DOCSLIB.ORG

Building the Foundation: Whole Numbers in the Primary Grades

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Building the Foundation: Whole Numbers in the Primary Grades New ICMI Study Series Maria G. Bartolini Bussi Xu Hua Sun Editors Building the Foundation: Whole Numbers in the Primary Grades The 23rd ICMI Study New ICMI Study Series Published under the auspices of the International Commission on Mathematical Instruction (ICMI) Series Editors: Jill Adler, ICMI President Ferdinando Arzarello, ICMI Past President Abraham Arcavi, ICMI Secretary General Information on the ICMI Study programme and on the resulting publications can be obtained at the ICMI website http://www.mathunion.org/ICMI/ or by contacting the ICMI Secretary General, whose email address is available on that website. More information about this series at http://www.springer.com/series/6351 Maria G. Bartolini Bussi • Xu Hua Sun Editors Building the Foundation: Whole Numbers in the Primary Grades The 23rd ICMI Study Editors Maria G. Bartolini Bussi Xu Hua Sun Department of Education and Humanities Faculty of Education University of Modena and Reggio Emilia University of Macau Modena, Italy Macao, China ISSN 1387-6872 ISSN 2215-1745 (electronic) New ICMI Study Series ISBN 978-3-319-63554-5 ISBN 978-3-319-63555-2 (eBook) https://doi.org/10.1007/978-3-319-63555-2 Library of Congress Control Number: 2017960195 © The Editor(s) (if applicable) and The Author(s) 2018. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Foreword The International Commission on Mathematical Instruction (ICMI) was established at the Fourth International Congress of Mathematicians held in Rome in 1908. It was initiated to support active interests in school education which were widespread among mathematicians at the time. ICMI is crucial for the International Mathematical Union (IMU), because education and research cannot be separated from each other. ICMI and the IMU function together for mathematics like the two wheels of a cart, since ICMI develops education systems which enable mathematics to prevail in society, while the IMU contributes to society through the development of pure and applied mathematical sciences. I have been serving the IMU as President since January 2015. Since I served the IMU previously (1995–2002), I have noticed that ICMI and the IMU are working together to establish collaborations given that mathematics education is a major preoccupation of most scientific organisations nowadays. My first physical involvement with ICMI as IMU President started with my par- ticipation in the ICMI Study 23 Conference in Macao in May/June 2015. The ICMI Study 23 was planned and run jointly by its local and foreign co-chairs, with support from the University of Macau. The International Program Committee (IPC) meeting of ICMI Study 23 in Berlin was also supported by the IMU Secretariat. I saw that the support and cooperation of ICMI and the IMU have been essential throughout these activities. I was very pleased to learn that ICMI Study 23 addressed, for the first time, mathematics teaching and learning in primary school (and pre-school as well) for all, and I believe that it will have a larger impact for later mathematics knowing. I hope the volume supports the whole of mathematics education. Kyoto University Shigefumi Mori Kyoto, Japan v Preface I am particularly proud of the publication of this 23rd volume in the series of ICMI Studies, not only since this means that a long scientific and organisational work has been happily accomplished, but also for the outstanding quality of its content and for the absolute relevance of the theme. ICMI Study 23 fully realises the objectives of ICMI ‘to offer a forum for promoting reflection, collaboration and the exchange and dissemination of ideas on the teaching and learning of mathematics from pri- mary to university level’. Concentrating on whole numbers in primary grades repre- sents a relevant signal of interest for a crucial moment in educational programmes everywhere in the world. Never as in this case it is truer that the study addresses a theme of particular significance to contemporary mathematics education. The content of the volume is in perfect consonance with the overall ICMI pro- grammes, according to which ‘ICMI works to stimulate the creation, improvement and dissemination of recent research findings and of the available resources for instruction (e.g curricular materials, pedagogical methods, the appropriate use of technology, etc.). The objective is of providing links among educational research- ers, curriculum designers, educational policy makers, teachers of mathematics, mathematicians, mathematics educators and others interested in mathematical edu- cation around the world’. The people of this study have worked on a project that is challenging both scien- tifically and culturally: the topics in the chapters and in the panel reports of the book, the commentaries on them written by eminent scholars, and the two appendi- ces face a large horizon of themes that go well beyond mathematics and show how focusing on the learning and teaching of whole numbers is an immensely demand- ing task that requires a wide range of competencies in addition to mathematics, from linguistics to ethnomathematics, to neuroscience and more. The processes according to which kids learn and elaborate whole numbers and their properties are incredibly rich and intermingled with the culture where they live and with which they can speak and think, as well as with the artefacts, which the tradition of their countries or the most recent technology allows them to use. vii viii Preface The book builds, in this sense, a real-world map of whole number arithmetic: even if it is far from being complete, it does cover many regions of the world, from East to West, from North to South, including many non-affluent countries. The con- tributions from the different cultures illustrate the fascinating enterprise of the mathematics teachers at the primary level, who from the one side speak the univer- sal language of mathematics, but from the other side can link it to the specific lan- guage and cultural environment of their own countries, in order to make it accessible to their students. This amazing synthesis emerges clearly from the research and practice described in the volume, which has the not-common capacity of intertwin- ing the rigour of mathematics, linguistics, cognitive sciences, etc. with the extraor- dinary different ways according to which numbers are alive in the different regions. It is particularly significant that the study and the volume also had the contribu- tions of invited people from another relevant IMU-ICMI programme, the ‘Capacity and Networking Project (CANP)’, aimed to enhance mathematics education at all levels in developing countries so that their people are capable of meeting the chal- lenges these countries face. Their inputs have been important, since they empha- sised the problems and the specificities of teaching whole numbers in those countries and contributed to the richness and variety of voices in the volume. The 536 pages of the book demonstrate the incredibly intense work of this study, which lasted almost five years, from the appointment of the International Program Committee and its two co-chairs at the end of 2012, to the preparation and organisa- tion of its Conference, which was held in Macao in June 2015, to the last intense work for the preparation and editing of this book, which, as it is usual for the ICMI Study volumes, does not consist of the proceedings of the meeting, but is a further elaboration of the discussions and results reached during the meeting itself. I followed all these phases, and I must say that without the incredible work of the IPC and particularly of the two co-chairs, Profs. Maria G. Bartolini Bussi and Xu Hua Sun, we could not have now so nice a book, which is really a reference and a source of inspiration for theory, research and practice to all the community of researchers, practitioners and policymakers in mathematics education, especially, but not only, those interested in mathematical education at the primary level. On behalf of the ICMI EC and of the ICMI larger family, I wish to thank all of them here for their remarkable work.
Load more

Recommended publications
  • Numbers - the Essence
    Numbers - the essence A computer consists in essence of a sequence of electrical on/off switches. Big question: How do switches become a computer? ITEC 1000 Introduction to Information Technologies 1 Friday, September 25, 2009 • A single switch is called a bit • A sequence of 8 bits is called a byte • Two or more bytes are often grouped to form word • Modern laptop/desktop standardly will have memory of 1 gigabyte = 1,000,000,000 bytes ITEC 1000 Introduction to Information Technologies 2 Friday, September 25, 2009 Question: How do encode information with switches ? Answer: With numbers - binary numbers using only 0’s and 1’s where for each switch on position = 1 off position = 0 ITEC 1000 Introduction to Information Technologies 3 Friday, September 25, 2009 • For a byte consisting of 8 bits there are a total of 28 = 256 ways to place 0’s and 1’s. Example 0 1 0 1 1 0 1 1 • For each grouping of 4 bits there are 24 = 16 ways to place 0’s and 1’s. • For a byte the first and second groupings of four bits can be represented by two hexadecimal digits • this will be made clear in following slides. ITEC 1000 Introduction to Information Technologies 4 Friday, September 25, 2009 Bits & Bytes imply creation of data and computer analysis of data • storing data (reading) • processing data - arithmetic operations, evaluations & comparisons • output of results ITEC 1000 Introduction to Information Technologies 5 Friday, September 25, 2009 Numeral systems A numeral system is any method of symbolically representing the counting numbers 1, 2, 3, 4, 5, .
    [Show full text]
  • On the Origin of the Indian Brahma Alphabet
    - ON THE <)|{I<; IN <>F TIIK INDIAN BRAHMA ALPHABET GEORG BtfHLKi; SECOND REVISED EDITION OF INDIAN STUDIES, NO III. TOGETHER WITH TWO APPENDICES ON THE OKU; IN OF THE KHAROSTHI ALPHABET AND OF THK SO-CALLED LETTER-NUMERALS OF THE BRAHMI. WITH TIIKKK PLATES. STRASSBUKi-. K A K 1. I. 1 1M I: \ I I; 1898. I'lintccl liy Adolf Ilcil/.haiisi'ii, Vicniiii. Preface to the Second Edition. .As the few separate copies of the Indian Studies No. Ill, struck off in 1895, were sold very soon and rather numerous requests for additional ones were addressed both to me and to the bookseller of the Imperial Academy, Messrs. Carl Gerold's Sohn, I asked the Academy for permission to issue a second edition, which Mr. Karl J. Trlibner had consented to publish. My petition was readily granted. In addition Messrs, von Holder, the publishers of the Wiener Zeitschrift fur die Kunde des Morgenlandes, kindly allowed me to reprint my article on the origin of the Kharosthi, which had appeared in vol. IX of that Journal and is now given in Appendix I. To these two sections I have added, in Appendix II, a brief review of the arguments for Dr. Burnell's hypothesis, which derives the so-called letter- numerals or numerical symbols of the Brahma alphabet from the ancient Egyptian numeral signs, together with a third com- parative table, in order to include in this volume all those points, which require fuller discussion, and in order to make it a serviceable companion to the palaeography of the Grund- riss.
    [Show full text]
  • Kūnqǔ in Practice: a Case Study
    KŪNQǓ IN PRACTICE: A CASE STUDY A DISSERTATION SUBMITTED TO THE GRADUATE DIVISION OF THE UNIVERSITY OF HAWAI‘I AT MĀNOA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THEATRE OCTOBER 2019 By Ju-Hua Wei Dissertation Committee: Elizabeth A. Wichmann-Walczak, Chairperson Lurana Donnels O’Malley Kirstin A. Pauka Cathryn H. Clayton Shana J. Brown Keywords: kunqu, kunju, opera, performance, text, music, creation, practice, Wei Liangfu © 2019, Ju-Hua Wei ii ACKNOWLEDGEMENTS I wish to express my gratitude to the individuals who helped me in completion of my dissertation and on my journey of exploring the world of theatre and music: Shén Fúqìng 沈福庆 (1933-2013), for being a thoughtful teacher and a father figure. He taught me the spirit of jīngjù and demonstrated the ultimate fine art of jīngjù music and singing. He was an inspiration to all of us who learned from him. And to his spouse, Zhāng Qìnglán 张庆兰, for her motherly love during my jīngjù research in Nánjīng 南京. Sūn Jiàn’ān 孙建安, for being a great mentor to me, bringing me along on all occasions, introducing me to the production team which initiated the project for my dissertation, attending the kūnqǔ performances in which he was involved, meeting his kūnqǔ expert friends, listening to his music lessons, and more; anything which he thought might benefit my understanding of all aspects of kūnqǔ. I am grateful for all his support and his profound knowledge of kūnqǔ music composition. Wichmann-Walczak, Elizabeth, for her years of endeavor producing jīngjù productions in the US.
    [Show full text]
  • PRINTEMPS 2018 5 Mot De L’Éditeur Et Président De La SHFQ
    MERCI À NOTRE PARTENAIRE 4- HISTOIRES FORESTIÈRES PRINTEMPS 2018 5 Mot de l’éditeur et président de la SHFQ La revue Histoires forestières du Québec consacre cette édition à l’innovation dans l’industrie québécoise des produits forestiers. L’histoire de l’industrie des produits forestiers s’établit sur un continuum d’innovations. En effet, au fil des périodes et des besoins changeants, notre industrie manufacturière des produits du bois s’est adaptée et elle a fait preuve de génie et d’audace. L’innovation se retrouve au sein de l’industrie des pâtes et papiers, des panneaux à base de bois, de procédés comme les structures lamellées-collées et de l’émergence de produits forestiers à valeur ajoutée. L’innovation, c’est l’ensemble du processus qui se déroule depuis la naissance d’une idée jusqu’à sa matérialisation. La foresterie scientifique se situe donc au cœur même de l’innovation, les articles traitant du Centre de recherche et développement de Cascades et du programme de baccalauréat en sciences du bois nous présentent entre autres cette composante importante de l’innovation. En entrevue, Monsieur Gilbert Tardif nous parle de sa carrière et de sa vision de l’industrie sous le signe de l’innovation. Un autre article présente l’histoire des Auger, une famille de marchands de bois s’étendant sur quatre générations et exerçant leurs activités jusqu’au début des années 1980, après plus de cent ans de commerce du bois. La revue Histoires forestières du Québec poursuivra la même thématique l’automne prochain, cette fois en traitant de l’innovation en aménagement et approvisionnement forestiers.
    [Show full text]
  • The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes
    Portland State University PDXScholar Mathematics and Statistics Faculty Fariborz Maseeh Department of Mathematics Publications and Presentations and Statistics 3-2018 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hu Sun University of Macau Christine Chambris Université de Cergy-Pontoise Judy Sayers Stockholm University Man Keung Siu University of Hong Kong Jason Cooper Weizmann Institute of Science SeeFollow next this page and for additional additional works authors at: https:/ /pdxscholar.library.pdx.edu/mth_fac Part of the Science and Mathematics Education Commons Let us know how access to this document benefits ou.y Citation Details Sun X.H. et al. (2018) The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes. In: Bartolini Bussi M., Sun X. (eds) Building the Foundation: Whole Numbers in the Primary Grades. New ICMI Study Series. Springer, Cham This Book Chapter is brought to you for free and open access. It has been accepted for inclusion in Mathematics and Statistics Faculty Publications and Presentations by an authorized administrator of PDXScholar. Please contact us if we can make this document more accessible: [email protected]. Authors Xu Hu Sun, Christine Chambris, Judy Sayers, Man Keung Siu, Jason Cooper, Jean-Luc Dorier, Sarah Inés González de Lora Sued, Eva Thanheiser, Nadia Azrou, Lynn McGarvey, Catherine Houdement, and Lisser Rye Ejersbo This book chapter is available at PDXScholar: https://pdxscholar.library.pdx.edu/mth_fac/253 Chapter 5 The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes Xu Hua Sun , Christine Chambris Judy Sayers, Man Keung Siu, Jason Cooper , Jean-Luc Dorier , Sarah Inés González de Lora Sued , Eva Thanheiser , Nadia Azrou , Lynn McGarvey , Catherine Houdement , and Lisser Rye Ejersbo 5.1 Introduction Mathematics learning and teaching are deeply embedded in history, language and culture (e.g.
    [Show full text]
  • Proquest Dissertations
    University of Alberta Qin Jiushao and His Mathematical Treatise in Nine Sections in Thirteenth-Century China by Ke-Xin Au Yong A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Master of Arts in History History and Classics ©Ke-Xin Au Yong Fall 2011 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de ('edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-81281-5 Our file Notre reference ISBN: 978-0-494-81281-5 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par I'lnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non­ support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats.
    [Show full text]
  • Deltas in the Anthropocene Edited by Robert J
    Deltas in the Anthropocene Edited by Robert J. Nicholls · W. Neil Adger Craig W. Hutton · Susan E. Hanson Deltas in the Anthropocene Robert J. Nicholls · W. Neil Adger · Craig W. Hutton · Susan E. Hanson Editors Deltas in the Anthropocene Editors Robert J. Nicholls W. Neil Adger School of Engineering Geography, College of Life University of Southampton and Environmental Sciences Southampton, UK University of Exeter Exeter, UK Craig W. Hutton GeoData Institute, Geography Susan E. Hanson and Environmental Science School of Engineering University of Southampton University of Southampton Southampton, UK Southampton, UK ISBN 978-3-030-23516-1 ISBN 978-3-030-23517-8 (eBook) https://doi.org/10.1007/978-3-030-23517-8 © Te Editor(s) (if applicable) and Te Author(s), under exclusive license to Springer Nature Switzerland AG, part of Springer Nature 2020. Tis book is an open access publication. Open Access Tis book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. Te images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
    [Show full text]
  • Datatype Defining Rewrite Systems for Naturals and Integers
    UvA-DARE (Digital Academic Repository) Datatype defining rewrite systems for naturals and integers Bergstra, J.A.; Ponse, A. DOI 10.23638/LMCS-17(1:17)2021 Publication date 2021 Document Version Final published version Published in Logical Methods in Computer Science License CC BY Link to publication Citation for published version (APA): Bergstra, J. A., & Ponse, A. (2021). Datatype defining rewrite systems for naturals and integers. Logical Methods in Computer Science, 17(1), [17]. https://doi.org/10.23638/LMCS- 17(1:17)2021 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:30 Sep 2021 Logical Methods in Computer Science Volume 17, Issue 1, 2021, pp. 17:1–17:31 Submitted Jan. 15, 2020 https://lmcs.episciences.org/ Published Feb.
    [Show full text]
  • Experimental Methods and Instrumentation for Chemical Engineers Experimental Methods and Instrumentation for Chemical Engineers
    Experimental Methods and Instrumentation for Chemical Engineers Experimental Methods and Instrumentation for Chemical Engineers Gregory S. Patience AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands First edition 2013 Copyright © 2013 Elsevier B.V. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in ­evaluating and using any information, methods, compounds, or experiments
    [Show full text]
  • Annual Report 2006 the Most Make Natural Elements, We of Their Energy }
    boralex 2006 Annual Report natural elements, wind, we make the most water of their energy and fire} 2006 Annual Report Wind, water, fire, natural elements, we natural elements, fire, water, Wind, make the most of their energy. www.boralex.com General Information Profile Head office Transfer agent and registrar Boralex inc. Computershare Investor Services Inc. 36 Lajeunesse Street 1500 University Street, Suite 700 Boralex Inc. (“Boralex” or the “Corporation”) is a major private electricity Kingsey Falls, Quebec J0A 1B0 Montreal, Quebec H3A 3S8 producer whose core business is the development and operation of power Canada Canada Telephone: (819) 363-5860 Telephone: stations that run on renewable energy. Employing close to 300 people in Fax: (819) 363-5866 1-800-564-6253 / (514) 982-7888 Québec, the northeastern United States and France, the Corporation owns Fax: 1-888-453-0330 / (514) 982-7635 Business office [email protected] and operates 21 power stations with a combined installed capacity of Boralex inc. 770 Sherbrooke Street West Shareholder information 333 megawatts (“MW”). Boralex is distinguished by its leading expertise Montreal, Quebec H3A 1G1 The annual Meeting of Shareholders and long experience in four types of power generation: Canada will be held at 11:00 a.m., Wednesday, Telephone: (514) 284-9890 May 16, 2007 at the: Fax: (514) 284-9895 Centre Mont-Royal >> Over the past five years, Boralex has become >> Boralex is the largest producer of wood- In addition to its own power stations, Boralex Website Room Mont-Royal I one of the biggest and most experienced wind residue energy in North America, operating six manages 10 power stations for the Boralex www.boralex.com 4th floor power producers in France, where it currently thermal power stations in the states of Maine Power Income Fund (the “Fund”), in which 2200, Mansfield Street operates six wind farms with a total installed and New York with a total installed capacity it holds a 23% interest.
    [Show full text]
  • Lecture 2: Arithmetic
    E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010-2017 Lecture 2: Arithmetic The oldest mathematical discipline is arithmetic. It is the theory of the construction and manip- ulation of numbers. The earliest steps were done by Babylonian, Egyptian, Chinese, Indian and Greek thinkers. Building up the number system starts with the natural numbers 1; 2; 3; 4::: which can be added and multiplied. Addition is natural: join 3 sticks to 5 sticks to get 8 sticks. Multiplication ∗ is more subtle: 3 ∗ 4 means to take 3 copies of 4 and get 4 + 4 + 4 = 12 while 4 ∗ 3 means to take 4 copies of 3 to get 3 + 3 + 3 + 3 = 12. The first factor counts the number of operations while the second factor counts the objects. To motivate 3 ∗ 4 = 4 ∗ 3, spacial insight motivates to arrange the 12 objects in a rectangle. This commutativity axiom will be carried over to larger number systems. Realizing an addition and multiplicative structure on the natural numbers requires to define 0 and 1. It leads naturally to more general numbers. There are two major motivations to to build new numbers: we want to 1. invert operations and still get results. 2. solve equations. To find an additive inverse of 3 means solving x + 3 = 0. The answer is a negative number. To solve x ∗ 3 = 1, we get to a rational number x = 1=3. To solve x2 = 2 one need to escape to real numbers. To solve x2 = −2 requires complex numbers. Numbers Operation to complete Examples of equations to solve Natural numbers addition and multiplication 5 + x = 9 Positive fractions addition and
    [Show full text]
  • Arabic Numeral
    CHAPTER 4 Number Representation and Calculation Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 1 4.4 Looking Back at Early Numeration Systems Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 2 Objectives 1. Understand and use the Egyptian system. 2. Understand and use the Roman system. 3. Understand and use the traditional Chinese system. 4. Understand and use the Ionic Greek system. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 3 The Egyptian Numeration System The Egyptians used the oldest numeration system called hieroglyphic notation. Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 4 Example: Using the Egyptian Numeration System Write the following numeral as a Hindu-Arabic numeral: Solution: Using the table, find the value of each of the Egyptian numerals. Then add them. 1,000,000 + 10,000 + 10,000 + 10 + 10 + 10 + 1 + 1 + 1 + 1 = 1,020,034 Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 5 Example: Using the Egyptian Numeration System Write 1752 as an Egyptian numeral. Solution: First break down the Hindu-Arabic numeral into quantities that match the Egyptian numerals: 1752 = 1000 + 700 + 50 + 2 = 1000 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 Now use the table to find the Egyptian symbol that matches each quantity. Thus, 1752 can be expressed as Copyright © 2015, 2011, 2007 Pearson Education, Inc. Section 4.4, Slide 6 The Roman Numeration System Roman I V X L C D M Numeral Hindu- 1 5 10 50 100 500 1000 Arabic Numeral The Roman numerals were used until the eighteenth century and are still commonly used today for outlining, on clocks, and in numbering some pages in books.
    [Show full text]