Physiological Concepts with Inbuilt Mathematics in Scratches on Ishango Bone by Upper Paleolithic People of Africa
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Migracijske Teme 4/1988
Migracijske teme 15 (1999), 1-2: 63-153 UDK: 809.45-0 Izvorni znanstveni rad Primljeno: 17. 11. 1998. Paolo Agostini University of Padova [email protected] LANGUAGE RECONSTRUCTION – APPLIED TO THE URALIC LANGUAGES* SUMMARY After pointing out the shortcomings and methodological weakness of the general theory of linguistic reconstruction, the author disputes the alleged antiquity of Uralic. Proto-Uralic as recon- structed by the scholars seems to be the sum of a set of features belonging to several distinct language families. The paper examines a number of lexical concordances with historically attested languages and comes to the conclusion that the Proto-Uralic word-stock is the result of a sum of borrowings that took place from the most disparate languages: Balto-Slavic, Old Swedish, several Turkic dialects, Mongolic, Tunguz, Aramaic, Hebrew, Arabic, late Middle Persian dialects, Byzantine Greek and Latin. Yet, other languages may also come into account: Chinese, Caucasian languages as well as lan- guages unknown in present day are possible candidates. A large number of bases of the Uralic word- stock can be easily identified by following a few phonological constraints. The linguistic features of the Uralic daughter-languages seem to show that they originated from a pidgin language spoken along the merchant routes that connected the Silk Road to North- and East-European trade. It is a well-known phenomenon that sometimes, when groups of people speaking different languages come into contact for the first time, a new restricted language system (lingua franca or pidgin) comes into being in order to cater to essential common needs. -
CULTURAL HERITAGE in MIGRATION Published Within the Project Cultural Heritage in Migration
CULTURAL HERITAGE IN MIGRATION Published within the project Cultural Heritage in Migration. Models of Consolidation and Institutionalization of the Bulgarian Communities Abroad funded by the Bulgarian National Science Fund © Nikolai Vukov, Lina Gergova, Tanya Matanova, Yana Gergova, editors, 2017 © Institute of Ethnology and Folklore Studies with Ethnographic Museum – BAS, 2017 © Paradigma Publishing House, 2017 ISBN 978-954-326-332-5 BULGARIAN ACADEMY OF SCIENCES INSTITUTE OF ETHNOLOGY AND FOLKLORE STUDIES WITH ETHNOGRAPHIC MUSEUM CULTURAL HERITAGE IN MIGRATION Edited by Nikolai Vukov, Lina Gergova Tanya Matanova, Yana Gergova Paradigma Sofia • 2017 CONTENTS EDITORIAL............................................................................................................................9 PART I: CULTURAL HERITAGE AS A PROCESS DISPLACEMENT – REPLACEMENT. REAL AND INTERNALIZED GEOGRAPHY IN THE PSYCHOLOGY OF MIGRATION............................................21 Slobodan Dan Paich THE RUSSIAN-LIPOVANS IN ITALY: PRESERVING CULTURAL AND RELIGIOUS HERITAGE IN MIGRATION.............................................................41 Nina Vlaskina CLASS AND RELIGION IN THE SHAPING OF TRADITION AMONG THE ISTANBUL-BASED ORTHODOX BULGARIANS...............................55 Magdalena Elchinova REPRESENTATIONS OF ‘COMPATRIOTISM’. THE SLOVAK DIASPORA POLITICS AS A TOOL FOR BUILDING AND CULTIVATING DIASPORA.............72 Natália Blahová FOLKLORE AS HERITAGE: THE EXPERIENCE OF BULGARIANS IN HUNGARY.......................................................................................................................88 -
The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars
Heavenly Mathematics: The Mathematics of the Chinese, Indian, Islamic and Gregorian Calendars Helmer Aslaksen Department of Mathematics National University of Singapore [email protected] www.math.nus.edu.sg/aslaksen/ www.chinesecalendar.net 1 Public Holidays There are 11 public holidays in Singapore. Three of them are secular. 1. New Year’s Day 2. Labour Day 3. National Day The remaining eight cultural, racial or reli- gious holidays consist of two Chinese, two Muslim, two Indian and two Christian. 2 Cultural, Racial or Religious Holidays 1. Chinese New Year and day after 2. Good Friday 3. Vesak Day 4. Deepavali 5. Christmas Day 6. Hari Raya Puasa 7. Hari Raya Haji Listed in order, except for the Muslim hol- idays, which can occur anytime during the year. Christmas Day falls on a fixed date, but all the others move. 3 A Quick Course in Astronomy The Earth revolves counterclockwise around the Sun in an elliptical orbit. The Earth ro- tates counterclockwise around an axis that is tilted 23.5 degrees. March equinox June December solstice solstice September equinox E E N S N S W W June equi Dec June equi Dec sol sol sol sol Beijing Singapore In the northern hemisphere, the day will be longest at the June solstice and shortest at the December solstice. At the two equinoxes day and night will be equally long. The equi- noxes and solstices are called the seasonal markers. 4 The Year The tropical year (or solar year) is the time from one March equinox to the next. The mean value is 365.2422 days. -
The Aztecs Gave the Tribe the Name Olmec
Mayan, Incan, and Aztec Civilizations: The Arrival of Man Alternate Version Download The Arrival of Man Crossing the Bering Strait Land Bridge Giant ice caps once covered both the Arctic and Antarc- tic regions of the earth. This was over 50,000 years ago. The levels of the oceans were lower than today. Much of the earth’s water was trapped in the polar ice caps. The lower water level showed a piece of land that connected Siberia to Alaska. To- day this area is once again under water. It is called the Bering Strait. Many scientists believe that early humans crossed over this land bridge. Then they began to spread out and settle in what is now North America. These people then moved into Central and South America. The Bering Strait land bridge was covered with water again when the ice caps thawed. This happened at the end of the Ice Age around 8,000 B.C. Today, we call the first people who settled in the West- ern Hemisphere Paleo-Indians. They are also called Paleo- Americans. Paleo is a prefix from the Greek language mean- As tribes migrated throughout North, ing “old.” The term Indian comes from the time of Columbus’ Central, and South America, they dis- voyages. He thought he had landed in India. Other names for covered agriculture and learned how to native people include Native Americans and First Nations. make stone tools and clay pottery. Each tribe or cultural group has its own name for its people. Hunting and Gathering The Paleo-Indians were hunters and gatherers. -
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 -
Beginnings of Counting and Numbers Tallies and Tokens
Beginnings of Counting and Numbers Tallies and Tokens Picture Link (http://www.flickr.com/photos/quadrofonic/834667550/) Bone Tallies • The Lebombo Bone is a portion of • The radius bone of a wolf, a baboon fibula, discovered in the discovered in Moravia, Border Cave in the Lebombo Czechoslovakia in 1937, and mountains of Swaziland. It dates dated to 30,000 years ago, has to about 35,000 years ago, and fifty‐five deep notches carved has 29 distinct notches. It is into it. Twenty‐five notches of assumed that it tallied the days of similar length, arranged in‐groups a lunar month. of five, followed by a single notch twice as long which appears to • terminate the series. Then Picture Link starting from the next notch, also (http://www.historyforkids.org/learn/africa/science/numbers.htm) twice as long, a new set of notches runs up to thirty. • Picture link (http://books.google.com/books?id=C0Wcb9c6c18C&pg=PA41&lpg=PA41 &dq=wolf+bone+moravia&source=bl&ots=1z5XhaJchP&sig=q_8WROQ1Gz l4‐6PYJ9uaaNHLhOM&hl=en&ei=J8D‐ TZSgIuPTiALxn4CEBQ&sa=X&oi=book_result&ct=result&resnum=4&ved=0 CCsQ6AEwAw) Ishango Bone • Ishango Bone, discovered in 1961 in central Africa. About 20,000 years old. Ishango Bone Patterns • Prime 11 13 17 19 numbers? • Doubling? 11 21 19 9 • Multiplication? • Who knows? 3 6 4 8 10 5 5 7 Lartet Bone • Discovered in Dodogne, France. About 30,000 years old. It has various markings that are neither decorative nor random (different sets are made with different tools, techniques, and stroke directions). -
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 -
Wood Identification and Chemistry' Covers the Physicalproperties and Structural Features of Hardwoods and Softwoods
11 DOCUMENT RESUME ED 031 555 VT 007 853 Woodworking Technology. San Diego State Coll., Calif. Dept. of Industrial Arts. Spons Agency-Office of Education (DHEA Washington, D.C. Pub Date Aug 68 Note-252p.; Materials developed at NDEA Inst. for Advanced Studyin Industrial Arts (San Diego, June 24 -Au9ust 2, 1968). EDRS Price MF -$1.00 He -$13.20 Descriptors-Curriculum Development, *Industrial Arts, Instructional Materials, Learning Activities, Lesson Plans, Lumber Industry, Resource Materials, *Resource Units, Summer Institutes, Teaching Codes, *Units of Study (Sublect Fields), *Woodworking Identifiers-*National Defense Education Act TitleXIInstitute, NDEA TitleXIInstitute, Woodworking Technology SIX teaching units which were developed by the 24 institute participantsare given. "Wood Identification and Chemistry' covers the physicalproperties and structural features of hardwoods and softwoods. "Seasoning" explainsair drying, kiln drying, and seven special lumber seasoning processes. "Researchon Laminates" describes the bending of solid wood and wood laminates, beam lamination, lamination adhesives,. andplasticlaminates."Particleboard:ATeachingUnitexplains particleboard manufacturing and the several classes of particleboard and theiruses. "Lumber Merchandising" outhnes lumber grades andsome wood byproducts. "A Teaching Unitin Physical Testing of Joints, Finishes, Adhesives, and Fasterners" describes tests of four common edge pints, finishes, wood adhesives, and wood screws Each of these units includes a bibhography, glossary, and student exercises (EM) M 55, ...k.",z<ONR; z _: , , . "'zr ss\ ss s:Ts s , s' !, , , , zs "" z' s: - 55 Ts 5. , -5, 5,5 . 5, :5,5, s s``s ss ' ,,, 4 ;.< ,s ssA 11111.116; \ ss s, : , \s, s's \ , , 's's \ sz z, ;.:4 1;y: SS lza'itVs."4,z ...':',\\Z'z.,'I,,\ "t"-...,,, `,. -
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. -
The Indian Luni-Solar Calendar and the Concept of Adhik-Maas
Volume -3, Issue-3, July 2013 The Indian Luni-Solar Calendar and the giving rise to alternative periods of light and darkness. All human and animal life has evolved accordingly, Concept of Adhik-Maas (Extra-Month) keeping awake during the day-light but sleeping through the dark nights. Even plants follow a daily rhythm. Of Introduction: course some crafty beings have turned nocturnal to take The Hindu calendar is basically a lunar calendar and is advantage of the darkness, e.g., the beasts of prey, blood– based on the cycles of the Moon. In a purely lunar sucker mosquitoes, thieves and burglars, and of course calendar - like the Islamic calendar - months move astronomers. forward by about 11 days every solar year. But the Hindu calendar, which is actually luni-solar, tries to fit together The next natural clock in terms of importance is the the cycle of lunar months and the solar year in a single revolution of the Earth around the Sun. Early humans framework, by adding adhik-maas every 2-3 years. The noticed that over a certain period of time, the seasons concept of Adhik-Maas is unique to the traditional Hindu changed, following a fixed pattern. Near the tropics - for lunar calendars. For example, in 2012 calendar, there instance, over most of India - the hot summer gives way were 13 months with an Adhik-Maas falling between to rain, which in turn is followed by a cool winter. th th August 18 and September 16 . Further away from the equator, there were four distinct seasons - spring, summer, autumn, winter. -
Tally Sticks, Counting Boards, and Sumerian Proto-Writing John Alan Halloran
Early Numeration - John Alan Halloran - August 10, 2009 - Page 1 Early Numeration - Tally Sticks, Counting Boards, and Sumerian Proto-Writing John Alan Halloran http://www.sumerian.org/ Work on the published version of my Sumerian Lexicon (Logogram Publishing: 2006) has revealed that: 1) the early Sumerians used wooden tally sticks for counting; 2) tally marks led to proto-writing; 3) the Sumerian pictograms for goats and sheep probably derive from tally stick notch conventions; 4) the Uruk period civilization used counting boards; 5) the authors of the proto-cuneiform tablets drew with animal claws or bird talons; 6) the historical Sumerians used clay split tallies as credit instruments; and 7) the Sumerians may sometimes have counted by placing knots or stones on a string. 1. Counting with Wooden Tally Sticks 1.1. The following text caught my attention. It is from The Debate between Sheep and Grain, ETCSL 5.3.2, lines 130-133: "Every night your count is made and your tally-stick put into the ground, so your herdsman can tell people how many ewes there are and how many young lambs, and how many goats and how many young kids." The Sumerian reads: 130 ud šu2-uš-e niñ2-kas7-zu ba-ni-ak-e ñiš 131 ŠID-ma-zu ki i3-tag-tag-ge - the Unicode version has ñiš-šudum- ma-zu 132 na-gada-zu u8 me-a sila4 tur-tur me-a 133 ud5 me-a maš2 tur-tur me-a lu2 mu-un-na-ab-be2 š is read |sh| and ñ is read |ng|.