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African Mathematics African Mathematics From Bones to Computers Mamokgethi Setati Abdul Karim Bangura UNIVERSITY PRESS OF AMERICA,® INC. Lanham • Boulder • New York • Toronto • Plymouth, UK 10_535_Setati.indb i 9/7/10 5:04 PM Copyright © 2011 by University Press of America,® Inc. 4501 Forbes Boulevard Suite 200 Lanham, Maryland 20706 UPA Acquisitions Department (301) 459-3366 Estover Road Plymouth PL6 7PY United Kingdom All rights reserved Printed in the United States of America British Library Cataloging in Publication Information Available Library of Congress Control Number: 2010934250 ISBN: 978-0-7618-5348-0 (paperback : alk. paper) eISBN: 978-0-7618-5349-7 ™ The paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z39.48-1992 10_535_Setati.indb ii 9/7/10 5:04 PM To the Afrikan teacher and learner! 10_535_Setati.indb iii 9/7/10 5:04 PM 10_535_Setati.indb iv 9/7/10 5:04 PM Contents 1 General Introduction 1 2 Beginnings: Mathematics of Bones 11 3 Geometry South of the Sahara 16 4 Numbers 33 5 Beginnings of Written Mathematics: Egypt 41 6 The Maghrebian Tradition 50 7 Combinatorics and African Applications 60 8 Vector Calculus and African Applications 84 9 The Fourier Transform and African Applications 96 10 Mathematical Tiling/Tessellation and African Applications 111 11 Bifurcations and African Applications 120 12 Fractals 145 13 African-centered Automated Generation of Metadata 156 14 General Conclusion: Access to Mathematics versus Access to the Language of Power: Lessons from the Struggle in South African Multilingual Mathematics Classrooms 188 Bibliography 203 v 10_535_Setati.indb v 9/7/10 5:04 PM 10_535_Setati.indb vi 9/7/10 5:04 PM Chapter One General Introduction Scholars and other professionals working in the field of Mathematics Educa- tion in Africa have identified a plethora of problematic issues in the endeavor. These issues include attitudes, curriculum development, educational change, instruction, academic achievement, standardized and other tests, performance factors, student characteristics, cross-cultural differences and studies, lit- eracy, native speakers, social class and differences, equal education, teaching methods, knowledge level, educational guidelines and policies, teacher asso- ciations, transitional schools, comparative education, other subjects such as Physics and Social Studies, skills development, surveys, talent, educational research, teacher education and qualifications, academic standards, teacher effectiveness, lesson plans and modules, teacher relationship, teacher charac- teristics, instructional materials, program effectiveness, program evaluation, African culture, African history, Black Studies, class activities, educational games, number systems, cognitive ability, foreign influence, inequalities, eth- nicities, and fundamental concepts (Adler, 1994; African-American Institute, 1976; Ginsburg, 1978; Hoadley, 2007; Howie, 1997; Howie and Hughes, 1998; Howie and Pietersen, 2001; Howie et al., 2000; Linder and Hudson, 1989; Jama, 1983; Le Roux et al., 1985; Masota, 1982; Mbiriru, Sallah, 1982; 1983; Williams, 1978; Zaaiman et al., 2000; Zaslavsky, 1970). While this book is not intended to serve as a panacea for these problems, for that requires a different work, it is our hope, however, that it will be of use to professionals in Mathematics Education in addressing at least a few of these issues. In its extensive 2007 study titled Developing Science, Mathematics, and ICT Education in Sub-Saharan Africa: Patterns and Promising Practices, au- thored by Wout Ottevanger, Jan van den Akker and Leo de Feiter, the World Bank in its effort to assist African countries south of the Sahara to overcome the difficulties encountered in Science, Mathematics, and Information and 1 10_535_Setati.indb 1 9/7/10 5:04 PM 2 Chapter One Communication Technology (SMICT) education in secondary schools of- fers a number of suggestions. The study is based on ten Sub-Saharan Af- rican nations: Botswana, Burkina Faso, Ghana, Namibia, Nigeria, Senegal, South Africa, Uganda, Tanzania, and Zimbabwe. The major challenges that the authors consider to impede SMICT education in those countries are poorly-resourced schools, large classes, a curriculum that is hardly relevant to students’ daily lives, a lack of qualified teachers, and inadequate teacher education programs (2007:v). The World Bank study goes on to say this about learning Science and Mathematics in Africa south of the Sahara: Science and mathematics have always been considered difficult subjects. Many science and mathematics concepts are counter-intuitive and therefore difficult to learn; in fact, often students do not succeed and are then stuck with miscon- ceptions . As science and mathematics learning and the curriculum spiral in the sense that at higher educational levels content is treated at higher levels of formalization and abstraction, SMICT education is built on shaky foundations and students experience problems in applying their knowledge in practical con- texts. Paradigmatic in this sense are well-known examples of university students who still have major problems with basic mathematical operations like fractions and decimal points, although they may have—more or less successfully—gone further with much more advanced topics (2007:40). Despite this clearly stated problem by the World Bank study, nowhere in the study is it mentioned that either African Mathematics or African Science can be part and parcel of the remedy. This book attempts to provide a comprehensive examination of African Mathematics, by tracing the subject from its early beginnings with bones on to the contemporary computer era. Perhaps the following excerpt from the synopsis of Ifeoma Onyefulu’s book, A Triangle for Adaora: An African Book of Shapes (2007), captures at least one aspect of the essence of African Mathematics in everyday life: In the center of Adaora’s slice of paw-paw is a perfect star shape. She doesn’t want to spoil it, so she and her cousin Ugo set off to find a different piece of fruit. As they walk, the children see all kinds of shapes: Uncle Eze’s rectangu- lar agbada, musicians playing circle-topped elephant drums, a crescent-shaped plantain, even plants with leaves in the shape of a heart. That Mathematics—generally defined as the systematic study of quanti- ties and relations through the use of numbers and symbols—pervades every branch of human knowledge is hardly a matter of dispute. It is a useful and 10_535_Setati.indb 2 9/7/10 5:04 PM General Introduction 3 fascinating field of inquiry, and it possesses the power to solve some of the deepest puzzles humans encounter. Mathematics is used in everyday life in such simple ways as telling time or counting the change returned by a cashier. A customer in a supermarket employs Mathematics when s/he buys groceries. A wife and husband use Mathematics to make their household budget or to keep track of their bank accounts. And children use Mathematics in many of the games they play. In business, Mathematics is prevalent in all exchange transactions. Busi- ness entities need Mathematics to establish and maintain their records. Bankers rely on Mathematics to handle and invest money. Many businesses rely upon accountants to keep their records and statisticians to analyze large aggregates of data. And insurance companies rely upon actuaries to compute the rates charged for insurance. In industry, almost all companies rely upon Mathematics for their research and development. Many major industrial firms hire trained mathematicians. All engineering projects rely upon Mathematics. For instance, designing a superhighway requires a great deal of mathematical analyses. Without exten- sive mathematical formulae and calculations, constructing a giant dam would be impossible. The fact that engineering students must take many Mathemat- ics courses highlights the importance of this field of study. In science, Mathematics pervades all aspects of the field. Without exact mathematical descriptions, formulae and observations, most scientists will not be able to perform their tasks. Many scientific problems have become so complex that without a sound training in Mathematics, one may not be able to solve them. The hard, behavioral and social sciences all depend on Math- ematics to advance their disciplines. Despite this extensive utilization of Mathematics, it is not obvious to many that it is also a tool used to teach students that Europeans are culturally supe- rior to Africans. Most books dealing with the history of Mathematics devote only a few pages to ancient Egypt and to northern Africa during the Middle Ages. These books generally ignore the history of Mathematics in Africa south of the Sahara, giving the impression that this history either did not exist or, at least, is not knowable, traceable, or, even more absurd, that there was no Mathematics at all in that part of Africa. To Eurocentric scholars, even the Africanity of Egyptian Mathematics is often denied. But contrary to the popular Eurocentric view, one can neither racially nor geographically sepa- rate Egyptian civilization from its Black African roots. That Africa was in the center of Mathematics history for tens of thousands of years is undisputable. From the civilizations across the continent emerged contributions which would enrich both ancient and modern understanding of 10_535_Setati.indb 3 9/7/10 5:04 PM 4 Chapter One nature through Mathematics. From the measurement
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