksal ex- emplarisch für die Vernichtung der jüdischen wissenschaftlichen Elite in Deutsch- land durch die Nationalsozialisten ist. Sein Einfluss in der Mathematik ist noch heute, besonders durch den Begriff der Toeplitzmatrizen, deutlich spürbar. Die kulturgeschichtliche Bedeutung von Toeplitz ist ebenso groß. Als Gründungs- herausgeber von zwei Zeitschriften hat er sein Engagement für die Didaktik und die Wissenschaftsgeschichte untermauert. Wir geben einen Einblick in sein Schicksal und seine Gedankenwelt unter dem Einfluss von David Hilbert und Felix Klein. Einleitung Die Mathematischen Semesterberichte wurden 1932 von Heinrich Behnke und Otto Toeplitz gegründet. Heinrich Behnke mit seinen herausragenden Verdiensten um die Mathematikdidaktik, die Lehrerfortbildung, seinem mathematischen Œuvre und seiner in der Nachkriegszeit so einflussreichen Schule ist im Hinblick auf die Semesterberichte schon gewürdigt worden [10, 11]. Mit dem vorliegenden Aufsatz wollen wir nun stattdessen einmal Otto Toeplitz gerecht werden. Es soll insbesondere herausgearbeitet werden, wie nahe die Idee der Semester- berichte zur Gedankenwelt von Toeplitz und der von Behnke ist. Toeplitz, der aus Breslau stammte, wurde zunächst in Göttingen von David Hilbert und Felix Klein stark beeinflusst. Ab den 1920er Jahren entwickelte er aber vollkommen eigene Ideen, sowohl in seiner Forschung als auch im Bereich der Wissenschaftsgeschichte und der Hochschuldidaktik. Die beiden von ihm gegründeten Zeitschriften, seine historischen Seminare sowie seine Reden und Schriften zeigen seine Individualität und seine mittlerweile erreichte inhaltliche Distanz zu seinen beiden Vorbildern, mit denen er auch eine umfangreiche Korrespondenz unterhielt. Für Toeplitz bedeutete Deutschland die Erfüllung seiner wissenschaftlichen Träu- me. Durch die Nationalsozialisten wurde Toeplitz aus seinem Amt entfernt und emigrierte 1939 nach Jerusalem, wo er Anfang 1940 starb. -
The Enjoyment of Math
The Enjoyment of Math The Enjoyment of Math BY HANS RADEMACHER AND OTTO TOEPLITZ TRANSLATED BY HERBERT ZUCKERMAN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY Published 1957 by Princeton University Press, Princeton, New Jersey In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved Library of Congress Card No.: 57-627 ISBN 0-691-07958-7 (hardcover edition) ISBN 0-691-02351-4 (paperback edition) This is a translation from Von Zahlen und Figuren: Proben Mathematischen Denkens fUr Liebhaber der Mathematik, by Hans Rademacher and Otto Toeplitz, second edition originally published by Julius Springer, Berlin, 1933. Chapters 15 and 28 by Herbert Zuckerman have been added to the English language edition. Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources First Princeton Paperback edition, 1966 Second paperback and eighth hardcover printing, 1970 Seventh printing, for the Princeton Science Library, 1994 Printed in the United States of America 15 14 13 12 II 10 9 8 7 Preface Otto Toeplitz, co-author of this book, died in Jerusalem on Fe bruary 19, 1940, after having left Germany in the Spring of 1939. Toeplitz began his academic career in Gottingen as a disciple of David Hilbert, was then professor in Kiel and later in Bonn. His scientific work is centered around the theory of integral equations and the theory of functions of infinitely many variables, fields to which he has made lasting contributions. The plan for this book arose at frequent meetings which the authors had, while Toeplitz was in Kiel and I was at the University of Hamburg. -
A Bibliography of Publications By, and About, Max Born
A Bibliography of Publications by, and about, Max Born Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 07 June 2021 Version 1.125 Title word cross-reference 1=r [DL08]. 137 [Bor35b]. $2.50 [Mor54]. $3.80 [Kle70c]. $4.95 [Kle70b]. 2 [Bor18g, Pac10]. n log n [ADO11]. 1 [Lor10]. 14.99/$25.00 [Ber04]. 1801 [vS21]. 1911 [Meh75]. 1916-1955 [Mar73]. 1926 [Bor54a]. 1928 [HKP+32a, HKP+32b]. 1933 [Kr¨o98]. 1945 [Bor45b]. 1947 [Bor63-28]. 1951 [Bor63b]. 1955 [For70]. 1957 [BBF+57a]. 1970 [KS71, Lin98]. 1995 [Kr¨o98]. 20 [Goe88]. 24.80 [Kle70a]. 240pp [Ber04]. 3.85 [Bro72]. 37th [Bor53c]. 4 [RSS+07]. 480pp [Ber04]. 4th [Bor50b]. 1 2 50th [Bor59b, Bor63v]. 55 [BB69a]. 60th [Ber15, Sho20]. 7 [Sho20]. 7th [Sho20]. 80th [Ano63, CS79b, Ros79]. 978 [Sho20]. 978-1-108-47743-7 [Sho20]. ˆın [Bor69f]. Abbildungsfehler [Bor32c]. aberrations [Bor32c]. Abhandlungen [BB22a, BB63g, Bor63c, Bor63d, Seg64]. Ableitung [Bor10b]. Ablenkung [vS21]. Abraham [BvL23, BvL63]. Absolute [BS35, BL18b, Bor18d, BL63b]. absoluten [Bor18d]. absorbance [BL11]. absorbierenden [BL11]. absorbing [BL11]. absorption [Bor32d]. Absorptionsbanden [Bor32d]. Absorptionsverm¨ogen [BL11]. Achievement [Bor68a]. Achtzehn [BBF+57b, ABH+55, ABH+87]. achtzigsten [Hei62]. Activity [Bor35c, Bor15d, Bor15e, Bor22e, Bor36a, Bor63-51, Bor63-52]. Adiabatenprinzip [Bor26b, Bor63j]. Adiabatensatzes [BF28, BF63b]. adiabatic [Bor26b, BF63b, BF28, Bor63j]. Adsorption [BW63a, BF30, BW31a, BW31b, BF63c].