Geometry from Africa: Mathematical and Educational Explorations

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Brand new. We distribute directly for the publisher. The peoples of Africa south of the Sahara desert constitute a vibrant cultural mosaic, extremely rich in its diversity. Among ... the peoples of the sub-Saharan region, interest in creating and exploring forms and shapes has blossomed in diverse cultural and social contexts with such an intensity that with reason, to paraphrase Claudia Zaslavsky's Africa Counts, it may be said that "Africa Geometrizes" as well. Paulus Gerdes demonstrates the influence of geometrical ideas on African Culture with dozens of stories and beautiful illustrations.In his first section, Gerdes presents examples of geometrical ideas in the work of wood and ivory carvers, potters, painters, weavers, and mat and basket makers. In the next section he uses examples from Senegal and Madagascar to illustrate how diverse African ornaments and artifacts may be used to lead students to discover the Pythagorean Theorem and to find proofs of it. Paulus also explores connections to Pappus' Theorem, similar right triangles, Latin and magic squares, and arithmetic modulon.In the third section of this book, Paulus analyzes geometrical ideas inherent in various crafts and explores possibilities for their educational use. Topics include symmetrical wall decorations in Lesotho and South Africa, house building in Mozambique, and Liberia and finite geometrical designs from the Lower Congo region.The theme of the fourth section of this book is the geometry of the south-central African sand drawing traditionthe drawings are called sona in the Chokwe language (predominately northeast Angola). As slavery and colonial domination disrupted and destroyed so many African traditions, the sona tradition with its strong geometrical component virtually disappeared. Paulus explains this traditional method by which folklore was passed from generation to generation via beautiful, often symmetric, designs made in the sand and he offers many suggestions by which the reader will find mathematical ideas imbedded in their work. Read more Show Less

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Selected by Choice Magazine as one of the Outstanding Academic Books of 2000

The peoples of Africa south of the Sahara constitute a vibrant cultural mosaic, extremely rich in its diversity.  Among these peoples interest in creating and exploring forms and shapes has blossomed in diverse cultural and social contexts with such an intensity that with reason it may be said that "Africa Geometrizes."

Gerdes presents examples of geometrical ideas in the work of wood and ivory carvers, potters, painters, weavers, and mat and basket makers.  He analyzes geometrical ideas inherent in various crafts and explores possibilities for their educational use. Using as examples African ornaments and artifacts, he shows how students may be led to discover the Pythagorean Theorem and to find proofs of it. He also explores connections to Pappus' Theorem, similar right triangles, and Latin and magic squares as well as the geometrical ideas inherent in mat and basket weaving, house building, and wall decoration.

The author presents the geometry of a sand drawing tradition called sona in the Chokwe language (predominantly northeast Angola). The knowledge of sona has been passed from generation to generation via beautiful, often symmetric, designs made in the sand. Gerdes uncovers mathematical ideas in sona and presents examples of how they may be used in teaching mathematics. He underscores the mathematical potential of the sand drawing tradition by developing the geometry of a new type of design pattern, which he calls Lunda-designs.

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Editorial Reviews

AAAS Science Books and Films
"Are you a mathematics teacher seeking new sources for ideas? This book may be just what you are looking for.... With copious illustrations, the author shows how geometrical ideas are manifested in the work of African artisans.... One does not have to be a mathematician to appreciate the illustrations in this book."
Choice
"The author expertly blends art, mathematics and lore, thereby giving the reader a greater appreciation of African culture.... Gerdes' volume is a significant contribution to the literature of non-European centric mathematics. All of the mathematical ideas are accessible to undergraduates."
John D. Barrow
"This beautifully illustrated book by the world's leading authority on African mathematics provides us with a wide-ranging introduction to mathematical intuition in sub-Saharan African cultures... Strongly recommended to mathematicians and teachers wanting to learn something fresh about traditional truths of geometry and symmetry."
PLUS Magazine
MAA Online
"This work by Paulus Gerdes of the Univeridade Pedagogica in Mozambique is a masterpiece. Not only are the illustrations within the Gerdes book plentiful, they have been well chosen so as to draw the reader into the mathematics of the artifacts and geometric figures...there has not yet been a book in English which unites the Geometric patterns and thought of Africa with the related mathematics. There has also been nothing so useful for student investigations and research on Sub-Saharan African mathematics. The mathematics educator will be pleased. Furthermore, Gerdes' book leads the reader to understand that Africa is a "vibrant cultural mosaic." There are many different peoples with a great diversity in cultural and social mores, leading to many different expressions of mathematical design. This book challenges its readers to open their eyes to a large continent where hundreds of languages are spoken, where people are urban and rural, where there are many different countries with varied cultures, and where geometry lives in each groups everyday life. This book will likely be an eye-opener for many that think of Africa as a continent where everyone looks and lives alike and shares the same language."
Telegraphic Reviews
"This treasure of African patterns and designs demonstrates their use in conveying geometrical ideas..Summarizes and extends previous work of this major contributor to ethnomathematics and further validates African mathematical traditions"
The Mathematics Teacher
"This book may be the most complete volume available on the ethnomathematics of Africa. Most important, Gerdes definitely establishes that Africa is indeed the "cradle of world mathematics"(p.3). Gerdes' clearly written book gives insight into the geometric thinking of African communities as demonstrated in their arts, crafts, architecture, ceremonies, and religion."
Booknews
Sub-Saharan African craft patterns are ingeniously mined by Gerdes (mathematics, Universidade Pedag<'o>gica, Mozambique) to introduce students to the Pythagorean Theorem and other math concepts. Replete with b&w illustrations of the decorative patterns that inspired this culturally-sensitive math. References are primarily in English, French, and Portuguese. Lacks an index. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780883857151
  • Publisher: Mathematical Association of America
  • Publication date: 4/1/1999
  • Series: Classroom Resource Materials Series
  • Edition description: New Edition
  • Pages: 210
  • Product dimensions: 9.10 (w) x 8.40 (h) x 0.60 (d)

Table of Contents

Foreword
Preface: Geometrical and educational explorations inspired by African cultural activities

1 On geometrical ideas in Africa south of the Sahara
Bibliography

2 From African designs to discovering the Pythagorean Theorem
1. From woven buttons to the Theorem of Pythagoras
2. From decorative designs with fourfold symmetry to Pythagoras
3. From a widespread decorative motif to the discovery of Pythagoras' and Pappus' theorems and an infinity of proofs
4. From mat weaving patterns to Pythagoras, and Latin and magic squares
Bibliography

3 Geometrical ideas in crafts and possibilities for their educational exploration
1. Wall decoration and symmetries
2. Rolling up mats
3. Exploring rectangle constructions used in traditional house building
4. A woven knot in starting point
5. Exploring a woven pyramid
6. Exploring square mats and circular basket bowls
7. Exploring hexagonal weaving: Part 1
8. Exploring hexagonal weaving: Part 2
9. Exploring finite geometrical designs on plaited mats
10. Decorative plaited strips
11. From diagonally woven baskets and bells to a twisted decahedron
Bibliography

4 The 'sona' sand drawing traditions and possibilities for its educational use
Introduction: The Chokwe and their sand drawing traditions
1. Some geometrical aspects of the sona sand drawing tradition
2. Examples of mathematical-educational exploration of sona
3. Excursion: Generation of Lunda-designs
Bibliography

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Foreword

I am honored to write the Foreword to this beautifully illustrated book, Geometry from Africa: Mathematical and Educational Explorations. In it, our Mozambican colleague, Paulus Gerdes elaborates and presents us a rare mathematical gift. Through him, we learn of the diversity, richness, and pleasure of mathematical ideas found in Sub-Saharan Africa. From a careful reading and working through this delightful book, one will find a fresh approach to mathematical inquiry as well as encounter a subtle challenge to Eurocentric discourses concerning the when, where, who, and why of mathematics.

            Besides being a distinguished mathematicians and mathematics educator, Paulus Gerdes is a productive researcher and prolific writer. He is a leading researcher in uncovering mathematical ideas embedded in African cultural practices and artifacts. He writes in English, French, German, and Portuguese. In just two decades, he has published well over one hundred journal articles and other books, several of which have won awards.

            Gerdes has also given service to African educational and professional institutions. From 1986 to 1995, he served as a member of the Executive Committee of the African mathematical Union and, from 1991 to 1995, as Secretary of the Southern African Mathematical Sciences Association. At present, he heads the Commission on the on the History of Mathematics in Africa of the African Mathematical Union and edits its newsletter (AMUCHMA). In an important position for furthering educational development in his country, for eight years (1988 to 1996), he was the Rector of the Universidade Pedagogica (formerly, Instituto Superior Pedagogico). Currently, continuing in the mathematics department, he devotes himself to teaching and leading a research team in ethnomathematics.

            With this publication, Gerdes brings to our awareness geometrical ideas encoded in cultural products of Sub-Saharan Africa, ranging from woven and tiled designs to carved patterns, created by women and by men. For as he states in the first chapter, “women and men all over Africa south of the Sahara, in diverse historical and cultural contexts, traditionally have been geometrizing.” He connects the encoded geometrical ideas of Africans to topics such as the Pythagorean Theorem, number theoretic notions, polyhedra, combinatorics, vector geometry, and trigonometry. Importantly, he goes even further to highlight geometrical ideas found in African artifacts. For instance, he draws fascinating and extraordinary connections between the geometry of hexagonal basket weaving and molecular chemistry as well as ideas of Smalley and Kroto for which they won the 1996 Nobel Prize in chemistry. Throughout the book, in the tradition of an engaging teacher, he stimulates our students and us with challenging reflections and questions.

            This publication raises absorbing and challenging questions regarding the origin of particular geometrical ides. It does so in ways reminiscent of two of his earlier books: African Pythagoras ([1992], 1994), for which one of the aims is to stimulate teachers on the African continent to Africanize mathematics teaching; and Sobre o despertar do pensamento geometrico ([1985], 1992), where he discusses directly questions concerning the origin of geometrical thinking. Implicitly, this book raises the possibility of African origins of the theorem that asserts the equality between the sum of square of the  lengths of the two sides of any right triangle and the square of the length of the two sides of its hypotenuse. Gerdes demonstrates how one can discover this and other important mathematical ideas by attending to quantitative, qualitative, and spatial features in a diversity of ancient and modern objects in the multicultural mix of African civilization. Is it not possible, then, to claim African originality of these mathematical ideas? Clearly, cultural groups in other geographical (perhaps even planetary) regions do and can make similar claims. It, therefore, is crucial to strive for a history of mathematics that is unfettered with nationalistic and ethnocentric bias, that acknowledges and valorizes multicultural manifestations of mathematical ideas, and that entertains no primacy claims.

            Indeed, the theoretical framework for questioning the origins of mathematical ideas, calling for a new history of mathematics, and research projects from which this publication emerges are all part of a new disciplinary paradigm: ethnomathematics. Gerdes is a major theoretical and empirical contributor to this field (see, for example, Gerdes, 1995), whose theoretical origins are owed to the Braziliam mathematician, Ubiratan D’Ambrosio (see, for instance, D’Ambrosio 1990). Theoretically, D’Ambrosio (1997) recognizes that.

[t]he complexity of every society, so different one from another, is responsible for the generation of codes, norms, rules and values in the direction of organizing, classifying, comparing, and ordering the action of its individuals. Instances of these codes, norms, rules and values are instruments of analyses, of explanations and of actions, such as more or less, small and big, few or many, near and far, and in and out. These codes, norms, rules and values-for instance, cardinality and ordinality, counting and measuring, and sorting and comparing-take different forms according to the cultures in which they were generated, organized and accepted. To recover these forms and behaviors in different cultural environments has been the main thrust of ethnomathematics… (p. xvi).

           

Ethnomathematics not only recognizes culturally shaped forms of cognition but also the notion that mathematical ideas develop from reflection on labor and other cultural activities. As human beings, we reflect on perceived quantitative, qualitative, and spatial forms in objects around us. Further, we attend to abstracted relations among objects and ides that connect different relations. The Egyptian-born mathematician and educational psychologist, Caleb Gattegno, succinctly states that the elements of reality upon which mathematics is built are “objects, relations among objects, and dynamics linking different relations” (1987, p. 14). The objects, relations, and dynamics may be concrete and contextual or abstract and decontextual. Mathematical ideas may arise from context or apply to contextual situations not necessarily related to the origin of the ideas. Ethnomathematics notes that the power of mathematical ideas often becomes manifest once they transcend their physical, tangible origins.

Different cultures attend to different aspects of reality and express themselves differently. We interact with the world and attempt to contend with and give meaning to what we encounter and perceive. We try to comprehend, interpret, and explain the many aspects and challenges that reality presents. The means by which we comprehend, interpret, and explain, using number logic, and spatial configurations are culturally shaped and are the ways we produce mathematical knowledge.

Gerdes examines culturally shaped products of mathematical knowledge expressed in African material culture as a search for meaning and beauty. In Chapter One, we learn that he is concerned with those spheres of African life in which “geometrical ideas, geometrical considerations, geometrical explorations, geometrical imagination are interwoven, interbraided, interplaited, intercut, intercoiled, intercised, interpainted….” He has found that frequently “geometrical exploration also is an expression of and develops hand in hand with artistical and esthetical exploration….” This widespread, seamless connection between beauty and geometrical explorations represents a cultural value common throughout Africa. Gerdes uncovers this idea through his methodological approach.

Consistent with the theoretical foundations of ethnomathematics, Gerdes has developed a research methodology. Gerdes (1986, [1988] 1997) explains an aspect of his research methodology in this way:

 

We developed a complementary methodology that enables one to uncover in traditional, material culture some hidden moments of geometrical thinking. It can be characterized as follows. We looked to the geometrical forms and patters of traditional objects like baskets, mats, pots, houses, fishtraps, and so forth and posed the question: why do these material products possess the form they have? In order to answer this question, we learned the usual production techniques and tried to vary the forms. It came out that the form of these objects is almost never arbitrary, but generally represents many practical advantages and is, quite a lot of time, the only possible or optimal solutions of a production problem. The traditional form reflects accumulated experience and wisdom. It constitutes not only biological and physical knowledge about the materials that are used, but also mathematical knowledge, knowledge about the properties and relations of circles, angles, rectangles, squares, regular pentagons and hexagons, cones, pyramids, cylinders, and so forth (p. 227-228).

 

To uncover the “accumulated experience and wisdom” that indicate, among other things, the mathematical knowledge of a culture requires respectful and attentive focus on encoded ideas of the material culture. Gerdes’s research perspective and outcomes inform his curriculum development and pedagogical work (Gerdes, 1998). This book represents the painstaking empirical-nearly archeological- spadework of a world-class ethnomathematician. Moreover, the empirical evidence within this book begets questions concerning whether creators of cultural artifacts and ornaments engage in mathematical thinking. Leaning on his methodology, Gerdes ([1988], 1997) answers these theoretical questions.

 

Applying this method, we discovered quite a lot of ‘hidden’ or ‘frozen’ mathematics. The artisan, who imitates a known production technique, is, generally, doing some mathematics. But the artisans, who discovered the techniques, did and invented quite a lot of mathematics, were thinking mathematically (p. 228, original emphasis).

 

Importantly, the justification for his claim rests on both theoretical and empirical premises. Expressions of mathematical thinking vary, s do modes of communicating mathematical ideas. On this point, Marcia Ascher, an important ethnomathematician, states that

 

[t]here has to be more understanding of their [people in particular fields] ideas, as they are culturally embedded, so that their mathematical aspects can be recognized. Take weaving for example. That surely involves geometric visualization. It not only requires the creation and conception of a pattern, but also requires knowing what moves to execute or colors to use to cause a pattern to emerge. In effect, the weaver is digitalizing the pattern. The weaver expresses the visualization through actions and materials (Ascher in Ascher and D’Ambrosio, 1994, p. 42).

 

The visualization and corresponding mental (and physical) actions of the weaver are akin to visual and mental functionings of the professional geometer. The weaver mathematizes her or his visual field and materials. These cognitive products and practices of a culture correspond to Gattegno’s criteria and are mathematical. Furthermore, the mathematical attribution of the weaver’s cognitive products and practices is independent of whether a professional mathematician expropriates them in their original form and transforms them into an academic, codified form. This, I believe, is an important point we reach from a reading of this book.

The mathematical and historical significance of this book proceeds from repercussions of Eurocentric projects predating modernity and ending hopefully with the start of the new millennia. African artisans, laborers, and professional mathematicians have much to learn from each other. Others can also learn from participating in this dialogue.

 

Arthur B. Powell

June 1998

Rutgers University, Newark, NJ

(1993-1994), Universidade Pedagogica, Mozambique

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