Multicultural History of Mathematics A Project Based ...webpages.charter.net/druby/Papers/MultiCulturalMath.pdf · A Multicultural History of Mathematics is a required Middle ...

Multicultural History of Mathematics

A Project Based Middle School Course

Douglas Ruby, Leo Tometich, Melinda Willis

Diversity

01.501

Roland A. Gibson

~

One can invent mathematics without knowing much, if any, of its history.

But one cannot have a mature appreciation of mathematics without a

substantial knowledge of its history.

- Abe Shenitzer (Kleiner1988)

~

Abstract

Many students at risk of failure in mathematics loose interest during the Middle

School years (Loveless). A Multicultural History of Mathematics is a required Middle

School course designed and targeted to capture the student’s interest in mathematics

through relational interest and current application. Through a project-based curriculum,

students are exposed to the historical foundations of mathematics and how these concepts

are now applied in daily life. The course is designed to target the times in history that

most interest each specific class of students, thereby enhancing the chance of catching the

student’s interest. Through project based studies, students learn about interesting “non-

traditional” aspects of mathematics, which will hopefully become a spring board for a

desire to further pursue the study of mathematics.

Why and How?

Once students are involved in Middle School Mathematics, teachers often hear the

mantra “ Why do I care about math? Will I ever use this? Who came up with this stuff

anyway?” These are great questions that need answers! Through this course students will

learn how math is part of their “every day” lives, how these uses connect to history, and

who some of the important people were the history of mathematics.

Over the past 30 years the academic progress of American students has been

tracked with the National Assessment of Educational Progress, NEAP, test. Results from

the years 1990 – 1996 show student performance as both gaining and loosing. An

analysis (Loveless) of these results shows that student’s ability to problem solve, analyze

data, and do geometry have improved from the 1990 levels. These same tests show that

student’s abilities in basic arithmetic are dropping. This is of great concern. As stated by

Loveless, “Only if students master arithmetic, can learning algebra follow.” Thus if

students are not mastering their arithmetic skills by 8th grade, how are they to be

successful in secondary mathematics and beyond? Subsequently many students feel they

can not do math and loose their interest and motivation, not seeking to learn beyond their

minimum requirements.

There are a number of steps that can be taken to turn this trend around. These

steps differ depending upon when in a student’s schooling they are applied. Certainly for

students still in elementary school, a renewed emphasis on mastering arithmetic skills is

necessary, and will hopefully reverse this trend. Middle school students need a plan that

not only improves their arithmetic skills to the level of mastery, but one that also includes

a component in the curriculum which awakens their interest in mathematics. We are

proposing that component of the curriculum be the course detailed within this paper. The

course is on the multicultural history of mathematics. It relates the history of math to

specific cultures around the world (possibly including their own) and how this history

relates to the math the students use in their every day lives. This course is designed to

help students gain an interest in further pursuing their mathematics education, to raise

their level of success in those courses, and additionally to teach them research and

presentation skills.

We are recommending a course which is offered one of two ways. The first would

be a stand-alone class offered once a week for 1 class period. The second option would

be to devote one class period a week from the standard mathematics curriculum to the

curriculum outlined in this paper. Our preference would be the stand-alone class so as to

not take away instruction time for the standard math class.

There are both sort-term and long-term desired results for the course. The short-

term desired results are three-fold; an increase in post-treatment affective scores, students

perceived connections between culture studied and math, and the students perceived

positive relationship between math and everyday interests. The desired long-term results

are also three-fold; higher enrollment levels in high school level math courses, higher

attainment in these courses, and higher scores on standardized tests.

In order to measure the success or failure of including such a course in the 7th

grade curriculum, both short-term and long-term assessments are necessary. We propose

that the short-term assessment be of an affective manner. Both a pre and a post-treatment

questionnaire, utilizing a 5-point Likert scale, would be given and analyzed to measure

the attitude of the students before and after taking the course. The long-term assessment

would be of an environmental manner. Included in the study would be the students

achievement scores (MCAS, NEAP, TIMSS), enrollment records in middle and high

school level math courses, attendance, and other achievement measures utilized by the

school.

Course Overview

The purpose of this course is for students to gain a new understanding of and

interest in mathematics by looking at modern mathematics through the lens of other

cultures in history. As students research a mathematical topic through a culture of their

interest and find a connection to mathematics in their everyday lives, it is the intent of

this course that the student will make a connection that entices them to further pursue the

study of mathematics.

The plan of study is for a 16-week course that meets once a week for 50 minutes.

The intent is not to replace 7th Grade math, but rather to supplement and increase student

interest levels. This class is designed to either be required as 1 out of 5 regular days of

mathematics instruction, or else as a stand alone “special” class such as art or music,

which meet once a week for a semester.

The course is designed to run mostly in a workshop fashion. Some lecture is

necessary in order to teach the required skills for preparing and writing the project and

report. Most of the classes will be composed of initial instruction, followed by workshop

time when students apply what they have just learned to their research. Four classes are

devoted solely to workshop time. The last four classes are student presentations of their

findings, with the final class ending with a class designed Math Holiday celebration.

Course Details – A Teacher’s Guide

Week 1 Course overview Describe to student’s purpose and format of class. Discuss their ancestral backgrounds and how they might chose that as their research topic. Discuss why would this might be interesting?

Assignment Read handouts, bring questions to next class (Handouts are brief overview sheets of cultures of interest, syllabus, rubric, assignment log, list of reference suggestions, Project schedule and scope)

Week 2 How do you research your topic? What is a good resource material? Conduct class in the school library. Discuss what it means to “research” a topic. What are the steps necessary? Discuss what a good resource material is. Show students how to use resource materials in library. Choose a topic and then show the students how to find information on that topic in books, magazines, and on the internet.

Assignment Choose your research topic Find 3 resources (a combination of text and internet) Bring them to class!

Week 3 Review resource materials How do you document your resources? Show students correct APA style for documenting their resources. Have students use class time to document the sources they have with them. This avails you as a resource as they have questions.

Assignment Find additional resources as necessary Document resources in use

Week 4 Taking research notes Bring packages of note cards and show students how to properly take and credit research notes, Explain plagiarism and how to make sure you don’t.

Assignment Document resources used

Collect a minimum of 10 research notes

Week 5 Review and organize research notes How to write an Outline Any questions on note taking? Suggest different methods for organizing research notes. Bring in examples. Discuss how to write an outline for a research paper of this style.

Assignment Collect a min. of 5 notes and organize Write outline

Week 6 Are you ready for more resources? How to write an Abstract Touch base that they are having success collecting the necessary resources and notes. Discuss how and why to write an abstract. What are the requirements, restrictions?

Assignment Assess resource need Find additional resources as necessary Collect and organize a min. of 5 notes Write Abstract

Week 7 Half way there! – How is your research progressing?

Project and presentation description Touch base with student progress, questions… Discuss requirements for project and presentation related to research paper.

Assignment Write a short description of your project Collect and organize more data as necessary Bring report related work to work on in class.

Week 8 Review project and presentation ideas Review student’s descriptions of their project and presentation. Guide them as necessary for a successful outcome. Have students begin body of their paper while they are in class.

Assignment Begin body of paper Collect materials for project

Week 9 Working session – bring paper or project to class How to write a Conclusion Discuss how to write a conclusion Facilitate as students work on paper or project in class.

Assignment Work on body of paper, write conclusion Collect materials for project

Week 10 Working session – bring paper or project to class Facilitate as students work on paper or project in class.

Assignment Work on final draft of paper Begin to assemble project


Assignment Complete final draft of paper Assemble project Begin to organize presentation


Assignment Complete project and presentation

Week 13 Final presentations 1 - 9 3 - 5 minutes per presentation



Week 16 Final presentations 28 – 30 (3 - 5 minutes each)

Math Holiday Celebration (Last 30 minutes of class) Students and teacher design celebration to celebrate the cultures studied during the semester.

Topic Summaries

Below are 2-3 page summaries of suggested cultures and mathematical concepts that the students could choose from to research. Each summary includes research background for the teacher, targeted questions for the teacher to choose from to help guide student research, and related resources (Appendix II) for both students and teachers.

~

Babylonian and Egyptian Math

The Babylonians and Egyptians were the first early civilizations (from

approximately 3000 BC to 300 BC) to contribute to modern day mathematics. While

mathematics as a formal discipline of study was not developed, many practical uses for

mathematics were developed and used by the Babylonians and Egyptians. As primitive

peoples settled into permanent communities, mathematics were developed to address

issues of defining property borders, building structures, making goods, exchanging

goods, taxation, and defining calendars for religious and agricultural uses.

Our understanding of Egyptian math is based upon the hieroglyphics found on

two papyri documents and hieroglyphic inscriptions found on stones. These indicate that

the Egyptians used a non-positional, base 10, number system. There were additional

symbols for 100, 1000, 10,000, etc. Numbers were formed by writing the appropriate

symbols from right to left. Egyptian arithmetic included the four basic arithmetic

operations, +, -, *, (, but they were all basically reduced to additive processes. Addition

and subtraction were done by combining or removing symbols to reach the proper result.

Multiplication and division were also reduced to additive processes. The Egyptians also

developed rudimentary use of fractions. Geometry and algebra were not developed as

separate mathematical studies, but part of arithmetic. Little symbolism was used and

geometry was only developed for concrete practical uses. Geometry was used to redraw

property boundaries after the annual flood of the Nile River, and in astronomy to

maintain a calendar. This calendar was crucial because it was used to determine when to

plant crops, when to harvest, and when to move to avoid the floods of the Nile River.

The calendar was also used to schedule religious ceremonies.

The primary sources of information about Babylonian math are clay tablets that

were inscribed while soft, and then baked to harden the tablet. The most highly

developed arithmetic of the Babylonian civilization is the Akkadian. This is a base 60

system with positional (or place value) notation. The Babylonians also used positional

notation to represent fractions. The Babylonian numeric system had symbols for

numbers 1 and 10. Combining fewer or more of these symbols formed numbers from 1

to 59. Similarly, addition and subtraction were done by adding or taking away symbols

to get the proper value. Multiplication of integers was performed by multiplying

positional values and then adding the results. For example, 37 * 5 would be done by

multiplying 30 * 5, and then 7 * 5. The result of these two operations would be added to

determine the product of 37 * 5. The Babylonians also performed integer division by

multiplying by the reciprocal. This also indicates the ability to manipulate fractions.

Some distinct Babylonian tablets exist which deal with algebraic and geometric

problems. The substance of Babylonian geometry was a collection of rules to determine

the area of simple plane figures, and the volumes of simple solids. Although neither the

Egyptians nor the Babylonians recognized negative numbers, Babylonian algebra was

more advanced than Egyptian algebra. The Babylonians had a solution very similar to

the quadratic formula we use today. More complicated algebraic problems were reduced

by transformations to simpler ones.

The Babylonians lived in Mesopotamia, which was along a major trade route. To

this end, mathematics were used to express lengths and weights, and to exchange money

and merchandise. The construction of dams and canals for irrigation projects required

calculations. The use of bricks to construct larger structures raised numerous numerical

and geometric problems. Computation of granary volumes and the area of the fields

required to fill them are more examples of how Babylonian mathematics were applied to

practical problems and affected many aspects of their lives.

Questions:

1. What did the Babylonian number system look like? How were the symbols for “1’

and “10” used to create numbers from 1 to 59?

2. Do the following math using the Babylonian number system:

a. 5 + 19 = 24

b. 47 - 21 = 26

c. 13 x 20 = 260

3. How did the Egyptians do multiplication?

4. The Egyptians made the following calculations. How far off is each calculation

according to our modern information?

a. Solar year = 365 days

b. pi = 3.1605

~

Greek Math

The Greek civilization conquered the Babylonians and Egyptians by about 300

BC. The Greeks developed mathematics into a field of study and created the foundation

on which modern mathematics is built. They created and developed mathematics to

understand the physical world around them. Mathematics were a result of their

investigation into nature and the key to comprehension of the universe. From

approximately 300 BC to 600 AD the Greeks developed their “rationalization of nature”

concept. They placed an emphasis on the power of the mind in understanding nature, so

they adopted principles that appealed to the mind or that could be rationalized by the

human mind. Mathematical abstractions must help us to understand the observable

world. Aristotle said that we must start from principles that are known and manifest to

the mind and then proceed to analyze things found in nature. We proceed, he said, from

universals to particulars, from man to men, just as children call all men father and then

learn to distinguish. Beyond the mathematic contributions of ancient Greeks, their belief

in the concepts of deduction and rational thought to understand our universe laid the

groundwork for future developments in mathematics and other fields of study.

Since the Greeks replaced the various hieroglyphic systems of writing with the

Phoenician alphabet around 775 BC we have a much better understanding of mathematical

developments during the Greek civilization. Greek mathematical development is divided

into two periods. The “classical period” that lasted from about 600 BC to 300 BC, and the

Alexandrian or Hellenistic period that was centered around the city of Alexandria and

lasted from about 300 BC to 600 AD.

The cream of the classical periods contributions are Euclid’s Elements and

Apollonius’ Conic Sections. Euclid’s Elements comprised thirteen books that were a

compilation of mathematical development up to that time. Euclid developed the form of

presentation of the entire work – the statement of axioms at the outset, the explicit

statement of all definitions, and the orderly chain of theorems. Moreover, the theorems

are arranged to go from simple to more complex ones. Apollonius compiled previous

work and his original material into the eight books that form the Conic Sections material.

This material was so complete and well organized it did not receive further study from

later Greek mathematicians.

The Alexandrian period of Greek mathematics expanded on the work of the

classical period and developed the fields of geometry, trigonometry, arithmetic, and

algebra. The Alexandrian mathematicians were often inventors who discovered practical

uses for math, such as in mechanics – determining the center of gravity of bodies, dealing

with forces on bodies, or pulleys and gears. As a broad generalization they severed their

relationship with philosophy and allied themselves with engineering. Arguably the

greatest mathematician in all of antiquity, Archimedes epitomized the character of the

Alexandrian age. He possessed both practical and theoretical mechanical skill and

founded the area of hydrostatics that deals with the equilibrium of bodies floating in

water. He also invented the catapult and other military devices to protect his home city

of Syracuse when it was besieged by the Romans.

This is but a small sampling or overview of the rich history of the mathematics

during the Greek civilization. Investigation into the contributions of other great Greek

mathematicians like Ptolemy, Diophantus, and many others are left to the reader. The

Greeks developed many arithmetic, geometric, trigonometric, and algebraic concepts that

are the foundation of modern mathematics. An understanding of the mathematical

principles developed by the Greeks is essential for today’s math student.

Questions:

1. What are the five postulates and five axioms from Euclid’s Elements?

2. What field of mathematics did Diophantus study and develop?

a. Algebra

3. Determine at least one flaw or limitation of Greek mathematics?

a. Unable to grasp the concept of irrational numbers

b. Failed to deal with the limit process in the infinitely small or large case.

~

Hindu Math

The Hindu civilization dates from at least 2000 BC, but there was no known

mathematics prior to 800 BC. Most of the early Hindu math was imported from the

Alexandrian Greeks. Smaller contributions came from China. The primary motivation

for Hindus to investigate and understand mathematics was tied to astronomy and

astrology. There is also evidence that the Hindus found pleasure in many mathematical

problems.

Between 800 BC and 200 AD the Hindus did produce some primitive

mathematics. Although there are not any separate mathematical writings, a few facts

about Hindu mathematics from 800 BC to 200 AD can be derived from other documents

and from coins and inscriptions. Separate number symbols for the numbers 1 through 9

appeared after the 3rd century BC. There was no positional notation, nor did the concept

of 0 exist. The geometry of the ancient Hindus is slightly better known because their

religious writings had instructions on how to build altars. The rules described the

conditions for the shapes and sized of altars. The three shapes most commonly used were

the square, circle, and semicircle. In general, the early Hindu geometry was a

disconnected set of approximate verbal rules for determining areas and volumes.

The second period of Hindu mathematics, the high period, may be roughly dated

from about 200 AD to 1200 AD. The Alexandrian Greeks had a significant influence on

the arithmetic and algebra in the early part of the high period. Varahamihira, a Hindu

astronomer, said, “The Greeks, though impure [anyone having a different faith is

impure], must be honored, since they were trained in the sciences and therein excelled

others. What, then, are we to say of a Brahman if he combines with his purity the height

of science?” The Hindus advanced both geometry and algebra. Although the methods of

writing numbers varied until about 600 AD, the Hindus converged on a positional base

10 standard. Zero was treated as a complete number rather than the Greek interpretation

of 0 as the absence of a number. Zero was used in arithmetic operations. For example,

multiplication of a number by 0 resulted in 0, and subtracting 0 from a number does not

diminish the number. Hindu fractions were denoted as a ratio of integers without the bar

between the two integers. The Hindu rule for division by a fraction was the same as ours

today, invert and multiply. Two other Hindu contributions to arithmetic and algebra

during the high period were the use of negative numbers to represent debts, and an

increased understanding of irrational numbers.

While the Hindus took pleasure in advancing arithmetic and algebra, there were

no notable advances in geometry between 200 AD and 1200 AD. They often used the

irrational 10 as the value of . Some interest and advancements in trigonometry took

place because it helped the Hindus in their study of astronomy. They also advocated the

sphericity of the earth in their teachings.

About the year 1200 AD scientific activity in India declined and progress in

mathematics stopped. The Hindu made more contributions to arithmetic and

computational activities of mathematics than to deductive reasoning or patterns as

applied to mathematics. Some evidence of this perspective can be derived from the

Hindu word for mathematics, ganita, which means “the science of calculation”.

Although the Hindus may not have realized it, their contributions to mathematics laid the

groundwork for further advances in mathematics. Concepts such as using a base 10

numbering system, negative numbers, and the recognition that quadratics have two roots

where important steps to the further development in mathematics.

Questions:

1. What did the Hindu symbols for the numbers 1 through 9 look like? Make a table

indicating the numbers 1 through 9 and their Hindu symbols.

2. Do the following math using the Hindu number system. Note inclusion of 0 and that

the arithmetic is the same as our system today.

a. 0 + 19 = 19

b. 12

(4/3) = 9

c. 13 x 0 = 0

3. Sometimes the Hindus used 10 for their value of . How far off from our current

value of

is 10?

4. Did the Hindus teach that the earth was round or flat?

~

Arabic or Islamic Math

Mohammed united the Arabs, and by 711 AD the Islamic armies had conquered

lands from India to Spain. These conquests brought the previous mathematical work of

the Greeks, Christians, Persians, and Jews under Arab rule. To their credit, the Arabs

allowed these infidel groups to function freely under Arab rule. Some say that in the

Islamic culture “secular knowledge” was part of religious or “holy knowledge”. This

religious connection made the Islamic mathematicians more sensitive to the everyday

needs of people, and resulted in more practical applications instead of purely theoretical

study.

Basically, the Arabs acquired and translated the major Greek mathematical works

directly from Greek manuscripts or from Syrian and Hebrew versions. They copied

Euclid’s Elements from the Byzantines around 800 AD, and Ptolemy’s Mathematical

Syntaxis was translated to Arabic in 827. Ptolemy’s Mathematical Syntaxis became a

preeminent work to the Arabs, earning the nickname Almagest or greatest work. Over

time the work of Aristotle, Apollonius, Archimedes, Heron, Diophantus, and of the

Hindus were translated to Arabic. Arab scholars improved the translations and added

their own commentaries. This translation and compilation of almost all prior math

knowledge into Arabic may be the greatest contribution Islam made to modern math.

After absorbing the Hindu symbols for numbers 1 through 9 the Arabs added the

modern zero from the Arabic word sifr, which means empty. They also improved the

idea of positional notation, and worked freely with negative and irrational numbers.

Other Arabic contributions are the addition of the bar between the whole numbers of

common fractions, and the words “algebra” and “algorithm” to the mathematical

vocabulary.

The Arabs conquered vast lands from about 500 A.D to 1300 AD. The people

that lived in each of these conquered regions were typically allowed to function with

autonomy in spite of the religious fanaticism that came with the Islamic conquests. This

allowed the Greek mathematical works to be translated to Arabic and improved. These

Arabic translations that were available in Europe during the Renaissance have formed the

basis of modern math, as we know it.

Questions:

1. What number symbol did the Arabs add to the Hindu symbols for the numbers 1

through 9?

2. How did the Arabic translation and compilation of previous mathematical works

affect western mathematical development?

3. Name at least three Arabic mathematicians and a summary of their contributions.

~

Renaissance Math (1200AD – 1700AD)

Almost one thousand years after the fall of the Roman Empire, during the 1300s

and 1400s, medieval western European civilization began to give way to an age of

enlightenment known as the Italian Renaissance. This period began with a rekindling of

interest in ancient arts and science. Patronage from wealthy Italian families engaged in

trade throughout the Mediterranean supported the studies of artists, poets, musicians, and

philosophers.

Ironically, much of what was learned about the great Greek and Roman

mathematics, philosophy, art, science, and culture, was gained by increasing trade with

the Byzantine Greek and Arab world of the time. Italian scholars, most of whom were

educated by the Church in Rome, studied the Roman and Greek masters through Arabic

texts and with Arabic scholars. It is through this exchange with the Arabic world that the

Roman numeral system then in use was replaced with the Arabic numeral system in use

today. Unfortunately, the 1400’s also became the period of the Black Death throughout

most of Europe, again slowing down the pace of academic and scientific development.

Major development in mathematics in Europe began again at the beginning of the

16th Century in algebra and trigonometry (O’Connor and Robertson 2002). However, as

time progressed, Renaissance scholars were increasingly unable to reconcile their new

ideas about science, particularly astronomy, with the teachings of the Catholic Church.

By the middle of the 16th century, as the Protestant Reformation took hold, the center of

academic development moved out of Italy into northern Europe, particularly into France,

Germany, and England. The increasing religious and academic conservatism in Church-

dominated southern Europe culminated in the famous trial of the mathematician and

astronomer, Galileo Galilei, in 1633. From the middle of the 16th century (1550 AD)

through the end of the 17th century (1700 AD), the development of trade, navigation,

astronomy, science, and mathematics are all closely intertwined, with a pace and quality

that had astonishing rapidity and results. This period culminated in the development of

calculus, a mathematical method that is used to describe scientific and physical realities

even today (Eves 1990 and Klein 1972).

Questions:

1. Prior to the introduction of the Arabic numeral system, medieval Europe used the

Roman numeral system.

a. What were the drawbacks of the Roman numeral system? What did it lack that

the Arabic System has? Do some arithmetic using the Roman numeral system

to show examples.

b. Who is Leonardo of Pisa? He is not the famous Leonardo of Venice, Leonardo

da Vinci. (Hint: We know him today by his nickname as “the son of

Bonnacci”). Why is he important to the introduction of the Arabic numeral

system? How did he come to be able to learn so much from the Arabs?

2. What do you believe is the difference between “rigorous” mathematics and

“empirical” mathematics? Why was trade with the Arabs and Byzantine Greeks

important to the redevelopment of rigor in mathematical thought?

3. Ancient Greek mathematics was based entirely on the visual characteristics of

Geometry with only a rudimentary number system and no symbolic Algebra.

Euclidean geometry was based on a rigorous approach that allowed only

constructions made with a compass and straight edge. Why is the introduction of

Arithmetic, Algebra, and Trigonometry important to further mathematical

development?

4. Algebra came to Europe from which culture? Who coined the term Algebra? What

other mathematical terms do we get from this scholar?

5. What important concept did Napier and Briggs develop in the early 1600’s? (Hint:

This concept led to the development of the slide rule.) Why was this important to

development of other pure and applied sciences such as astronomy and navigation?

6. Why were astronomy and mathematics so closely intertwined during the

Renaissance? Try to trace the relationships between some of the famous

mathematician/scientists (Gallileo being one) from 1550-1700AD.

7. Who “invented” calculus? (Hint: Two scholars are credited) Why has this been a

topic of such controversy? Why (and how) do you think two different scholars could

develop such as sophisticated concept in different locations?

~

Mayan Math

The Maya of the Classic Period (roughly 300AD to 900AD) had a very well

developed number system. Unknown to European civilization until uncovered much later

by the Spanish in the early 16th century, the Maya used several concepts more advanced

than those in European civilization.

The Maya used a positional (or place value) number system that was essentially

vigesimal (i.e. base 20). The normal Mayan population is believed to have used a pure

vigesimal system, however the priest class used the notion of 360 days in the calendar

year and thus their number system only went up to 360 (18 x 20) instead of 400 (20 x 20)

in its second position. In addition to a positional (or place value) number system, the

Maya understood and used the concept of zero (0). Both place value and zero in the

Mayan system made it more advanced than the European number system inherited from

the Romans during the Dark Ages.

While archeological records in Central America date pre-Mayan civilization back

to 10,000BC, the Maya reached their peak in the 1st one thousand years AD. The Mayan

civilization collapsed significantly after 800 AD, having been overtaken by the Aztecs (to

the north) and the Inca (to the South), by the time of the Spanish Conquistadors invasion.

Most of our current knowledge of Mayan culture comes from the stone

inscriptions from Mayan temples. These inscriptions were written by the “Priest Class”

and tell us a lot about religious and political life with the Maya.

The Mayan calendar begins at 3114BC (the beginning of the Maya “Long

Count”) and ends in 2012 AD. In addition to the 360 (+ 5 “extra”) day solar calendar, the

Maya used a 260 day religious calendar. The priests carved circular calendar stones that

interlocked. One stone represented the solar year while the other stone represented the

religious year. The intersection point as the two stones were turned would tell the priests

what day it was with respect to each calendar.

While the Maya had a very advanced number system and extremely advanced

astronomical and celestial knowledge, they appear to have had no formal geometry or

trigonometry comparable to the Greeks. Yet, the Maya still were able to develop an

extremely sophisticated ceremonial architecture of stone temples, sacrificial alters, and

other buildings, all perfectly aligned with solar, lunar, and celestial occurrences tied to

their religious calendar.

Questions:

1. What did the Mayan number system look like? How were the “Bars”, “Dots” and

“shell” (representing zero) used to create a number between 0-19? Between 20 and

359?

2. Do the following math using the Mayan number system:

a. 5 + 19 = 24

b. 47 – 21 = 26

c. 13 x 20 = 260

3. How did the Maya do multiplication and division?

4. The Maya made the following astronomical calculations for their religious calendars.

How far off is each calculation according to our modern information:

a. Synodic Revolution of Venus: = 584 days

b. Solar year: = 365.242 days

c. Lunar Month: = 29.5302 days

Asian Math

History tells us that Chinese culture along the Hwang Ho and Yangtze rivers is

probably as old as the ancient Babylonian culture from the “Fertile Cresecnt” of the

Tigris and Euphrates rivers. Unfortunately, we have less written record of ancient

Chinese civilization, since writing was done on bamboo tablets instead of in clay,

papyrus, or stone as was done with the Babylonians and Egyptians. Because these

bamboo tablets were made by taking one section of bamboo, slitting it vertically, and

then pounding it out flat, the writing of the Chinese has always evolved in a vertical

manner instead of horizontal as in other cultures (Eves 1991).

Chinese history is divided into “dynasties”. The earliest recorded dynasty for

which we have a mathematical record is the Shang Dynasty, dating from 1500 BC. From

this era, we have “counting rods” that show that the Chinese used a decimal number

system. These counting rods (or bamboo sticks) later evolved into the familiar abacus.

By 1100 BC, it appears that the Chinese knew the Pythagorean Theorem as can be

seen from the following picture (Stapleton and Tripodi 1996):

One of the most important mathematical texts of the Chinese dates from the Han

Dynasty (206 BC to 221 AD). This text, known as Arithmetic in Nine Sections, contains

knowledge believed to be even older than the Han Dynasty. It describes a collection of

246 problems related to agriculture, business, engineering, surveying, solving right

triangles, and other mathematical problems.

While the western Arabic number system is used throughout academia and in

much of regular Chinese life, the traditional Chinese number system is also in use today.

This number system is decimal (Base 10), but is not a strict “place value” system like the

Arabic system. In Chinese, in addition to symbols for the digits zero (0) through nine (9),

different symbols are used to represent the quantity 100, 1000, 10000, and 10,000,000.

Then, a number is made up of a combination of addition and multiplication of the

words/symbols used to write it. See the table below for the actual symbols.

If one wanted to say the number 11, one would say the words (or write the

symbols) for “ten” “one” implying (10 + 1). Twenty-one is “two” “ten” “one”, meaning

(2 x 10) + 1. Unlike western languages such as English or any of the Romantic

0 1 2 3 4 5 6 7 8 9 10 100

1000

10000

100000000

Traditional

Simplfied

Formal Trad. (Daxie)

Formal Simp. (Daxie)

Pinyin ling2

yi1

er4

san1

si4

wu3

liu4

qi1

ba1

jiu3

shi2

bai3

qian1

wan4 yi4

languages, there are no vestiges of the ancient Egyptian “vigesimal” (base 20) system in

the Chinese language.

Questions:

1. Research more completely the Chinese number system using one of the resources

below. Show (using cut-and-paste) the Chinese numerals for the following numbers

and explain how we would translate this into English: For example: 22,010,010 is

??????????

which translates as “two-thousand-two-hundred-zero-one

ten-thousands plus zero-one tens” and as an Algebraic expression as: (2201 x 10,000)

(01 x 10) = 22,010,010

a. 1,122,010

b. 68

c. 122,334,257

d. 47001

2. What do you believe are the advantages and disadvantages of the Chinese number

system?

3. How did the Counting Rods evolve into the modern Abacus? How do the abacus and

counting rods work?

4. Over 300 years before Pascal is credited with developing Pascal’s triangle, the

Chinese developed a similar concept in Number Theory. What is Pascal’s Triangle?

How does it describe natural occurring phenomenon? Who was Blaise Pascal? When

did he develop his concept of triangular numbers? When did the Chinese develop the

similar concept and who was responsible for it? What did the Chinese use triangular

numbers for?

5. What were the significant mathematical developments in the major dynasties of the

Chinese Empire through 1700? Develop a timeline (not just by “cut and paste”) of the

development of mathematics in China. Pick at least one significant mathematical

thought or development from each major Chinese dynasty and write a paragraph

describing it.

6. When did Japan develop sophisticated mathematics? How much of Japan’s traditional

mathematics came form China? Show a timeline of the development of significant

Japanese alongside the timeline from China.

~

African Math

When we discuss the development and history of mathematics in Africa, we

normally look at two distinct topics. The first is the development of math in Egypt, dating

back to the ancient Egyptian Pharaohs over 4000 years ago. The second topic area is the

development of mathematics in sub-Saharan Africa (typically thought of as tribal Africa).

We get much of our knowledge of Egyptian mathematics from just two surviving

ancient papyrus. These are known as the “Rhind Papyrus” and the “Moscow Papyrus”.

Between these two papyri there are a total of 122 mathematical problems that show that

the Egyptians had sophisticated methods of solving arithmetic, geometric, and algebraic

problems (Williams, 2002). We already knew that Egyptians had to have some

mathematical knowledge to design and construct their ancient architecture such as the

pyramids, monoliths, and other structures; however, the Egyptian economy was also

dependent on farming in the Nile River Valley. Thus the Egyptians had sophisticated

means for measuring and surveying land. Therefore, both the written and archeological

record point to a significant mathematical development long before Alexander the Great

and the Greeks arrived in 332BC. After the arrival of the Greeks, the great library at

Alexandria, Egypt became the center of mathematical knowledge in the “developed”

world up until its destruction in the 600’s AD.

Many have claimed that mathematics was not developed at all in sub-Saharan

Africa due to the “hunter-gatherer” tribal nature of the native populations. However, the

archaeological record shows artifacts known as “counting sticks” that demonstrate

mathematical concepts by their owners. The Lebombo Bone was discovered in

Swaziland. Dating from 35,000BC. It has 37 distinct notches used for some form of

counting. The Ishango Bone, a mere 8000 years old, is believed by some to be the oldest

table of prime numbers.

More recent study of pre-colonial sub-Saharan Africa has shown that Yoruba (of

Nigeria) had a well-developed base-20 vigesimal number system (similar to the Mayan)

and used mathematics for astronomy, astrology and other mystical arts prior to the arrival

of European colonial powers. Other studies (Zaslavsky 1973) discuss the written, spoken

and gesture counting and numeration systems used by African tribes.

Questions:

1. What is the difference between “rigorous” mathematics and “empirical”

mathematics? What form was Egyptian math before the arrival of the Greeks?

2. What did the Egyptian number system look like? See if you can find evidence of

another African number system by searching the web sites referenced below.

3. What was math used for in sub-Saharan Africa? (more details than those above!)

4. Is there a relation between math and art in rock drawings and sand paintings in tribal

Africa? If so, what do the scholars believe?

~

Native American

The peoples of the Native American Nations of North America developed their

understanding of mathematics through agricultural, religious, and trade needs. Native

American Nations through out North America had different calendars, counting systems,

and monetary systems. To describe them as one is not only impossible, but also incorrect.

A student who chooses Native American math as their topic of study, must first

narrow their study to a specific region of North America and then, depending on the

amount of information available may need to choose a specific Native American Nation

or a few Nations. There is little information available for each Nation partially because

the first Native American written language was invented in 1809 by a Cherokee named

Sequoyah. Prior to this, knowledge was passed orally and through architecture, drawings

and weavings. No written number systems are evidenced.

Some Nations had 13-month lunar calendars, while other Nations based their

calendars on the four seasons. Most of the Nations had rituals revolving around the four

solstices, and had methods and calendars for determining when the solstices would occur

(Heizer, 1978), (Carlson and Judge, 1983),(McCoy, 1992), (Mails, 1996). Some

California Native Americans had calendars for days, the seasons, and years. This

information was recorded through notches on sticks, the collection of rocks in baskets, or

verbally (Heizer 1978). Many of the Californian, Southwestern, Southeastern, and

Northeastern Native American Nations used this information for both ritual purposes and

to aid in growing crops. Some built buildings which were specifically oriented to have

the sun pass through certain windows for different solstices, while others are known to

have kept verbal records as to when the solstices would occur. The Great Basin, Plains,

and Plateau Nations were primarily hunter/gather Nations and did not have buildings or a

written record which was passed on describing their knowledge of the solstices.

Most coastal tribes had a monetary system for trade, usually using seashells, made

into beads, as currency. Seashells were more often made into belts or strings for

descriptions of rituals and treatise, than used in a monetary sense. The use of seashells as

currency increased as trading began with the Europeans (Heizer 1978), (Mails, 1996).

Different tribes had different counting systems. The Yurok used a decimal and a

duodecimal system for determining a woman’s fertile time, and for determining the

expected delivery date of an unborn child (Heizer 1978:151). The Karok used the

numbers 5 and 10 for ritual purposes and as part of their counting system (Heizer

1978:188). The Wintu had a no word for fractions, but did use a base 20 counting system.

They had 3 systems, one for “ordinary purposes”, one for monetary purposes, and one for

arrowheads. The Wintu were able to count into the thousands using their base 20 system

(Du Bois 1935:70). The Coast Miwok people had decimal numeration, using the number

4 as a ritual number. They counted to 25 using sticks to record the days and they had

names for times of the day, days, seasons and cardinal directions (Heizer 1978:423). The

Salinan peoples used shells for currency, with values determined by the color and

geographical origin of the shell. They charged high interest rates on loans, but are also

known for their generosity (Heizer 1978:502).

Questions:

1. What about the Native American culture explains why they did not have a need for

more rigorous math, as other cultures did at the same time?

2. How did the Southwest Native Americans use buildings to mark the dates of the

equinoxes?

3. What form of currency did the Algonguins use? How is this similar or different from

currencies used by other Native American Nations?

4. How did the need for and use of currency evolve over time for one Native American

Nation of your choice? Was this similar for other Native American Nations? Why?

~

Women in Math

As in other disciplines, the contribution of women to the development and history

of mathematics has been much overlooked. Despite the lack of historical focus, women

have made major contributions to mathematical development since classical times. For

example:

1. Theano, along with her two daughters, carried on the work of her husband,

Pythagoras, after his death in the 5th century, BC. In addition to maintaining the

center of learning now known as the Pythagorean School, Theano contributed her

own unique work in what is now known as Number Theory with her contributions

to the “Golden Mean” (Riddle, L. 2002).

2. Hypatia was a leading Greek philosopher, mathematician and scientist from 5th

century AD Alexandria. Her life and tragic death was marked by major

contributions to astronomy, philosophy, and math. Her work on conics was later

studied and enhanced by major names such as Leibniz, Newton, and Descartes.

(O’Connor and Robertson 2002)

Much later, on June 25, 1678 Elena Lucrezia Cornaro Piscopia, daughter of a

noble Venitian family, received her Doctorate of Philosophy degree after twice being

denied the right to stand for examination by the University of Paduato. In so doing, at age

thirty-two, Elena Piscopia became the first woman in the world to receive a doctorate

degree. Despite the enormous contributions made by other women mathematicians such

as Mary Somerville, Ada Byron, and Florence Nightengale, it was not until 1888 that

Winifred Edgerton became the first woman to be awarded a Ph.D. in Mathematics in the

United States or Canada. Since then, women have made major academic and practical

contributions to mathematical science. Since Madame Marie Curie was awarded her first

of two Nobel Prizes, eleven women have been Nobel laureates in scientific disciplines,

and another nineteen women have received Nobel prizes for literature or for peace. Yet as

recently as the late 1960’s, Linda Rothschild, Ph.D. in Mathematics from M.IT. and

former vice president of the American Mathematical Society, was rejected for acceptance

into the graduate program at Princeton University on the grounds that the university only

accepted men (Riddle, L. 2002).

Questions:

1. It was not until 1943 that the first black woman was awarded a Ph.D. in Mathematics

in the United States. By 1950, only three black women had been awarded Ph.D’s.

Who were these three women and what has been there contribution?

2. Who is Grace Hopper? Tell us about her life. What was her contribution to math and

computer science?

3. Ada Byron, Lady Lovelace, and daughter of Lord Byron, the poet, worked with

Charles Babbage in the early 19th century. Both are credited with work that led to

modern computers. What was this work? What part did Lady Lovelace play?

4. In addition to being a famous nurse during the Crimean War, Florence Nightengale is

also known for developing important techniques used in mathematical analysis. What

kind of technique did Florence Nightengale develop? What is it called and how does

it work? How has Florence Nightengale’s work changed behavioral and social

sciences?

5. Sophie Germain was a self-taught mathematician who grew up in revolutionary Paris

in the late 1780’s. She was a revolutionary herself, leaving a lasting contribution to

number theory. Sophie studied at the Ecole Polytechnique in Paris under a man’s

assumed name. Her work is foundational to proving Fermat’s Last Theorem. Tell us

more about Sophie Germain. If we were to visit Paris, how might we be reminded of

Sophie Germain?

Conclusion

The course, Multicultural History of Mathematics, is designed to create

connections with various cultures and every day math. The purpose of this course is to

increase student interest and performance in mathematics beyond the required 8th grade

level. Through integrating this course into a 7th grade curriculum, all middle school

students will be exposed to a new and different way to view mathematics before reaching

the critical 8th grade year. Desired outcomes for and assessment of the course are both

based upon both short-term and long-term goals and measurables. It is the desire of this

team that with positive measurables, this course would be implemented beyond their

district.

~

I hope that you have enjoyed your mathematical tour. Like any tour, it includes only a

few highlights, for the history of mathematics is a vast subject. I hope that I have wetted

your appetite for further study of this fascinating subject.

Loretta Kelley

~

Appendix I

Student Handouts

Course Syllabus………………. pg. AI:1 Project Scope………………… pg. AI:1

Course Schedule and Outline… pg. AI:2-3

Assignment Log……………… pg. AI:4 Rubric………………………… pg. AI:5

Course Syllabus

Multicultural History of Mathematics Wednesday 1:00 - 1:50 p.m.

Fall Semester, 2003 Interesting Middle School

Your Town, U.S.A.

How does our modern math relate to ancient math? Where did the math we use today come from? Let’s find out. In this project-based class you will choose a culture to study and learn about their contributions to modern day mathematics. How does knowledge of this math effect our everyday life? You figure this out and then share it with the class. Then we can all better understand not only the importance of math in our everyday lives, but also why it is interesting and how it improves our daily lives!

Recommended Texts: See Resource list for recommended texts for your research topic.

Project Scope

d. Choose a culture. e. Choose a math topic in that culture. f. How does this topic relate to modern day use?

g. Patterns & Symmetry h. The use of zero i. Imaginary numbers j. Algebra k. Etc.

l. Write a 3 – 5 page research paper explaining your findings. m. Prepare a project demonstrating what you have learned. n. Include the number system used and do a few example math problems. o. Present your project and finding in a 3 – 5 minute presentation.

Course Schedule and Outline

Week 1 Course overview Wednesday, Sept. 3 Assignment Read handouts, bring questions to next class

Due Sept. 10

Week 2 How do you research your topic? Wednesday, Sept. 10 What is a good resource material?

Assignment Choose your research topic Due Sept. 17 Find 3 resources - Bring to class!

(a combination of text and internet)

Week 3 Review resource materials Wednesday, Sept. 17 How do you document your resources?

Assignment Find additional resources as necessary Due Sept. 24 Document resources in use

Week 4 Taking research notes Wednesday, Sept. 24 Assignment Document resources used

Due Oct. 1 Collect a minimum of 10 research notes

Week 5 Review and organize research notes Wednesday, Oct. 1 How to write an Outline

Assignment Collect a min. of 5 notes and organize Due Oct.8 Write outline

Week 6 Are you ready for more resources? Wednesday, Oct. 8 How to write an Abstract

Assignment Assess resource need Due Oct. 15 Find additional resources as necessary

Collect and organize a min. of 5 notes Write Abstract

Week 7 Half way there! How is your research progressing? Wednesday, Oct. 15 Project and Presentation description

Assignment Write a short description of your project Due Oct. 22 Collect and organize more data as necessary

Bring report related work to class to work on.

Week 8 Review project and presentation ideas Wednesday, Oct. 22 Assignment Begin body of paper

Due Oct. 29 Collect materials for project

Week 9 Working session – bring paper or project to class Wednesday, Oct. 29 How to write a Conclusion

Assignment Work on body of paper, write conclusion Due Nov. 5 Collect materials for project

Week 10 Working session – bring paper or project to class Wednesday, Nov. 5 Assignment Work on final draft of paper

Due Nov. 12 Begin to assemble project

Week 11 Working session – bring paper or project to class Wednesday, Nov. 12 Assignment Complete final draft of paper

Due Nov. 19 Assemble project Begin to organize presentation

Week 12 Working session – bring paper or project to class Wednesday, Nov. 19 Assignment Complete project and presentation

Due Nov. 26

Week 13 Final presentations 1 - 9 Wednesday, Nov. 26 3 - 5 minutes per presentation

Week 14 Final presentations 10 - 18 Wednesday, Dec. 3 3 - 5 minutes per presentation

Week 15 Final presentations 19 - 27 Wednesday, Dec. 10 3 - 5 minutes per presentation

Week 16 Final presentations 28 – 30 (3 - 5 minutes each) Wednesday, Dec. 17 Math Holiday Celebration (Last 30 minutes of class)

Appendix II

Resources

Student Resources…………….. pgs. AII:1 – 5

Babylonian and Egyptian…….…pg. AII:1

Greek…...…………………….… pg. AII:1

Hindu…………………….….… pg. AII:1

Arabic or Islamic……………… pg. AII:1

Renaissance …………………… pg. AII:2

Mayan……………………...… pg. AII:2

Asian………………………...… pg. AII:3

African……………………...… pg. AII:3

Native American….…………… pg. AII:4

Women……………………..… pg. AII:5

Teacher Resources…………….. pg. AII:6

Student Resources

Babylonian and Egyptian Resources

Kline, Morris. (1990) Mathematical Thought From Ancient to Modern Times, Oxford

University Press, Vol. 1, Chaps. 1 and 2.

Katz, Victor J. (1998) A History of Mathematics: An Introduction, 2nd ed., Addison-

Wesley Educational Publishers, Inc., Chap. 1.

Greek Resources


University Press, Vol. 1, Chaps. 3 - 8.


Wesley Educational Publishers, Inc., Chaps. 2 – 5.

Hindu Resources


University Press, Vol. 1, Chap. 9.



Arabic Resources


University Press, Vol. 1, Chap. 9.



Renaissance Resources

Eves, H. (1990) An introduction to the history of mathematics, Saunders College

Publishing.

Klein, M. (1972) Mathematical thought from ancient to modern times, Oxford University

Press.

O’Connor, J. and Robertson, E. (2002), The MacTutor History of Mathematics archive,

School of Mathematics and Statistics, University of St. Andrews, Scotland,

retrieved from http://www-gap.dcs.st-and.ac.uk/~history/

Mayan Resources

Calleman, C. J. (2002) About the Venus Passages and the completion of the Mayan

Creation Cycles or Transits retrieved on October 28, 2002 from

http://hem.passagen.se/alkemi/venus-eng.htm


Publishing, 19-20

Labrosse, P (2002) Mathematics in Mesoamerica, retrieved on October, 28, 2002 from

http://www.hbhs.k12.nh.us/

Unknown, (2002) Mayan History Homepage, One World Journeys, retrieved on October

28, 2002, from http://www.oneworldjourneys.com/jaguar/history.html

http://www-gap.dcs.st-and.ac.uk/~history/

http://hem.passagen.se/alkemi/venus-eng.htm

http://www.hbhs.k12.nh.us/

http://www.oneworldjourneys.com/jaguar/history.html

Asian Resources

Eves, H. (1990) An introduction to the history of mathematics, Saunders College Pub.

Joyce, D. (2002) History of Mathematics in China, retrieved on November 26, 2002 from

http://aleph0.clarku.edu/~djoyce/mathhist/china.html

Klein, M. (1972) Mathematical thought from ancient to modern times, Oxford Univ.

Press.

Peterson, E. (2002) Chinese Number System, retrieved on November 26, 2002 from

http://www.mandarintools.com/numbers.html

Saxakali (2002) History of Mathematics in Asia, Cultural Online Learning Organization

& Resource (COLOR) After school Program, Saxakali Organization, retrieved on

November 26, 2002 from http://www.saxakali.com/COLOR_ASP/history.htm

Stapleton, R. & Tripodi, T. (1996) Ancient Chinese Mathematics, retrieved on November

26, 2002 from http://www.unisanet.unisa.edu.au/07305/chinese.htm

African Resources

Eves, H. (1990) An introduction to the history of mathematics, Saunders College Pub.19-

20

Gerdes, P. (1991) On the history of mathematics south of the Sahara, paper presented at

Third Pan-African Congress of Mathematicians, Nairobi, 20-28 August 1991,

retrieved November 17, 2002, from

http://www.math.buffalo.edu/mad/AMU/amu_chma_09.html - pioneer.

Williams, S., (2002) Mathematics of the African Diaspora, retrieved on November 17,

2002, from http://www.math.buffalo.edu/mad/00.INDEXmad.html

http://aleph0.clarku.edu/~djoyce/mathhist/china.html

http://www.mandarintools.com/numbers.html

http://www.saxakali.com/COLOR_ASP/history.htm

http://www.unisanet.unisa.edu.au/07305/chinese.htm

http://www.math.buffalo.edu/mad/AMU/amu_chma_09.html

http://www.math.buffalo.edu/mad/00.INDEXmad.html

Zaslavsky, C. (1973a): Africa Counts: number and pattern in African culture, Prindle,

Weber and Schmidt, Boston, 328 p.(out of print); paperback edition (available):

Lawrence Hill Books.

Native American Resources

Heizer, R. (1978) Handbook of North American Indian, Volume 8 California, The

Smithsonian Institution

Capaldi, G. and Rife, D. (2001) Pilgrims – Garments, History, Legends, and Lore, Good

Apple

Weinstein-Farson, L. (1991) The Wamanoag, Indians of North America Series, Chelsea

House

Mails, T. (1996) The Cherokee People - The Story of the Cherokees from Earliest Origins

to Contemporary Times, Marlowe & Company.

McCoy, R. (1992) Archaeoastronomy – Skywatching in the Native American Southwest,

Museum of Northern Arizona

Carlson, J.B. and Judge, W. J. (1983) Astronomy and Ceremony in the Prehistoric

Southwest, Maxwell Museum of Anthropology, Anthropological Papers No. 2

Trigger, B. (1978) Handbook of North American Indians, Volume 15 Northeast, The

Smithsonian Institution.

Malville, J.M. and Putnam, C. (1989) Prehistoric Astronomy in the Southwest, Johnson

Books

D’Azevedo W. (1986) Handbook of North American Indians, Volume 11 Great Basin,

The Smithsonian Institution

Women Resources


Publishing

Klein, M. (1972) Mathematical thought from ancient to modern times, Oxford University

Press

Riddle, L. (2002) Biographies of Women Mathematicians, Mathematics Department,

Agnes Scott College, Atlanta, Georgia, retrieved on November 17, 2002 from

http://www.agnesscott.edu/lriddle/women/women.htm

O’Connor, J. and Robertson, E. (2002), The MacTutor History of Mathematics archive,

School of Mathematics and Statistics, University of St. Andrews, Scotland,

retrieved on November 17, 2002 from http://www-gap.dcs.st-and.ac.uk/~history/

Vitulli, M. (2002) Women in Math Project, Department of Mathematics, University of

Oregon, Corvallis, Oregon, retrieved on November 17, 2002 from

http://www.uoregon.edu/~wmnmath/index.html

Williams, S., (2002) Mathematics of the African Diaspora, retrieved on November 17,

2002, from http://www.math.buffalo.edu/mad/00.INDEXmad.html

_________. (2002) History of Black Women in Mathematics, retrieved on November 19,

2002, from http://www.math.buffalo.edu/mad/wohist.html

http://www.agnesscott.edu/lriddle/women/women.htm

http://www-gap.dcs.st-and.ac.uk/~history/

http://www.uoregon.edu/~wmnmath/index.html

http://www.math.buffalo.edu/mad/00.INDEXmad.html

http://www.math.buffalo.edu/mad/wohist.html

Teacher Resources

Furinghetti, F. (1997) History of mathematics, mathematics education, School practice:

Case studies in linking different domains. For the Learning of Mathematics, 17,1

55-61. (case studies of 4 different teachers usage of mathematics history in the

classroom)

Jardine, R. (1997) Active learning mathematics history, Primus, 7, 115-21

(An integrated history approach)

Kelley, L. (2000) A mathematical history tour, The Mathematics Teacher, 93,1 14-17

Kleiner, I. (1988) Thinking the unthinkable; the story of complex numbers (with a

moral).

Mathematics Teacher, 81, 583-592

Loveless, T. (2000) The Brown Center report on American education, Brookings

Institute, 13-19

McBride, C. & Rollins, J. (1977) The effects of history of mathematics on

Attitudes toward mathematics of college algebra students, Journal for

Research in Mathematics Education, 1, 57-61. (Seminal work on the use of 5

minute historical vignettes)

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