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ETHNOMATHEMATICS The Study of People, Culture, and

Mathematical Anthropology

Dr. Eduardo Jesús Arismendi-Pardi

Orange Coast College

January, 2002

Preliminary Draft


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Dedicated to my son,

Mikhail “Mischa” Andrej Arismendi-Knutson

To the Memory of my maternal grandfather

Don Antonio José Pardi Del Castillo

(1903-1978)

“Thinking is the hardest work there is, which is the probable reason why so few engage in it.”

--Henry Ford

“There is nothing which can better deserve our patronage than the promotion of science and

literature. Knowledge is in every country the surest basis of public happiness.”

--George Washington

address to Congress, January 8, 1790


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Table of Contents

Acknowledgements 5

Chapter 1: Mathematics and its Role in the Development of Civilization 6

Theoretical Foundations of Ethnomathematics 7

Interactions Between Culture and Mathematics 9

The Anatomy of Eurocentric Bias 12

Historical Analysis of the Cultural Development of Mathematics 14 Chapter 2: Multicultural Mathematical Knowledge 17

The Mathematics of the Navajo 19

The Mathematics of the Aztec 20

The Mathematics of the Maya 22

The Mathematics of the Inca 32

The Mathematics of Africa 39

The Mathematics of Arabia 46

The Mathematics of China 53

The Mathematics of India 58

Chapter 3: Ancient Egyptian Mathematics 67

An Overview of the Development of Egyptian Mathematics 70

The Golden Section or the Golden Ratio 78

The Mathematical Contributions of the Legendary High Priest Imhotep 82


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Chapter 4: The Role of Women in the Development of Mathematics 86

The First Woman Mathematician 88

A Biographical Overview of Women in Mathematics 89

A Biographical Overview of Women in Mathematics in Modern Society 98

Reflections About the Role of Women in Mathematics 99

Chapter 5: Mathematics, Democracy, Civil Rights and Social Issues 101

Quantitative Literacy, Mathematics and Democracy 102

Sustaining a Democratic Society Through Quantitative-Based Citizenship 105

The Sociology of Mathematics 108

Mathematics and Civil Rights 112

Societal Factors Affecting Mathematical Thought 116

The Mathematics of Karl Marx 120

Distribution of Wealth and the Lorenz Curve 124

The Social Efficiency of Ethnomathematics 126

Reflections on the Past, Present, and Future 127

Chapter 6: The Politics of Mathematics Education 129

Sociopolitical Implications of Mathematics 131

Paulo Freire’s Epistemology in Relation to Ethnomathematics 132

Historiographical Foundations of Eurocentrism in Mathematics 135

The Theory of Antidialogical Action 136

Mathematics of Politics and Politics of Mathematics 138


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Acknowledgements

I wish to thank Cheryl “Cher” Annette Knutson-Arismendi for all her support during this

endeavor. She is my best friend—she was my editor and made many significant suggestions

during the creation of this preliminary text. I also wish to thank my family because without their

support and encouragement this project would have been impossible. I want to particularly

thank my mother, Mrs. Cecilia Pardi de Ramia, my sisters: Angela María Arismendi-Pardi, and

María de los Angeles “Mao” Arismendi-Pardi-Wong, and my brother Anthony “Tony” Joseph

Arismendi-Pardi. I also wish to acknowledge my students because without their support this

course, Anthropology 140—Ethnomathematics, would have not been offered at Orange Coast

College. They provided many valuable suggestions and recommendations. I want to personally

thank the following students from whom I learned a great deal: Mesdames Brenda Bermudez,

Stacey Boynton, Sabrina Carmona, Heather Fellows, Alejandra Larson, Jennifer Peet, Laurie

Wagner; and Messrs. John King, Russell Miller, Ricardo Quintanilla, and Phillips Roggow. I

also wish to thank my friend, colleague, and academic mentor Professor Dennis Kelly from

Orange Coast College who has for the past several years provided tremendous support and

constructive scholarly advice.

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Dr. Eduardo Jesús Arismendi-Pardi

32

Orange Coast College


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Chapter 1

Mathematics and its Role in the Development of Civilization

The field of ethnomathematics examines the epistemology and genesis of mathematical

ideas from a global and cultural perspective. Ethnomathematics is a relatively new term, first

coined by the Brazilian mathematician Ubiratan D’Ambrosio in the early 1980s. The field

focuses on how mathematical ideas are manifested in non-western cultures. Ethnomathematics

lies at the intersection of cultural anthropology and mathematics, and is also referred to as

mathematical anthropology. The formal development of ethnomathematics may have been

slowed due to the pervasive and Eurocentric perspective that mathematics is universal and

culture-free.

Ethnomathematics can also be defined as the application and use of mathematics by various

cultural groups described in terms of gender, occupation, age, ethnicity, or ideology. The prefix

“ethno” refers to an identifiable cultural group and their jargon, codes, symbols, myths, and

specific ways of reasoning and making inferences about the world around them. D’Ambrosio

points out that ethnomathematics investigate the underlying structure of inquiry by considering

the following questions:

1. How are ad hoc practices and solution to problems developed into methods?

2. How are methods developed into theories?

3. How are theories developed into scientific invention?

Ethnomathematical knowledge demonstrates and proposes that the Greek foundations of

European knowledge are themselves founded upon the ancient Black Egyptian civilization. The

mathematical traditions of many non-western cultures focus on the applications and usefulness of


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mathematics. For example, the Indian mathematician Srinivasa Ramanujan (1887-1919) style of

doing mathematics was very different from that of the conventional mathematician trained in the

Western tradition of deductive axiomatic method of proof. In the non-western tradition of

mathematics there is no strong need to justify results. Importance lies in the results being true.

The cultural mathematical traditions of many ancient civilizations such as China and India were

concerned with the generalization of results. In other words, the great ancient mathematicians

merely stated results to problems, leaving their students an opportunity to provide oral

demonstrations or written commentaries. The ethnomathematical literature provides an abundant

body of mathematical knowledge that was developed in Mesopotamia, Egypt, China, pre-

Colombian America (Aztec, Maya, and Inca civilizations), India, and the Arab-Islamic world

(present-day Iran, Turkey, Afghanistan, and Pakistan).

Section 1: Theoretical Foundations of Ethnomathematics

Mathematics is a social product and the construct of the human mind. Mathematical

knowledge has been created by human beings in societies throughout the world. These ideas

germinated from human and social activities such as counting, measuring, locating, designing,

explaining, playing, and inferring. With the passage of time, these social activities link with

each other to provide an understanding of the socio-cultural and natural environment. The

theoretical foundations of ethnomathematics are dependent upon the cognitive practices of

culturally differentiated groups such as engineers, architects, peasants, children, and computer

scientists. These and other cultural groups have a unique and distinct way of reasoning, coding,

measuring, classifying, and inferring. Each group has its own ethnomathematics. The

mathematical ideas of these cultural groups are sometimes maintained, evolved, modified, or

simply disappear according to the dynamics of the groups and its relationship with other groups.


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The theoretical foundation of ethnomathematics is based on the cultural expressions of

mathematical ideas.

A desire to understand and decipher the world around us is a human universal endeavor.

The desire to create new knowledge is also common to all civilizations and is what gave origin to

what we now call western science. The theoretical foundations of ethnomathematical research

are rooted in the following principles:

1. Design Principle: A careful examination of the mathematical practices of non-western

cultures within their social domain is neither trivial nor haphazard; the practices reflect evidence

of a cohesive structure of knowledge for that society. For example, the 4-fold symmetry in

Native American designs is analogous to the 4-direction concept of the Cartesian coordinate

system.

2. Anti-primitivism Principle: By showing sophisticated mathematical practices, not just

trivial and isolated examples, ethnomathematics challenges the stereotypes most damaging to

women and minority ethnic groups. For example, early in the twentieth century, W. S.

Routledge, a British investigator, attested to the excellent knowledge of engineering of the

Kikuyu of Kenya. He reported that the suspension bridges built by tree trunks could not be

improved by the knowledge of engineers who built suspension bridges in England. From an

intuitive perspective the Kikuyu must have used some form of mathematical concepts in a

particular order to achieve a stable structure well suited for their societal needs: Transportation

of man and beast.

3. Translation Principle: Non-western mathematical practices are often analyzed from a

Eurocentric view. The analysis of these mathematical practices from an ethnomathematical view

uses relations between the non-western conceptual framework and the mathematics embedded in


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related non-western designs. In other words, the application of mathematical modeling is used as

a tool to provide a “translation” from non-western knowledge systems to western knowledge

systems. This principle provides a sense of cultural ownership of mathematics that is inclusive

and universal in nature.

4. Dynamic Principle: The body of anthropological evidence of the independent

development of non-western mathematics is of critical importance in challenging Eurocentrism

and the elitist view that non-western mathematics is primitive in nature. It is also important to

avoid and challenge the stereotype of non-western people as historically isolated, alive only in

the static past of a museum display. This is the reason why ethnomathematics includes the

vernacular practices of the contemporary descendents of non-western cultures. For example, the

geometric design of fishing nets on fishing boats, the intricate patterns of Black hairstyles, and

the knowledge of “street mathematics” of Latino street venders.

Section 2: Interactions Between Culture and Mathematics

Ethnomathematics recognizes that all cultures engage in mathematical ability and that no

single individual, group of people, culture, or even gender, has a monopoly on mathematical

achievement. The critics of the interaction between mathematics and culture, that is, the

mathematization of culture, argue that such interaction is necessary to raise the self-esteem of

ethnic minority cultures and to improve mutual understanding and respect among ethnic and

cultural groups. This attitude is misconceived and patronizing. The cultural affirmation of

mathematics is a key factor in struggles against mathematical underdevelopment caused by

racism, sexism, and imperialism. For mathematics to become emancipatory and liberatory in

nature, it is necessary to stimulate confidence in the creative powers of every person and every


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culture to understand, develop, and use mathematics. It is important to realize that cultures are

able to develop mathematical ideas via a language code of a given sociocultural group.

The study of mathematics as a cultural element is not a new concept in that anthropologists

have done so, but since their knowledge of mathematics is generally limited, their reactions and

conclusions have traditionally consisted of scattered remarks concerning the arithmetic found in

non-western cultures. In the study of culture there are many elements and artifacts that are

considered mathematical. For example, The Ishango Bone, which was found at the fishing site

of Ishango on Lake Edward, in Zaire (formerly, the Republic of Congo). This mathematical

artifact dates back to a period between 23,000 B.C. and 18,000 B.C. The discoverer of the

artifact, Dr. Jean de Heinzelin, suggests that it may have been used for engraving or writing.

Further examination of the artifact suggests knowledge of multiplication by two and of prime

numbers. The markings on the Ishango Bone have also suggested that the artifact may have been

used as a lunar calendar. Is it plausible that the development of such lunar calendar may have

started the beginning of the science of astronomy?

Each culture has its own mathematics, which evolves and dies with the culture. The

primary reason for the truncation and circumvention, or reversal of most, but not all, forms of

intellectual, mathematical, scientific, and technological activities of advanced ancient

civilizations and cultures was the European domination, enslavement, and colonization of Africa,

Asia, and the Americas. The traditional argument that mathematics should be used only for the

study of numerical and spatial concepts for their own sake, rather than for their applications is

very restrictive in nature. This restrictive view of mathematics has its roots in the Greek

tradition of mathematical thought. The view of mathematics with a cultural bent can certainly

lead one to an appreciation of the intellectual endeavors of others. Perhaps the reason why the


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tradition of mathematics and science has never been strong in the Black community is because

the traditional Eurocentric view of mathematics disregards the historical contributions of

Africans and African Americans to mathematics. This Eurocentric view ignores the structural

role that the political economy of the United States plays in the mathematical underachievement

of African Americans. However, in spite of the social and institutional racism that is prevalent in

America, there are examples of many African Americans that have contributed to the

mathematical sciences. Some examples include: Benjamin Banneker (1731-1806); Marjorie Lee

Browne (1914-1979); David Blackwell (1919-); and Evelyn Boyd Granville (1924-).

In August of 1791, Benjamin Banneker wrote to the Secretary of State, Thomas Jefferson,

to challenge the notions of black inferiority. In his correspondence, Banneker included his

almanac of astronomical observations in manuscript form. Jefferson forwarded the almanac to

the Royal Academy of Sciences in Paris along with a letter citing as evidence the equal scientific

talents of blacks and described the author of the almanac as a respectable mathematician and

astronomer who had been employed as a surveyor in the District. Jefferson also alluded to

Banneker’s elegant solutions of many geometrical problems.

Marjorie Lee Browne (1914-1979) financed her education at Howard University with a

combination of scholarships, loans, and odd jobs and graduated in 1935. Browne received her

Ph.D. in mathematics in 1950. Her dissertation area was in topological groups. From 1949 until

her death in 1979, Dr. Browne taught at Carolina Central University.

David Blackwell (1919-) was a professor of mathematics at Howard University and later at

the University of California at Berkeley where he taught statistics from 1954 until his retirement

in 1990. Dr. Blackwell is best known among statisticians for the Rao-Blackwell Theorem. Dr.

Blackwell received his Ph.D. in mathematics from the University of Chicago and was elected to


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membership in the National Academy of Sciences. He has written over 80 articles in various

mathematical journals.

Evelyn Boyd Granville (1924-) received her Ph.D. in 1949 in the area of Complex Infinite

Series from Yale University. After a year of postdoctoral research at NYU, she applied for

several academic positions and was not seriously considered; one hiring committee member

laughed at her application. Dr. Granville taught at Fisk College from 1950 to 1952. She spent

16 years in government and industry. As an employee for IBM, she worked on orbit

computations and computer procedures for the first US space project and Project Mercury, the

first manned earth orbital project. She conducted research in celestial mechanics, trajectory and

orbit computations, numerical analysis and computer techniques for the Apollo Project. Dr.

Granville taught at California State University at Los Angeles from 1967 until her retirement in

1984. In 1989, she received an honorary doctorate from Smith College.

Section 3: The Anatomy of Eurocentric Bias

The standard western treatment of the development of non-European mathematical

knowledge has been rooted in historiographical bias in relation to the selection and interpretation

of facts. This Eurocentric bias also suggests that mathematical activity outside Europe is of little

consequence. In fact there is widespread acceptance that mathematical discovery can only

follow from a rigorous application of a form of deductive axiomatic logic, which is perceived to

be a unique product of the Greeks. One byproduct of this view is that empirical methods are

dismissed. Another commonly expressed view is that there was no mathematics before the

Greeks because pre-Greek mathematics had no well-defined idea of proof. To better understand

the anatomy of Eurocentric bias in the development of mathematical knowledge requires an

understanding of Gheverghese-Joseph Models of Eurocentrism as outlined below:


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1. The Classical Eurocentric Trajectory;

2. The Modified Eurocentric Trajectory;

3. The Alternative Trajectory for the Dark Ages.

The Classical European Trajectory proposes European superiority and ignores the research

evidence that points to the development and use of mathematics in Mesopotamia, Egypt, China,

pre-Colombian America, India, and the Arab-Islamic world. This model assumes that a period

of intellectual inactivity known as the Dark Ages took place between Greek knowledge and the

discovery of this knowledge by the Europeans. Furthermore, under the assumptions of this

model the Europeans saw themselves as the true inheritors of this great intellectual wisdom—the

accumulation of two thousand years of scientific, philosophical, mathematical, and scientific

knowledge. The conclusion about the development of mathematics under this model is the

existing perception in the western world that mathematics is an exclusive product of white

European civilizations. It is important to point out, however, that according to Aristotle (350

B.C.), Egypt was the cradle of mathematics. In fact, Eudoxus—a noteworthy mathematician—

and Aristotle’s teacher is said to have studied in Egypt prior to teaching in Greece.

The Modified Eurocentric Trajectory proposes some acknowledgement to the Arab-Islamic

world and cultures as—only—custodians and guards of Greek learning during the Dark Ages in

Europe. The role as transmitters and creators of knowledge of China, India, and pre-Colombian

America is completely ignored or simply dismissed as uneventful. The foundation of this model

assumes that Greek knowledge and mathematical development was kept alive by the Arab-

Islamic cultures. This particular model is quite popular in many of the books dealing with the

history of mathematics.


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The Alternative Trajectory for the Dark Ages is an unbiased alternative to the Classical and

Modified Trajectories in that this model highlights mathematical activity and exchange between

a number of cultural centers that took place while Europe was in deep intellectual slumber. This

model points out that scientific knowledge originated in India and China as well as the

Hellenistic world. The knowledge was then sought out by Arab scholars and it was translated,

refined, synthesized, and augmented at different centers of learning starting with Jund-i-shapur in

Persia around the sixth century and then moving to Baghdad, Cairo, and finally Toledo and

Cordoba in Spain. From Spain the knowledge spread into Western Europe. Spain and Italy were

at the nearest points of contact with Arab science. Two very significant pieces of writings took

place under the Alternative Trajectory Model for the Dark Ages. These writings authored by

Muhammad ibn Musa Al-Khwarizmi are Calculation by Restoration and Reduction (Algebra),

and Calculation With Indian Numerals (Arithmetic). The translation of the Arithmetic book into

Latin corrupted the name of Al-Khwarizmi into algorism and later into the word we commonly

use today in mathematics and computer science: Algorithm.

The byproduct of a careful, fair and equitable analysis of the Alternative Trajectory for the

Dark Ages is that it is not only bias, but also dangerous to characterized mathematical

development in terms of European superiority. The period of the Dark Ages in Europe for one

thousand years before the illumination that came with the Renaissance, did not, in any way

interrupted nor affected mathematical activity elsewhere.

Section 4: Historical Analysis of the Cultural Development of Mathematics

Anthropological research has shown evidence of mathematical activities and practices that

are common among many cultures. These activities and practices include counting, ordering,

sorting, measuring, weighing, ciphering, coding, decoding, etc. These activities are quite


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different from the activities that students learn in a traditional western curriculum. The research

and observations made by anthropologist when studying culture has encouraged a few studies of

the concepts of mathematics in a cultural and anthropological framework. The western reference

in relation to the development and use of mathematics has two clearly disjointed branches:

Scholarly mathematics and “practical mathematics.”

The genesis of scholarly mathematics is rooted in the traditional and ideal education of the

Greeks. Practical mathematics on the other hand, was disseminated to the working class—the

manual workers of ancient Egypt. In relation to the ancient cultural traditions, the study of

ciphering, arithmetic, and astronomy was reserved for a minute elite societal class. The

dichotomy between scholarly mathematics and practical mathematics was carried on by the

Romans and was reserved for different social classes. The education of the elite few—the liberal

man—was comprised of the “trivium” and the “quadrivium” which consisted of the seven liberal

arts and sciences: Grammar, rhetoric, logic, arithmetic, geometry, music, and astronomy.

Practical training was strictly reserved for the labor class.

During the Middle Ages, there was a convergence of both branches of mathematics in one

direction. Practical mathematics began to adopt aspects of scholarly mathematics in the study of

geometry. In fact, practical geometry becomes a subject in its own right during the Middle Ages.

The convergence of practical to theoretical geometry follows the translation of Euclid’s Elements

in the 12th century by Abelard of Bath.

During the Renaissance a new labor structure emerged and changes take place in the field

of architecture because plans and designs are accessible to bricklayers, and machines can be

drawn and reproduced by the labor class. Scholars began to write in a non-technical language


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and style thus creating access for non-scholars. The best example of this kind of writing is that

of Galileo and Newton’s Optiks.

During the industrial era the convergence of practical mathematics to scholarly

mathematics increases its pace for reasons related to complex machinery, instruction manuals

and social concerns. This convergence begins to permeate the school system and by the

twentieth century the question of what kind mathematics should be taught in a mass educational

systems is posed. The answer to this early inquiry was that the kind of mathematics that needed

to be taught in the mass educational system should be the kind of mathematics that supports the

economic and social structure of the aristocracy. The rationale behind this answer was that a

good training in mathematics was of paramount importance for the preparation of the elite

class—as advocated by Plato—to assume effective management over the labor or productive

sector. The adaptation of the convergence of practical to theoretical mathematics gave birth to

“scholarly practical mathematics” which became what is known today as academic

mathematics—the mathematics taught and learned in today’s institutions. In contrast to this brief

historical account, ethnomathematics is the kind of mathematics practiced by various identifiable

cultural groups including tribal societies, children, labor groups, members of a certain

professional class, etc. The concept of ethnomathematics also includes much of the mathematics

practiced by engineers, scientist, and business professionals. For instance, calculus, which is an

essential tool for the engineer does not respond nor does it adhere to the rigor and formalism of

academic calculus courses. The same can be said about the mathematics used by actuaries and

business professionals. On the other hand, academic mathematics is very much a part of

academic mathematicians who are also members of a certain professional class.


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Chapter 2

Multicultural Mathematical Knowledge

This chapter will focus on a brief overview of multicultural mathematical knowledge and it

is therefore important to keep in mind that the presentation herein is only introductory. Before

one can begin to discuss the particulars of multicultural mathematical knowledge it will be

indispensable to explore the concept of multiculturalism. Though the concept of

multiculturalism is somewhat fuzzy, one can certainly argue that multiculturalism is about the

development and appreciation of knowledge germinated from different world perspectives. In

the context of ethnomathematical knowledge, this concept will be directly related to non-western

cultures. Multiculturalism should cause an individual to develop an appreciation in concert with

respect and tolerance towards how cultures developed, used, and applied mathematical principles

to meet their economical and social needs. Multiculturalism is important in the creation of a just

society because understanding of one’s own culture is dependent upon knowledge of other

cultures.

The non-western epistemological position on the nature of mathematics is quite different

from the European tradition in that these cultures—Chinese, Indian, Arab-Islamic, Pre-

Colombian American, and African—have used demonstrations of truths that are not formulated

in the formal deductive system of logic. For example demonstrations of the Pythagorean

Theorem involve—at least in the Indian and Chinese tradition—the use of visual methods. The

aim of these cultures is not to build up an imposing edifice of a few self-evident axioms but

rather to validate the results by any method, including—of course—visual demonstrations. The

argument of the critics is rather ethnocentric in nature. The flaw of the critic’s argument is the


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unwillingness to view the contributions of non-European cultures via the culture itself. So, from

a purely Eurocentric perspective the argument is “You see they had no proofs, as we know them

in modern mathematics. Were their contributions really mathematics?” The fact is that the

notion of what a proof is has changed over time and there is no consensus on what constitutes

proof. For example, consider the solution for a quadratic equation in a Babylonian tablet from

about 3,500 years ago (expressed here in modern language).

The length of a rectangle exceeds its width by 7 units. Its area is 60 square units. Find the

length and the width. Using symbolic algebra the problem is written as 2 7 60x x where “x”

is the width of the rectangle. The solution indicated in the Babylonian text (here in modern

language) is as follows: Halve the quantity by which the length exceeds, that is, 3.5. Square 3.5.

To this result add the area, that is, 60. Find the square root of this sum. Subtract 3.5 from this

square root, that is, 8.5 to get the width as 5 units. Add 3.5 to the square root to get the length as

12 units. The modern symbolic form of the solution for the width can be expressed as follows:

If 2x bx c , then 2

2 2

b bx c

. Can one claim that the Babylonians were not aware of

this general form of the solution, even if they did not express it in symbolic terms?

It is plausible that a non-symbolic, rhetorical argument or proof can be quite sophisticated

and rigorous when given a particular value of the variable. The condition for the sophistication

or rigor is that the particular value of the variable should be typical and the generalization of the

result to any value should be immediate.

This chapter will present an overview of the mathematics of the Navajo, Aztecs, Mayas,

Incas, Africa, Arabia, and China. The chapter will also provide a brief commentary on the use of


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mathematical principles and concepts used in the building of European cathedrals during the

Middle Ages.

Section 1: The Mathematics of the Navajo

The Navajo are the Native American society whose population and cultural center extends

on the high mountainous plateau that covers the regions of northwestern New Mexico into

northeastern Arizona and slightly above southern Utah. The Navajo believe in a dynamic

universe unlike the western mode of thought. For example while we believe that time is

continuous we frequently talk about time in terms of discrete units or point in time. The Navajo

mode of thought is made up of processes. The reason why the Navajo react negatively when

fences are placed upon their reservation land is rooted in their belief that time and space should

not be segmented in an arbitrary and static fashion.

When we speak of a boundary line dividing a surface into two parts or a line being divided

by a point, we are indeed describing a static situation—a situation in which time plays no role

what so ever. Moreover, when we discuss the idea of the limit of a rational or polynomial

function where “a” is in the domain of the function “f” we write that the lim ( ) ( )x a

f x f a

which

is a direct substitution property called continuous at a. Naturally not all limits can be evaluated

by direct substitution. However, when one studies calculus this concept is often taken for

granted which results in the mechanic evaluation of the limit of a function as a discrete and static

situation.

Around the year 1690 the Navajo adapted into their life the art of weaving wool. Today,

the Navajo rugs are highly admired around the world. The art of producing Navajo rugs has now

developed into thirteen regional styles with varying materials and techniques. The design of the

rugs and estimation of materials needed for their production is a task that is carried out by the


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Navajo women. The form of mathematics found in the Navajo rugs is unlike the European style

developed in the seventeenth century. This form of mathematics has developed into an abstract

theory in imitation of Euclid’s classical geometry. The Navajo mathematics is a creative process

visualized in the mind of the weaver. The results of such art can be beautifully contemplated in a

completed rug.

All the mathematics used in the creative process of weaving the rugs is based on some

intuitive idea or concept such as counting, the idea of the limit, the geometric notion of the circle,

symmetry, and the algebraic understanding of the slope of the line. The Navajo women use

creative talents from similar intuitions found in contemporary modern mathematics. The

classical diamond-shaped elements of a rug are the intuitive ideas that form the algebraic concept

of the slope. Geometry is perhaps the most unique characteristic of the Navajo rug. In fact, the

geometric shapes and symmetrical relations is the trademark of the Navajo rug. It is important to

contemplate that what is usually unnoticed is the tremendous spatial visualization that Navajo

women possess in that there are no written patterns nor plans to be followed. Each weaver

visualizes the complex design of the entire rug and executes the art from mental conception

alone. The weaving process is the interrelation of many processes that emanate from a center,

spread out from the center, and return back along the same path.

Section 2: The Mathematics of the Aztec

The Aztecs of Central America developed a system of numerals quite similar in principle to

the Egyptian number system from 3500 BC. The Aztecs migrated to Mexico from the north in

the early thirteenth century AD and founded a large empire, ruled from their capital city of

Tenochtitlán, reaching the height of its power in the fifteenth and early sixteenth centuries. The

prosperity of the Aztec Empire had its foundation on a highly centralized agricultural system in


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which land was intensively cultivated, irrigation systems were built, and swampland was

reclaimed. The Aztec Empire’s dependence on an agricultural system is evident in some of the

symbols used in their vigesimal (base-20) number system.

In 1519 AD Hernán Cortés arrived in Mexico with about 450 conquistadores which was the

year One Reed in the Aztec calendar, which was the seventeenth year of the reign of Montezuma

II, the powerful Aztec emperor. Under his leadership forty-four cities of Mexico were

conquered. The capital of Tenochtitlán was a fortress where vast amounts of rich tributes took

place and thousands of human beings were sacrificed at the altar of the Aztec god

Huitzilopochtli. According to Aztec legend, the year One Reed was the year in which the god

Quetzalcoatl would return to Mexico to re-establish his kingdom. This was a fearful year when

Montezuma would begin to ponder his fate. Was Hernán Cortés the god Quetzalcoatl? In the

period since his arrival in 1519 AD Cortés along with the conquistadores and the help of a few

Nahuatl Indian allies, crushed the great Aztec Empire.

The Aztec culture demanded a reliance on meticulous record keeping of cities conquered as

well as religious practices that were based on their vigesimal number system. One of the most

complex forms of known Aztec mathematics is the Códice de Santa María Asunción, which is a

16th century record from the village of Tepetlaoztoc. The ancient numerical record demonstrates

procedures for obtaining statistical information such as census of households and other

mathematical procedures for the calculation of taxes, and other particulars related to land

holdings. The Aztecs also had a special symbol for zero that was represented as a corn glyph.

The geometrical patterns used by the Aztecs were defined, according to Texcocan cadastral

recordings, as the micocoli field and the tlahuelmantlii area. The micocoli field is a rectangular

or quadrilateral region whose sides are measured in quahuitls (2.5 meters).


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Following are two examples illustrating tlahuelmantlii area problems that were used by the

Aztecs. These two problems fall within the purview of optimization theory problems commonly

seen in finite mathematics courses. The first problem deals with a tlahuelmantlii area with a

constraint of at most 400 square quahuitls. The second problem illustrates a tlahuelmantlii area

with a constraint of least 400 square quahuitls. In modern notation, the first problem is

symbolically represented as follows:

Area < 400 square quahitls

20

0 20

0 20

0

A y x

x

y

z

The second problem, in modern notation, is as follows:

Area > 400 square quahitls

20

0 20

1

0

A z x

x

z

y

Finally, it is important to point out that at the time the Spaniards arrived to the land that is

now known as Mexico, they found large buildings and impressive engineering and architectural

accomplishments. In fact, the Aztec measures of land were more accurate than those used by the

Spaniards. The Aztec square quahitl was a standard unit, in contrast to the Spanish caballería—

a non-standard unit of measurement—that varied from farm to farm.

Section 3: The Mathematics of the Maya

The Maya Empire of Central America was isolated from any center of mathematical

activity and yet their achievements in the particulars of numeration and calendar construction


23

were and still are quite remarkable. The Maya society covered much of the region of what are

today the Republics of Belize and Guatemala, parts of Honduras and El Salvador, and several

Mexican states. This region where the Maya and their ancestors have lived, and still live is often

referred to as Meso-America (Middle America). In contrast to all other Native American

mathematical systems, it is safe to argue that that of the Maya is the most complex. The Maya

number system consisted of three symbols: The dot for the number one, the bar for the number

5, and the shell for the number zero. The Maya made used a quinquavigesimal (a system based

on groupings of fives and twenties) number system. The Maya numerals were written in vertical

columns and the system had place value at the bottom of the column. According to many of the

accounts from the Spanish Conquistadores, there were Maya books containing intricate

astronomical calculations. Sadly—and unfortunately, only a few of these books or records

survived the destruction by either the conquistadores or the missionaries. The latter of the two

believed that the knowledge and information contained in these books was the work of the

devil—a concept that is strictly European in nature. What is known today about the mathematics

of the Maya was primarily concerned with calendar dates; these were used for religious

purposes. Though the Maya were able to arrive at accurate answers concerning astronomical

calculations, the details and algorithm concerning any systematic form of logic or procedure is

unknown. The only algorithms that exist are strictly speculative in nature and are—only at

best—just a generalization of the kind of algorithms that they could have used. Victor Katz, a

mathematician from the University of the District of Columbia suggests that the Maya may have

used an algorithm involving the mathematical definition of the ring of integers of modulo n.

This definition can be stated as follows:


24

Let n be a fixed integer greater than 1. If a and b are integers such that a – b is divisible by

n, we say that “a is congruent to be modulo n,” and indicate this by writing (mod )a b n . As an

example suppose that n = 5, then we have 18 3(mod5) since 18 – 3 is divisible by 5. Other

examples include 2 8(mod5) , 4 4(mod5) , and 1342 2(mod5) . The following example

will serve as a prelude to appreciate the nature of the speculative Maya algorithm for the

calculation of calendrical dates.

Let n = 13, then 20 7(mod13) since 20 – 7 is divisible by 13. Similarly,

18 20 4(mod13) , 20 18 20 2(mod13) , and 20 20 18 20 1(mod13) . If a is an

integer and r is the remainder in the division of a by n, then (mod )a r n . Also notice that if

a qn r , then a r qn and hence (mod )a r n . The following is the speculative algorithm

in modern notation that may have been used by the Maya in the calculation of calendrical dates.

The algorithm states that if a specified number of dates is denoted in vigesimal (i.e., base-20)

calendrical notation by m, n, p, q, and r, where 0 , , , , 19m n p q r and 0 17q , and an initial

date is given by 0 0 0( , , )t v y , then a new date ( , , )t v y which is m, n, p, q, and r days later can be

determined by the following equations:

0 2 4 7 (mod13)t t m n p q r

0 (mod 20)v v r

0 190 100 5 20 (mod365)y y m n p q r

As an example that can be verified by the reader is as follows: If the given Maya date is given

by (4, 15, 120) then the new date (t, v, y) which is 0, 2, 5, 11, and 18 days later is (10, 13, 133).


25

The Maya developed a 52-year calendar, named a Calendar Round, comprising two cycles.

The first of these cycles was a 260-day Sacred Round and the second was a 365-day Vague Year,

an approximate solar year. The Sacred Round consisted of integers 1 through 13 paired to the 20

named days. The names of these 20 dates are Imix, Ik, Akbal, Kan, Chicchan, Cimi, Manik,

Lamat, Muluc, Oc, Chuen, Eb, Ben, Ix, Men, Cib, Caban, Etz’Nab, Cauac, Ahau. The Vague

Year consisted of 18 months of 20 nameless days plus a final month named Uayeb with unlucky

nameless days. The names of the month glyphs are Pop, Uo, Zip, Zotz’, Zec, Xul, Yaxkin, Mol,

Ch’en, Yax, Zac, Ceh, Mac, Kankin, Maun, Pax, Kayab, Cumku, and Uayeb. The two cycles

worked as meshed gears. Any particular date in the 52-year cycle could be uniquely specified by

four symbols as in the Calendar Round date of 4 Lamat 15 Xul. After 18,980 days, the same

date occurred again and thus time repeated itself. In order to specify dates, the Maya counted the

number of days between events. In fact, a detail representation from a Mayan ceramic vessel

depicts two mathematicians, one of them a female with number scrolls emanating from her

armpit [Clarkson, P. B. (1978). Classic Maya pictorial ceramics: A survey of content and theme

in R. Sidrys, (Ed.)., Papers on the economy and architecture of ancient Maya (pp. 86-141). Los

Angeles, CA: University of California at Los Angeles Institute of Archeology]. According to M.

P. Closs (1986, 1992) [Native American mathematics, Austin, TX: University of Texas Press;

and I am a Kahal; my parents were scribes. Research Reports on Ancient Maya Writing, 39.

Center for Maya Research, Washington, DC.], once the name of this female mathematician is

deciphered she will be known as perhaps the earliest mathematician known in history. The text

on this ceramic contains a statement concerning the parentage of the mathematician in question,

“Lady Scribe Sky or Lady Jaguar Lord, the scribe. Maya scribes used many different glyphs to

express the same idea. One can definitely appreciate that the particulars relating to the


26

calculation of calendrical Maya dates is quite complex and requires in-depth knowledge about

the Maya culture as well as mathematical proficiency.

The Maya defined the first day of history by postulating a first Calendar Round date of 4

Ahau 8 Cumku and hence estimating their existence thousands of years back. A transcription of

the date 4 Ahau 8 Cumku corresponds to August 13th, 3114 B.C. Given this date as the first date

of history defined, the Maya could specify any important date in Maya history. The Maya had

the ability to forecast the Calendar Round date and the long count, and the number of days that

elapse from the beginning of history to the current date.

In addition to the accomplishments related to the particulars of calendar calculation and

dates, the Maya attained great heights in the fields of art, sculpture, architecture, mathematics,

and numeration (arithmetic). The Maya understanding of astronomical observation was much

more precise than the methods available in Europe at that time. For example, they arrived at

accurate estimations of the duration of solar, lunar, and planetary movements. The astronomers

of the Maya culture were able to calculate the synodic period of Venus, in other words, the time

between one appearance at a given point in the sky and its next appearance at that exact point.

The Maya calculation of the synodic period of Venus was estimated to be 584 days—an

underestimate of 0.08 days by using the today’s astronomical calculation methods. Modern

astronomers have calculated the synodic period of Venus to be 583.92166 days. What is truly

amazing about the Maya is that they were able to achieve astronomical discoveries without the

knowledge of glass or optical devices. The Maya were also able to measure the passage of time

without the aid of clocks or time measuring devices.

One probable reason for the lack of physical records describing the algorithms or

procedures used by the Maya in their astronomical calculations is that these records could have


27

been made of perishable material such as wood. Also, as mentioned earlier, during the invasion

to the Americas by the Spaniards, Roman Catholic priest destroyed the Maya libraries leaving

only four hieroglyphic books or codices. These records possess only the end results of the

calculations and methods that were used by the Maya. These codices are: The K’iche’ Codex,

the Paris Codex, the Dresden Codex, and the Madrid Codex.

The K’iche’ Codex contains astronomical as well as calendrical as well as ritual

information. This codex contains a plethora of signs corresponding to each day. The days of the

260-day cycle are organized as groups of five, with dates in a given group spaced at 52-day

intervals. The K’iche’ Codex makes direct reference to Venus as the morning star. The excerpt

of the text that alludes to Venus is “Utzilaj q’ij tikb’al, awexab’al, saq amaq’ kawexik, chaqan

nima ch’umil, eqo q’ij, cha’om q’ij” which in modern English it translates as follows: “Good

days for planting, sowing, in peace one plants, when the great star [Venus] rises, Bringer of Day

[morning star], beautiful day.”

The point of greatest interest in the K’iche’ text is that the appearance of Venus as the

morning star is given a positive interpretation. However, among the Mayas, this planet has

always had a bad omen because of its connection to warfare as indicated in the Dresden Codex.

Very little is know about the way in which the Maya astronomers might have used the fixed stars

or features of the Milky Way to measure time, whether by observing their rises and sets or by

using them to track the movements of the Sun, Moon, and planets.

The Paris Codex has long been interpreted as a series of zodiacal creatures. The text is

built around a table that divides each of the five idealized years into thirteen periods of 28 days

each, for a total of 364 days per year. The Paris Codex could have been used to give the

locations of possible solar eclipses with respect to constellations. Astronomical observation by


28

the Maya by systematic, narrative, or sequential methods was quite accurate in reference to the

passage of time.

The Dresden Codex revealed information concerning the movement of Venus. The

Dresden Codex was designed to cover a 200-year period beginning three centuries after the

Palace of the Governor was constructed, but its preface gives an origin date that falls before the

construction of the palace. The Dresden Codex was a good instrument for the prediction of

eclipses over a long period of time. The Maya may have probably read the eclipse glyph as a

cross-reference device.

The Madrid Codex is similar to the Dresden Codex in that it also contains the 260-day

cycle organized as group of five, with the dates in a given group spaced at 52-day intervals.

Finally, there is epigraphic evidence found in the Maya codices that consists almost in its entirety

of patterns related to architectural alignments. The ancient Maya civilization has proved to be

much more complex than it is usually supposed by the critics of the ethnomathematics

movement.

In reference to the particulars related to Maya geometry, it is known that this knowledge

was developed and integrated for the collective societal good. According to the oral tradition of

the Maya priests such knowledge came from the corn. In fact, the shape of the temples was

derived from ears of corn. Many of the calculations on the Maya calendar came from a period of

cultivation and its various stages such as planting, banking up of soil around the corn stalks,

weeding, and others. Many of Maya temples are truncated tetrahedrons, rectangular prisms, and

cylindrical structures such as those found at the archeological site of Ceibal. The relationship

that exits between these engineering and architectural edifices and astronomy suggests careful

and meticulous planning before their construction.


29

Archeological excavations of Maya sites have provided physical evidence of objects that

reveal a great deal of geometric knowledge. The physical evidence is comprised of jugs, vessels,

plates, and bowls adorned with geometrical figures and curves. It is plausible that the Maya had

understood the concepts of curves and lines. A non-exhaustive list of the expressions found in

the Maya languages of K’ekchí and Chortí include line, align, side, edge, place horizontally,

cylindrical, square, to square, quadrilateral, measure, half-measure, equal, under, and along side.

The thought of geometry within the western European perspective often relates in direct

association with the geometry of Euclid. When most people think geometry, they do not

associate this body of knowledge with the knowledge inherited from the Americas, in

particular—the Maya. Maya geometry has its roots in the geometry of nature in that it has its

foundation in the Canamayté Quadrivertex which is the central square in the row of squares on

the back of the Yucatecan rattlesnake, that is, the Crotalus Durissus Tzabcan Yucateco.

The Maya culture began with the formative period around 500 B.C. The classical period,

on the other hand, goes back between 300 and 800 A.D. and around 700 A.D. the Maya

civilization was at its height. The decline of the Maya civilization began between 800 A.D. and

925 A.D. The Maya erected great ceremonial centers in the Southeast region (i.e., Mexico) of

their empire. The motivation behind the success of the Maya in relation to mathematics and

astronomy according to Eric Thompson’s (1966) book, The Rise and Fall of the Maya

Civilization published by the University of Oklahoma Press, was based on careful and patient

observation over hundreds of years. These observations were transmitted from one generation to

the next whereupon the future generations were willing to discard inaccurate calculations.

The cosmogonic idea behind the Canamayté Quadrivertex is evident in the Popol Vuj, a

sacred book of the Maya. A passage from this book states that “it is with great detail that the


30

description and narration of how everything—Heaven and Earth—was formed is given, how the

four corners and four sides were made and governed by the square of the Crotalus Durissus

Tzabcan Yucateco.” According to the Maya, the creation was carried out in accordance to the

geometric principle of the rattlesnake. The Maya creator is a mathematician and a geometer—

the creator of all things. The god Quezalcoatl was reputed as the inventor of the science dealing

with the measure of time. The mathematical insignia used by the wise priests and scribes of the

Maya culture was a represented by the head of a snake. The ceremonial rods adorned with the

head of a snake that are found in many artifacts represent the wise priest and scribes that ordered

the construction of their temples.

The Canamayté Quadrivertex is a geometrical model that predates all archeological and

historical cultures and offered its mathematical foundation to all the pre-Columbian cultures. As

it moves, the rattlesnake produces a dynamic geometry in that the squares are transformed into

rhombuses that immediately return to being what they were. This dynamic geometry reveals

geometrical, arithmetical, cosmological, and architectural knowledge. Geometry was the soul of

the terrestrial and celestial thought of the Maya in the same fashion that mathematics was the

soul of Greek culture. The Canamayté Quadrivertex provided the bases for many geometric

constructions such as the pentagon and the star. The Canamayté Quadrivertex inserted into

another square provided the cross of the octants of the moon and its faces. Other geometrical

constructions include:

1. The proportion of a flower;

2. The facial profile of the Maya;

3. The proportions of the human face;


31

4. The proportion of the human body exactly as it appears in the well-known drawing of

Leonardo Da Vinci that illustrates the Pythagorean theory of the golden section;

5. The front view of the first Maya temple; and

6. The aerial view of a pyramid.

The Canamayté Quadrivertex also indicates the four cardinal directions with its four

vertices: North, South, East, and West.

It is remarkable that the figures of Euclidean geometry are implicit in the rattlesnake—that

is—in nature. The Crotalus Durissus Tzabcan Yucateco also expresses the scientific bases of

disciplines such as architecture, arithmetic, and cosmology. Finally, a comparison between

Euclidean geometric propositions and Maya geometric propositions is presented below.

The Euclidean geometric propositions are:

Euclidean Proposition 1: Two points determine a line.

Euclidean Proposition 2: A line can be extended from each end.

Euclidean Proposition 3: Given a point and a center it is possible to draw a circle.

Euclidean Proposition 4: All right angles are congruent.

Euclidean Proposition 5: Two lines are parallel if they do not intersect as they approach infinity.

The Maya geometric propositions are:

Maya Proposition 1: A point is used for infinity; a line is used to indicate five units; the

Canamayté Quadrivertex is formed by line segments.

Maya Proposition 2: The lines that calculate the solstice angles can be extended.

Maya Proposition 3: For the Maya, the Earth is the center of a circle; the center of the universe

is the sun; the radius of such circle was the segment given by the distance from the Earth to the


32

Moon; The circle was observed in the phases of the Moon indicated in the Canamayté

Quadrivertex.

Maya Proposition 4: In the cross inserted on the Canamayté Quadrivertex all angles are right

angles; the four interior squares and all the interior angles are congruent.

Maya Proposition 5: The Canamayté Quadrivertex is a geometrical structure with opposite sides

parallel, such that if they are extended toward infinity, they will never intersect.

If the had axiomatized their observations, what knowledge would we have today? Finally,

as a point of interest it would be worth while to provide some brief detail concerning the Maya

educational system. At the top of the educational ladder was the leader who was both a high

priest (Ahau Can) and a Maya noble. Under the leader were the master scribes, priests, and

teachers as well as writers. This social class taught sciences and wrote books about them.

Mathematics was recognized as such an important discipline that depictions of scribes who were

adept at it appear in the iconography of Maya artists. The mathematical identity of these

individuals was depicted with either the Maya bar-and-dot numeral coming out of their mouths

or with a number of scrolls carried under the armpit.

Section 4: The Mathematics of the Inca

The Inca (also spelled Inka) empire encompassed more than 375,000 square miles high in

the Andes Mountains and included what is now all of Peru, Ecuador, Bolivia, and portions of

Chile and Argentina. The Inca society was a complex culture comprised of three to five million

people that existed from around 1400 to 1560 A.D. Four brothers and four sisters conquered this

new homeland—they called themselves Inca. They were lords appointed by their god Viracocha

to thus bring civilization to the world. The Inca set upon their neighbors with great tact and

political savvy. Using deceit, bribery, and military force the Inca were able to secure the rich


33

agricultural valley of Cuzco. Though the Inca remained a modest political power until the

warrior king Pachacuti ascended the Inca throne in 1438. Under his leadership there was great

engineering and architectural expansion projects which included, highways, fortresses,

storehouses, and schools. Pachacuti’s successors continued his work and with a century the Inca

Empire became the largest pre-Columbian kingdom in the Americas. The Inca achieve

consolidation by the overlay of a common state religion and a common language—Quechua. All

in all the Inca can be described as a very efficacious culture, civilization, and state. When the

Spanish arrived to conquer, the Inca Empire had already existed for about 100 years. Within

thirty years—the number of years used to designate one human generation—the Inca state was

completely destroyed.

The culture of the Spaniards at the time of the conquest was quite remote from our present

culture. In fact many of us do not share with them, for instance, a fear for the Devil, even though

we are a part of the tradition that invented the concept. Now just imagine if for a moment, what

if anything could the Spaniards could have had in common with the Inca—absolutely nothing.

What we know about the Inca is provided only within a Spanish framework where many of the

accounts and historical records could have been distorted in the process interpretation process

from one culture, the Inca, via another culture, the Spanish. Now over four hundred and fifty

years have passed since the destruction of the Inca and the interpretation process is again

transmitted from perhaps a second culture, the Spanish, to a third or fourth culture, North

American. It is quite possible that there are some things in one culture for which there is no

counterpart elsewhere. Because the Inca did not have writing in the western sense, we know

very little about them. What we know about them is information that is secondary in that we are

unable to know about them in their own words. The existing information is fragmentary and


34

some of this information comes from European chronicles written by those who were part of the

very group that justified the Inca destruction on the basis of cultural superiority.

At the height of its power during the sixteenth century, the Inca Empire created a state that

controlled its population. The Inca society can be characterized as methodical, highly organized,

concerned with detail, and intensive data users. The economic organizational structure of Inca

society was based on a heavy reliance on the use of statistics. The sociopolitical structure of the

Inca Empire was lead by bureaucrats who supervised and regulated the minutest details in the

lives of its citizens. Clothing, food, housing, and employment were monitored by their central

government. The Inca Empire was possibly one of the most controlled societies in history. The

Inca government was quite efficacious. How did they administered and managed such a vast and

tightly controlled society without a form of writing? The answer to this question is simply—the

Quipu.

Researchers today agree that the writings of Cieza de León—the first person to write about

the particulars of the quipu—are probably the most reliable. Cieza de León thought of the Incas

as victims. His feelings are clearly documented in his writings; he wrote, “wherever the Spanish

have passed, conquering and discovering, it is as though a fire had gone, destroying everything it

passed.” Cieza de León was a 29-year old soldier who was basically told where and when to go.

In one of his journal entries, he writes about the superb road system built by the Inca and points

out that those roads were far better than the roads knew as a child in Spain—he was struck by

intricacy, beauty, and craftsmanship of the buildings and irrigation system. He also wrote about

the surplus of goods that were distributed to the poor and elderly.

Quipu is a Quechua word meaning knot. A quipu resembles a mop that has had better

days. A quipu is a collection of knotted strings made of a variety of colors. Initially, the quipu


35

was thought of as a primitive artifact with little aesthetic appeal. The quipu, therefore, was a

logical-numerical data-recording instrument for the Inca. The quipu was to carry messages that

had to be clear, concise, and portable from one region to another region that may have been

separated by hundreds or even thousands of miles. Quipu makers were responsible for coding

and de-coding messages. The quipus were carried from post-house to post-house by well trained

runners that used the extensive road system that was engineered by the Inca. These trained

runners were called chasquis who were stationed about a mile apart along the highway. The

chasquis running at top speeds in mountanous terrain delivered messages from place to place.

The post-houses were stationed by two runners all of whom ran a grand total of about 300 to 350

miles in 24 hours. So a message from the capital of Cuzco could have been transmitted within a

24-hour period.

The information on a quipu was recorded in a base-10 positional number system. The

definitions used in the quipu system consisted of three types: A single knot representing powers

of 10, that is, 10n ; a figure eight knot which always denoted the number 1; and a 5-turn slip knot.

The absence of knots denoted zero. In combination these knots represented the numbers 2, 3, 4,

5, 6, 7, 8, and 9. The quipu consisted of the:

1. Main cord,

2. Pendant cord,

3. Top cord,

4. Subsidiary cords of various hierarchical levels, and

5. Dangle end cord.

The quipu worked as a statistical instrument in that it maintained information that was

classified in terms of categorical and hierarchical data. On the quipus accounts were kept of


36

each town: How many people lived there, how many in each family, what each person

produced, and so on. According to Garcilaso de la Vega, an Inca historian, the Inca king

maintained two official quipu translators capable of receiving and sending complex messages by

quipu, analogous to the way a modern computer sends messages. The quipu varied in size from

only 3 attached cords to about 3,000.

Some examples or scenarios of the kind of problems that could be solved by using the

mathematics of the quipu might include the following scenarios, the first one of which could be

represented in a quipu scheme.

Scenario 1

Three sheds are built. They are different in sizes but each has walls made of cinder block,

a floor and roof made of wooden boards. The materials used for the first shed are 284 cinder

blocks, 100 pounds of mortar, 28 boards, and 200 pounds of nails. For the second and third shed

respectively the materials consists of 244 cider blocks, 85 pounds of mortar, 24 boards, 170

pounds of nails; and 364 cinder blocks, 150 pounds of mortar, 51 boards, and 400 pounds of

nails. In this scenario the quipu could be shown schematically along with cord placement, cord

color, knot types, and relative knot placement. For more information on this particular scenario,

the reader should consult the book Mathematics of the Incas: Code of the Quipu by Ascher and

Ascher (1997).

Scenario 2

A small community consists of four families. The families have the following members:

Family 1: Man, woman, man’s mother, man’s father, two children

Family 2: Man, woman, woman’s older sister, four children

Family 3: Man, woman


37

Family 4: Man, woman, three children

One hundred potatoes and 242 ears of maize (corn) are to be distributed among the families.

Each person in the community is to get the same share. However no potatoes or ears of maize

are to be chopped in pieces. The distribution among families is to be carried out with whole

items. Any excess is to be divided between the two largest families who will distribute it to the

oldest members of their household. (a) Calculate the number of potatoes and the number of ears

of maize to be distributed to each family, (b) for each family, calculate the number of potatoes to

be distributed to adults and the number to be distributed to children. Do the same for the ears of

maize. Write the results in tabular form.

There are 20 people in the community. Since there are 100 potatoes and each person is to

get the same share, each person receives 5 potatoes. Family 1 has six people and they get 30

potatoes, Family 2 with seven people receives 35 potatoes and Family 3 with only two people

gets 10 potatoes. Twenty-five potatoes will go to Family 4 with 5 people. There are 242 ears of

maize to be divided equally among twenty people. Each person’s share is 12. However leaving

two ears of maize. The two largest families are Families 1 and 2 so each gets one additional ear

of maize. Family 1 with six people gets 6 12 1 ears of maize, Family 2 with seven people

receives 7 12 1 ears of maize, Family 3 gets 2 12 ears of maize, and Family 4 gets 5 12 ears

of maize.

TABLE 1

(Potatoes and Maize Distribution for Four Families)

Family 1 Family 2 Family 3 Family 4 Potatoes 30 35 10 25 Maize 73 85 24 60


38

TABLE 2

(Potatoes)

Family 1 Family 2 Family 3 Family 4 Adults 20 15 10 10

Children 10 20 -- 15

TABLE 3

(Maize)

Family 1 Family 2 Family 3 Family 4 Adults 49 37 24 24

Children 24 48 -- 36

Since there were 100 potatoes and 242 ears of maize to be distributed in equal proportions

to each of the four families then the distribution of proportions are, therefore, the same within

that limitation or constraint. The probability distribution of potatoes and maize for Table 1

above is:

TABLE 4

(Probability Distribution Table for Families)

Family 1 Family 2 Family 3 Family 4 Potatoes 0.300 0.350 0.100 0.250 Maize 0.302 0.351 0.099 0.248

The following two scenarios illustrates the logical structure and cross-categorization of

information that can be easily coded in a quipu.

Scenario 3

Suppose there are two storehouses. The first one contained 375 potatoes, 520 ears of

maize, and 400 measures of beans. The second storehouse contained 423 potatoes, 305 ears of


39

maize, and 180 measures of beans. By color coding each commodity, the information could be

easily displayed in into two groups, one for each storehouse. The reader may wish to illustrate

the information in tabular form. This scenario illustrates how the Inca used categorical data to

obtain or represent information needed at that point in time.

Scenario 4

This scenario demonstrates a hierarchical structure. Suppose an Inca supervisor was in

charge of overseeing training programs for weavers in three towns. In the first town, the master

weaver’s instructions to the apprentices resulted in 21 and 23 blankets completed in one month.

At the second town, two apprentices produced 24 and 20 blankets in the same month. Finally, at

the third town, the apprentices produced 25 and 31 blankets. This information could be simply

coded in a quipu network.

After the Spanish conquest of the Inca, the system of quipus was lost. Finally, in relation

to the quipu, cord placement, color coding, and the representation of numbers constitutes feature

that are combined, re-combined, and defined in such a way to convey a logical, systematic, and

coherent information or data concerning the demands and needs of a large population. The quipu

was at the center of the highly organized society of the Inca.

Section 5: The Mathematics of Africa

The continent of Africa has many interesting countries with a variety of beautiful customs.

The mathematics of Africa is often quite practical in nature because farming, raising animals,

and the trading of commodities. All of these activities have been of great importance to most

Africans. For example, the Asante people of the Gold Coast, which today is modern Ghana,

were a wealthy and powerful nation that used gold as a form of money. The Asante invented a

special scale and system of weights to measure gold in their business dealings. The designs of


40

the scales were artistic representations of birds, animals, geometric shapes, and men. The

purpose of the designs was to remind members of society to be generous and to treat others

fairly.

Drums have for centuries been an important mode of communication in Africa. A by-

product of this particular mode of communication is that the use of drums has greatly affected

the development of speech. This is perhaps the reason why Africans often use many finger

gestures when speaking. The genesis of these gestures may have their roots in the movement of

the beat of the drums. It is not uncommon for Africans to simultaneously pronounce and gesture

numbers. The Africans have developed a variety of games that sharpen arithmetic skills. Some

of these games include Bao (Tanzania), Gabatta (Ethiopia), Omweso (Uganda), and Oware—a

board game of strategy played by kings and chiefs. The interested reader may want to learn

more about the particulars of African games in Claudia Zaslavsky’s book Africa Counts:

Number and Pattern in African Culture.

It is interesting to point out that African number words, in general, express arithmetic

operations explicitly and the expansion of the number system grew as a result of economic

expansion and other societal demands. As the African economy grew more complex it required

more sophisticated ways of counting, collecting of taxes, and taking a population census. For

example the Yoruba people of southwest Nigeria use an interesting principle of subtraction to

represent, for instance, the number forty-six. In fact, the number forty-six is literally “twenty in

three ways less ten less four” or (20 3) 10 4 . Another example is the Igbo number word for

three hundred which is “ohu iri noohu ise” or (20 10) (20 5) . Besides the tradition of oral

history, many Africans kept all sorts of numerical records such as the passage of time, score of

games, and financial transactions on tally sticks as well as on a variety of knotted strings.


41

It is worthwhile to mention that during the beginning of the twentieth century W. S.

Routledge, a British investigator, provided a description of the excellent engineering used by the

Kikuyu of Kenya in the building of suspension bridges. He also pointed out that England could

make no improvement in the structural design of these bridges. From an intuitive perspective

one can certainly appreciate that the Kikuyu must have had used certain mathematical principles

in order to achieve a stable engineering structure suitable for the transportation of man and beast.

Many African societies have used problem-solving strategies that are based on

mathematical reasoning. For example, the Kepple children of Liberia recount a familiar story

that requires the use of logic in its solution. The scenario deals with a man who has a leopard, a

goat, and a pile of cassava leaves to be transported across a river. The boat cannot carry more

than one at a time, besides the man himself. Obviously the goat cannot be left alone with the

leopard. The goat on the other hand will eat the cassava leaves if the man does not guard them.

There is no mutual attraction between the leopard and the cassava leaves. How can the man take

himself, the leopard, the goat and the cassava leaves across the river?

Logical problems are closely related to games of strategy but often require much shorter

path to success. These kinds of problems like the one of the Kepple of Liberia present a logical

challenge because the story presents a goal and specifies constraints on how the goal can be

achieved. River-crossing puzzles have been found not only in western cultures, but also in many

African cultures. These kinds of puzzles are not exclusive to the Kepple alone. The western

origin of these puzzles is attributed to a set of 53 problems designed to challenge youthful minds.

The text Propositiones ad Acuendos Iuvenes containing the set of puzzles was circulated around

the year 1000. It is generally agreed upon that the author of the text was a theologian named

Alcuin of York who lived from the year 735 to around the year 805. The origins of such puzzles


42

may not be as important as the fact that the puzzles are found in various cultures. A similar

puzzle to that of the Kepple is found in a folk story on the Cape Verde Islands just off the

western coast of Africa.

The general structure of the river-crossing puzzle can be stated as follows: A, B, and C

must be transported across a river by a human who can only transport two of A, B, and C at one

time. Neither A nor C can be left alone with B on either shore. The following tables illustrate

three possible solutions.

TABLE 1

(Solution Number 1)

---- Side 1 In transit Side 2 ---- Man, A, B, C ---- ----

Round trip 1 B Man, A, C Man

---- A, C

Last trip ---- Man, B A, C ---- ---- ---- Man, A, B, C

TABLE 2

(Solution Number 2)


Round trip 1 A, C Man, B Man

---- B

Last trip ---- Man, A, C B ---- ---- ---- Man, A, B, C


43

TABLE 3

(Solution Number 3)


Round trip 1 C Man, A, B Man, B

---- A

Last trip ---- Man, B, C A ---- ---- ---- Man, A, B, C

Games and puzzles of another culture provide one with an unusual window into the ideas

of others; one can certainly appreciate, in some limited sense, these ideas. Though the context

and ambiance of the problem or puzzle may be different, it cannot be denied that if one engages

in the problem solving process it can be concluded that as humans we share the same or at least

similar mental processes.

Finally, problems dealing with the particulars of graph theory have been evident in the sand

tracing among the Bushoong children from the Republic of Zaire. From a geometric point of

view graph theory is concerned with arrays of points called vertices and interconnected lines

called edges. The field of graph theory has been very important in our culture in that it provides

many solutions, applications, and approaches useful in the study of flows through networks. For

example, the study of traffic flow involves intersections or vertices interconnected by roads

which are considered to be edges. In 1905 a European ethnologist was challenged to trace

certain sand figures with the instructions that each line be traced once and only once without

lifting his finger from the ground. The ethnologist was unable to meet the challenge at hand. To

better appreciate the particulars related to sand tracing it is necessary to briefly discuss some of

the particulars related to graph theory. Hence, in reference to graph theory, a connected graph is

one in which each vertex is joined to every other one via some set of edges. A planar graph is


44

one that lies entirely on the plane; that is, it not need be depicted as a graph that rises outside of,

say, this flat piece of paper. A freeway overpass and the road beneath it, for example, would not

represent a planar figure. A classical question in graph theory is: For a connected planar graph,

can a continuous path be found that covers each edge once and only once? And if such path

exists, can it end at the same vertex as it started? According to mathematical historians, this is

the question that is said to have inspired the beginnings of graph theory by the mathematician

Leonhard Euler. According to the story, there were seven bridges in Konigsberg (East Purssia),

where he lived. The bridges spanned a forked river that separated the land into four distinct

pieces of land. The locals were interested in knowing if, on their Sunday walks they could start

from home cross the bridges once and only once and end at home. Euler demonstrated that for

that particular situation such a route was impossible and also began to consider the more general

question. Between Euler in 1736 and Hierholzer some 130 years later, a complete answer was

found. In order to state this result another definition is needed, that is, the degree of a vertex.

The degree of a vertex is the number of edges emanating from it—a vertex is odd if its degree is

odd and even if its degree is even. The answer to the question is that not all connected planar

graphs can be traced continuously covering each edge once and only once. However, if such a

path can be found, it is therefore called—in honor of Leonhard Euler—an Eulerian path.

It is important to discuss some sociological factors underlying the growth of mathematical

pursuits in the continent of Africa. The first requirement for the pursuit of mathematical

activities is the existence of a society that is well structured and organized in such a way that an

economic surplus could support the division of labor. The commercial interaction with other

cultures is likely to bring about a stimulus for scientific growth. This commercial interaction is

based greatly on cultural exchange in that regions of the world that are culturally isolated are


45

unlikely to provide the necessary stimulus or impetus for greater mathematical growth. For

example the Greeks lived in an atmosphere of freedom from political and religious despotism

and hence were probably the first one to develop a logical mathematical system. However, it is

worthwhile to point out that there was a time when in Europe, mathematics was looked upon

with great fear because of its supposed magical power of numbers. In fact, during the latter part

of the seventeenth century, at the time of Galileo’s persecution, mathematicians were denounced

as heretics. This of course was probably the result of ignorance on the part of religious officials.

Outside of Egypt, there were a few cultural and commercial centers in Africa. For example

Timbuktu, in ancient Mali was an international city—a center of commerce and culture. In the

early sixteenth century, Malian society was comprised of many learned members of society.

According to researchers, there has yet not been found any evidence of original work on

pure mathematics in the African centers of the Islamic world in that Islam does not make

distinctions based on ethnicity. There are several reasons that have caused obstruction to the

development of mathematical pursuits in Africa. These reasons are listed as follows:

1. Surviving documents have not been translated nor published.

2. In many cultures the only history that exists is oral history.

3. The geography of Africa is impenetrable with few natural harbors, non-navigable rivers,

arid desert, high mountains, and rain forest—an inhospitable continent discouraging growth of

stable agricultural and urban communities in support of world trade and commerce.

4. For many societies the economy consisted of little more of just survival.

5. Slave trade and destruction of African culture by the Europeans in the 1900—

Colonization.


46

6. Introduction of European religious values and principles as they relate to loyalty to

government and their masters—Adherence to Christian views as the only means to salvation.

7. European use of African labor for the purpose of maximizing profit while minimizing

cost.

8. European destruction and looting of archaeological sites.

Section 6: The Mathematics of Arabia

The classic designs from the decorative pages of the Koran and beautifully tiled mosques

proclaim not only the artistic discipline of the Arab people but also their appreciation and skill in

mathematics. This skill and appreciation is rooted in their inspiration and desire to contemplate

the mystery of the mercy and greatness of Allah. The Arab-Islamic calligraphers have produced

magnificent patterns based on the particulars of geometry, complex symmetries, and algebraic

relations.

During the eight century A.D., under caliph al-Mansur (712-774 A.D.), the city of Baghdad

dominated the world of mathematics and science. The great Hindu works of astronomy,

mathematics, and medicine were translated to Arabic and were carefully studied. In fact, the

ancient Greek classics were saved for our present time by the work of many prominent Arab

scholars.

The intellectual revolution that was begun by al-Mansur culminated in the tradition of

learning and inquiry at Baghdad. The successors of al-Mansur produced a golden age for Arabic

mathematics and science during the 9th and 10th centuries. In fact, some of the western

contributions to mathematics and science have their genesis in the Arab-Islamic world.

Unfortunately in our western tradition we do not acknowledge nor recognize the contributions of

people from other cultures. The early description of pulmonary circulation of the blood by ibn


47

al-Nafis is usually attributed to Harvey even though there are records that go as far back as the

time of ancient China. Also, the first known statement about the refraction of light credited to

Isaac Newton was known by ibn al-Hayatham. Also credited to Newton was the discussion and

argumentation of gravity by al-Khazin. In the field of biology, the concept and origination of the

theory of evolution credited to Darwin was also known and formulated by ibn-Miskawayh, and

the scientifc method credited to Francis Bacon was also evident in the scientific works of ibn

Sina, ibn al-Haytahm, and al-Biruni. Details about the specific Arab-Islamic contributions to

science and mathematics can be found in C. C. Gillespie (Ed.). (1969). Dictionary of Scientific

Biographies, a 15 volume reference published in New York by Charles Scribner’s Sons; and R.

M. Savory (1976). Introduction to Islamic Civilization published in England by Cambridge

University Press.

The eminent mathematicians and scientists at the time of al-Mansur came to Baghdad to

acquire a scientific and mathematical education and to study subjects such as geometry, number

theory, algebra, and trigonometry. Many renowned astronomical observatories were built and

medicine was skillfully and successfully practiced. This great body of knowledge and intellect

was preserved and passed on to Europe. In fact, it was the prodigious intellectual Arab-Islamic

tradition that made possible the advances in mathematics and the sciences in our present time.

Perhaps, the most well known influence of Arabic mathematics on western mathematics is

found in the word ALGEBRA derived from the famous Arab mathematician Mohammed ibn

Musa al-Khwarizmi (i.e., Mohammad, the father of Jafar and the son of Musa, from Khwarizmi).

His book, Hisab al-jabar w’al-mukabala (825 A.D.) which can be translated into Calculation by

Restoration and Reduction was the first unified organization of the subject of algebra.


48

One of the least known contributions of Arab mathematics that was fundamental to the

development of western mathematics was made during the fifteenth century in that scholars of

that time made use of decimal fractions long before their introduction in Europe. In the work

entitled Circumference (1430 A.D.) by al-Kashi, Director of the Observatory at Samarkad

approximated the value of to sixteen decimal places. He wrote this approximation in the

innovative decimal form without the use of a period as 3 1415926535898732.

In relation to major Arab mathematicians little is known about the life of al-Khwarizmi

other than the fact that he was born around the year 780 and died in the year 850. In addition to

the unified work of algebra, al-Khwarizmi also wrote Algorithmi de Numero Indorum, that is,

Calculation with Indian Numerals whose original Arabic version no longer exists. Other

contributions made by al-Khwarizmi included: The measurement of the length of one degree at

the latitude of Baghdad which was 91 kilometers—an accurate result; and the use of

astronomical observations to find the latitude and longitude of 1200 important places on the

Earth’s surface, including cities, rivers and lakes.

The second half of al-Khawarizmi’s algebra book contains a series of problems related to

the laws of inheritance in accordance to Islamic tradition. The problems presented in his book

are straightforward and easy to follow. One such problem presents a scenario where a woman

dies and leaves a husband, a son, and three daughters. She also leaves a bequest consisting of

1 1

8 7 her estate to a stranger and according to Islamic law, per al-Khawarizmi, the shares of her

estate that go to each of her beneficiaries is as follows: The stranger receives 1 1 15

8 7 56 of the

estate leaving 41

56 to be distributed among her heirs. Her husband receives one-quarter of what


49

remains, that is, 1 41 41

( )4 56 224

. The son and the three daughters receive their shares in the

ratio 2 :1:1:1. In other words, the son’s share is 2

5 of the estate after the stranger and her

husband were given their individual bequests. If the estate is divided into 5 224 1120 equal

parts, then the shares received by each beneficiary will be as follows:

1. Stranger inherits 15

1120 30056

parts

2. Husband inherits 41

1120 205224

parts

3. Son inherits 2

(1120 505) 2465 parts

4. Each daughter inherits 1

(1120 505) 1235 parts

Al-Khwarizmi also identified and provided six different types of equations. He also

provided rules for solving various types of equations. The general forms of equations outlined

by al-Khawarizmi are:

1. Roots equal squares, 2bx ax ;

2. Roots equal numbers, bx c ;

3. Squares equal numbers, 2ax c ;

4. Squares and roots equal numbers, 2ax bx c ;

5. Roots and numbers equal squares, 2bx c ax ; and

6. Squares and numbers equal roots, 2ax c bx .


50

In a particular example, al-Khawarizmi demonstrated how to solve a common problem,

illustrated here in English and in modern notation that is usually encountered in intermediate

algebra courses. The problem at hand is to solve the equation 2 10 39x x whose solution is

2 10 39x x

2( 5) 39 25 64x

5 64 8x

3x

The negative root 13x is ignored. Variations of the rules to solve the above type of

equations are found in Babylonian and Indian mathematics, and it is probable that the algorithm

may have come from either or both of these sources.

Another eminent Arab-Islamic scholar was Abul Hassan Thabit ibn Qurra Marwan al-

Harrani who was born in Harran, northern Mesopotamia and lived between 836 until his death in

the year 901. He translated many of the Greek works including Euclid’s Elements as well as

several works by Apollonius’ Conics and Ptolemy’s Almagest. These works were in turn

translated into Latin by Gherardo of Cremona during the 12th century and from that point on the

work had a momentous impact in medieval Europe. His commentaries on the particulars of the

quadrature of the parabola has been described as one of the most innovative approaches known

to human kind prior to the emergence of integral calculus.

Omar Khayyam’s The Rubaiyat is one of the best known works of literature that has been

translated to various languages. Edward Fitzgerald was the translator of many of the quatrains

into the English language in the middle of the nineteenth century. However, what is not widely

known outside of the Arab-Islamic world is the fact that the well-known poet Omar Khayyam


51

was also an accomplished mathematician, astronomer, and philosopher. Abul-Fath Umar ibn

Ibrahim al-Khayyami was born around the year 1040 at Nishapur in Khurasan which is now part

of Iran. The name Khayyami suggests that either Omar or his ancestry were tent-makers. He

wrote impressive works in algebra including the classification of equations according to their

degree. He also provided rules similar to the ones used in today’s algebra courses, to solve

quadratic equations. According to Dr. David Henderson of Cornell University, Omar Khayyam

and not the Italian mathematician Gerolamo Cardano (1501-1576) who first found a general

solution for the cubic equation. Dr. Henderson wrote a detailed exposition of Khayyam’s

geometric solution of the general cubic equation in a textbook for a course, in Arabic, at the

University of Birzeit in 1981.

Omar Khayyam’s solution to the cubic equation of the form 3x cx d is what he referred

to as the equation for the case in which a cube and sides equal to a number. Khayyam was

careful in his adherence to the Greek idea of homogeneity. In other words, he considered the

cubic equation as an equation between solids. Since the variable x represents a side of a cube,

then c must represent an area that can be expressed as a square so that term cx is a solid, while d

itself represents a solid. To construct the solution, Khayyam sets AB equal in length to a side of

the square c, that is, AB c . He then constructs BC perpendicular to AB so that

2( )BC AB d , or d

BCc

. He then extends AB in the direction Z opposite to AB and

constructs a parabola with vertex B, axis BZ, and parameter AB. In modern notation, this

parabola has the equation 2 ( )x c y . In a similar fashion he then constructs a semicircle on the

line BC. Its equation is


52

2 2 2( ) ( )2 2

d dx y

c c

2( )d

x x yc

The circle and the parabola intersect at the point D. It is the x coordinate of this point, here

represented by the line BE, which provides the solution to the equation. In addition to the

general solution for the cubic equation, Omar Khayyam also wrote about the triangular array of

binomial coefficients known as Pascal’s Triangle as illustrated below.

1 1 2 1

1 3 3 1 1 4 6 4 1

. . . . . . . . . . Thabit Ibn Qurra is another Arab scholar who provided a generalization and a proof of the

Pythagorean theorem. The Arab-Islamic contributions to the study of trigonometry include the

work by Abul Wafa in his book Zij almagesti where he provides a systematic treatment of the six

trigonometric functions (i.e., sine, cosine, tangent, secant, cosecant, and cotangent). The familiar

trigonometric identity that one encounters in trigonometric courses is also found in his work.

This identity is:

sin( ) sin cos cos sin

Nasir al-Din al-Tusi also provided the following rule, here in modern notation, for the

solution of any triangle with angles ABC and sides a, b, c.

sin

sin

b r B

c r C

In addition to the development of the sine tables for various angles, the Arabs, in particular al-

Kashi, provided the relationship that for any angle , the following identity holds to be true:

3sin(3 ) 3sin 4sin


53

The above identity is also usually found in many trigonometry courses either as an illustration on

how to prove identities or as an exercise.

Section 7: The Mathematics of China

The background and sources pertaining to the particulars of Chinese mathematics begin

with dawn of Chinese civilization to the end of the Ming dinasty in 1260 to 1644 A.D. and their

contact with European civilization. The oldest source of Chinese mathematics comes from the

Chou Pei Suan Ching, that is, The Arithmetical Classic of the Gnomon and the Circular Paths of

Heaven. The book probably dates back to the Shang dinasty. The text contains details related to

the particulars of right-angle triangles and the Pythagorean triples along with a geometric proof.

The most influential mathematical text is the Chiu Chang Suan Shu, that is, The Nine Chapters

on the Mathematical Arts authored by Yang Hui in the year 1261. This particular text is as

revered by the Chinese in the same manner, and with the same respect, as Euclid’s Elements in

western society. The subjects covered in the Yang Hui’s book include arithmetic progressions,

decimal fractions, and quadratic equations with negative coefficients. Archeological

investigation has provided a text that could possibly be older than Yang Hui’s The Nine Chapters

on the Mathematical Arts. This investigation yielded a text on arithmetic—that may be older

than the Chou Pei Suan Ching—authored by the mathematician Suan Shu Shu.

During the Chin and Han dynasties there were noteworthy mathematicians including Hsu

Yue who wrote the Shu Shu Chi Yi, that is, Manual on the Traditions of the Mathematical Arts

discussing the particulars of calendar construction and magic squares which shall be presented

later in this section. Other mathematicians of stature include Sun Tsu who did much work in the

intermediate analysis and Tsu Chung Chih who probably lived during the Liu dinasty in the

period from 420 to 479. The book entitled The Ten Mathematical Manuals written by Shih Shu


54

remained as an important text in Chinese mathematics for several centuries. In the year 1247,

Chin Chiu Shao wrote the book entitled Shu Shu Chiu Chang which in English translates to Nine

Sections of Mathematics—a different text from the The Nine Chapters on the Mathematical Arts.

Chin Chiu Shao provided solutions to equations of various degrees as well as a discussion and

comments based on the work on intermediate analysis by Sun Tsu. Li Yeh’s book entitled Tshe

Hai Ching or The Sea Mirror of the Circle Measurements provided discussions on the

constructions of equations of various degrees from a given set of data. Between the years 1261

and 1275 Yang Hui wrote a series of works from which the most important and influential was

the Hsiang Chieh Chiu Chang Suan Fa Tsuan Lei which in English is entitled Detailed Analysis

of the Mathematical Methods in the Nine Chapters. The text exposes original work in a series of

topics including series, quadratic equations, and higher-order equations. Another eminent

Chinese mathematician was Chu Shih Chieh who wrote two books. The first book entitled Suan

Shu Meng or Introduction to Mathematical Studies written in 1299. The second book Szu Yuen

Yu Chien or The Precious Mirror of the Four Elements was written in 1303. The texts provide

details concerning Pascal’s triangle as well as the solution to simultaneous equations by the

method that we now called matrix operations. The Chinese also made impressive contributions

in trigonometry. However, their work was probably based on the work of the Arabs.

The Chinese used counting rods colored in red to represent positive and black colored rods

to represent positive quantities to solve systems of linear equations. One such problem from the

Chiu Chang Suan Shu is the following system of three equations with three unknowns—the type

of problem that is normally encountered in an intermediate algebra or college algebra course.

2 3 8 32

6 2 62

3 21 3 0

x y z

x y z

x y z


55

The reader with an intermediate algebra background is invited to solve the above system of

linear equations.

The next topic of discussion in Chinese mathematics is the magic square that unfortunately

is of marginal interest in today’s mathematics curriculum. The following illustration is magic

square, i.e., Lo shu, of order three:

4 9 2 3 5 7 8 1 6

In reference to the above Lo shu, notice the following:

4 + 9 + 2 =15

3 + 5 + 7 = 15

8 + 1 + 6 = 15

4 + 3 + 8 = 15

9 + 5 + 1 = 15

2 + 7 + 6 = 15

4 + 5 + 6 = 15

2 + 5 + 8 = 15

The construction of the magic square or Lo shu of order three, according to Yang Hui is as

follows:

1. Arrange the numbers 1 to 3, 4 to 6, and 7 to 9 with a slant downward to the right as

follows:

1 4 2

7 5 3 8 6

9


56

2. Interchange 1 on the top with 9 at the bottom and 7 on the left with 3 on the right as

follows:

9 4 2

3 5 7 8 6

1

3. Move the 9 down to fill in the position between 4 and 2 and move the 1 up to fill the

position between 8 and 6 as follows:

4 9 2 3 5 7 8 1 6

There are more complex magic squares or Lo shu of higher orders, but Yang Hui provides no

explanation as to how higher order, namely five or greater, magic squares or Lo shu are

constructed. In 1650 Chang Chao produced a magic square of order 10 and in 1880 Pao Chi

Shou provided constructions for three dimensional magic cubes, spheres, and tetrahedrons. The

following are examples of magic squares or Lo shus of order 4, 5, and 7.

MAGIC SQUARE (LO SHU) OF ORDER 4

2 16 13 3 11 5 8 10 7 9 12 6 14 4 1 15


1 23 16 4 21 15 14 7 18 11 24 17 13 9 2 20 8 19 12 6 5 3 10 22 25


57


46 8 16 20 29 7 49 3 40 12 14 18 41 47 44 37 33 23 19 13 6 28 15 11 25 39 35 22 5 24 31 27 17 26 45 48 9 38 36 32 10 2 1 42 34 30 21 43 4

Yang Hui’s book the Chiu Chang or The Nine Chapters on the Mathematical Arts consists

of the following nine chapters:

1. Land Surveying

2. Percentages and Proportions

3. Distributions by Proportions

4. Extraction of Square and Cube Roots

5. Engineering Mathematics

6. Fair Taxation

7. Excess and Deficiency (topics in linear algebra and determinants)

8. Solutions of Simultaneous Equations and the Method of Rectangular Arrays

9. Right-angle Triangles

A few of the accomplishments and historical facts from ancient China are listed below:

1. Estimation of by Liu Hui and later by Tsu Chung Chih as

3.1415926 3.1415927

2. Development of Pascal’s triangle and the binomial coefficients which cannot be credited

to any single individual and its used are found in India and China.


58

3. Development of Horner’s Method named after the British mathematician William

George Horner who published his work in 1819 as a method to find the roots of equations of the

type 10 1 1( ) 0n n

n nf x a x a x a x b . The procedure existed and was used in China five

hundred years prior to Horner’s rediscovery.

4. The use of the concept of determinants to solve systems of equations—a concept that

was later expanded by the Japanese mathematician Seki Kowa in 1683—ten years before

Leibniz, to whom mathematical historian usually attribute the discovery of determinants.

5. The earliest proof of the Kou Ku Theorem (i.e., the Pythagorean Theorem) by the

mathematician Chou Pei Suan Chin.

6. Development of the Rule of Three used for ratio and proportions

7. The use of zero

By far, the most revered mathematician coming out of China is Chin Chiu Shao. Finally,

during the seventeenth and eighteenth centuries, Europe became aware of the Chinese

intellectual heritage and the man of science and mathematics such as Voltaire, Leibniz, and

others were influenced by this Chinese heritage via translation of the texts by the Jesuits.

Section 8: The Mathematics of India

The mathematics of India is comprised of six periods that will be briefly outlined as

follows:

1. First Period (3000-1500 B.C.): This period is characterized by mathematical

developments related to the measurement of weights which included artistic designs of Hindu

scales. The brick technology of the time provided the impetus for the building of Vedic altars.


59

2. Second Period (1500-500 B.C.): This period is characterized by the particulars related

to problems in astronomy, arithmetic, and Vedic geometry. Notable mathematicians of this

period included Budhayana, Apastamba, and Katyayana.

3. Third Period (500-200 B.C.): This period is characterized by advances in number

theory, combinatorial mathematics, the binomial theorem, astronomy.

4. Fourth Period (200 B.C.-400A.D.): Jaina mathematics and rules of mathematical

operations including decimal place notation were characteristics of this period. Evidence of the

first use of zero is also particular to this period. There were also notable contributions to the

study of algebra, namely, solutions to simple equations, simultaneous and quadratic equations,

extraction of square roots, and details of how to represent unknown quantities and negative signs.

5. Fifth Period (400-1200): The fifth period is also known as the Classical Period of

Indian Mathematics. During this period there were a number of notable mathematical

manuscripts that included the following: The Bakhshali, Pancha Siddhantika, Araybhatiya

Bhasya, Maha Bhaskariya, Brahma Sputa Siddhanta, Patiganiata, Ganita Sara, Samgraha,

Gantilaka, Lilavati, and Bijaganita. Notable mathematicians of this period included Aryabhata I,

Varahmihira, Bhaskara I, Brahmagupta, Sridhara, Mahavira, and Bhaskara II (also known as

Bhaskaracharya)

6. Sixth Period (1200-1600): This period experienced a decline of mathematics and

learning in the north of India. However, the period is characterized by the rise of the Kerala

School of Mathematics and Astronomy. There were impressive contributions made in the

particulars of infinite series and mathematical analysis. The notable mathematicians of this

period included Narayana, Madhava of Kerala, and Nilakantha.


60

The number theorist Srinivasa Ramanujan (1887-1920) also made impressive and well

respected contributions to mathematics in his notebooks. Some of these contributions included

infinite series, trigonometric and circular functions. The writing style of the Indians was in the

form of sutras. This kind of writing is characterized by brevity and use of poetic form designed

to capture the essence of an argument or result. The sutra form of writing was adopted by

schools of philosophy and the sciences. This form of writing was also by authors of books on

statecraft and sex manuals such as the well-known Kamasastra. The three characteristics found

in the Sulbasutras geometry were:

1. Geometric results and theorems explicitly stated;

2. Procedures for constructing different shapes of altars; and

3. Algorithmic devices contained in the geometric results and theorems, and the

procedures for the construction of altars.

The mathematician Rama who lived in the middle of the fifteenth century provided an

approximation for 2 that was found on three different sulbasutrases. That approximation is

1 1 1 1 12 1

3 (3)(4) (3)(4)(34) (3)(4)(34)(33) (3)(4)(34)(34) .

The Amuyoga Dwara Sutra provides the following lists of successive squares and square

roots of numbers:

2 2 2 2 2 2, ( ) , (( ) ) ,...a a a

, , , ,...a a a a

The Bakhshali Manuscript provides the following rule for the extraction of roots:


61

2

2( )2

2 2( )2

rr aA a r a

ra aa

where 2a is the perfect square nearest to A and 2r A a . For example:

25( )5 1241 6 6.4031

512 2(6 )12

The reader is invited to evaluate 2 and 3 by using the rule from the Bakhshali Manuscript.

The mathematician Aryabhatiya I wrote about a number of rules dealing with the methods

for solving simple as well as quadratic equations. He also provided correct general rules for

computing the sum of natural numbers and of their squares and cubes. He also provided a detail

exposition pertaining to the study of trigonometry. The first Indian artificial satellite that was

built and designed in 1975 by Indian scientists was appropriately named the “Arybhata.” The

work of Brahmagupta includes a discussion of mathematical series as well as the method for

generating the sines of intermediate angles. Mahavira was the best-known mathematician of the

ninth century. Unlike his predecessors his work was confined to mathematics—he was not an

astronomer. His contributions include:

1. Examination of operations with fractions, methodology for the decomposition of

integers and fractions into unit fractions;

2. Systematization of the Jaina work on combinatorial mathematics, namely, combinations

and permutations;

3. Solutions for various types of quadratic equations; and

4. Study of right-angle triangles from a geometric perspective.


62

The mathematician Sridhara wrote about the topics of arithmetic and measuration.

Bhaskara II provided insight in to the particulars related to combinations and permutations. He

also provided a method for solving intermediate equations of the form 2ax bx c y that was

later discover in the west by William Brouncker in 1657. The laws of signs and mathematical

operations with zero were the contribution of the mathematician Narayana Pandit.

The Bakhshali Manuscript proposes that 2 4

2

b ac bx

a

is a solution to the equation of

the form 2 0ax bx c . The mathematician Mahavira obtained the following general

solution—in modern notation—for an equation of the form 2ax bx c . His result was:

4( )

2

b cb a b

baax

The following trigonometric identities are normally studied in trigonometry courses and

unfortunately no recognition is ever given to their authors. In fact, these trigonometric identities

are usually presented without a multicultural perspective. The reader with a background in

trigonometry may wish to verify the validity of the identities.

2

16 ( )sin

5 4 ( )

(Bhaskara I)

1sin( 1) sin( ) sin( ) sin( 1) (sin( ))

225n n n n n (Aryabhata I)

cos sin( )2

(Varahamihira)

2 2sin cos 1 (Varahamihira)


63

2 2 21 1sin (sin (2 ) sin (2 )) (1 cos(2 ))

4 2ver (Varahamihira)

2 2 21 sin cos sin ( )2

(Brahmagupta)

1 sinsin( )

4 2 2

(Arybhata II)

sin( ) sin cos cos sin (Bhaskaracharya)

The formula for the power series for the inverse tangent that is presented in calculus

courses is generally known as the Gregory series after the Scottish mathematician James Gregory

derived it in 1667. However, the formula for the inverse tangent is provided in various

manuscripts, in particular, Narayana’s text entitled Kriyakramakari. Madhava of Kerala is

usually credited for the power series for 1tan x . The power series for the inverse tangent—

referred to here as Madhava-Gregory is

3 5 71tan

3 5 7

x x xx x for 1x

Madhava of Kerala also discovered the power series for the sine and cosine functions about three

hundred years before Isaac Newton. The series made their first appearance in Europe in 1676 in

a letter written by Newton to Secretary of the Royal Society, Henry Oldenburg. The series are:

3 5 7

sin3! 5! 7!

x x xx x

2 4 6

cos 12! 4! 6!

x x xx


64

In addition to the above series, Madhava also provided the following approximation for :

2 2 3

1 1 112(1 )

3 (5)(3 ) (7)(3 )

The following formulas for the summation of integers were derived by Nilakantha, who provides

an explanation of the relationship to piling rectangular strips of unit width. Nilakantha proves

that

( 1) ( 1)( 2)

2 6

n n n n n

The mathematician Narayana provided the following two well-known trigonometric

identities:

2 2sin cos sin( )sin( )A B A B A B

2 2sin sin sin ( ) sin ( )2 2

A B A BA B

A rule found in the work of Paramesvara for obtaining the radius “r” of a circle in which a

cyclic quadrilateral of sides a, b, c, and d is inscribed:

( )( )( )

( )( )( )( )

ab cd ac bd ad bcr

a b c d b c d a c d a b d a b c

A detail demonstration of this result is found in Bhaskaracharya’s Kriyakramakari. The result

makes its first appearance in Europe in 1782 in the work of L’Huilier. Bhaskaracharya also

contributed to the development of calculus and provided the results in relation to the differential

of the sine function, that is, (sin ) cosd x xdx . Finally, Bhaskaracharya was aware that when a

variable attained a maximum value, its differential vanishes. In fact, there are traces of the well-

known calculus result called the Mean Value Theorem—stated below—that is generally derived

or proved from Rolle’s Theorem.


65

The Mean Value Theorem states that if f is a function that satisfies the following

hypothesis:

1. f is continuous in the closed interval [a,b]

2. f is differentiable on the open interval (a,b)

Then there is a number c in (a,b) such that

( ) ( )'( )

f b f af c

b a

or ( ) ( ) '( )( )f b f a f c b a

The following two examples will provide the reader with an appreciation of how the Mean

Value Theorem works. These examples are limited in that they are illustrations of the Mean

Value Theorem with specific functions.

Consider the following specific function defined on the closed interval [-1,1] by

2( ) 3 2 5f x x x . The function is differentiable on the open interval (-1,1) and continuous on

the closed interval [-1,1] because the function is a polynomial and all polynomials are continuous

and differentiable over real numbers, . The derivative of the function is '( ) 6 2f x x .

Therefore, the hypothesis of the Mean Value Theorem is satisfied. The theorem states that if the

hypothesis is indeed satisfied, then there is a number c in (a,b), which in this specific instance,

the number c is between (-1,1) such that (i.e., applying the Mean Value Theorem):

( ) ( )'( )

f b f af c

b a

(1) ( 1) 10 66 2 2

1 ( 1) 2

f fc

6 0 0c c

0 ( 1,1)c


66

Suppose that another specific function defined on the closed interval [0,2] is 3( )f x x x .

Again, this function is a polynomial and hence it is both differentiable on the open interval (0,2)

and continuous on the closed interval [0,2]. The derivative of the function f(x) is 2'( ) 3 1f x x .

The hypothesis of the Mean Value Theorem is satisfied and therefore there is a number c in (0,2)

such that

( ) ( ) '( )( ) (2) (0) '( )(2 0)f b f a f c b a f f f c

Now 2'( ) 3 1f x x (2) 6, (0) 0f f , and 2'( ) 3 1f x x , so the equation above becomes

2 26 0 (3 1)(2 0) 6 6 2c c

Solving the equation 26 6 2c yields 2

3c , but the value of c must be in (0,2). Therefore,

this value is 2

3c . The meaning of the result is that there is a tangent line at the point

2

3c

in (0,2) that is parallel to the secant line that intersects the function 3( )f x x x at two distinct

points. The reader is invited to geometrically investigate this situation by sketching the function

along with a secant line and the tangent line at the point 2

3c .


67

Chapter 3

Ancient Egyptian Mathematics

The objective of this chapter is to provide a fair and accurate overview of the particulars

related to the development of the mathematics of ancient Egypt. This chapter will also challenge

the Eurocentric bias of historians and other scholars who have adversely criticized the

mathematics of ancient Egypt on the basis that their mathematics was primitive because of the

absence of formal axiomatic proofs. Most histories of mathematics treat the particulars of

ancient Egyptian mathematics very briefly. For example, many scholarly works of hundreds of

pages only dedicate or treat the mathematics of Egypt in less than at most three to ten pages even

though the ancient Egyptian culture has existed for at least three thousand years. Much of the

mathematics from China and India have also been ignored by many well-respected

mathematicians on the basis of cultural bias in that the work produced by other cultures has

traditionally been examined from a western perspective only. One can only conclude that many

authors and scholars of the histories of mathematics have not fully informed themselves of the

extent and nature of non-western mathematics.

It is true that the Egyptians did not demonstrate exactly how results, rules, and formulas

were established. However, the Egyptians nearly always proved that the numerical solution to

the problem at hand was always indeed correct for a particular value or set of values chosen. For

the Egyptians, this approach to mathematics constituted both, a method and a proof. Western

scholars and students alike argue that Egyptian mathematics is primitive and non-consequential

because they are inclined to use the argument that a proof must follow a logic and symbolic

structure in order for the mathematics to be considered rigorous. This argument has several


68

flaws in that a non-symbolic argument or proof can be quite rigorous when given a particular

value of a variable; the conditions for rigor are that the values given for a variable should be

typical, and that a further generalization to any values should be immediate. The rigor of ancient

Egyptian mathematics is implicit in the methodology.

The Egyptians did not reason as the Greeks did—this is a fact. If they found that a method

worked, regardless of how it was discovered, they were not concern with why it worked. They

did not seek to establish its universal truth by means of a symbolic or axiomatic argument. What

they did was to define in an ordered style the necessary sequence of steps required in the

procedure and at the conclusion they added a verification or proof that the steps lead to the

correct solution of the problem. This was mathematics, and science for that matter, as they knew

it. Is it proper or fitting for us in the twenty-first century to compare—critically and

ethnocentrically—the methods of the Egyptians with those of the Greeks or other cultures of

latter emergence? Many of these cultures, in this case the Greeks, probably stood on the

shoulders of the Egyptians as it has been clearly alluded to by the Greek philosopher Herodotus

in his writings about the accomplishments of the Egyptians in relation to geometry, metallurgy,

chemistry, and astronomy. So who were the Egyptians? The ancient Egyptians were, according

to Herodotus, people of dark skin and woolly hair who live in the northeastern region of the

African continent. So by geographic definition the Egyptians were and still are Africans. As a

group of people they were moral and very religious and lived their lives according to the

principles of Maat—an Egyptian goddess who stood for justice, truth, harmony, and balance.

The Egyptians were a highly civilized society whose access to the Nile was responsible for much

of their advancements in science, applied mathematics, politics, arts, architecture, and

socioeconomic development.


69

It is important that in the, rather brief, treatment of the mathematics of ancient Egypt that

the African roots of the Egyptian civilization be emphasized so as to challenge Eurocentrism in

mathematics education. There is a biased view among many in academia that the ancient

Egyptians were ethnically, linguistically, and geographically separated from Africa. There are

many scholars who still persist in regarding ancient Egypt as a separate entity replanted in the

middle of the Mediterranean Sea. The term “ethnically” used in the previous sentence is, often

but, incorrectly used interchangeably with “racially” even though ethnicity does not have the

same meaning as race in that the human race is unique, but comprised of various ethnic groups.

Up to the year 1350 B.C., the territory of Egypt also covered parts of modern Israel and Syria.

Control of this wide region required ancient Egyptians to have an efficacious system of

administrative control over taxation, censuses, maintenance of the logistical aspects of armies,

agricultural control and administration including not only irrigation but also drainage and flood

control systems. At the high point of its legacy the Egyptians are best known for the building of

the pyramids—A mathematical endeavor that has stood the test of time.

One commonly used Eurocentric argument to suggest that Egyptian mathematics was

inferior to that of the Greeks is that the Egyptians did not have the concept of zero. This claim is

absolutely false and unfounded. The symbol for the ancient Egyptian zero was a trilateral

hieroglyph with consonant sounds “nfr.” The mathematics of the Egyptians was utilitarian in

nature. However, generalizations about the volume of a truncated square pyramid are most

evident in Egyptian mathematics. The Egyptian approach to finding the volume of a square

truncated pyramid is found in Problem 14 of the Moscow Papyrus. The problem states that “you

are told: A truncated pyramid of six cubits for vertical height by 4 cubits on the base by 2 cubits


70

on the top.” The Egyptian approach to solving the problem is equivalent to the modern symbolic

representation of the formula given by 2 21( )

3V h a ab b .

Evidence suggesting that ancient Egyptian mathematics was both a methodology and a

proof is supported by the fact that upon the completion of problems, the Egyptians used

sentences that conveyed not only a result, but also a generalization that was an immediate

consequence. These sentences are evident in the Ahmose, or Rhind Mathematical Papyrus as

well as in the Moscow Mathematical Papyrus. These sentences include:

1. The producing of the same

2. The Manner of reckoning it

3. The correct procedure for this type of problem

4. Manner of working out

5. Behold! Does one according to the like for every uneven fraction that may occur

6. Thus findest thou the area

7. These are correct and proper proceedings

8. Do it as it occur

9. That is how you do it or shalt do thou according to the like in relation to what is said to

thee, all like example this.

Section 1: An Overview of the Development of Egyptian Mathematics

The number system of the ancient Egyptian was based on powers of ten. The sources of

Egyptian mathematics are the Ahmose Papyrus, named after the scribe who composed it in about

1650 B.C. The Ahmose Papyrus is also known as the Rhind Mathematical Papyrus, after the

British collector who acquired it in 1858 and afterwards donated it to the British Museum.


71

Together, the Ahmose and the Moscow Papyri contain a collection of 112 problems with

solutions. Vladimir S. Golenischev acquired the Moscow Mathematical Papyrus in 1893 from

one of the brothers of the Abd-el-Rasul family who had stolen it from a king’s coffin found in a

burial ground of the Pharaoh-Queen Hatshepsut’s Temple. In 1912 Golenischev sold it to the

Moscow Museum of Fine Arts in exchange for a yearly sum of money to be paid to him for the

remainder of his life. The author of the Moscow Papyrus is still unknown. Other mathematical

records include the Berlin Papyrus, the Reisner Papyrus, and the Kahun Papyrus. The Moscow

Papyrus contains the formula for the volume of a truncated square pyramid as well as the

solution to a problem dealing with the finding of the curved surface area of a hemisphere. In

modern notation the volume of a truncated pyramid and the volume of the curved surface area of

a hemisphere are respectively given by:

2 21( )

3V h a ab b

where h is the height of the pyramid with upper and lower length base a and b respectively and

21

2A d

where d is the diameter of the hemisphere.

In relation to the particulars of arithmetic, the Egyptians were able to use a method of

multiplication or division that only required prior knowledge of addition and the two times table.

This method is known as the duplication and mediation method. To better understand the

Egyptian procedure, the product of 225 by 17 and the quotient of 696 by 29 are presented below.

The multiplication method is also known in the west as the “Russian Peasant Multiplication

Method.”


72

Multiply 225 by 17

225 17

112 34

56 68

28 136

14 272

7 544

3 1088

1 2176

Inspecting the left-hand column for odd numbers and adding the corresponding terms in the

right-hand column yields the result 17 544 1088 2176 3825 .

Divide 696 by 29

1 29

2 58

4 116

8 234

16 464

The scribe would stop at the number 16 on the left-hand column because a doubling of 16 would

give a divisor greater than 29. On the right-hand column, the sum of 234 and 464 gives 696,

which is the exact value of the dividend. Adding the corresponding values on the left-hand

column, that is, 16 8 24 , which is the result of dividing 696 by 29.


73

Another interesting arithmetical problem can be found in the Ahmose Papyrus, Problem

Number 33. The problem in English is stated as follows: “The sum of a certain quantity

together with its two-thirds, its half, and one-seventh becomes 37.” The solution to the problem

in modern notation is as follows:

2 1 1(1 ) 37

3 2 7x

216

97x

A Eurocentric perspective concerning Egyptian mathematics is the claim that Egyptian

mathematics consisted of only a few arithmetical rules. This perspective is unfounded on the

basis that Egyptian algorithms are based on the following principles or properties within their

sociocultural context:

1. Algorithms are clear and simple, laying out step-by-step procedures

2. Algorithms emphasize the general character of applications by pointing out its

appropriateness to a group of similar problems

3. Answers are clearly obtained after following a set of prescribed procedures

An examination of the Ahmose Papyrus Problem Number 72 will demonstrate that the

scribe’s reasoning in solving problems adheres to the above principles or properties. Problems

of this kind are found in various papyri and are examples of Rhetorical Algebra or Proto-

Mathematics. Problem 72 of the Ahmose Papyrus can be classified as “pesu problems” in that a

pesu is an Egyptian measure of the strength of beer or bread, after either of them is made.

Problem 72 states that “100 loaves of pesu 10 are to be exchanged for a certain number of loaves

of pesu 45. What is this certain number?” The Egyptian reasoning is as follows:

1. One hundred loaves of pesu 10 exchange for loaves of pesu 45. How many of these

loaves are there?


74

2. Find the excess of 45 over 10. The result is 35. Divide the 35 by 10 and the result is

13

2

3. Multiply the quantity 1

(3 )2

by 100. The result is 350. Add 100 to this amount (350)

to obtain 450

4. Say then that the exchange is 100 loaves of pesu 10

5. For 450 loaves of pesu 45

An algebraic examination of the scribe’s reasoning is as follows:

1. If x loaves of pesu p are exchanged for y loaves of pesu q, find y if x, p, and q are

known.

2. Find the excess of q over p. It is (q – p). Divide this (q – p) by p. You get ( )q p

p

.

3. Multiply this ( )q p

p

by x. The result is ( )

q px

p

. Add x to this. You get

( )q p

x xp

4. Say then that the exchange is x loaves of pesu p

5. For ( )q p

x xp

loaves of pesu q. Then,

( )q p

y x xp

( 1)q

y x xp

qy x x x

p


75

( )q

y xp

The last step above is the scribe’s formula or method used for Problem 72 of the Ahmose

Papyrus.

The Berlin Papyrus contains two problems that appear to involve systems of non-linear

simultaneous equations. The papyrus is damaged and thus the solution was afterwards

reconstructed. One of these problems is stated as follows: “It is said to thee that the area of a

square of 100 square cubits is equal to that of two smaller squares. The side of one is 1 12 4

of the other. Let us find the sides of the two smallest squares.” The translation of this rhetorical

algebraic problem into modern notation is equivalent to solving the following non-linear system

of equations:

2 2 100

4 3 0

x y

x y

The reader is invited to solve the above system of equations by the method of substitution and to

confirm that the result is either the ordered pair (6,8) or (-6,-8).

A series is defined as the sum of a sequence of terms. The most common types of series

are geometric and arithmetic. Problem Number 64 of the Ahmose Papyrus deal with the sum of

n terms of an arithmetic series. The following definitions are necessary before the discussion of

Problem 64:

Definition 1

A sequence of the form 1 2, ,..., ,...na a a is an arithmetic series if there is a real number d

such that for every positive integer k, 1k ka a d . The number d is called the common

difference and is defined by 1k kd a a


76

Definition 2

If 1 2, ,..., ,...na a a is an arithmetic series with common difference d, then the nth partial sum

nS , that is, the sum of the first n terms is given by 1[2 ( 1) ]2n

nS a n d .

Problem Number 64 from the Ahmose Papyrus might be restated as follows: “Divide 10

hekats of barley among 10 men so that the common difference is 18 of a hekat of barley. A

hekat is equivalent to 292.24 cubic inches, or 4.75 liters, or roughly 18 bushel. The Egyptian

method is outlined as follows:

1. Average value: 10

110

2. Total number of common differences: 10 1 9

3. Find half the common difference: 1 1 1

2 8 16

4. Multiply 9 by 1

16: The result is

1 1

2 16

5. Add this average value to get the largest share: 1 1

12 16

6. Subtract the common difference 1

( )8

nine times to get the lowest share: 1 1 1

4 8 16

7. Other shares are obtained by adding the common difference to each successive share,

starting with the lowest. The total is 10 hekats of barley.

The corresponding modern algebraic method is as follows:

First Term (lowest)= a

Last Term (highest) = l


77

Common Difference = d

Number of Terms = n

The sum of n Terms = S

The average value of n terms is S

n and the number of differences is one less than the

number of terms, that is, ( 1)n . Half of the common difference is 2

d. The product of ( 1)n by

2

d yields the result ( 1)

2

dn . Now consider the following two cases.

Case1: For this case the result of ( 1)2

dn will be added to the average value of

S

n

yielding the expression ( 1)2

S dn

n . This is the highest term l. Then it follows that

( 1)2

S dl n

n

( 1)2

S dl n

n

1[2 ( 1) ]

2

Sl n d

n

[2 ( 1) ]2

nS l n d

Case 2: For this case the result of ( 1)2

dn will be subtracted from the average value of

S

n yielding the expression ( 1)

2

S dn

n . This is the lowest term. Then it follows that the lowest

term a is

( 1)2

S da n

n


78

( 1)2

S da n

n

1[2 ( 1) ]

2

Sa n d

n

[2 ( 1) ]2

nS a n d

1[2 ( 1) ]2n

nS a n d

Finally, one can certainly appreciate the Greek dependence on Egyptian mathematics. If

one is able to recognize the myth of the Greek miracle, then one can no longer sustain the

Eurocentric view that undermines the contributions of earlier civilizations. In fact, when

Alexandria became the center of learning and mathematical activity it produced the great

mathematicians of antiquity. These ancient scholars germinated from the synthesis of Classical

Greek mathematics in concert with the strong geometric, deductive, algebraic, and empirical

traditions of the Egyptians thus producing scholars of the stature of Archimedes, Ptolemy,

Diophantus, Pappus, and Heron.

Section 2: The Golden Section or the Golden Ratio

The mathematical mystery of the pyramids was known by Herodotus—he obtained the

information from the Temple priests. These priests informed Herodotus that the pyramid was

designed in such a way that the area of each of the sides of its faces was equal to the square of its

height. The pyramids were designed to incorporate not only the proportion for but also

another and even more useful constant proportion known during the Renaissance as the Golden

Section—referred to here as the constant , or the number 1.618. This constant can be obtained

by dividing the 356 cubits of the pyramid’s apothem—the distance from the apex down one face


79

to the center of a base side—by half the base, or 220 cubits resulting in 89

1.61855

. Like the

number , the number can not be worked out arithmetically. The number can be easily

obtained by using the straightedge and compass alone. To the Egyptians the number was

considered much more than a number. They considered this constant to be the symbol of the

creative function, or of reproduction in an endless series. To them it represented the fire of life,

the male action of sperm, the logos of the Gospel of St. John.

The Golden Section or is obtained by dividing a line AB at a point c as follows: Given

the line AB, divide the line at point c in such a way that the whole line is longer than the first

part, that is, AB c in the same proportion as the first part is longer than the remainder, cB.

This means that 1.618AB Ac

Ac cB . The problem is illustrated as follows:

A-----------------------------------------------B

A--------------------------------c--------------B

A------------------------------------------------B

A--------------------------------c

A--------------------------------c

c-------------B

The equation 1.618AB Ac

Ac cB , which appears so simple, turns out to be loaded with

meaning. Plato went so far as to consider it, and wrote about it in his Timaeus. In the Great

Pyramid of Giza the rectangular floor of the King’s Chamber consists of two equal squares, or a

1 by 2 rectangle, and in it one can obtain the Golden Section. Splitting one of the two squares in

half and swinging the diagonal down the base, the point where the diagonal touches the base is


80

or 1.618 in relation to the side of the square that has the value 1. Using the Pythagorean Triples

of the Kemetic Theorem, 2 2 2c a b , one can obtain the Golden Section as 1 5

2 . There

is an odd, but unique consequence in relation to , in that 21 and 1

1

. This

consequence leads to a series known as the Fibonacci series, in which the new number is the sum

of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, . . . and the limit of the ratio of any term by the

previous term of this sequence approaches 1 5

2 .

Leonardo Bigollo Fibonacci, known also as Leonardo de Pisa (1170-1240), was one the

greatest mathematicians of the Middle Ages. He traveled to Algiers with his father, who acted as

a consul for the Pisan merchants. From the Arabs, Fibonacci learned the Hindu system of

numerals that we use today (0, 1, 2, . . ., 9) which he is credited for having introduced in Europe,

where the calculations were still being made by clumsy means of Roman numerals and Greek

letters.

In Egypt Fibonacci learned about the series 1, 1, 2, 3, 5, 8, 13, 21, 34, … , etc. He

popularized the mystical qualities of the series that now bears his name by bringing to Europe the

famous rabbit problem during the 12th century. This problem posses the following question:

“How many pairs of rabbits can be produced each month from a single pair in a year if each

adult pair gives birth to a new pair every month, each new pair reproduces from the second

month on, and no rabbit dies?” The solution to the problem posed by Fibonacci is as follows:

Start with a pair of rabbits born in January, one male and one female. If the rabbits begin

to bear young two months after their own birth, and after reaching an age of two months each


81

pair gives birth to another pair each month thereafter, then how many pairs will be there at the

end of each month?

Month Pairs Number of Pairs January 00 1 February 00 1 March XX 00 2 April XX 00 00 3 May XX XX 00 00 00 5 Jun XX XX XX 00 00 00 00 00 8

Note: XX indicates a pair that is of breeding age

From the table above, the number of pairs for the remaining months of the year are 1, 1, 2, 3, 5,

8, 13, 21, 34, 55, and so on. Since the quotient resulting from the ratio of any term divided by

the previous term approaches 1 5

2 as previously stated, then one can conclude that if

1 2, ,..., ,...na a a is a Fibonacci Sequence then the following result is true:

1 1 5lim

2k

kk

a

a

The Golden Section or Golden Ratio occurs in nature in the ways leaves grow on many

plants, in the spiral of the seed arrangement of sunflowers as well as in the Chamber Nautilus

and in numerous other surprising places. It is also what the pentagram, the Parthenon, and the

Great Pyramid of Giza have in common.

In the Renaissance, the Golden Section—a name that was used by Leonardo Da Vinci—

served as the hermetic structure on which some of the great masterpieces were composed. Da

Vinci illustrated a book on the Golden Section known as The Monk Drunk on Beauty. The book

was later published in 1509.

The conclusion that the ancient Egyptians were acquainted with the Fibonacci sequence

and Golden Section or Golden Ratio is startling in relation to the Eurocentric bias concerning the


82

assumptions about the level of Egyptian mathematics. This knowledge could hardly be accepted

on Herodotus’ statements alone, or on the fact that happens to be incorporated in the Great

Pyramid of Giza. There is also architectural and archaeological evidence that the Egyptians had

worked out a relationship that exists between the values of the constants and in that

26

5 . The reader is invited to verify this result.

In the tomb of Rameses IX there is a strange figure of a royal mummy with one arm raised

and an erect phallus. The mummy is lying at the hypotenuse of the sacred 3-4-5 right-angled

triangle indicated by a snake. The length of the body according to Egyptologists is 5 cubits and

that of the upper right arm is one more cubit, for a total of 6 cubits. At the same time the body is

divided by the phallus in the proportion of 1 and 1 5

2 or 1 , for a total of

2 21 5( )

2

. This makes the outstretched arm give a value for of 6

5 of the body, or

2 21 5( ) 3.1416

2

. In other words, the king is shown as 2 , split into a 1 proportion

by the phallus. The reader is invited to verify that 21 . The king’s raised arm gives a 6

5,

or 21.2 , proportion which is exactly 3.1416.

Section 3: The Mathematical Contributions of the Legendary High Priest Imhotep

Imhotep was a high priest that lived during the reign of King Zoser of the Third Dinasty in

2650 B.C. Egyptologists consider Imhotep as a national hero in that he was appointed by Zoser

to design the building of the step pyramid at Saqqara. The high priest Imhotep devoted his life to

various activities. He held the office of Grand Vizier to the ruling pharaoh and his responsibility


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was one of high dignity whose jurisdiction comprised various departments of the state. Some of

these departments included the treasury, judiciary, army and navy, interior, agriculture, and

general executive. As a judiciary officer he was regarded as one who could not do wrong. As

the Chief Architect, Imhotep was responsible for all the engineering and architectural works of

the kingdom. He designed the step pyramid, which was destined to become the tomb of King

Zoser, at Saqqara near the city of Memphis. Imhotep also held the office of Chief Lector and

had important functions that related to the liturgy of funerary offerings as well as the particulars

related to the ritual of embalmment and mummification. As a Sage and Scribe, Imhotep is

regarded as a literary scholar and physician who produced works on medicine, poetry, and the

philosophy of life. Imhotep had a keen interest in astronomy in that he studied the movements of

the heavenly bodies, eclipses, and the procession of equinoxes. Magic and medicine were

closely allied in the time of the pharaohs. Although Imhotep was a noted magician, it appears

that medicine was the mistress he most zealously wooed. His reputation as a healer of affliction

and human malady elevated him to the office of Court Physician. Imhotep was not of royal

descent nor was he a noble—he was a common man who eventually moved in the highest social

circles of ancient Egypt. The Egyptians described Imhotep as a sort of the Leonardo Da Vinci of

Egypt—a mathematician, a scientist, and an engineer. A few years after his death he was made

into a demigod, Son of Ptah—The god of craftsmen and technicians.

As a mathematician, Imhotep used the concept of the slope of the line. The idea of the

slope of the line was used at the pyramid site to construct ramps to pull blocks of stone weighing

more than two tons, onto the pyramid to put them in place. The formula for the slope of the line

originated from the gradient calculated from the ramps used by the workmen during pyramid


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construction that was supervised by Imhotep. In modern algebraic notation the following

formula for the slope of a line is well known by algebra students:

2 1

2 1

y y ym

x x x

The above formula was also known as the tangent and cotangent ratios. These ratios were

called the “seket” (sometimes spelled “seqt”). The seket was defined as the run over the rise or

simply as

2 1

2 1

cotx x

seket Ay y

In the design of the pyramids, Imhotep used grids to sketch plans for the Step Pyramid. He

also divided the area under the curve into rectangles to determine the correct dimensions of the

Step Pyramid. Today this process is referred to as the Riemann sum named after the German

mathematician Bernhard Riemann (1826-1866). Imhotep probably used—here in modern

notation—the following formula behind his reasoning for the calculation of the area under the

curve.

1 2lim lim[ ( ) ( ) ( ) ]n nn n

A R f x x f x x f x x

The above formula can be explained as follows: The area A of a region that lies under a

curve [a graph] that is a continuous function named f is equal to the limit as the number of

rectangles increase without bound of the sum of the areas of approximating rectangles.

The reader with a calculus background might be familiar with the mathematical definition

of the Riemann sum stated here as follows:

If f is a continuous function defined for a x b and if the interval [a,b] is divided into n

subintervals of equal width b a

xn

. Let 0 1 2( ), , ,..., ( )nx a x x x b be endpoints of these


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subintervals and choose sample points * * *1 2, ,..., nx x x in these subintervals, so that *

ix lies in the ith

subinterval 1[ , ]i ix x . Then the definite integral of the function f from a to b is

*

1

( ) lim ( )b n

in

ia

f x dx f x x

Finally, in the humble beginning of a geometric problem, the Riemann sum has its roots in

ancient Egypt 4000 years before Sir Isaac Newton (1642-1727) and Gottfried Wilhelm Von

Leibniz (1646-1716), 2363 years before Archimedes, and definitely thousands of years before

the birth of Bernhard Riemann.

The reader with a calculus background is invited to evaluate (the answer is ½) the

following limit problem that may have been used by Imhotep among the many calculations that

he carried out in his design efforts for the building of the Step Pyramid.

1 0 1 2 1lim ( )n

n

n n n n n


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Chapter 4

The Role of Women in the Development of Mathematics

The history of science in general, and of mathematics in particular tends to minimize, and

in some cases ignore, the sociocultural atmosphere and motivation behind the scientific and

mathematical achievements of women. The societal factors related to issues of gender equity in

relation to the development of mathematics have not been seriously considered when trying to

understand and explain the process of scientific and mathematical creativity. The role of women

in the development of mathematics has not been seriously considered resulting in the

disenfranchisement of women in the mathematical sciences. The reason for this

disenfranchisement can be attributed to the fact that young girls tend to view mathematics as a

cut-and-dried, esoteric subject that arose from the minds of a few white men. It is therefore not

surprising that many students, in particular young women, may sometimes view mathematics to

be irrelevant or unimportant. Malecentrism in mathematics education may be the most

significant root cause associated with the fear and anxiety of women with respect to the subject

matter. Most mathematics taught in schools in the United States is Eurocentric and malecentric

in that the mathematical practices of other cultures as well as those of women are not identified

as being important. This is compounded with the sad reality that many textbooks abound with

images of white European males. The teaching of mathematics in schools is not only insensitive

to gender equity issues but it is also aligned with cultural imperialism in that the histories of

mathematics are presented as the intellectual monopoly of white academic mathematical

scholars. Linking mathematics with the history of women mathematicians and their


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achievements is likely to add more meaning to the subject matter in addition to adding a positive

female perspective.

The role of women in mathematics has not been recognized probably because most

mathematicians are men. Also, mathematics has always been seen in our society as a masculine

endeavor. In relation to issues of gender equity, mathematics has been historically presented as a

less attractive career option for women. Another problem that has added a negative impact to the

development of mathematics by women is the myth and unfounded claim that males are

biologically superior to women when it comes to mathematical ability. There is absolutely no

credible scientific evidence to support this claim. If this were true, then one should certainly ask

how is it possible for women, for example, Hypatia or Sofia Kovalevskaya, to have made so

many significant contributions to mathematics. Malecentrism in mathematics is evident in the

teaching of mathematics in this country in that it is likely that women have at one time or another

been faced with comments alluding to high performing girls as being—only—hardworking while

poorly achieving boys could be understood as “bright.” These hidden messages about gender are

definitely rooted in malecentrism and/or male superiority. In order to become emancipated from

malecentrism, women must develop confidence in their capacities to understand, develop, and

use mathematics. It is therefore important for academic women and for those who support

gender equity in the mathematical sciences to take a stand against male imperialism and

domination by being agents of change if progress is to be made for gender equity in the

mathematical sciences.

The male mathematical domination has its genesis in the male domination of social

institutions many of which are closely associated with the employment of professional

mathematicians, operations researchers, management scientists, and statisticians. The funding of


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these institutions creates an economic system that is controlled by corporate sources headed by

mostly white males of European descent.

Section 1: The First Women Mathematician

The earliest anthropological evidence or reference to the first mathematician in recorded

history comes from a Mayan ceramic vessel depicting two mathematicians. One of the

mathematicians is believed to be a woman with number scrolls emanating from her armpit. The

name of the woman depicted in the ceramic has not been deciphered. For now, the woman

depicted in the vessel is referred to as “Lady Scribe Sky” or “Lady Jaguar Lord.” Once her name

is deciphered she will perhaps be the earliest mathematician known in history.

According to a reference from two bibliographical sources made by Dr. Claudia Zaslavsky

[Zaslavsky, C. (1992, January). Women as the first mathematicians, International Study Group

on Ethnomathematics, (7), 1, p. 1], “women were the first mathematicians” [as claimed by Dena

Taylor’s (Winter 1991) “The Power of Menstruation” from Mothering]. The article by Dr.

Zaslavsky also provides following direct quote: “The cyclical nature of menstruation has played

a major role in the development of counting, mathematics and the measuring of time… Lunar

markings found on prehistoric bone fragments show how early women marked their cycles and

thus began to mark time. Women were possibly ‘the first observers of the basic periodicity of

nature, the periodicity upon which all later scientific observations were made’ (quoted by Dr.

Zaslavsky from page 97 of William Irwin Thompson’s (1981) The Time Falling Bodies Take to

Light published by St. Martin Press).”

Dr. Zaslavsky argues that women would be the only ones who would have had a need to

keep records of menstrual cycles. These records could have been kept on bone fragments or as

in many Native American cultures on walls made of rocks. As perhaps the first agriculturalists,


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women would have needed to retrieve information about the periodicity of the seasons for

purposes of successful cultivation. The role of men during the early days of humanity and

civilization was hunting.

Section 2: A Biographical Overview of Women in Mathematics

“Lady Scribe Sky” or “Lady Jaguar Lord” from the Maya civilization could very well be

the first woman mathematician in the Americas. Nothing is known about her and unfortunately

the ceramic vessel that identifies her cannot be read by modern-day Mayans since they cannot

read ancient hieroglyphs. The Spaniards destroyed most of the documents they found, and the

few that remain do not provide much information.

Hypatia (360-415) was the daughter of Theon of Alexandria who was a professor at the

University of Alexandria. She was a distinguished scholar in mathematics, philosophy, and

medicine. She wrote commentaries on Diophantus’ Arithmetica and Apollonius’Conic Sections.

She is the first woman mathematician to be mentioned in the history of mathematics. Hypatia

received her education from her father—he held an administrative post at the University of

Alexandria. She traveled extensively for a number of years and afterwards lectured on

mathematics and philosophy, perhaps at the university and in public. Her lectures attracted wide

attendance and praise. Most of Hypatia’s writings are lost, except a copy of her commentaries

on Diophantus that was discovered at the Vatican library in the fifteenth century. As a leader of

the Neo-Platonic School of Philosophy, Hypatia played an important role in the defense of

paganism against Christianity. This caused rage on the part of the new patriarch, Cyril of

Alexandria, who, with frenzied zeal, opposed and oppressed all “heretics.” The fact that Hypatia

was a student of various religions made her relationship with Cyril, who was a religious zealot,

much worse. In March of 415, her life ended at the hands of a fanatical Christian mob. She was


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drag from her chariot, her hair pulled out; her flesh scrapped from her body with oyster shells,

and what remained of her body was burnt. This horrific act of intolerance and fanaticism ended

the creative days of the University of Alexandria.

Marquise Emilie Du Chatelet (1706-1749) was born in Paris. She studied, translated and

extended the work of Isaac Newton and other eminent mathematicians and scientists. She was a

mathematician, physicist, linguist, and musician who performed skillfully on the clavicembalo,

an early form of a piano. She wrote numerous scholarly papers on philosophy and religion that

were posthumously published. Du Chatelet was married to a Marquis and was a long time

companion of Voltaire. She authored the books Institutions de Physique in 1740 and Elements

de la Philosophie de Newton in 1756. The latter of the two was an introductory book in physics.

Du Chatelet’s writings on Newton’s Principia have not yet been translated from French to

English and it is considered by some to be her most important contribution to mathematics.

Almost all—with a few exceptions, of course—of the existing histories of mathematics briefly

allude to the work of Marquise Emilie Du Chatelet.

Maria Gaetana Agnesi (1718-1799) was born in Milan. She was the first of her father’s

twenty-one from three marriages. At an early age, she mastered Latin, Greek, Hebrew, French,

German, and Spanish. At the age of nine she wrote a discourse—which was later published—

proposing a defense in favor of higher education for women. Her father, a professor of

mathematics at the University of Bologna, hosted many gatherings of the intelligentsia at which

Maria had the opportunity to converse with learned professors in their respective languages

about a gamut of topics that included logic, mechanics, elasticity, gravitation, celestial

mechanics, chemistry, botany, zoology, and mineralogy. In 1748, at the age of thirty, Maria

published a two-volume work entitled Instituzioni Analitiche. The first volume of her book was


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comprised of a course on elementary and advanced mathematics geared for the young minds.

This first volume deals with the particulars of arithmetic, algebra, trigonometry, analytic

geometry, and calculus. The first volume of her book is considered to be the first calculus text

written for young people. The second volume deals with infinite series and differential

equations. Together, the 1070-page document constitutes a remarkable contribution to

mathematics education. Maria Gaetana Agnesi is best known for the curve3

2 2

ay

x a

, which is

called the Curve of the Witch of Maria Agnesi. When Maria wrote the Instituzioni Analitiche

she confused the word versoria or versorio meaning “free to move in every direction” with the

word versiera, which in Latin means “Devil’s grandmother” or “female goblin.” Later, when

Agnesi’s text was translated into English the word versiera was translated as “witch.” The curve

has ever since in English been called the “Witch of Maria Agnesi.” The word versiera was also

an abbreviation for the Italian word avversiera, meaning “wife of the Devil.” The famous

Cartesian equation of the curve, here in present day notation, 3

2 2

ay

x a

was at one time of

interest to Pierre de Fermat who did not name the curve. This curved was later studied by the

Italian mathematician Guido Grandi (1672-1742), who named it versoria. It is not clear why

Grandi gave the curve this name. The Curve of Agnesi, as it is known in other languages, also

has the following parametric form that may be familiar to the reader who has studied calculus.

The set of parametric equations of the Curve of Agnesi are defined as follows:

2

2 cot

2 sin

x a

y a

The preceding set of equations can be found by taking any chord of the circle that goes through

the origin O and some other point A on the circle and extending the chord until it intersects the


92

line y = 2a at the point C. The point P is the intersection of a line parallel to the y-axis through

the point C and a line parallel to the x-axis through the point A.

Sophie Germain (1776-1831) was born in Paris and as a woman was denied admission to

the Ecole Polytechnique. Despite this act of malecentrism and gender discrimination she

persisted and secured lecture notes from various professors and began to write mathematical

commentaries under the male pseudonym of M. Leblanc. Her commentaries won high praise

from the well-known Italian born mathematician Joseph Louis Lagrange (1736-1813). In 1816,

Sophie was awarded a prize from the French Academy of Sciences for a paper on the

mathematics of elasticity. In the 1820s, she proved that for each odd prime 100p , Fermat’s

equation, p p px y z , has no solution in integers not divisible by p. She made significant

contributions in the field of differential geometry. She introduced the concept of the mean

curvature 12 ( ')M k k of a surface at a point P of the surface. A discussion on this topic is

beyond the mathematical scope of this section. Sophie Germain corresponded with Karl

Friedrich Gauss (1777-1855) and Lagrange. Shortly after her death in 1831, at the request of

Gauss, she was awarded an honorary doctorate degree from the University of Gottingen. Gauss

and Germain never met. Sophie Germain is often referred to as the Hypatia of the nineteenth

century.

Mary Fairfax Somerville (1780-1872) was born in Scotland who was self-taught and

studied in great detail Pierre-Simon Laplace’s (1749-1827) Traité de Mécanique Céleste. She

was almost fifty years old and had no formal mathematical training when she wrote the

meritorious book entitled The Mechanics of the Heavens in 1830. Her book became a standard

text for students in mathematics and astronomy in all the British universities. Her book contains

full mathematical explanations and diagrams of Laplace’s difficult work in that she made the


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technical context comprehensible to the students and academic readers of that time. She married

at 24 to a man who had little interest in academic pursuits of women. After three years her

husband died leaving her a substantial estate that afforded her the opportunity to purchase

mathematics literature. She was married a second time to a man that unlike her previous

husband was supportive of her academic pursuits. She continued to work until the day she died

at the age of 92. Today, Somerville College is one of the five women’s colleges at Oxford

University. There is an anecdote that alludes to a young woman named Mary Somerville who

wanted a copy of Euclid’s Elements but had to get a male friend to purchase it at a bookstore

because such book was considered improper reading for a young lady.

Christine Ladd-Franklin (1847-1930) completed her work for a Ph.D. degree in

mathematics at Johns Hopkins University in 1882. Her degree was not awarded until a few years

before her death in 1926. Her dissertation on the Algebra of Logic was included in a collection

edited by C. S. Pierce in 1883. Ladd-Franklin published over 200 scholarly articles in general

mathematics, symbolic logic, and color theory. She only taught part-time at both Johns Hopkins

and Columbia Universities.

Sophia Korvin-Kovalevskaya also known as Sonja Kovalevsky (1850-1891) was born into

a family of Russian nobility. When she was seventeen she traveled to St. Petersberg and studied

calculus there from a teacher of the Russian Naval Academy. She was withheld from pursuing

advanced mathematical studies in Russia because she was a woman. She then married out of

convenience with the sympathetic Vladimir Kovalekskaya, who later became a notable

paleontologist, to thus be free from her parental objections to studying abroad. After their

marriage in 1868, the newly wed couple traveled to Heidelberg where Sophia attended lectures

in mathematics delivered by Leo Koningsbeger (1837-1921), lectures in physics delivered by Du


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Bois Reymond (1831-1889). Sophia was not only a great mathematician, but also a writer and

an advocate for women’s rights in the nineteenth century. Her struggle to obtain the best

education possible opened the doors at universities to women. Her groundbraking work in

mathematics made her male counterparts reconsider the archaic and scientifically unfounded

notion that women were mathematically inferior to men. Sofia was exposed to mathematics

from an early age prior to learning calculus in St. Petersberg, Sofia taught herself the particulars

of trigonometry so that she could read a book on optics. She studied under the renowned

mathematician Karl Weierstrass at the University of Berlin. At first Weierstrass did not take her

seriously, but after evaluating a set of problems that he had given her, he realized that the caliber

of her work was superb. He agreed to tutor her for four years. Among the numerous papers that

Sophia published was “On the Theory of Partial Differential Equations” and in July of 1874 she

was granted, in absentia, a Ph.D. degree in mathematics from the University of Gottingen.

Despite her accomplishments she was unable to secure employment as a mathematician. Later

with the aid and influence of Weierstrass she was able to secure a position at the University of

Stockholm. Sofia received an illustrious award granted by the French Academy of Sciences after

submitting a scholarly paper in which she developed the theory for the motion of an

unsymmetrical rigid body where the center of mass is not an axis on the body. Prior to the

submission of this paper the only solutions to the motion of a rigid body about a fixed point had

been developed for the case where the body is symmetric. Sophia Kovalevskaya became ill with

depression and pneumonia and died on February 10, 1891. The President of the French

Academy of Sciences once said: “Our co-members have found that her work [Sophia

Kovalevskaya] bears witness not only to profound and broad knowledge, but to a mind of great

inventiveness.”


95

Charlotte Angas Scott (1858-1931) was the first British woman to receive a doctorate, in

any field, by the University of London. Charlotte received the D.Sc. degree in mathematics after

having passed the qualifying examinations at First-Class level. She spent nine years at

Cambridge University. The university at that time did not grant degrees to women until 1948.

Charlotte was a superb teacher who insisted on high academic standards. From 1899 until 1926,

Charlotte served as co-editor of the American Journal of Mathematics. She played an active role

in founding the New York Mathematical Society, which, in 1894 was reorganized as the

American Mathematical Society. Gender discrimination and bias in the mathematical sciences

that was typical of the nineteenth and early twentieth centuries was broken by women of the

caliber of Charlotte Angas Scott.

Winifred Edgerton-Merrill (1862-1951) was the first woman ever to be awarded a degree at

Columbia University, she was also the first American woman to receive a Ph.D. in mathematics.

She also studied mathematical astronomy and received a Ph.D. in 1886. At Wellesley College

she declined a position as a professor of mathematics because she was engaged to be married,

but was instrumental in the Foundation of Barnard College for Women.

Grace Chisholm-Young (1868-1944) studied mathematics at Girton College (Cambridge),

but could not receive the degree for which she was qualified. She traveled to Gottingen,

Germany, where she studied with renowned mathematician Felix Klein and passed the doctoral

qualifying examination to earn the first official doctorate granted to a woman in any subject. She

married her former tutor William Young, and was the mother of six children. William and Grace

wrote many scholarly papers together, often published under his name, and published the first

textbook in set theory.


96

Amalie Emmy Noether (1882-1935), one of the most outstanding mathematicians in the

field of abstract algebra, was born in Erlangen, Germany, in 1882. Her father Max Noether

(1844-1921) was a distinguished mathematician at the University of Erlangen. Emmy completed

her doctoral dissertation in algebra at the University of Erlangen in 1907. She took an unpaid

lectureship at the University of Gottingen in 1915, and remained there until she was forced to

leave Germany in 1933. Emmy Noether accepted a position at Bryn College in Pennsylvania for

a couple of years and died suddenly from surgery complications in 1935 at the age of fity-three.

At the time of her death, Albert Einstein wrote in the New York Times, “… Noether was the

most significant creative mathematical genius thus far produced since the higher education of

women.”

Grace Brewster Murray Hopper (1906-1992) is one of the world’s foremost pioneers in the

field of computer science. She graduated from Vassar College in 1928 with a degree in

mathematics and physics, and received both a Master of Science and Ph.D. degrees in

mathematics from Yale University. She then returned to Vassar College to teach mathematics.

In 1943 she joined the Naval Reserve, and was commissioned as a lieutenant in 1944. She work

for the United States Department of Navy, where she developed software programs, compilers,

and the COBOL language. She was asked to resign several times, then was called back for

military duty. After a 43-year career in the US Navy, Grace Hopper retired in 1986 with the rank

of rear admiral.

Marjorie Lee Browne (1914-1979) was born in Memphis, Tennessee. Her father was a

railroad postal clerk who completed only two years of college and was known for his mental

arithmetical/mathematical quickness. She financed her college education at Howard University

with a combination of scholarships, odd jobs, and student loans. She graduated in 1935. She


97

taught mathematics for one year at Gilbert Academy—a private high school in New Orleans—

and it was there that she learned about the affordability of the University of Michigan. She

received a master’s degree in 1939 from the University of Michigan and began teaching at Wiley

College in Texas. She studied during the summers toward her Ph.D. and in 1950 she received a

doctorate in mathematics with a specialization in Topological Groups. From 1949 until her

death she taught at North Carolina Central University. She was the only Ph.D. in her department

for 25 years. She taught undergraduate and graduate mathematics courses and was department

chair from 1951 to 1970. Nine out of ten students master’s thesis that she supervised later went

on to earn Ph.D.s in mathematics.

Evelyn Boyd Granville (1924-present) was of very modest means and was Black. She

grew up in an all Black neighborhood of very modest economic means. She graduated from

Smith College in 1945 and received no financial assistance for the first year of studies. She

earned an M.A. in both mathematics and physics in one year from Yale University. In 1949, she

also completed a Ph.D. degree at Yale in mathematics with a specialization in functional

analysis. As a woman, Evelyn was one of the first African Americans to have earned a doctorate

in mathematics in the United States. In the same year a Dr. Marjorie Browne (1914-1979),

another African American woman also received her Ph.D. from the University of Michigan. At

that time neither Evelyn nor Marjorie knew of each other. Evelyn taught for three years at NYU

and Fisk, the she began a career in government research. She worked on the Mercury and

Apollo Space projects, and became a senior mathematician at IBM. In 1967 she took a position

at California State University, Los Angeles (CSULA), where she became known as a

mathematics educator. After her retirement from CSULA in 1984, she moved to Texas, and


98

shortly afterward, began teaching at Texas College in Tyler. She still lives in Texas with her

husband of 31 years, Ed Granville.

Section 3: A Biographical Overview of Women and Mathematics in Modern Society

Lenore Blum (1942-present) was born in New York and lived most of her child and

teenage years in Caracas, Venezuela where she attended the American School of Campo Alegre.

When Lenore arrived in Caracas, an attempted revolution was taking place and five years later

the Colonel Marcos Peréz Jimenez, was finally overthrown. Mathematics was Lenore’s favorite

subject and she excelled and hoped to some day pursuit a career in mathematics—her teacher

advised her against it. Lenore was valedictorian of her high school graduating class. Lenore

studied architecture at Carnegie Tech and in her second year she changed her major to

mathematics. Lenore was married to her boyfriend, Manuel, whom she had met in Venezuela.

Manuel studied at MIT and worked at the university’s Neuro-Physiology Lab. Lenore earned

her doctoral degree in mathematics from MIT. She received a post-doctoral fellowship at UC

Berkeley, where her husband Manuel also received an offer in the Mathematics Department.

During her post-doctoral degree Lenore received numerous offers from Yale, MIT, and

Berkeley. Lenore took a job at Berkeley since Manuel already had a job there. The position that

Lenore took was the lowest rank—Lecturer in Mathematics. She was, after two years, not re-

hired. Lenore and other women mathematicians together formed the Association for Women in

Mathematics. In 1973, Lenore was hired to teach a college algebra course at Mills College in

Oakland, California. She later became the Head of a new Department of Mathematics and

Computer Science at Mills College. She is the co-founder of Math for Girls. In 1990 Lenore

Blum was invited to speak to international mathematical community at the International

Congress of Mathematicians in Kyoto, Japan. In 1992 Lenore began a new career as deputy


99

Director of the Mathematical Sciences Research Institute in Berkeley, one of the foremost

mathematics research institutes in the world. Finally, Lenore also assited in the formation of the

Mathematics and Science Network, a San Francisco-based group that encourages women to

study mathematics in order to qualify for careers in engineering, medicine, and computer

science.

Fanya Montalvo (1948-present) was born in Monterrey, Mexico and her father was a radio

and TV repairman. The Montalvo family migrated to Chicago and Fanya had to learn English on

her own. At times she would struggle and it took her a long time to finish a book in her new

adopted language. She tested poorly in high school entrance exams and was placed in the lowest

academic track. She began to move up in science and mathematics and ended up taking honors

courses in high school. She attended Loyola University and majored in physics. She was the

only Latino woman in many of her mathematics and physics courses. She went on to study

mathematical psychology. She later enrolled in a masters program at the University of Illinois in

Chicago and took courses in electrical engineering, systems theory, and signal processing. Fanya

earned her doctorate in computer science from University of Massachusetts. She was one of two

women out of eighteen to have completed her doctorate. She took a position in California at the

Lawrence Laboratory at Berkeley. She has made significant contribution in the field of artificial

intelligence. Fanya presently lives in a house near the beach in Boston.

Section 4: Reflections About the Role of Women in Mathematics

It is true that the work of women have been ignored, robbed of credit, or perhaps forgotten.

However as women endeavor to enter mathematics related professions they will continue to

make great strides in the pursuit of scientific truth. Women will probably face many great

challenges in attaining gender equity, admiration, and respect in the mathematical sciences.


100

These challenges can only be conquered by making connections with others who have

succeeded. It is therefore, imperative for women who are pursuing mathematical related

professions to keep in mind the great examples of the remarkable women that have been

presented in this chapter.


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Chapter 5

Mathematics, Democracy, Civil Rights and Social Issues

This chapter will present ideas related to mathematics and social issues in modern society.

For example, the relationship that exists between the struggle for civil rights—in particular—the

right to vote on the part of African Americans during the 1960s and the case for quantitative

literacy as a tool of liberation in modern society. Other issues of social interests will be

presented. These include defining what is quantitative or mathematics literacy in the context of

democracy and freedom from academic and social oppression. The chapter will also explore the

sociology of mathematics and its relationship with sources of patronage and professionalism.

Mathematics is a social product and as such it serves the interests of social groups and

entities whose products and services are based on the particulars of mathematical knowledge.

Economic access to the American dream in modern society in relation to labor is based on the

kind of worker or professional who has the technical background to meet the current needs of

society. It appears that algebra is the gatekeeper for success in advanced mathematics. Not

addressing the problem of quantitative literacy or mathematics—for all—in modern American

society will lead to a social structure of the half and the half-not. Mathematical models are

socially significant in that the applications of mathematics, and statistics in particular, legitimize

public policy and opinion.

One of the goals of ethnomathematical thought is to provide a view of knowledge about the

particulars of mathematics that is anti-racist, anti-sexist, anti-all the other dehumanizing

totalitarian institutional structures and attitudes. From a Freirean perspective, the

ethnomathematics movement is committed to justice and liberation in that, students are not just


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an accidental presence. Students are active participants capable of advancing the theoretical

understanding of others as well as themselves. From a sociological perspective

ethnomathematics recognizes the reality of the so called “mathematics anxiety.”

Ethnomathematics differs from the white male European perspective of mathematical

understanding in academia in reference to students in that it does not blame the victims of

mathematical anxiety, but rather looks into the psychological reasons for the anxiety that exists

in many American students. The philosophy of ethnomathematics seeks to understand the

political dimension of mathematics by demonstrating and teaching the students how to use and

learn mathematics to demystify the institutional structures of society.

The underlying social reasons for the poor mathematical literacy of many Americans is that

students often reflect on the real reason why they have not done well in mathematics and often

believe that it is because of their lack of mathematical ability. Most students who have not done

well in mathematics internalize their lack of mathematical ability as a primary reason for not

pursuing mathematics any further or they just simply avoid it altogether. The real social reasons

are poor schooling in concert with the stereotypical assumptions of who can do mathematics.

Student avoidance or postponement of mathematics is a conspiracy of silence about racism in our

society. When women, for example, participate in mathematics, does their presence change the

discipline?

Section 1: Quantitative Literacy, Mathematics and Democracy

A successful democracy can only exist in a society where individuals are able to think for

themselves and judge independently between good and bad information. We live in a society

that abounds with quantitative information and to make decisions about the information

presented requires quantitative literacy. The necessary prerequisite for a life of freedom and


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competence for each and every individual of our society is at the heart of quantitative literacy.

An individual that lacks quantitative literacy in modern society is equivalent to an individual

who lacks reading and writing in Gutenberg’s time. In today’s modern society, quantitative

literacy is analogous with the reading and writing competencies of over one hundred years ago.

These competencies were the traditional core of literacy at that time. Quantitative information is

essential to the discourse of public life and if put to good use, the access to numerical

information will place more power in the hands of individuals and serve as a stimulus to

democratic and civic decision-making.

Contrary to popular belief, a rigorous education in mathematics along traditional lines does

not lead an individual to a high degree of quantitative literacy. Only a miniscule part of the

education needed to attain control, and understanding over numbers and the overwhelming

quantitative information found in daily life can be found in a typical mathematics curriculum.

The reason for this is because the skills in complex analysis of data and interpretation of numeric

or graphic information are rarely found in traditional calculus courses. Once students leaves

arithmetic, the curriculum moves on to more abstract concepts that are most appropriate for a

limited number of technical professions that require, in many cases, advanced degrees. In fact,

when a professional mathematician is most engaged in his work, he must shut out the real world.

The application and extension of mathematics to other fields can be thought of as a kind of

quantitative literacy. The argument here is not that calculus and the traditional mathematics

curriculum is deficient. The argument is that the traditional curriculum does not do enough to

include the kind of mathematics that is likely to surface in the lives of modern professionals.

Many educated adults remain functionally “innumarate” and thus unable to fully participate in

democratic society. Even those who have studied trigonometry, pre-calculus, and calculus


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remain unaware of the abuses of data and are often unable to comprehend and articulate the

peculiarities related to the particulars of quantitative inferences. The key to understanding a

society immersed with numerical information is quantitative literacy and not calculus.

Quantitative literacy is a tool that empowers members of society to analyze and question the

decisions of elected officials and those who are in positions of authority. Quantitative literacy

allows the members of modern society to confront leaders of our communities confidently and

intelligently.

The lack of quantitative literacy in American society has resulted in the every day practices

and misuses by newspapers and politicians alike of charts, graphs, and quantitative information.

There is no argument about the fact that American society believes that quantitative literacy is an

integral part of civic life and citizenship. However, members of academia have not yet agreed on

exactly what constitutes quantitative literacy. The traditional curriculum in pure and applied

mathematics has in the past several years expanded in to a gamut of fields dependent on

mathematical approaches to solving problems that arise in society. These fields include

statistics, biostatistics, biomathematics, financial mathematics, managerial mathematics,

management science, operations research, actuarial science, and bioinformatics. All of these

fields share a solid foundation in mathematics, but differ in character, methodology, standards,

and objectives. Quantitative literacy is not watered-down mathematics nor is it statistics.

Quantitative literacy is an approach to solving problems and judging decisions that use and

enhance mathematics and statistics. Quantitative literacy has to do with the logic of certainty.

Mathematics on the other hand is more about a Platonic realm of abstract structures and statistics

on the other hand, is about uncertainty and making sense out of data. The traditional

decontextualized mathematics curriculum has failed many students including ethnic minorities


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and women in that upon completion of high school they do not have the numeracy skills nor the

quantitative confidence that is required for complete and successful participation in a democratic

society. The compartmentalization of academia coupled with weak and incompetent

administrative leadership and micro-management is partly responsible for the lack of initiative to

see the possible connections across the disciplines. The result of this sad reality is that when

students learn skills or ideas in one class they are totally forgotten or remain unconnected when

similar problems arise in a different context. Hence, knowledge transfer from one academic

discipline to another does not take place.

Full participation in a democratic society requires its citizens to be literate and numerate.

These two characteristics will be the indispensable and inseparable qualities of an educated

person in the twenty-first century. Though quantitative literacy is closely linked to mathematics,

it is important to understand that both concepts are slightly different in the context of

ethnomathematics. For example, what is needed for life is quantitative literacy and what is

needed for education is mathematics. Similarly, what is needed for general school subjects is

quantitative literacy and what is needed for engineering and the physical sciences is

mathematics.

Section 2: Sustaining a Democratic Society Through Quantitative-Based Citizenship

Quantitative-based citizenship can be defined as an individual’s capacity to identify and

understand the role that mathematics plays in the world. This capacity is based on well founded

decisions or judgements that engage an individual in mathematics in such ways that they would

meet the needs of the individual’s current and future life as a constructive, concerned, reflective,

law-abiding citizen.


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Sustaining a democratic society through quantitative literacy requires an individual to have

the ability to not only appreciate, but also to understand the information that is presented in

mathematical terms. The sum total of skills, knowledge, beliefs, disposition, habits of mind,

communication capabilities, and problem solving skills that people need in order to participate

effectively in the quantitative alternatives that arise at work and in daily life are at the root of

quantitative-based citizenship.

In order to sustain democracy and freedom from oppression, a citizen in the twenty-first

century must have the following competencies in addition to those competencies that were

required over one hundred years ago:

1. Confidence with mathematics, that is, being comfortable with quantitative ideas and at

ease with applying quantitative methods for decision-making. This confidence is diametrically

opposite to “math anxiety” in that it makes numeracy as natural as ordinary spoken or written

language. Citizens with this confidence are able to use mental estimates to quantify, interpret,

and validate information or claims.

2. Ethnomathematical and multicultural appreciation, that is, understanding the nature of

accurate history and culture in which mathematical inquiry developed. This means to understand

the role that culture plays in the development of scientific inquiry. The understanding of culture

is likely to induce the individual to appreciate and ponder social and political issues that affect

society and knowledge.

3. Interpreting data, that is, the individual’s ability to reason with data, and graphical

representation of scenarios and alternatives, and to recognize confounding factors and sources of

error. This competency differs from traditional mathematics in that data—not mathematical

formulae—are at front and center of issues that affect society in general.


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4. Logical thinking, that is, an individual’s ability to analyze evidence by reasoning and

understanding arguments, questioning assumptions, detecting fallacies, and evaluating risks as

well as opportunities. Individuals with this competency accept little at face value; these

individuals are likely to look beneath the surface and line of inquiry, and often demand

appropriate information to get at the root cause of issues and concerns.

5. Making decisions, that is, using mathematics or ethnomathematical approaches to make

decisions and solve problems that arise in daily life. For individuals with this competency,

mathematics goes beyond the classroom and into the realm of an individual’s powerful tool for

survival. This competency is sometimes outside the context of academic mathematics as it is in

the case of many African cultures that use mathematics as a means of survival and meeting the

needs of a given culture.

6. Mathematical context is an individual’s ability to use and apply mathematical tools in

specific situations to provide meaningful results as a prerequisite for making decisions. These

strategies depend on the specific context of the issues at hand.

7. Numeracy is and individual’s accurate intuition about the meaning of numbers in

relation to confidence in estimation and common sense in using numbers as a measure of things.

8. Mathematical practicability, that is, an individual’s ability and/or knowledge of how to

solve problems that are often encounter at home or at work. Citizens with this competency are

usually comfortable with using elementary mathematics in wide variety of scenarios.

9. Prerequisite mathematical knowledge, that is, an individual’s ability to use a wide

variety of algebraic, geometric, and statistical tools that are required in many fields of post-

secondary education.


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10. Symbolism, that is, being comfortable with the use of proper mathematical notation

and at ease with reading, interpreting, and exhibiting adequacy concerning the syntax and

grammatical characteristics of mathematical symbols.

The above competency skills in relation to sustaining a democracy in contemporary society

is analogous to verbal literacy. The fundamental reading, writing, and arithmetic that are taught

in an individual’s early life are no longer sufficient to ensure full participation in democratic

society and in the emancipation from social or economic oppression. Today’s educated citizens

are required to be literate and numerically sophisticated. The success of American freedom and

society will be in the hands of the generations of individuals who can think through subtle issues

that are communicated by our elected officials in a collage of verbal, symbolic, and graphic

forms. The citizens of this great country—if they are to succeed and enjoy freedom from

oppression—will need the confidence to express themselves in these modern forms of

communication. People who have never experienced the power of quantitative thinking often

underestimate its importance—especially for tomorrow’s society.

Section 3: The Sociology of Mathematics

From an ethnomathematical perspective, the sociology of mathematics is closely linked to

many aspects of civic responsibility. Many sophisticated expressions of quantitative reasoning

have become an integral part of mainstream modern data-driven society. To some extent some

of these expressions of quantitative reasoning serve personal needs, while others serve the goals

of a free democratic society. These quantitative expressions are comprised of the following

broad general categories: Citizenship, culture, education, profession, personal finance, personal

health, management, and work.


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In relation to citizenship, virtually every public and/or policy issue, ranging from health

care to social security, from international economics to welfare reform, depends on data,

projections, inferences, and probability of outcomes. This kind of systematic thinking is at the

heart of decisions based on some kind of mathematics or ethnomathematics such as analyzing

economic or demographic data to oppose or support a policy issue. For instance, does the

population at large really understand the social and economic implications of the privatization of

schools or of social security? Another example is recognizing how apparent bias in hiring or

promotion may be a fact of how data are aggregated or understanding quantitative arguments

made in voter information pamphlets about school budgets or tax cut proposals.

The category of culture is directly related to an individual’s basic understanding of history,

literature, and art—at least in general terms. This category is most commonly articulated in the

liberal education component of a college education. For example, understanding that

mathematics is a deductive process of thinking in which conclusions are universally true, if and

only if, the assumptions are satisfied. Understanding how the role that mathematics has played

in the development of culture in relation to ethnomathematics. Recognizing that mathematical

power is a double edge sword in that it can empower its citizens or endanger the ideas of equity,

privacy, and freedom of liberties in the shaping contemporary public policy. Understanding how

the history of mathematics relates to the development of culture and society is at the genesis of

this category.

In relation to education, mathematics, engineering, physics, and other mathematical fields

have always required a strong background in calculus. However, many academic fields are now

requiring competency in the mathematical sciences including statistics, discrete mathematics,

and quantitative methods. For example the field of biology now requires computer mathematics


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for the mapping of genomes, probability for the study of heredity, and calculus to determine the

rates of change of populations. The field of medicine requires the understanding of statistics to

assess the effectiveness of clinical trials, and calculus to better understand the body’s electrical,

biochemical, and cardiovascular systems. In the social sciences the understanding of statistics is

as important as the study of calculus is to engineers and other physical scientists.

In light of the fact that interpretation of evidence and data has become increasingly

important in the decisions that affect people’s lives, professionals in practically every filed are

expected to be quantitatively competent. For example attorneys rely on the particulars of logic to

build a case on subtle arguments concerning the probability to establish or refute “reasonable

doubt.” Physicians and surgeons need to understand statistical evidence to clarify patient’s

informed consent about medical decisions and procedures. Architects use geometry and

computer graphics to design structural plans and probability theory to assess structural risks as

well as calculus to understand engineering principles.

Personal finance is the single most neglected issue in which ordinary people are faced with

on a regular basis. These personal financial matters can have serious consequences ranging from

harassment of collection agencies to ultimately declaring personal bankruptcy. The area of

personal finance has long been neglected in the traditional academic track of the mathematics

curriculum in American education. Some examples in the category of personal finance include

the understanding of depreciation and its effect on the purchase of cars or computer equipment.

For instance, is it better for a particular individual to lease or buy a car? Another example is the

understanding of the long-term cost of making lower monthly payments on credit cards.

Designing and understanding plans for the purchase of a home as well as understanding the


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various factors affecting a mortgage (e.g., principal, points, fixed versus variable rates, monthly

payments, and duration).

Personal health has been for some time a hot topic of debate in American politics. Many

patients have become partners with their physicians and health care professionals in making

decisions about medical services as a result of the continually increasing cost of medical care.

Quantitative skills have become necessary in the medical decisions affecting patients’ lives. For

example, interpreting medical statistics and formulating relevant questions about different

treatment options in relation to known risks and a specific person’s condition is of outmost

importance for consumers in modern democratic society. Also understanding the impact of

outliers on medical summaries of medical research is of paramount importance to both patients

and medical professionals.

Many people are now interested in managing small businesses or running their own

companies. For example, looking for patterns in data to identify trends in costs, sales, and

demand are of importance to executive board members as well as business owners. Equally

important to these individuals is also the analysis of data to improve profits.

Finally, practically everyone in the work environment uses quantitative tools of one form

or another to determine simple things such as wages and benefits. Some examples include the

production of a tree diagram to schedule the completion of a complicate project. Researching,

interpreting, and developing work-related formulas is also common in the work place. Equally

important in the work environment is the use and application of managerial statistics to evaluate

alternatives and make executive briefings to internal and external customers.


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Finally, for most students, skills learned free of context are skills devoid of meaning and

utility. To be effective, quantitative skills must be presented in settings that are meaningful and

memorable.

Section 4: Mathematics and Civil Rights

In today’s society the most urgent social issue affecting poor people and people of color is

economic access. As articulated in the previous sections of this chapter, full participation in

democratic society and citizenship is rooted in quantitative literacy, mathematics, and its

applications. The absence of quantitative literacy, mathematics, and its applications in today’s

urban and rural communities throughout this country is an issue as urgent as the lack of

registered Black voters in Mississippi was in the 1960s. Politicians and elected officials propose

bad-aid solutions to these kinds of societal problems—build more jails, privatize education,

provide educational vouchers, dismantle public education, implement testing of standards, put

more police on the streets, and so forth. This is working out the problem from the back end.

There are, of course, no barriers in terms of voting rights, but to take advantage of the technology

available as well as economic opportunity will require and demand the political mentality of the

struggle required in the 1960s during the civil right’s movement.

The most important external societal factor affecting the long-term production of

mathematical scientists is the inadequacy of primary and secondary school mathematics

programs. The traditional function of mathematics education in this country has been to identify

bright young potential mathematicians and steer them into programs at various university

campuses. Before one can get to anything that is interesting a student needs to absorb a lot of

abstract mathematics. Unlike the social sciences, or even English, at the hands of creative

teachers can be presented in memorable and interesting ways through literature, stories, or


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current events. These subjects need not be boring. However, from a student perspective.

Mathematics is expected to be a cut and dried subject that is often seen as boring. In our culture

illiteracy in mathematics is acceptable in the same way that illiteracy in reading and writing is

unacceptable. Forty to fifty percent of students taking freshmen calculus in American colleges

will fail it—not being good in mathematics simply confirms sameness for everyone else and does

not necessarily imply inferiority. It is interesting that a large percentage of delinquent children

and criminal adults come from severely undereducated families or families where literacy skills

are strikingly low. So today, how do the people at the bottom, that is, Blacks, Latinos, and poor

white students, get into the mix or the mainstream? This question is the same question that was

posed when the Mississippi Freedom Democratic Party made its challenge to Mississippi

Democrats in Atlantic City in 1964. How do we close the knowledge gap? How does society

stabilize itself?

Mathematics can be seen as a tool of liberation in that quantitative literacy, mathematics,

and economic access is how society can give hope to the young generation of disadvantaged

members of society. Not only is literacy in reading and writing essential—literacy in

mathematics and science is a prerequisite for full participation in democracy, society, and

citizenship. In light of this dilemma, algebra has become an enormous barrier. The debate over

algebra as a gatekeeper has been the subject of debate in the California Community College

system. The Northern and Southern Chapters of the California Community College Mathematics

Council has responded somewhat negatively to the infusion and practice of ethnomathematical

strategies to alleviate the existing problem of student failure in intermediate algebra. Despite the

arguments against ethnomathematics on the great majority of mathematics educators

representing many California community colleges, a resolution authored by Professor Jacqueline


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Dodds of Pasadena City College was passed during the Spring 2000 Plenary Session of the

Academic Senate for California Community by a small margin.

Resolution 9.07 entitle “Mathematics--Global Approach” by Jacqueline Dodds states:

Whereas ethnomathematics or multicultural global mathematics has been recognized at

conferences, nationally, internationally, and statewide; and

Whereas mathematics [algebra] remains a gatekeeper/gateway course for many of our

students; and

Whereas mathematics classes have a high dropout or attrition rate (some say actually 30%

actually finish the classes); and

Whereas the Academic Senate for California Community Colleges has certainly always

supported a variety of intellectual pursuits and student success is a key component of the

Academic Senate activity;

Therefore be it resolved that the Academic Senate for California Community Colleges

support efforts to improve the retention rate in mathematics and to study the effects of

mathematics as a gatekeeper/gateway course; and

Be it further resolved that the Academic Senate for California Community Colleges support

research into a variety of approaches to mathematics, including a multicultural, global

approach, such as ethnomathematics.

Algebra as a gatekeeper for higher mathematics is now the gatekeeper for citizenship; and

people who do not have it are like the people who could not read or write in the industrial age.

Algebra was organized in the educational curriculum of the industrial era, its place in society

under the old jurisdiction has become a barrier to full participation in democratic society and

civic responsibility. In France, for example, geometry is the driving force of mathematics and


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technology. So there is nothing that says that it is algebra or geometry, what is clear is that either

one or the other or a mix of the two must be in place for full participation in a free society.

Equality for minority, the poor, and the disadvantaged is linked to mathematics, quantitative

literacy, and science. The traditional training of mathematicians and educators of mathematics

does not prepare them for leadership in terms of literacy of the masses. The literacy effort that is

required cannot succeed unless it enlists active participation from activists in the mathematical

community. The times have changed; the traditional role of mathematics or science education

has been to train an elite class of few bright students and to bring them into research. The

traditional role of mathematics education has not been a literacy effort. Instead of weeding out

students out of advanced mathematics courses, schools must commit to everyone gaining of

quantitative literacy as they have in the past committed to everyone having a reading and writing

literacy. An application of what was learned from the civil rights movement is that the poor and

oppressed needs to face a system that does not lend itself to their needs (e.g., poor and

disenfranchised) and to device a means to change that system so that it serves the individual’s

needs.

The lessons that have been learned from the civil rights movement in Mississippi is that it

takes demands to take people from the bottom of the academic and economic ladder to the

mainstream. These are demands that the poor and oppressed must make on themselves and on

the system. This powerful attitude is what will lead to the most significant changes to level the

plane field. No matter how great Dr. Martin Luther King, Jr. was he could not, alone go and

challenge the seating of the Mississippi Democrats at Atlantic City. He could advocate and

support them; the only people who could do that were the people from Mississippi. People will

not organize that kind of sentiment and effort on somebody else’s agenda. The agenda has to be


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internalized from the grass roots. In Mississippi the voiceless found a voice, and once it was

raised, it could no longer be ignored.

Quantitative literacy and mathematics in relation to citizenship and full participation in

democratic society has a steep learning curve and it is something that will evolve over

generations as those who are activists in this social endeavor through study and practice begin to

address the problem. Young people may speed this up as the youth once did in the civil rights

movement. As with voting rights four decades ago, we most carve out a consensus on

quantitative literacy. Without it, moving the country into systematic change around mathematics

education reform will become impossible.

There are politics at stake. Just consider the following questions: “Who is going to gain

access to the new technology?” “Who is going to control it?” What do we have to demand from

educational entities and institutions to prepare for the technological era?” “What opportunities

will be available for our children?” These questions will ultimately hold accountable those who

are in power or leadership positions as the civil rights movement did in the 1960s even though

that earlier movement was more about ballots.

Section 5: Societal Factors Affecting Mathematical Thought

Mathematical knowledge is not neutral and it is shaped by cultural influences within a

given societal group. Mathematics is a product of society and as such it can serve the interests

and needs of any particular group within the social system. From an ethnomathematical

perspective, mathematical practices and developments can be examined by looking at the social

construct of mathematical ideas and by analyzing the culture from which these mathematical

ideas arose. Mathematical truths are universal and pure while the applications of mathematics

can be used for the good of society or the destruction of society or human life. This is certainly


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true in the case of science in that scientific knowledge is objective and its applications can be

used for either good or evil, emancipation or oppression, improving the human condition or

destroying human dignity. For example, during the Holocaust the Nazis made impressive

scientific advancements in medicine and science at the cost of millions of lives of innocent

people. This is a clear example of applying scientific principles and methods to destroy human

dignity.

Mathematical inquiry and curiosity is also shaped by the politics of society. For example,

former president Ronald Reagan during his speech campaign in 1980 stated “why should we

subsidize intellectual curiosity?” while on January 8, 1790, George Washington addressed the

United States Congress by stating that “there is nothing which can better deserve our patronage

than the promotion of science and literature. Knowledge is in every country the surest basis of

public happiness.” The reference to science and literature in the statement made by George

Washington in today’s modern society is the equivalent to quantitative literacy as discussed in

the previous sections of this chapter. Public happiness in the context of modern society can be

thought of as our guaranteed freedom to exercise full participation in democratic society. Is the

statement made by Ronald Reagan a statement of liberatory emancipation or is it a statement of

oppression in relation to the pursuit of full and complete participation of modern democratic

society? In the previous sections of this chapter it was discussed that access to freedom and

participation in democratic society is at the heart of education in general and quantitative literacy

in particular.

A definition of ethnomathematics in relation to the sociology of knowledge is that

ethnomathematics is the use and practice of mathematical concepts to serve the societal needs of

human kind. Scientific praxis is rooted in the ethnomathematics of cultural or social groups.


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From an ethnomathematical perspective mathematics is a tool of liberation and emancipation

from oppression in that mathematics and quantitative literacy allow members of society to

intellectually inquire about nature, society, politics, decisions, and many other internal as well as

external factors that impact the relationship between individual and society. Members of a free

democratic society must keep in their minds the well-known Latin proverb, “ubi dubium ibi

libertas” or “where there is doubt, there is freedom.” Certainly, the key political motive behind

mathematics and science is profit and social control. So, is there a relationship between

mathematics and society? The relationship between mathematics and society can be better

understood in terms of societal factors that affect mathematical thought. These factors include

the following: Funding sources, possible applications, areas of study, male domination,

interpretation of constitutes mathematical knowledge, and the influences of professional

organizations.

1. Funding sources ultimately shape the areas and focus of mathematical research and its

applications. Most of these sources come from the government and large corporations and as a

result the formulation of mathematical applications is directly tied to these sources of financial

patronage. Most mathematical applications have had their genesis in military concerns and are

usually thought of as being interesting. The implication of funding military applications is that

societal and humanitarian applications of mathematics end up being under-funded.

2. Possible applications of mathematics can affect the creation of new fields in the

mathematical sciences. For example, operations research grew out of military applications and

needs during World War II. This field of mathematics is still sought after by the military and its

contractors. From a corporate perspective, operations research became what is now known as

management science. The field of management science deals with the optimization of profits,


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costs, and control of labor and capital. Management science can be considered, from a Marxist

perspective, as a capitalist application of mathematics. Another example is game theory. This

branch mathematics or field of study deals with conflict situations where the key players have a

number of choices that are followed by payoffs. Game theory is built around the determination

of optimal strategies for making the choices. The competition and individuality of this area of

mathematics is built in the theory of games. Most of the applications of game theory have been

in the corporate sector as well as in the military.

3. The areas of study must needed in a modern free democracy that sustains capitalism are

directly related to computational mathematics. Economic productivity is tied to the growth of

applications in computational mathematics.

4. Male domination of the mathematical sciences is both an internal as well as external

societal factor that affects mathematical thought and development in that mathematics is

perceived as a male subject by society in general. This false perception provides little impetus

for women to want to pursuit careers in the mathematical sciences. The male domination in

mathematics is probably linked to the male dominated corporations and atmosphere that exists in

most research groups.

5. Interpretation of what constitutes mathematical knowledge is rooted in culture because

mathematics is the creation of humans and can be therefore be thought of as a cultural product.

What constitutes mathematical knowledge also depends on whose interpretation that knowledge

is being evaluated. Mathematical knowledge from a sociological point of view may have more

to do with quantitative literacy than with academic mathematics. From this perspective

mathematical knowledge is comprised of activities that include the following functions:


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Counting, measuring, designing, playing, explaining, estimating, and making inferences based on

quantitative factors.

6. The influence of professional organizations, to a certain degree, shapes the kind of

mathematics that is endorsed. For example, many organizations such as the American

Mathematical Association for Two Year Colleges (AMATYC) is very pessimistic and critical

about their inclusion of ethnomathematics in their journals. In fact, some of their editors follow

a Eurocentric perspective in their evaluation of articles. The main purpose of mathematics in the

1700s was to obtain results. In the 1800s and perhaps until recently the aim of mathematics has

been rigor.

Section 6: The Mathematics of Karl Marx

Out of the massive and complex writings of the philosopher G. W. F. Hegel, and the

derivative writings of the neo-Hegelians who flourished in Germany during Marx’s youth,

certain key concepts of Marxism developed. What Karl Marx derived from Hegel’s writings was

that the way to understand the world was not to see it as a collection of things but as an evolving

process. Marx was a believer in abstraction, systematic analysis, and successive approximations

to a reality too complex to grasp directly. It was precisely the complexity and ever-changing

phenomena of the real world that made systematic analytical procedures—science—necessary.

According to Marx, “all science would be superfluous if the appearance, the form, and the nature

of things were wholly identical.”

Marx was more interested in the foundation of the calculus than in the mere procedures,

rules, or techniques of the calculus. Marx’s preparation in mathematics during his time was

considered adequate. He received his training at the Gymnasium of Trier in 1835. Marx’s

fascination and curiosity with the foundations of differential calculus led him to the study and


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comparative methods of the differential calculus. He studied the methods of Newton and

Leibnitz—he called it “the mystical differential calculus.” He also studied and compared the

methods of D’Alembert and Lagrange. These he called respectively—“the rational differential

calculus” and “the purely algebraic differential calculus.”

Marx wrote about 900 pages of his mathematical studies and he never intended to publish

them despite of the fact that Engels did considered it worthwhile. In his manuscripts, Marx

derived the formula for the derivative of y uz where u and z are functions of x. He arrived at

the following formula that is familiar to many students of calculus for finding the derivatives of

the product of two functions:

dy du dzz u

dx dx dx

As a prelude to the continuing discussion of Marx’s interest in the foundation of calculus, it

will be necessary to establish some basic understanding of the derivative.

The mathematical limit definition of the derivative is therefore given as follows:

0

( ) ( )'( ) lim

h

dy f x h f xf x

dx h

An equivalent form of the preceding mathematical definition is that the derivative of a

function f at a number a is denoted by f’(a) and if the limit exists it is then given by:

( ) ( )'( ) lim

x a

f x f af a

x a

Both definitions above are concerned with a point that approaches a limit. The definition

given by 0

( ) ( )limh

dy f x h f x

dx h

is concerned with 0h , that is, h approaching zero. The

definition given by ( ) ( )

'( ) limx a

f x f af a

x a

is concerned with x a , that is, x approaching a.


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Marx’s concern has to do with whether or not h is actually zero or in the case of the equivalent

definition, whether or not x is actually a. He is inquisitive about the mechanics of the derivative

in that it seems to him that the calculation of the derivative treats the case of 0h and x a

as if they were actually either zero or x a . Marx wanted a process for finding the derivative in

which the process itself would be defined as 00 , and in this instance be denoted by the symbol dy

dx .

Marx provided an example for finding the derivative of a function. He used 3( )f x x to

illustrate the method. His methodological suggestion is presented below:

3( )y f x x

1 1( ) ( )f x f x y y

3 31 1y y x x

3 3 2 21 1 1 1( )( )x x x x x xx x

3 3 2 22 21 1 1 1 1 11 1

1 1 1 1

( ) ( ) ( ) ( )( )f x f x y y x x x x x xx xx xx x

x x x x x x x x

when 1x x , or 1 0x x , or 1x x , then the result is:

2 2 203

0

dyx xx x x

dx

According to Marx, “we obtain a first preliminary derivative, namely 2 21 1x xx x , and this

passes by 1x x into a definite derivative . . . its shows that the derivative is actually 00 ,

obtained when 1x x is actually zero. The reader is encouraged to use the illustrative method of

Marx to find the derivatives of 2( )f x x , 4( )f x x , or 5( )f x x . After finding a few

derivatives of various powers, it will be evident that the derivative of a function of the form


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( ) ny f x x is 1' '( ) ndyy f x nx

dx . This formula is known as the Power Rule and is stated,

mathematically as follows: If n is a positive integer, then 1( )n ndx nx

dx . The proof of the

Power Rule is outlined below:

If ( ) nf x x and ( ) ( )

'( ) limx a

f x f af a

x a

, then it follows that the derivative of the function

( ) nf x x is defined as:

( ) ( )'( ) lim lim

n n

x a x a

f x f a x af a

x a x a

From algebra it follows that the formula

1 2 2 1( )( )n n n n n nx a x a x x a xa a

can be established by multiplying out the right hand side.

1 2 2 1( )( )'( ) lim lim

n n n n n n

x a x a

x a x a x x a xa af a

x a x a

1 2 2 1'( ) lim( )n n n n

x af a x x a xa a

1 2 2 1'( ) n n n nf a a a a aa a

1 1 1 1'( ) n n n nf a a a a a

1'( ) nf a na

Marx believed that the derivative should be derived by a differentiation process and not

from the Binomial Theorem. A second proof for the Power Rule that uses the Binomial

Theorem, that is, 0

( ) ( )n

n n k k

k

na b a b

k

is illustrated below.


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0 0

( ) ( ) ( )'( ) lim lim

n n

h h

f x h f x x h xf x

h h

0

0 0

( )( )

'( ) lim lim

nn k k n

n nk

h h

nx h x

kx h xf x

h h

1 2 2 1

0

( 1)2

'( ) lim

n n n n n n

h

n nx nx h x h nxh h x

f xh

1 2 2 1

0

( 1)2'( ) lim

n n n n

h

n nnx h x h nxh h

f xh

1 2 2 1

0

( 1)2

'( ) lim

n n n n

h

n nh nx x h nxh h

f xh

1 2 2 1

0

( 1)'( ) lim

2n n n n

h

n nf x nx x h nxh h

Since every term has and h and h approaches zero, that is, 0h . Then, it follows that

1'( ) ndyf x nx

dx

Finally, the above result is the general version of the Power Rule.

Section 7: Distribution of Wealth and the Lorenz Curve

Economists use a cumulative distribution called the Lorenz curve to describe the

distribution of income or wealth between different households in a given country. In 1905,

Lorenz suggested a mathematical method for measuring the concentration of wealth in a given

population. A Lorenz curve represents the cumulative proportion of the population ranging from

the poorest to the wealthiest on the x-coordinate axis. The cumulative proportion of total wealth

held by the population in question is represented on the y-coordinate axis. The Lorenz curve is


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constructed on the basis that the upper and lower extremities are respectively (1,1) and (0,0).

The points on the Lorenz curve are determined by ranking all households by income and then

computing the percentage of households whose income is less than or equal to a given

percentage for the total income for the country. For example, a point, such as, (0.4,0.2) on a

Lorenz curve means that the bottom 40% of the households in that population receive 20% of the

total income available. On the other hand, the point (0.8,0.6) indicates that the bottom 80% of

the households receive 60% of the total available income. The line y x is used to indicate how

much the income distribution for a given population or country differs from absolute equality.

More precisely, the coefficient of inequality of income distribution is defined as the ratio of the

area between a Lorenz curve and the line y x to the area under the line y x . Since the area

under the line y x from 0x to 1x is 12 , it follows that the coefficient of inequality of

income distribution is simply twice the area between the two curves. The mathematical

definition of the coefficient inequality of income distribution for a Lorenz curve is: If ( )y f x

is the equation of a Lorenz curve, then the Coefficient of Inequality of Income Distribution is

given by

1

0

. . 2 ( )C I x f x dx

The calculation of the coefficient of inequality requires that the reader be familiar with the

properties of basic integration. To understand the mechanics and meaning behind the coefficient

of inequality, consider the case for a Lorenz curve defined by 23 2( )

5 5f x x x . Applying the

definition of 1

0

. . 2 ( )C I x f x dx it follows that


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1 1 1

2 2

0 0 0

3 2 3 3. . 2 ( ) 2 ( ) 2 ( )

5 5 5 5C I x f x dx x x x dx x x dx

2 3 10

3 1 3 1 1. . 2( ) | 2( )

10 5 10 5 5C I x x

The coefficient of inequality is 15 or 0.2. This number provides a relative measure of the

income distribution in a given country. For example, if the coefficient of inequality for a second

country equal to 0.3, then it would be concluded that income is less equally distributed in this

second country.

Section 8: The Social Efficiency of Ethnomathematics

The ethnomathematics developed by different social groups is likely to be more efficient

than academic mathematics in that the practical application of quantitative techniques is

somehow linked to the problems faced by that specific social group. When the solution to

problems require a mathematical treatment within a societal context, the solution to the problem

contributes to the ethnomathematical development of the cultural group. The by-product of this

ethnomathematical development will be more efficacious, over time, than the mathematical

models presented in textbooks. Ethnomathematics should, therefore, not be interpreted as

“second class” or watered-down mathematics—it is simply different. This difference is a

cultural expression based on the sociocultural praxis of mathematics. Culture influences the way

members of society see and understand the world. For example, the concept of privacy as

practiced in the United States, is rarely experienced in communal societies.

The social efficiency of ethnomathematics is based on the idea of how people relate to each

other and to the world. Meanings of signs and codes change from cultural group to cultural

group. These changes are comprised of social activities such as counting, estimating, inferring,

measuring, modeling, and classifying. The activities ultimately shape the social mathematical


127

practices of cultural groups within a given society. Ethnomathematics in terms of the needs of

society are more efficacious than academic mathematics in that the latter usually train students to

solve pseudo-problems based on memorized techniques or algorithms in order to achieve high

grades. This kind of teaching places the student in a vulnerable position in terms of his or her

relationship with society because pseudo-problems teach little about what is necessary for full

participation in democratic society—it fails to give students what is needed for life and civic

responsibility—Quantitative literacy. With that in mind Paulo Freire proposes an educational

content for critical social consciousness. In other words, searching with the students for

experiences that give meaning for understanding the world within society.

From a sociocultural perspective, a dialogue between a teacher and students where the

teacher speaks through the ethnomathematics learned in college (formal academic training) and

students speak through theirs, is not neutral in that teaching and the interaction of ideas is a

cultural exchange. Such a dialogue creates opportunities for students to strengthen their own

sociocultural roots because the ethnoknowledge of both teacher and students is recognized as

valuable in the educational process.

Section 9: Reflections on the Past, Present, and Future

At the turn of the twentieth century four percent of all Americans went to college. Today

nearly eighty percent of Americans go on to college or at least expect to acquire education

beyond high school. At the turn of the century, only a small percentage of Americans spent their

professional lives engaged in knowledge-based work that required quantitative literacy. In

today’s complex society, a worker is expected to be intelligently creative in all facets related to

the productivity of the individual’s chosen profession. Also at the turn of the century decisions

about key issues that impacted society were made by a small leading group economically


128

prosperous white men and our nation had a separate and sometimes not-so-equal societal

standard for people of color and for women. Today, the United States celebrates diversity and an

emphasis that all citizens—including those who have been historically excluded—understands

and is an active participant in a democratic society that affects the quality of life. This

argumentation will only be a platitude unless every citizen, especially those who have been

disenfranchised develop the facility to deal with complex questions that require quantitative

literacy.

Finally, in spite of the claim made by educational institutions and those who govern and

control them that they encourage free and critical thinking, such institutions and organizations as

a general rule discourage critical consciousness and free thinking or expression, in particular

when that consciousness touches important societal issues. Those who question events and seek

new interpretations are seen as dangerous. Teaching people to think for critical consciousness, to

question to doubt, to argue, to experiment, and to be critical is seen by some as a threat to the

establishment as well as to the beliefs and authorities every where and of every kind. In fact,

Socrates was accused of this in ancient “democratic” Athens. Perhaps, Socrates was as much of

a threat as Dr. Martin Luther King was in relatively modern times. Mathematics is a political

activity in that it creates attitudes and intellectual models that will ultimately help members of

society to grow, develop critical consciousness, be more aware, become more involved, and

therefore more confident. Thus being able to go beyond existing parameters. The antithesis of

the view of mathematics as a political activity that does not cause the creation of critical

consciousness is likely to produces citizens who are rigid, timid, and alienated. Mathematics is a

human product and should be seen a bridge between the gap that often exists in science and

technology in relation to sociocultural concerns. From an ethnomathematical perspective


129

mathematics should cause citizens to fight against the evils of modern time—intolerance,

ignorance, and rigid absolutism.


130

Chapter 6

The Politics of Mathematics Education

This chapter will discuss the political ramifications of mathematics. Education in general

is political in nature and has become an issue of debate in the political landscape of modern

society. Despite the opinions of many mathematicians and mathematics educators alike,

mathematics has everything to do with the politics of modern society in that those in positions of

power dictate and in many instances prescribe how much mathematics and the kind of

mathematics that is to be taught in schools. Mathematics is directly related to the educational

process of our citizens. Funding sources ultimately affect the future knowledge and education of

free democratic society. Consider for instance that while many Americans seem to show little

reluctance to spend $25,000 a year on a prisoner, we do to a certain degree resist spending

$5,000 a year on a student in school. If the benefits of social and political power—particularly

the right to be educated—reside only in the hands of a few elite individuals, then how can a

society hope to produce the continuous advancements and innovations that sustain the

mathematical and quantitative sciences?

Hypocrisy is one of the most daunting and pervasive aspects of democratic politics. While

politicians talk openly as if they want people to be more educated at the same time they fear

intellectuals and are suspicious of experts. Our national heroes and heroines are not scholars,

teachers or scientists. When was the last time a famous movie star of either gender was cast as

an experimental scientist or theoretician who did something good as opposed to unleashing some

evil and monstrous virus?


131

No one seems to be satisfied with school mathematics. The on-going theme in politics is

that scores are always too low, students are bored and unprepared, teachers need to be more

accountable, and corporate America feels students don’t learn what they need to know. If one

thinks about the political ramifications behind this single subject one might be inclined to think

about who really owns school mathematics. In our modern society, the academy needs to

provide students with the tools of productivity and democratic freedom. In terms of mathematics

this means software, use of spreadsheets, and the ability to quantify problems. These are only

few of the necessary things that can be taught and should be taught. Mathematics is a universal

language and without the mastery of skills in algebra students will never understand how

mathematics is used. This lack of quantitative skill puts a society at risk in terms of the possible

political oppression of those who are in power. This is a reality in a number of developing

countries in that the majority who lack education are easily persuaded by an elite few—those

who control the government and make decisions that ultimately affect the individual.

Government and employers often express the need for graduates who are better prepare to

meet the challenges of the world of work. The problem is that university faculty, especially

mathematicians, often decry the open-ended problems that dominate emerging curricula,

believing that this approach undermines the technical fluency crucial to success in later courses.

Academic mathematicians are concerned that by stressing exploration and multiple approaches

to solving problems, the new curricula undermine important characteristics that give

mathematics its distinctive power—accurate answers and rigorous proof. In light of the

legitimate concerns of government and employers in addition to the concerns from academic

mathematicians in relation to maintaining the distinctive power of mathematics, then how does

one address each of the following questions from a social and political perspective?


132

1. What are the real mathematical skills need for today’s workplace? How are they likely

to change?

2. What mathematical skills are suited for bridging science and technology?

3. Are the mathematical skills needed for work significantly different from those skills

needed in academia?

4. Is school mathematics really one isolated subject?

5. Is mathematics for the workplace the same as mathematics for education?

Section 1: Sociopolitical Implications of Mathematics

The multicultural nature of American society offers a unique opportunity to quantitatively

prepare its young citizens to compete in an international economy. However, the reality that

concerns quantitative literacy in relationship to society is that mathematics is the one subject that

has the poorest record of equity. The poor performance of this subject has cascade into many

areas of education and employment, frequently leading to withdrawal from school and failure to

obtain employment. Politicians talk about “going back to basic” as a panacea to a great social

concern. The political debate over “going back to basic” is based on confusion between

mathematics, arithmetic, and quantitative literacy. Though they are certainly related, they are

different in many aspects.

The sociopolitical implications of mathematics can be classified under the following

general broad categories:

1. Access is directly related to creating opportunities for all students to gain entrance to

higher mathematics. Access is an external factor societal factor that is affected by the political

role of federal, state, and local policy makers.


133

2. Expectations relate to society’s conflicting demands for the goals and

pedagogy/andragogy (means) of academic mathematics. The issues of expectations deal with the

relationship between skills and understanding, and about the connection with other disciplines as

well as preparation for employment.

3. Articulation which can be thought of as the creation of a smoother transition between

education and work, between high school and college, and between college and life as a member

of society.

4. Integration deals with issues concerning the opportunities and impediments for

connecting mathematics and quantitative approaches to solve problems that affect society in

general.

5. Numeracy is closely related to the idea of quantitative literacy—the level of

mathematical knowledge and skill that sustains citizen’s participation in a free democratic

society.

Section 2: Paulo Freire’s Epistemology in Relation to Ethnomathematics

Paulo Freire’s theories about the nature of knowledge provide a foundation for the

emerging field of ethnomathematics in that to understand mathematical thought one must also

understand and reconsider what constitutes mathematical knowledge. Freire’s epistemology

provides a theoretical basis for ethnomathematics. To understand different mathematical ways

of thinking we must learn about how culture—language, praxis, ideology, and tradition—

interacts with a student’s view of mathematics. Learning about these views and ways of thinking

are likely to deepen mathematical and pedagogical knowledge. From a Freireian perspective we

must reclaim the hidden and distorted history concerning the mathematical contributions from

other cultures. Freire proposes that freedom can only be acquired by conquest—it is not gift. It


134

must be pursued constantly and responsibly. Freedom is neither an ideal located outside

humankind nor is it an idea that becomes myth. It is rather the indispensable condition for the

quest of human completion. The epistemology of Paulo Freire sustains a natural concern for

humanization in that such concern immediately leads one to the recognition of dehumanization,

not as an ontological possibility but rather as an historical reality.

Education in general and ethnomathematics education in particular as a practice of freedom

is diametrically opposed to the domination that is often found in the practice of education in the

American academy. Liberatory education denies that humankind is abstract, isolated,

independent, and disjointed from the world. Education as a practice of freedom also denies that

there exists as a reality apart from humankind. An educational experience that sustains freedom

is characterized, according to Freire, as one in which the teacher is no longer the one-who-

teaches, but one who is himself taught in dialogue with the students, who in turn while being

taught also teach.

The practice of mathematics education suffers from narration sickness because the teacher

talks and lectures about reality as if it were motionless, static, compartmentalized, and

predictable. Under the traditional system of education the teacher expounds on a topic that is

completely alien to the existential experience of the learner. This method of argumentation and

disputation is characteristic of an oppressive education in that students are treated as empty

containers that are to be filled by the teacher. Sadly, but often true, the more completely these

containers are filled the better the teacher is. Under this oppressive educational model, the more

meekly these receptacles permit themselves to be filled by the teacher, the better the students are.

Oppressive education becomes the act of depositing—the students are the depositories while the

teachers are the depositors. The ideology of oppression negates education and knowledge as a


135

process of inquiry and self-discovery. True knowledge emerges only through invention, re-

invention, restless, and impatient inquiry about the nature of knowledge and truth. Oppressive

education sustains, encourages, and mirrors oppressive society as a whole. Oppressive education

is rooted in the following principles:

1. The teacher teaches and the students are taught;

2. The teacher knows everything and the students know nothing;

3. The teacher thinks and the students are thought about;

4. The teacher lectures and the students listen—meekly;

5. The teacher chooses and enforces his choices—student comply;

6. The teacher acts and the students have the illusion of acting through the action of the

teacher;

7. The teacher chooses the program of instruction no matter how antiquated—students

must adapt to it;

8. The teacher confuses the authority of knowledge with his/her professional authority,

which may be in opposition to the freedom of the students;

9. The teacher is the “subject” of the learning process while the students are mere

“objects” or receptacles to be filled.

Authentic education from an ethnomathematical perspective in relation to the epistemology

of Paulo Freire sustains the investigation of thinking by encouraging dialogue and exchange of

ideas. Unlike traditional education, authentic education causes the student to become a critical

thinker in that only dialogue—which requires critical thinking—is capable of generating critical

thinking. The absence of dialogue results in lack of communication, and without communication

there can be no true education. From a Freireian perspective, education is a human act rooted in


136

the praxis of freedom and cultural exchange. Any situation that prevents others from engaging in

the process of inquiry is a violent act in that self-sufficiency is incompatible with dialogue.

Education as a practice of freedom should cause liberation and emancipation. Educational plans

are intertwined with the political process. Unfortunately, these plans often fail because the

authors of such plans design them in accordance to their own personal views of reality, never

taking into account how these plans may affect those for whom the plans are designed.

Politicians and educators are not understood because their language is not attuned to the concrete

situation of the populace.

Section 3: Historiographical Foundations of Eurocentrism in Mathematics

The historiographical foundations of Eurocentrism in mathematics are based on an elitist,

racist, and sexist curriculum. Institutionalized Eurocentrism reinforces ethnic, cultural, and

sexual inferiority complexes among people of color and women in that from this perspective, it is

believed that mathematics originated among white men and was further developed by their North

American descendants. Eurocentric scholarship has not acknowledged any contributions to

mathematics from non-European cultures. Eurocentrism in mathematics are based in the

following themes:

1. Negligence to recognize mathematics from a materialistic perspective in relation to its

development with economic, political, and cultural changes;

2. Confinement of mathematical activities and pursuits to an elite few who are believed to

have certain qualities and gifts denied to the vast majority of humanity;

3. Widespread acceptance of the view that mathematics discovery and development can

only occur from a rigorous form of deductive axiomatic logic believed to be the unique product


137

of the Greeks and thereby dismissing empirical or intuitive methods as being mathematically

irrelevant;

4. Belief that the presentation of mathematical results must conform to the formal style

devised by the Greeks over 2,000 years ago

5. Belief that mathematical knowledge can only be undertaken by an elite few self-

selecting group of individuals who have control over the acquisition and dissemination of such

knowledge which in many cases has a high Eurocentric character.

The Eurocentric education permeates and influences all aspects of the academy and

proposes that mathematics is a neutral subject free from human concern and unaffected by

culture. Ethnomathematics should not be understood as second class mathematics or vulgar

mathematics, but rather as a different cultural expression. The genesis of ethnomathematical

ideas depends on the cognitive practices of culturally differentiated groups. The ideas generated

by the groups are either maintained, evolved, or simply disappear in accordance to the dynamics

of the group and their relationship to the environment and/or other groups.

Section 4: The Theory of Antidialogical Action

The theory of antidialogical action is based on cultural invasion in that its application is

parallel to the tactics and manipulative goals that serves the purposes of a conquest as was the

case during the European invasion of the Americas. In the context of this social phenomenon,

the invaders penetrate the cultural boundaries of another group, and in without regard toward the

group’s potential and existing creations; they impose their own views of the world upon those

whom they invade. The result of such intervention is inhibition of creativity on the part of the

invaded group as well as stifling the creative powers of the invaded culture. Cultural invasion is

a violent act against the persons being invaded because through the process of invasion—


138

colonization or conquest—the culture invaded loses or are at best facing the threat of losing the

culture’s originality. Cultural invasion is a modality of antidialogical action—the invaders are

the authors and actors while those whom they invade are the mere objects of the invasion

process. The invaders mold while the invaded are molded in accordance to the traditions and

customs of the invaders or oppressors. The invaders choose; those who are invaded follow the

choices in that the invaded has—only—the illusion of acting independently.

From an ethnomathematical perspective, the mathematical traditions of non-European

cultures are invaded in that their contributions to the development of mathematics is usually

ignored or not recognized as important in the development of mathematical truths. All

domination involves some degree of invasion. This is invasion can be physical, overt, or

camouflaged, with the invader playing the role of a helping friend. Invasion functions as from of

cultural, economic, and educational domination. In the context of ethnomathematics, for cultural

invasion to succeed, it is indispensable for those invaded to be convinced of their intrinsic

inferiority.

Everything has its opposite; so if those who are invaded consider themselves inferior, then

they must—by default—recognize the superiority of the invaders. The stronger the invasion the

more alienated the invaded group will become thus resulting in the invaded to desire to be like

the invader—walking, talking, and dressing like them. Invaders are making use of science,

technology, and mathematics in an effort to improve and refine their actions. It is therefore

important for the invaders to know the past and present of those invaded in order to guide the

social evolution in favor of the invader’s interests. The theory of antidialogical action is cultural

invasion—steering and conquering—its application is both the instrument as well as the result of

domination. The theory antidialogical action transforms the dominating pronoun “I” into the


139

dominated and conquered “thou”—not “I.” In order for cultures to be emancipated from the

invaders and/or oppressors, they must unite and severed the umbilical cord of magic and myth

that binds the invaded to a world of cultural oppression. Unity links the members of the culture

to each other in a revolutionary process that can be defined as cultural action. In light of this, the

object of ethnomathematics is to challenge Eurocentrism in mathematics education. This is

perhaps the reason why ethnomathematics is bold, consistent, and radical—not sectarian.

Finally, there can be no freedom or truth without authority, just as there can be no authority

without freedom or truth.

Section 5: Mathematics of Poltics and Politics of Mathematics

To understand the mathematics of politics, citizens need to develop critical consciousness

based on ethnomathematical practices and quantitative literacy that allow humankind to

understand institutional structures of society. In essence, understanding the mathematics of

politics is about the ability to “critically” read the world we inhabit. This can occur in many

different forms including, but not limited to, the understanding of numerical descriptions of the

world (e.g., graphical representation of data, percentages, and fractions). Clear and accurate

understanding of the mathematics behind political knowledge also requires citizens to have the

ability to, quantitatively, verify the arguments made by others as well as being able to restate

information to further understand and analyze how data are collected and transformed into

numerical descriptions of society. The purpose of this critical consciousness is to develop a keen

sense of inquiry and understanding concerning decisions and choices that affect society as a

whole. For example consider that in 1989 the top 1% of Americans (934,000 households)

combined a net worth of $5.7 trillion while the bottom 90% (84,000,000 households) had a net

worth of about $4.8 trillion. According to government statistics, 1% of the wealthiest


140

Americans, in 1989, was worth more than the combined total of bottom 90% of Americans. The

politics of mathematics is closely associated with mathematical distortions about the world. For

example consider the Mercator map which greatly enlarges the continent of Europe while

shrinking the size of Africa when in reality Europe is smaller than 20% the area of Africa. The

Mercator map, created in 1569, distorts land areas as well as the relative importance of various

regions of the world. The politics of mathematics dictates that the intellectual activity of those

without power is always labeled as non-intellectual.

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