The Calculus: a Genetic Approach / Otto Toeplitz ; with a New Foreword by David M
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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. -
On the Standard Lengths of Angle Bisectors and the Angle Bisector Theorem
Global Journal of Advanced Research on Classical and Modern Geometries ISSN: 2284-5569, pp.15-27 ON THE STANDARD LENGTHS OF ANGLE BISECTORS AND THE ANGLE BISECTOR THEOREM G.W INDIKA SHAMEERA AMARASINGHE ABSTRACT. In this paper the author unveils several alternative proofs for the standard lengths of Angle Bisectors and Angle Bisector Theorem in any triangle, along with some new useful derivatives of them. 2010 Mathematical Subject Classification: 97G40 Keywords and phrases: Angle Bisector theorem, Parallel lines, Pythagoras Theorem, Similar triangles. 1. INTRODUCTION In this paper the author introduces alternative proofs for the standard length of An- gle Bisectors and the Angle Bisector Theorem in classical Euclidean Plane Geometry, on a concise elementary format while promoting the significance of them by acquainting some prominent generalized side length ratios within any two distinct triangles existed with some certain correlations of their corresponding angles, as new lemmas. Within this paper 8 new alternative proofs are exposed by the author on the angle bisection, 3 new proofs each for the lengths of the Angle Bisectors by various perspectives with also 5 new proofs for the Angle Bisector Theorem. 1.1. The Standard Length of the Angle Bisector Date: 1 February 2012 . 15 G.W Indika Shameera Amarasinghe The length of the angle bisector of a standard triangle such as AD in figure 1.1 is AD2 = AB · AC − BD · DC, or AD2 = bc 1 − (a2/(b + c)2) according to the standard notation of a triangle as it was initially proved by an extension of the angle bisector up to the circumcircle of the triangle. -
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. -
Advanced Euclidean Geometry
Advanced Euclidean Geometry Paul Yiu Summer 2016 Department of Mathematics Florida Atlantic University July 18, 2016 Summer 2016 Contents 1 Some Basic Theorems 101 1.1 The Pythagorean Theorem . ............................ 101 1.2 Constructions of geometric mean . ........................ 104 1.3 The golden ratio . .......................... 106 1.3.1 The regular pentagon . ............................ 106 1.4 Basic construction principles ............................ 108 1.4.1 Perpendicular bisector locus . ....................... 108 1.4.2 Angle bisector locus . ............................ 109 1.4.3 Tangency of circles . ......................... 110 1.4.4 Construction of tangents of a circle . ............... 110 1.5 The intersecting chords theorem ........................... 112 1.6 Ptolemy’s theorem . ................................. 114 2 The laws of sines and cosines 115 2.1 The law of sines . ................................ 115 2.2 The orthocenter ................................... 116 2.3 The law of cosines .................................. 117 2.4 The centroid ..................................... 120 2.5 The angle bisector theorem . ............................ 121 2.5.1 The lengths of the bisectors . ........................ 121 2.6 The circle of Apollonius . ............................ 123 3 The tritangent circles 125 3.1 The incircle ..................................... 125 3.2 Euler’s formula . ................................ 128 3.3 Steiner’s porism ................................... 129 3.4 The excircles .................................... -
4 3 E B a C F 2
See-Saw Geometry and the Method of Mass-Points1 Bobby Hanson February 27, 2008 Give me a place to stand on, and I can move the Earth. — Archimedes. 1. Motivating Problems Today we are going to explore a kind of geometry similar to regular Euclidean Geometry, involving points and lines and triangles, etc. The main difference that we will see is that we are going to give mass to the points in our geometry. We will call such a geometry, See-Saw Geometry (we’ll understand why, shortly). Before going into the details of See-Saw Geometry, let’s look at some problems that might be solved using this different geometry. Note that these problems are perfectly solvable using regular Euclidean Geometry, but we will find See-Saw Geometry to be very fast and effective. Problem 1. Below is the triangle △ABC. Side BC is divided by D in a ratio of 5 : 2, and AB is divided by E in a ration of 3 : 4. Find the ratios in which F divides the line segments AD and CE; i.e., find AF : F D and CF : F E. (Note: in Figure 1, below, only the ratios are shown; the actual lengths are unknown). B 3 E 5 4 F D 2 A C Figure 1 1My notes are shamelessly stolen from notes by Tom Rike, of the Berkeley Math Circle available at http://mathcircle.berkeley.edu/archivedocs/2007 2008/lectures/0708lecturespdf/MassPointsBMC07.pdf . 1 2 Problem 2. In Figure 2, below, D and E divide sides BC and AB, respectively, as before. -
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. -
Icons of Mathematics an EXPLORATION of TWENTY KEY IMAGES Claudi Alsina and Roger B
AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 45 Icons of Mathematics AN EXPLORATION OF TWENTY KEY IMAGES Claudi Alsina and Roger B. Nelsen i i “MABK018-FM” — 2011/5/16 — 19:53 — page i — #1 i i 10.1090/dol/045 Icons of Mathematics An Exploration of Twenty Key Images i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page ii — #2 i i c 2011 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2011923441 Print ISBN 978-0-88385-352-8 Electronic ISBN 978-0-88385-986-5 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER FORTY-FIVE Icons of Mathematics An Exploration of Twenty Key Images Claudi Alsina Universitat Politecnica` de Catalunya Roger B. Nelsen Lewis & Clark College Published and Distributed by The Mathematical Association of America i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Books Frank Farris, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Jeremy S. Case Rosalie A. Dance Tevian Dray Thomas M. Halverson Patricia B. Humphrey Michael J. McAsey Michael J. Mossinghoff Jonathan Rogness Thomas Q. Sibley i i i i i i “MABK018-FM” — 2011/5/16 — 19:53 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As- sociation of America was established through a generous gift to the Association from Mary P. -
Foundations of Euclidean Constructive Geometry
FOUNDATIONS OF EUCLIDEAN CONSTRUCTIVE GEOMETRY MICHAEL BEESON Abstract. Euclidean geometry, as presented by Euclid, consists of straightedge-and- compass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions” to “constructive mathe- matics” leads to the development of a first-order theory ECG of “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. Using Gentzen’s cut-elimination theorem, we show that when ECG proves an existential theorem, then the things proved to exist can be constructed by Eu- clidean ruler-and-compass constructions. In the second part of the paper we take up the formal relationships between three versions of Euclid’s parallel postulate: Euclid’s own formulation in his Postulate 5, Playfair’s 1795 version, which is the one usually used in modern axiomatizations, and the version used in ECG. We completely settle the questions about which versions imply which others using only constructive logic: ECG’s version im- plies Euclid 5, which implies Playfair, and none of the reverse implications are provable. The proofs use Kripke models based on carefully constructed rings of real-valued functions. “Points” in these models are real-valued functions. We also characterize these theories in terms of different constructive versions of the axioms for Euclidean fields.1 Contents 1. Introduction 5 1.1. Euclid 5 1.2. The collapsible vs. the rigid compass 7 1.3. Postulates vs. axioms in Euclid 9 1.4. The parallel postulate 9 1.5. Polygons in Euclid 10 2. -
The Postulates of Neutral Geometry Axiom 1 (The Set Postulate). Every
1 The Postulates of Neutral Geometry Axiom 1 (The Set Postulate). Every line is a set of points, and the collection of all points forms a set P called the plane. Axiom 2 (The Existence Postulate). There exist at least two distinct points. Axiom 3 (The Incidence Postulate). For every pair of distinct points P and Q, there exists exactly one line ` such that both P and Q lie on `. Axiom 4 (The Distance Postulate). For every pair of points P and Q, the distance from P to Q, denoted by P Q, is a nonnegative real number determined uniquely by P and Q. Axiom 5 (The Ruler Postulate). For every line `, there is a bijective function f : ` R with the property that for any two points P, Q `, we have → ∈ P Q = f(Q) f(P ) . | − | Any function with these properties is called a coordinate function for `. Axiom 6 (The Plane Separation Postulate). If ` is a line, the sides of ` are two disjoint, nonempty sets of points whose union is the set of all points not on `. If P and Q are distinct points not on `, then both of the following equivalent conditions are satisfied: (i) P and Q are on the same side of ` if and only if P Q ` = ∅. ∩ (ii) P and Q are on opposite sides of ` if and only if P Q ` = ∅. ∩ 6 Axiom 7 (The Angle Measure Postulate). For every angle ∠ABC, the measure of ∠ABC, denoted by µ∠ABC, isa real number strictly between 0 and 180, determined uniquely by ∠ABC. Axiom 8 (The Protractor Postulate). -
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 Steiner-Lehmus Theorem an Honors Thesis
The Steiner-Lehmus Theorem An Honors ThesIs (ID 499) by BrIan J. Cline Dr. Hubert J. LudwIg Ball State UniverSity Muncie, Indiana July 1989 August 19, 1989 r! A The Steiner-Lehmus Theorem ; '(!;.~ ,", by . \:-- ; Brian J. Cline ProposItion: Any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite sIde) is isosceles. (Steiner-Lehmus Theorem) ·That's easy for· you to say!- These beIng the possible words of a suspicIous mathematicIan after listenIng to some assumIng person state the Steiner-Lehmus Theorem. To be sure, the Steiner-Lehmus Theorem Is ·sImply stated, but notoriously difficult to prove."[ll Its converse, the bIsectors of the base angles of an Isosceles trIangle are equal, Is dated back to the tIme of EuclId and is easy to prove. The Steiner-Lehmus Theorem appears as if a proof would be simple, but it is defInItely not.[2l The proposition was sent by C. L. Lehmus to the Swiss-German geometry genIus Jacob SteIner in 1840 with a request for a pure geometrIcal proof. The proof that Steiner gave was fairly complex. Consequently, many inspIred people began searchIng for easier methods. Papers on the Steiner-Lehmus Theorem were prInted in various Journals in 1842, 1844, 1848, almost every year from 1854 untIl 1864, and as a frequent occurence during the next hundred years.[Sl In terms of fame, Lehmus dId not receive nearly as much as SteIner. In fact, the only tIme the name Lehmus Is mentIoned In the lIterature Is when the title of the theorem is gIven. HIs name would have been completely forgotten if he had not sent the theorem to Steiner. -
View This Volume's Front and Back Matter
Continuou s Symmetr y Fro m Eucli d to Klei n This page intentionally left blank http://dx.doi.org/10.1090/mbk/047 Continuou s Symmetr y Fro m Eucli d to Klei n Willia m Barke r Roge r How e >AMS AMERICAN MATHEMATICA L SOCIET Y Freehand® i s a registere d trademar k o f Adob e System s Incorporate d in th e Unite d State s and/o r othe r countries . Mathematica® i s a registere d trademar k o f Wolfra m Research , Inc . 2000 Mathematics Subject Classification. Primar y 51-01 , 20-01 . For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mbk-47 Library o f Congres s Cataloging-in-Publicatio n Dat a Barker, William . Continuous symmetr y : fro m Eucli d t o Klei n / Willia m Barker , Roge r Howe . p. cm . Includes bibliographica l reference s an d index . ISBN-13: 978-0-8218-3900- 3 (alk . paper ) ISBN-10: 0-8218-3900- 4 (alk . paper ) 1. Geometry, Plane . 2 . Group theory . 3 . Symmetry groups . I . Howe , Roger , 1945 - QA455.H84 200 7 516.22—dc22 200706079 5 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapter fo r us e in teachin g o r research .