The First Africans: African Archaeology from the Earliest Tool Makers to Most Recent Foragers Lawrence Barham and Peter Mitchell Index More Information
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The Qadan, the Jebel Sahaba Cemetery and the Lithic
PL ISSN 0066–5924 INDEX 32597x (VAT 8%) DOI: 10.23858/APa58.2020 Institute of Archaeology and Ethnology Polish Academy of Sciences Archaeologia Polona vol. 58: 2020 Special theme: PREHistorY OF North-EAST AFRICA Volume dedicated to Prof. Michał Kobusiewicz on his 80th birthday Archaeologia Polona Volume 58 : 2020 Editor: DAGMARA H. WERRA Associate Editor MARZENA WOŹNY International Advisory Prof. PETER BOGUCKI – Princeton University, USA Board: Prof. MARTIN GOJDA – University of Western Bohemia, Pilsen, Czech Republic Prof. ANTHONY HARDING – University of Exeter, Great Britain Prof. MARIA IACOVOU – University of Cyprus, Nicosia, Cyprus Prof. CAROLA METZNER-NEBELSICK – University of Munich, Germany Prof. BJØRNAR OLSEN – University of Tromsø, Norway Prof. GUIDO VANNINI – University of Florence, Italy Editors of the volume: PRZEMYSłAW Bobrowski and MirosłAW MASOJć Linguistic consultation: PAUL BARFORD Cover illustration: MACIEJ JÓRDECZKA Reviewers: Mirosław Furmanek (Wroclaw), Elena Garcea (Cassino), Maria Gatto (Leicester), Bolesław Ginter (Cracow), Tomasz Herbich (Warsaw), Karla Kroeper (Berlin), Alice Leplongeon (Leuven), Maria Kaczmarek (Poznan), Andrea Manzo (Naples), Arkadiusz Marciniak (Poznan), Henryk Paner (Gdansk), Tomasz Płonka (Wroclaw), Włodzimierz Rączkowski (Poznan), Andrzej Rozwadowski (Poznan), Jiří Svoboda (Brno), Philip Van Peer (Leuven), András Zboray (Budapest). Editorial office: Institute of Archaeology and Ethnology Polish Academy of Sciences, 00-140 Warsaw; Al. Solidarności 105 Tel. (4822) 6202881 ext. 154; Fax. (4822) 624 01 00 E-mail: [email protected] Website: http://journals.iaepan.pl/apolona DOI: 10.23858/APa DOI: 10.23858/APa58.2020 Archaeologia Polona Copyright © 2020 by Institute of Archaeology and Ethnology Polish Academy of Sciences. All articles in this volume are published in an open access under the CC BY 4.0 license (https://creativecommons.org/licenses/by/4.0/). -
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 -
Mathematics in African History and Cultures
Paulus Gerdes & Ahmed Djebbar MATHEMATICS IN AFRICAN HISTORY AND CULTURES: AN ANNOTATED BIBLIOGRAPHY African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) Mathematics in African History and Cultures Second edition, 2007 First edition: African Mathematical Union, Cape Town, South Africa, 2004 ISBN: 978-1-4303-1537-7 Published by Lulu. Copyright © 2007 by Paulus Gerdes & Ahmed Djebbar Authors Paulus Gerdes Research Centre for Mathematics, Culture and Education, C.P. 915, Maputo, Mozambique E-mail: [email protected] Ahmed Djebbar Département de mathématiques, Bt. M 2, Université de Lille 1, 59655 Villeneuve D’Asq Cedex, France E-mail: [email protected], [email protected] Cover design inspired by a pattern on a mat woven in the 19th century by a Yombe woman from the Lower Congo area (Cf. GER-04b, p. 96). 2 Table of contents page Preface by the President of the African 7 Mathematical Union (Prof. Jan Persens) Introduction 9 Introduction to the new edition 14 Bibliography A 15 B 43 C 65 D 77 E 105 F 115 G 121 H 162 I 173 J 179 K 182 L 194 M 207 N 223 O 228 P 234 R 241 S 252 T 274 U 281 V 283 3 Mathematics in African History and Cultures page W 290 Y 296 Z 298 Appendices 1 On mathematicians of African descent / 307 Diaspora 2 Publications by Africans on the History of 313 Mathematics outside Africa (including reviews of these publications) 3 On Time-reckoning and Astronomy in 317 African History and Cultures 4 String figures in Africa 338 5 Examples of other Mathematical Books and 343 -
Lake Turkana and the Lower Omo the Arid and Semi-Arid Lands Account for 50% of Kenya’S Livestock Production (Snyder, 2006)
Lake Turkana & the Lower Omo: Hydrological Impacts of Major Dam & Irrigation Development REPORT African Studies Centre Sean Avery (BSc., PhD., C.Eng., C. Env.) © Antonella865 | Dreamstime © Antonella865 Consultant’s email: [email protected] Web: www.watres.com LAKE TURKANA & THE LOWER OMO: HYDROLOGICAL IMPACTS OF MAJOR DAM & IRRIGATION DEVELOPMENTS CONTENTS – VOLUME I REPORT Chapter Description Page EXECUTIVE(SUMMARY ..................................................................................................................................1! 1! INTRODUCTION .................................................................................................................................... 12! 1.1! THE(CONTEXT ........................................................................................................................................ 12! 1.2! THE(ASSIGNMENT .................................................................................................................................. 14! 1.3! METHODOLOGY...................................................................................................................................... 15! 2! DEVELOPMENT(PLANNING(IN(THE(OMO(BASIN ......................................................................... 18! 2.1! INTRODUCTION(AND(SUMMARY(OVERVIEW(OF(FINDINGS................................................................... 18! 2.2! OMO?GIBE(BASIN(MASTER(PLAN(STUDY,(DECEMBER(1996..............................................................19! 2.2.1! OMO'GIBE!BASIN!MASTER!PLAN!'!TERMS!OF!REFERENCE...........................................................................19! -
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today
Writing the History of Mathematics: Interpretations of the Mathematics of the Past and Its Relation to the Mathematics of Today Johanna Pejlare and Kajsa Bråting Contents Introduction.................................................................. 2 Traces of Mathematics of the First Humans........................................ 3 History of Ancient Mathematics: The First Written Sources........................... 6 History of Mathematics or Heritage of Mathematics?................................. 9 Further Views of the Past and Its Relation to the Present.............................. 15 Can History Be Recapitulated or Does Culture Matter?............................... 19 Concluding Remarks........................................................... 24 Cross-References.............................................................. 24 References................................................................... 24 Abstract In the present chapter, interpretations of the mathematics of the past are problematized, based on examples such as archeological artifacts, as well as written sources from the ancient Egyptian, Babylonian, and Greek civilizations. The distinction between history and heritage is considered in relation to Euler’s function concept, Cauchy’s sum theorem, and the Unguru debate. Also, the distinction between the historical past and the practical past,aswellasthe distinction between the historical and the nonhistorical relations to the past, are made concrete based on Torricelli’s result on an infinitely long solid from -
Lukenya Hill, Kenya)
Late Pleistocene age and archaeological context for the hominin calvaria from GvJm-22 (Lukenya Hill, Kenya) Christian A. Tryona,1, Isabelle Crevecoeurb, J. Tyler Faithc, Ravid Ekshtaina, Joelle Nivensd, David Pattersone, Emma N. Mbuaf, and Fred Spoorg,h aDepartment of Anthropology, Harvard University, Cambridge, MA 02138; bUnité Mixte de Recherche 5199, de la Préhistoire à l’Actuel: Culture, Environnement, et Anthropologie, Centre National de la Recherche Scientifique, Université de Bordeaux, 33615 Talence, France; cArchaeology Program, School of Social Science, University of Queensland, Brisbane, QLD 4072, Australia; dDepartment of Anthropology, New York University, New York, NY 10003; eCenter for the Advanced Study of Hominid Paleobiology, Department of Anthropology, The George Washington University, Washington, DC 20052; fNational Museums of Kenya, Nairobi, Kenya 00100; gDepartment of Human Evolution, Max Planck Institute for Evolutionary Anthropology, D-04103, Leipzig, Germany; and hDepartment of Cell and Developmental Biology, University College London, WC1E 6BT London, United Kingdom Edited by Erik Trinkaus, Washington University, St. Louis, MO, and approved January 16, 2015 (received for review September 19, 2014) Kenya National Museums Lukenya Hill Hominid 1 (KNM-LH 1) is Hominid 1 (KNM-LH 1) partial calvaria from site GvJm-22 at a Homo sapiens partial calvaria from site GvJm-22 at Lukenya Hill, Lukenya Hill, Kenya, the only eastern African fossil hominin Kenya, associated with Later Stone Age (LSA) archaeological de- from a Last Glacial Maximum [LGM; 19–26.4 kya (19)] LSA posits. KNM-LH 1 is securely dated to the Late Pleistocene, and archaeological context. We construct a revised accelerator mass samples a time and region important for understanding the origins spectrometry (AMS) radiocarbon chronology built on 26 new of modern human diversity. -
The Growth of Population in the Province of the Western Cape
Southern Africa Labour and Development Research Unit A Tapestry of People: The Growth of Population in the Province of the Western Cape by Dudley Horner and Francis Wilson WORKING PAPER SERIES Number 21 About the Authors and Acknowledgments Professor Francis Wilson and Dudley Horner are both SALDRU Honorary Research Fellows and were previously respectively director and deputy-director of the research unit. We acknowledge with thanks the Directorate for Social Research & Provincial Population in the Department of Social Development within the Provincial Government of the Western Cape, and particularly Mr Gavin Miller and Dr Ravayi Marindo, who commissioned this study as part of the project on the state of population in the Western Cape Province. We thank, too, Mrs Brenda Adams and Mrs Alison Siljeur for all their assistance with the production of this report. While we have endeavoured to make this historical overview as accurate as possible we would welcome any comments suggesting appropriate amendments or corrections. Recommended citation Horner, D. and Wilson, F. (2008) E A Tapestry of People: The Growth of Population in the Province of the Western Cape. A Southern Africa Labour and Development Research Unit Working Paper Number 21. Cape Town: SALDRU, University of Cape Town ISBN: 978-0-9814123-2-0 © Southern Africa Labour and Development Research Unit, UCT, 2008 Working Papers can be downloaded in Adobe Acrobat format from www.saldru.uct.ac.za. Printed copies of Working Papers are available for R15.00 each plus vat and postage charges. Contact: Francis Wilson - [email protected] Dudley Horner - [email protected] Orders may be directed to: The Administrative Officer, SALDRU, University of Cape Town, Private Bag, Rondebosch, 7701, Tel: (021) 650 5696, Fax: (021) 650 5697, Email: [email protected] A Tapestry of People: The Growth of Population in the Province of the Western Cape by Dudley Horner & Francis Wilson Long Before Van Riebeeck. -
Biography of Richard G. Klein BIOGRAPHY
Biography of Richard G. Klein BIOGRAPHY s long as 160,000 years ago, and the Cold War was a very real thing. people who looked like mod- There wasn’t much exchange between ern humans roamed Africa. Russian scientists and those in the A For more than 100,000 years, United States or France.’’ these populations remained small in Howell advised Klein to prepare for number and were confined largely to working in Russia by studying Neander- Africa. Approximately 50,000 years ago, thal artifacts in France. After complet- despite no apparent physical change, a ing his master’s degree in 1964, Klein subset of these people dramatically al- went to the University of Bordeaux to tered their behavior, producing the first study with Franc¸oisBordes, a renowned artifacts unequivocally deemed to be professor of prehistory. Bordes had jewelry and inventing new technologies, worked out a system of 62 categories for such as projectile weapons, that allowed classifying Neanderthal tools (2). ‘‘His them to fish and to hunt dangerous work had an enormous influence on prey. Over the next 15,000 years, these me,’’ Klein said. Stone Age hunter–gatherers spread Bordes took Klein to see sites that from Africa to Europe, and, wherever had been occupied first by the Neander- they appeared, the Neanderthals, who thals and then by the Cro-Magnons, in- previously inhabited Europe, rapidly cluding La Quina and La Ferrassie disappeared. caves in southwest France. The Cro- For the last 35 years, Stanford paleo- Richard G. Klein Magnon layers lay directly on top of the anthropologist Richard G. -
Curriculum, Assessment, and Instruction Dr
May 2018 Curriculum, Assessment, and Instruction Dr. Fatima Morrell, Assistant Superintendent Our Vision: Through a commitment to equity and excellence, all students will receive a rigorous instructional program which prepares them to compete successfully, and contribute responsibly in a global society. Assistant Superintendent Highlights “Mathematics is Everywhere” Take a look around and math is everywhere. From finding the best deal while online shopping to calculating our students’ grades to cooking tonight’s dinner. When we think about math we tend to think about numbers and operations, fractions and decimals, standard algorithms and complicated formulas. But when was the last time you stopped to think about the stories behind the math? The who, the what, the where, the when? Stories connect us. They help us to create a deeper understanding and discover where we fit into the big picture. Have your students ever asked, or maybe you, yourself have wondered, “Why does this matter? When am I ever going to use this?” It’s these questions that show us how critical it is for students to understand both the history of math and the future of math. There are so many people that label the way students today are learning math as the “new way of doing math.” But how “new” is this math? The Ishango bone from Ancient Africa is one of the earliest artifacts of arithmetic, dating back 20,000 years ago. Many similarities can be found when comparing our base-10 number system to the base-20 number system Ancient Mayans used. Using shells to represent zero, dots to represent ones, and sticks to represent a groups of five, Mayans were able to calculate mathematical problems such as the length of the solar year. -
Proceedings Chapter
Proceedings Chapter Kaddanarti, a lower pleistocene assemblage from Northern Sudan CHAIX, Louis, et al. Abstract Un assemblage datable du Pléistocène inférieur du nord du Soudan est présenté ici. A côté des restes de mammifères, particulièrement Palaeoloxodon recki, ont été retrouvé plusieurs outils attribuables à l'Oldowayen. La datation entre 1.6 Ma et 0.5 Ka, indique une occupation très ancienne de cette portion de la vallée du Nil. Reference CHAIX, Louis, et al. Kaddanarti, a lower pleistocene assemblage from Northern Sudan. In: Recent research into the stone age of Northeastern Africa. Poznań : Poznań archaeological museum, 2000. p. 33-46 Available at: http://archive-ouverte.unige.ch/unige:106520 Disclaimer: layout of this document may differ from the published version. 1 / 1 Recent Research Into the Stone Age of Northeastem Africa Sûrdies in African ArchaeologY 7 PoznaÉ Archaeological Museum 2000 Louis Chaix, Martine Faure, Claude Guerin and Mathieu Honegger Kaddanarti, a Lower Pleistocene assemblage from Northern Sudan Résumé Un assemblage datable du Pléistocène inférieur du nord du Soudan est présenté ici. A côté de restes de mammifères, particulièrement Palaeoloxodon recki, ont été retrouvé plusieurs outils attribuables à I'Oldowayen. La datation, entre 1.6 Ma et 0.5 Ka, indique une occupation très ancienne de cette portion de la vallée du Nil. lntroduction During the winter of 1991, the archaeozoologist of the Swiss Archaeologi- cal Mission in Kerma (LC), was alerted by Mr. Hassan Ibrahim who found during a trip along the Nile near his birth place some large fossilized bones which he brought to us. After an initial examination, this material seemed to us very ancient and interesting enough to be more closely examined. -
Human Origin Sites and the World Heritage Convention in Eurasia
World Heritage papers41 HEADWORLD HERITAGES 4 Human Origin Sites and the World Heritage Convention in Eurasia VOLUME I In support of UNESCO’s 70th Anniversary Celebrations United Nations [ Cultural Organization Human Origin Sites and the World Heritage Convention in Eurasia Nuria Sanz, Editor General Coordinator of HEADS Programme on Human Evolution HEADS 4 VOLUME I Published in 2015 by the United Nations Educational, Scientific and Cultural Organization, 7, place de Fontenoy, 75352 Paris 07 SP, France and the UNESCO Office in Mexico, Presidente Masaryk 526, Polanco, Miguel Hidalgo, 11550 Ciudad de Mexico, D.F., Mexico. © UNESCO 2015 ISBN 978-92-3-100107-9 This publication is available in Open Access under the Attribution-ShareAlike 3.0 IGO (CC-BY-SA 3.0 IGO) license (http://creativecommons.org/licenses/by-sa/3.0/igo/). By using the content of this publication, the users accept to be bound by the terms of use of the UNESCO Open Access Repository (http://www.unesco.org/open-access/terms-use-ccbysa-en). The designations employed and the presentation of material throughout this publication do not imply the expression of any opinion whatsoever on the part of UNESCO concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. The ideas and opinions expressed in this publication are those of the authors; they are not necessarily those of UNESCO and do not commit the Organization. Cover Photos: Top: Hohle Fels excavation. © Harry Vetter bottom (from left to right): Petroglyphs from Sikachi-Alyan rock art site. -
Introduction 7 the Ishango Bone 10 210 Holy Days 11 210 Frustrum 12
The Josephus Problem 69 239 The Miner 104 256 The Explorer’s Problem 70 239 The Blind Abbot 105 256 Monkey Nuts 71 240 The Captive Queen 106 257 The Book of Precious Things 72 240 A Spiral Walk 107 257 A Medieval Riddle 73 241 Eight Queens 108 258 Introduction 7 The Luo River Scroll 39 224 The Mariner 74 241 The Dinner Party 109 258 The Ishango Bone 10 210 Buridan’s Ass 40 224 The Memory Wheel 75 242 The Monkey and the Pulley 110 259 Holy Days 11 210 Hi Shi’s Third Paradox 41 225 Jia Xian’s Triangle 76 242 Kirkman’s Schoolgirls 111 259 Frustrum 12 211 The Zero Proof 42 225 The Old One 77 243 The Counterfeit Bill 112 260 Triangles of Babylon 13 211 Crocodile Tears 43 226 The Trouble With Rabbits 78 243 The Travelling Salesman 113 260 Ahmes' Loaves 14 212 The Ladder of Horus 44 226 The Ring Game 79 244 Ethiopian Mathematics 114 261 As I was going to Amenemhet III's 15 212 The Sieve of Eratosthenes 45 227 The Well 80 244 Cantor’s Infinities 115 261 A Question of Quantity 16 213 Archimedes’ Revenge 46 228 Tartaglia’s Wine 81 245 Nobody 116 262 A Fractional Issue 17 213 The Nine Chapters 47 229 Topsy-Turvy 82 245 Tesseract 117 262 Strong Grain 18 214 The Cistern Problem 48 229 The Wanderer 83 246 Bertrand’s Box 118 262 Progressive Loaves 19 214 Dog and Hare 49 230 The Hound 84 246 Nothing Lost 119 263 Dates 20 215 The Chickens 50 230 Regiomontanus’ Angle 85 246 Hilbert’s Hotel 120 263 The Rule of Three 21 215 Leg and Thigh 51 231 The Problem of Points 86 247 Wine/Water Problem 121 264 Progressive Shares 22 216 Men Buy a Horse 52 231 Modesty