Numeracy Works for Life
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Annual Report 2007
4 THE CASH CHANGEOVER IN CYPRUS AND MALTA Six years after the successful introduction of the Monnaie de Paris minted 200.0 million euro coins euro banknotes and coins in 12 euro area countries with a face value of €56.1 million for Malta. at the same time and one year after Slovenia joined the Eurosystem, the euro banknotes and The Central Bank of Cyprus began frontloading coins were successfully introduced in Cyprus and euro coins to credit institutions on 22 October 2007; Malta on 1 January 2008. After a dual circulation the frontloading of euro banknotes began on period of one month, the euro had fully replaced 19 November 2007. Sub-frontloading to retailers the Cyprus pound and the Maltese lira. and the cash-operated machine industry started at the same time as the frontloading operation. As in the previous cash changeovers, timely and A widespread predistribution of euro coins was comprehensive preparation was crucial to the supported by making some 40,000 pre-packed smooth introduction of the euro banknotes and coin starter kits, worth €172 each, available to coins. Both Cyprus and Malta had established businesses and retailers from 3 December 2007. changeover plans at the national level long 250,000 mini-kits, worth CYP 10 each, went on before €-Day. These plans were based on, among sale to the general public on the same date. other things, the legal framework adopted by the Governing Council in July 2006 in relation to In Malta the frontloading of euro coins started certain preparations for the euro cash changeover in late September 2007, while the frontloading and the frontloading and sub-frontloading of euro of euro banknotes began in late October. -
Ethnomathematics and Education in Africa
Copyright ©2014 by Paulus Gerdes www.lulu.com http://www.lulu.com/spotlight/pgerdes 2 Paulus Gerdes Second edition: ISTEG Belo Horizonte Boane Mozambique 2014 3 First Edition (January 1995): Institutionen för Internationell Pedagogik (Institute of International Education) Stockholms Universitet (University of Stockholm) Report 97 Second Edition (January 2014): Instituto Superior de Tecnologias e Gestão (ISTEG) (Higher Institute for Technology and Management) Av. de Namaacha 188, Belo Horizonte, Boane, Mozambique Distributed by: www.lulu.com http://www.lulu.com/spotlight/pgerdes Author: Paulus Gerdes African Academy of Sciences & ISTEG, Mozambique C.P. 915, Maputo, Mozambique ([email protected]) Photograph on the front cover: Detail of a Tonga basket acquired, in January 2014, by the author in Inhambane, Mozambique 4 CONTENTS page Preface (2014) 11 Chapter 1: Introduction 13 Chapter 2: Ethnomathematical research: preparing a 19 response to a major challenge to mathematics education in Africa Societal and educational background 19 A major challenge to mathematics education 21 Ethnomathematics Research Project in Mozambique 23 Chapter 3: On the concept of ethnomathematics 29 Ethnographers on ethnoscience 29 Genesis of the concept of ethnomathematics among 31 mathematicians and mathematics teachers Concept, accent or movement? 34 Bibliography 39 Chapter 4: How to recognize hidden geometrical thinking: 45 a contribution to the development of an anthropology of mathematics Confrontation 45 Introduction 46 First example 47 Second example -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Leonhard Euler Moriam Yarrow
Leonhard Euler Moriam Yarrow Euler's Life Leonhard Euler was one of the greatest mathematician and phsysicist of all time for his many contributions to mathematics. His works have inspired and are the foundation for modern mathe- matics. Euler was born in Basel, Switzerland on April 15, 1707 AD by Paul Euler and Marguerite Brucker. He is the oldest of five children. Once, Euler was born his family moved from Basel to Riehen, where most of his childhood took place. From a very young age Euler had a niche for math because his father taught him the subject. At the age of thirteen he was sent to live with his grandmother, where he attended the University of Basel to receive his Master of Philosphy in 1723. While he attended the Universirty of Basel, he studied greek in hebrew to satisfy his father. His father wanted to prepare him for a career in the field of theology in order to become a pastor, but his friend Johann Bernouilli convinced Euler's father to allow his son to pursue a career in mathematics. Bernoulli saw the potentional in Euler after giving him lessons. Euler received a position at the Academy at Saint Petersburg as a professor from his friend, Daniel Bernoulli. He rose through the ranks very quickly. Once Daniel Bernoulli decided to leave his position as the director of the mathmatical department, Euler was promoted. While in Russia, Euler was greeted/ introduced to Christian Goldbach, who sparked Euler's interest in number theory. Euler was a man of many talents because in Russia he was learning russian, executed studies on navigation and ship design, cartography, and an examiner for the military cadet corps. -
On Mathematics As Sense- Making: an Informalattack on the Unfortunate Divorce of Formal and Informal Mathematics
On Mathematics as Sense Making: An InformalAttack On the Unfortunate Divorce of Formal and Informal 16 Mathematics Alan H. Schoenfeld The University of California-Berkeley This chapter explores the ways that mathematics is understood and used in our culture and the role that schooling plays in shaping those mathematical under standings. One of its goals is to blur the boundaries between forma] and informal mathematics: to indicate that, in real mathematical thinking, formal and informal reasoning are deeply intertwined. I begin, however, by briefly putting on the formalist's hat and defining formal reasoning. As any formalist will tell you, it helps to know what the boundaries are before you try to blur them. PROLOGUE: ON THE LIMITS AND PURITY OF FORMAL REASONING QUA FORMAL REASONING To get to the heart of the matter, formal systems do not denote; that is, formal systems in mathematics are not about anything. Formal systems consist of sets of symbols and rules for manipulating them. As long as you play by the rules, the results are valid within the system. But if you try to apply these systems to something from the real world (or something else), you no longer engage in formal reasoning. You are applying the mathematics at that point. Caveat emp tor: the end result is only as good as the application process. Perhaps the clearest example is geometry. Insofar as most people are con cerned, the plane or solid (Euclidean) geometry studied in secondary school provides a mathematical description of the world they live in and of the way the world must be. -
Goldbach's Conjecture
Irish Math. Soc. Bulletin Number 86, Winter 2020, 103{106 ISSN 0791-5578 Goldbach's Conjecture: if it's unprovable, it must be true PETER LYNCH Abstract. Goldbach's Conjecture is one of the best-known unsolved problems in mathematics. Over the past 280 years, many brilliant mathematicians have tried and failed to prove it. If a proof is found, it will likely involve some radically new idea or approach. If the conjecture is unprovable using the usual axioms of set theory, then it must be true. This is because, if a counter-example exists, it can be found by a finite search. In 1742, Christian Goldbach wrote a letter to Leonhard Euler proposing that every integer greater than 2 is the sum of three primes. Euler responded that this would follow from the simpler statement that every even integer greater than 2 is the sum of two primes. Goldbach's Conjecture is one of the best-known unsolved problems in mathematics. It is a simple matter to check the conjecture for the first few cases: 4=2+2 6=3+3 8=5+3 10 = 7 + 3 12 = 7 + 5 14 = 11 + 3 16 = 13 + 3 18 = 13 + 5 20 = 17 + 3 ········· But, with an infinite number of cases, this approach can never prove the conjecture. If, for an even number 2n we have 2n = p1 + p2 where p1 and p2 are primes, then obviously p + p n = 1 2 (1) 2 If every even number can be partitioned in this way, then any number, even or odd, must be the arithmetic average, or mean, of two primes. -
Mathematics in Informal Learning Environments: a Summary of the Literature
MATHEMATICS IN INFORMAL LEARNING ENVIRONMENTS: A SUMMARY OF THE LITERATURE Scott Pattison, Andee Rubin, Tracey Wright Updated March 2017 Research on mathematical reasoning and learning has long been a central part of the classroom and formal education literature (e.g., National Research Council, 2001, 2005). However, much less attention has been paid to how children and adults engage with and learn about math outside of school, including everyday settings and designed informal learning environments, such as interactive math exhibits in science centers. With the growing recognition of the importance of informal STEM education (National Research Council, 2009, 2015), researchers, educators, and policymakers are paying more attention to how these experiences might support mathematical thinking and learning and contribute to the broader goal of ensuring healthy, sustainable, economically vibrant communities in this increasingly STEM-rich world. To support these efforts, we conducted a literature review of research on mathematical thinking and learning outside the classroom, in everyday settings and designed informal learning environments. This work was part of the NSF-funded Math in the Making project, led by TERC and the Institute for Learning Innovation and designed to advance researchers’ and educators’ understanding of how to highlight and enhance the mathematics in making experiences.1 Recognizing that the successful integration of mathematics and making requires an understanding of how individuals engage with math in these informal learning environments, we gathered and synthesized the informal mathematics education literature, with the hope that findings would support the Math in the Making project and inform the work of mathematics researchers and educators more broadly. -
Effective Strategies for Promoting Informal Mathematics Learning
Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning Prepared for by Lynn Dierking Michael Dalton Jennifer Bachman Oregon State University 2008 with the generous support of This material is based upon work supported by the National Science Foundation under grant Number DRL-0714634. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the Oregon Museum of Science and Industry. © OREGON MUSEUM OF SCIENCE AND INDUSTRY, September 2011 Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning Prepared for the Oregon Museum of Science and Industry Oregon State University Lynn Dierking * Michael Dalton * Jennifer Bachman Executive Summary The Oregon Museum of Science and Industry (OMSI) received a five- year grant from the National Science Foundation (NSF) to develop the Access Algebra: Effective Strategies for Promoting Informal Mathematics Learning project. Access Algebra will include a traveling exhibition and a comprehensive professional development program for educational staff at host museums. Access Algebra will target middle- school age youth and engage visitors in design activities that promote creativity and innovation and build mathematics literacy by taking a non-traditional, experiential approach to mathematics learning. Oregon State University (OSU) faculty from the College of Education and the College of Science with expertise in mathematics teaching, administrative experience at K–20 levels and within the Oregon Department of Education, and in free-choice learning settings and research are collaborating on research and development for the exhibit prototypes and professional development components of the project. -
Goldbach, Christian
CHRISTIAN GOLDBACH (March 18, 1690 – November 20, 1764) by HEINZ KLAUS STRICK, Germany One of the most famous, still unproven conjectures of number theory is: • Every even number greater than 2 can be represented as the sum of two prime numbers. The scholar CHRISTIAN GOLDBACH made this simple mathematical statement to his pen pal LEONHARD EULER in 1742 as an assumption. (In the original version it said: Every natural number greater than 2 can be represented as the sum of three prime numbers, since at that time the number 1 was still considered a prime number.) All attempts to prove this theorem have so far failed. Even the offer of a prize of one million dollars hardly led to any progress. CHEN JINGRUN (1933-1996, Chinese stamp on the left), student of HUA LUOGENG (1910-1985, stamp on the right), the most important Chinese mathematician of the 20th century, succeeded in 1966 in making the "best approximation" to GOLDBACH's conjecture. CHEN JINGRUN was able to prove that any sufficiently large even number can be represented as the sum of a prime number and another number that has at most two prime factors. Among the first even numbers are those that have only one GOLDBACH decomposition: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 12 = 5 + 7. For larger even numbers one finds a "tendency" to increase the number of possibilities, but even then there is always a number that has only a few decompositions, such as 98 = 19 + 79 = 31 + 67 = 37 + 61. See the graph below and The On-Line Encyclopedia of Integer Sequences A045917. -
WINTER 2015 the Brick Architecture
B B R R I I C C K K B B U U L L L L E E T T I I N N The brick architecture of Kirkland Fraser Moor | First person: Alex Gordon of Jestico & Whiles Masonry masterpieces: 2015 Brick Awards | Satish Jassal Architects in London; SO-IL in New York WINTER 2015 Sutherland Hussey Harris in St Andrews | Acme’s prefabricated ‘pleated’ brick panels in Leeds 2 • BB WINTER 2015 BriCk Bulletin winter 2015 Contents 4 NEWS/FIRST PERSON New brick projects by Sergison Bates and Herzog &deMeurOn; First Person –Alex Gordon of Jestico &Whiles on brick’s ability to harmonise modern interventions with traditional contexts. 6 BRICK AWARDS 2015 Showcase of all 15 category winners. 12 PROJECTS Diego Arraigada Arquitectos, Alma-nac, Feilden Fowles, PollardThomas Edwards, Sutherland Hussey Harris, SO-IL,Make, Satish Jassal Architects, and TDO Architecture. 20 PROFILE David Kirkland discusses Kirkland Fraser Moor’s fascination with clay building products and vernacular design. 26 PRECEDENT Geraint Franklin on HKPA’sHouses for Visiting Mathematicians at the University of Warwick. 28 TECHNICAL The wave-likefacade of Marlies Rohmer’s Sportblok in Groningen, Holland, is constructed from brick-slips. 30 TECHNICAL Prefabricated ‘pleated’ brick panels articulate the exterior of amajor retail-led development in Leeds by Acme. extraordinary from the ordinary The ubiquity of brick means that it is all too easy to overlook its aesthetic qualities, performance benefits and historic importance. Not so David Kirkland of Kirkland Fraser Moor (p20-25), who marvels at the cleverness of being able to takeclay from the ground and, by way of making bricks, produce architecture. -
Vol 21, No 3, July
THE LINNEAN N e wsletter and Pr oceedings of THE LINNEAN SOCIETY OF LONDON Bur lington House , Piccadill y, London W1J 0BF VOLUME 21 • NUMBER 3 • JULY 2005 THE LINNEAN SOCIETY OF LONDON Burlington House, Piccadilly, London W1J 0BF Tel. (+44) (0)20 7434 4479; Fax: (+44) (0)20 7287 9364 e-mail: [email protected]; internet: www.linnean.org President Secretaries Council Professor Gordon McG Reid BOTANICAL The Officers and Dr John R Edmondson Dr Louise Allcock President-elect Prof John R Barnett Professor David F Cutler ZOOLOGICAL Prof Janet Browne Dr Vaughan R Southgate Dr J Sara Churchfield Vice-Presidents Dr John C David Professor Richard M Bateman EDITORIAL Prof Peter S Davis Dr Jenny M Edmonds Professor David F Cutler Dr Aljos Farjon Dr Vaughan R Southgate Dr Michael F Fay COLLECTIONS Dr D J Nicholas Hind Treasurer Mrs Susan Gove Dr Sandra D Knapp Professor Gren Ll Lucas OBE Dr D Tim J Littlewood Dr Keith N Maybury Executive Secretary Librarian & Archivist Dr Brian R Rosen Mr Adrian Thomas OBE Miss Gina Douglas Dr Roger A Sweeting Office/Facilities Manager Deputy Librarian Mr Dominic Clark Mrs Lynda Brooks Finance Library Assistant Conservator Mr Priya Nithianandan Mr Matthew Derrick Ms Janet Ashdown THE LINNEAN Newsletter and Proceedings of the Linnean Society of London Edited by B G Gardiner Editorial .................................................................................................................... 1 Society News ........................................................................................................... -
The Philosophy of Mathematics Education
The Philosophy of Mathematics Education The Philosophy of Mathematics Education Paul Ernest © Paul Ernest 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without permission in writing from the copyright holder and the Publisher. First published 1991 This edition published in the Taylor & Francis e-Library, 2004. RoutledgeFalmer is an imprint of the Taylor & Francis Group British Library Cataloguing in Publication Data Ernest, Paul The philosophy of mathematics education. 1. Education. Curriculum subjects: Mathematics. Teaching. I. Title 510.7 ISBN 0-203-49701-5 Master e-book ISBN ISBN 0-203-55790-5 (Adobe eReader Format) ISBN 1-85000-666-0 (Print Edition) ISBN 1-85000-667-9 pbk Library of Congress Cataloging-in-Publication Data is available on request Contents List of Tables and Figures viii Acknowledgments ix Introduction xi Rationale xi The Philosophy of Mathematics Education xii This Book xiv Part 1 The Philosophy of Mathematics 1 1 A Critique of Absolutist Philosophies of Mathematics 3 Introduction 3 The Philosophy of Mathematics 3 The Nature of Mathematical Knowledge 4 The Absolutist View of Mathematical Knowledge 7 The Fallacy of Absolutism 13 The Fallibilist Critique of Absolutism 15 The Fallibilist View 18 Conclusion 20 2 The Philosophy of Mathematics Reconceptualized 23 The Scope of the Philosophy of Mathematics 23 A Further Examination of Philosophical Schools 27 Quasi-empiricism