In my first- and second-year college classes, I infuse the standard curriculum with a small number of historical and cross-cultural activities closely linked to traditional topics. The activities, which I developed and have used since 2001, take the form of written, self-paced lessons or modules. Each activity focuses on a mathematical technique used in the medieval Arab or Islamic world. My students look at the cultural and historical context of the technique and its links to other methods, most often from the mathematics of medieval India and China. They explore the theory behind the technique, and they see how it is used to solve “real-world” problems, including the type of problem that prompted its discovery.

On the pages that follow, I will describe:

- how the presence and interests of my culturally diverse students prompted me to find ways to learn and to teach about the mathematical contributions of Muslim, Indian, Chinese, and other peoples (page 2);
- the goals I had in mind as I created my classroom activities (page 3);
- how these goals are met by the activity, “Combinations and Their Sums,” based on counting tassels of colored silk (pages 4 and 5);
- reactions from my students (page 6);
- creating a calculus activity, “Using the Derivative to Solve an Optimization Problem,” for which the challenge of modernization was greater than for the activity on combinations (page 7); and
- more ideas for activities and projects based on Islamic mathematics (page 8).

The five modules may be downloaded from this page and also at other convenient points throughout the article:

- Combinations and Their Sums (Elementary Statistics, Finite Mathematics)
- Binomial Coefficients and Subsets (Finite Mathematics)
- Using the Derivative to Solve an Optimization Problem (Calculus for Business and Social Sciences)
- The Rule of Double False Position (Calculus for Business and Social Sciences)
- The Rule of Double False Position (Linear Algebra)

In my first- and second-year college classes, I infuse the standard curriculum with a small number of historical and cross-cultural activities closely linked to traditional topics. The activities, which I developed and have used since 2001, take the form of written, self-paced lessons or modules. Each activity focuses on a mathematical technique used in the medieval Arab or Islamic world. My students look at the cultural and historical context of the technique and its links to other methods, most often from the mathematics of medieval India and China. They explore the theory behind the technique, and they see how it is used to solve “real-world” problems, including the type of problem that prompted its discovery.

On the pages that follow, I will describe:

- how the presence and interests of my culturally diverse students prompted me to find ways to learn and to teach about the mathematical contributions of Muslim, Indian, Chinese, and other peoples (page 2);
- the goals I had in mind as I created my classroom activities (page 3);
- how these goals are met by the activity, “Combinations and Their Sums,” based on counting tassels of colored silk (pages 4 and 5);
- reactions from my students (page 6);
- creating a calculus activity, “Using the Derivative to Solve an Optimization Problem,” for which the challenge of modernization was greater than for the activity on combinations (page 7); and
- more ideas for activities and projects based on Islamic mathematics (page 8).

The five modules may be downloaded from this page and also at other convenient points throughout the article:

- Combinations and Their Sums (Elementary Statistics, Finite Mathematics)
- Binomial Coefficients and Subsets (Finite Mathematics)
- Using the Derivative to Solve an Optimization Problem (Calculus for Business and Social Sciences)
- The Rule of Double False Position (Calculus for Business and Social Sciences)
- The Rule of Double False Position (Linear Algebra)

For the past 40 or more years, factories in southeastern Michigan have attracted immigrant workers from Middle Eastern and other countries. As a result, this area now has the largest population of people of Arab background in the Western hemisphere (approximately 300,000). In my own classes at a community college in metropolitan Detroit, students of Arab descent constitute the largest distinct ethnic and cultural minority (9%), followed by those of Indian and Chinese heritage (7% and 3%, respectively). Detroit proper is overwhelmingly African-American. Such rich diversity heightens both the necessity and the opportunity to explore the global history of science and other human endeavors, a history in which knowledge has been interwoven from many cultural strands.

Unfortunately, the cultural horizons of a classroom don’t automatically expand just because there are students from diverse backgrounds. Despite the fact that immigrants have played a key role in driving American industrial and scientific enterprise, the scientific contributions made by non-Western peoples are scarcely acknowledged in our classrooms (Joseph 1987).

The presence and interests of my culturally diverse students prompted me to find ways to learn and to teach about the mathematical contributions of Muslim, Indian, Chinese, and other peoples. Arab mathematicians, in particular, have been dismissed as mere copyists:

The Arabs made no significant advance in mathematics. What they did was absorb Greek and Hindu mathematics, preserve it, and […] transmit it to Europe. (Kline 1972: 197)

Although few writers nowadays would state views as extreme as Kline’s, the same underlying approach is often still at work, albeit more subtly. For instance, in David Burton’s widely used textbook on the history of mathematics, most recently issued in 2007, the discussion of medieval Islamic (as well as Chinese) contributions is relegated to the chapter, “The Twilight of Greek Mathematics: Diophantus.”

Far from simply preserving and transmitting ancient knowledge, the Arab people tremendously enriched mathematics and other sciences. Medieval scholars broke whole new ground, especially in these subfields of mathematics:

• algorithms of arithmetic

• algebra and the theory of polynomials

• number theory

• combinatorics

• plane and solid geometry

• plane and spherical trigonometry.

Some of these breakthroughs were incorporated into European mathematics centuries ago, while others can and should be resuscitated for classroom use today.

In my classes, I infuse the standard curriculum with a small number of historical and cross-cultural activities closely linked to traditional topics. The activities, which I developed and have used since 2001, take the form of written, self-paced lessons, or “modules.” The students begin the activity in class, working either individually or in small groups for about 30 minutes. They take the work home to complete it, and later submit it to me for grading and comments.

Each activity focuses on a mathematical technique used in the medieval Arab world. My students look at the cultural and historical context of the technique and its links to other methods. They explore the theory behind the technique, and they see how it is used to solve “real-world” problems, often including the type of problem that prompted its discovery.

The major goals of these activities are:

- to encourage students to appreciate the contributions of many cultures and peoples
- to enhance the appeal of mathematics lessons
- to foster the understanding that mathematics is not merely a collection of abstract ideas, but is a product of people trying to solve practical problems
- to round out students’ technical skills, and other skills and attitudes important for functioning in an interdependent world
- to compensate for the Eurocentric bias of the standard mathematics curriculum.

Creating new instructional materials along these lines is challenging. Most of my students are very practical-minded young people who are training for positions in the business, health, and engineering professions. My approach has been to recast information so that it is comprehensible and appealing to them, while preserving the basic integrity of the mathematics, cultures, and histories involved. Based on my experience, I have formulated a five-point strategy for developing cross-cultural activities:

- Select interesting and appropriate applications.
- Streamline the discussion and modernize the notation found in the original sources.
- Explore each concept from a number of different perspectives.
- Guide the students from easier to more difficult tasks.
- Compare and contrast examples drawn from different cultures.

Much of the remainder of this article will be devoted to illustrating my five-point strategy by describing classroom activities that I have developed.

A Moorish problem dealing with colored threads of silk makes a great starting point for learning about certain combinatorial relationships. I present the module, "Combinations and Their Sums," to students in Statistics and Finite Math courses after we’ve already covered the basic facts about factorials, permutations and combinations in the standard course syllabus.

The silk problem was posed by Ahmad ibn Ibrāhīm ibn ‘Alī ibn Mun‘im al-‘Abdarī (d. 1228), a physician and mathematician born in Spain who lived much of his life in Marrakech. During this period, what is now Spain and Morocco were united under the Almohad dynasty. This was a major silk-producing region, and ibn Mun‘im began by listing 10 colors that were available for dyed silk thread. He then asked: in how many different ways can threads in 3 of the 10 colors be combined to form a silk tassel? Breaking the possibilities down into mutually exclusive cases, ibn Mun‘im reasoned that one could select either the 3rd color in the list together with both of the colors above it; or the 4th color in the list together with any 2 of the 3 colors above it; or the 5th color together with any 2 of the 4 colors above it; and so on, up to the 10th color together with any 2 of the 9 colors above it. Written in the modern notation of binomial coefficients (“combinations”), his conclusion was as follows:

\[\binom{10}{3}=\binom{2}{2}+\binom{3}{2}+\binom{4}{2}+\binom{5}{2}+\binom{6}{2}+\binom{7}{2}+\binom{8}{2}+\binom{9}{2}.\]

In this way, ibn Mun‘im was able to express the number of combinations of colors taken 3 at a time as a sum of combinations of colors taken 2 at a time, for which he had already derived the formula:

\[\binom{n}{2}=\frac{n(n-1)}{2}.\]

My students have also previously learned this formula, and I ask them to complete the tassel calculation in the same way as did ibn Mun‘im:

\[\binom{10}{3}=1+3+6+10+15+21+28+36=120\,\,{\rm{different}}\,\,{\rm{tassels.}}\]

The students go on to learn that ibn Mun‘im generalized this pattern so that he could expand any given binomial coefficient as a sum of coefficients of lower order. To facilitate the use of this rule, he organized the results on successive rows of an arithmetical triangle that I present (see Figure 1). The calculation above is summarized on the third-lowest row of numerals in ibn Mun‘im’s triangle. Note the similarity between this triangle and “Pascal’s Triangle” that appeared centuries later in France (Lamrabet 1994: 215-216, 300-2; Katz 2009: 292-294).

**Figure 1a.** Ibn Mun‘im’s arithmetical triangle. (Image from *Actes du Huitième Colloque Maghrébin sur l’Histoire des Mathématiques Arabes, Tunis, les 18-19-20 Décembre 2004, *Tunis: Tunisian Association of Mathematical Sciences, 2006.)

**Figure 1b.** An adaptation of ibn Mun‘im’s arithmetical triangle for classroom use.

The tassel problem exemplifies the first of the five points of my strategy because it captures the interest of students, and its concreteness aids them in understanding the concepts involved. Going through this problem allows them to grasp the resulting pattern more durably than if it were derived—or simply stated—as an abstract formula. To test and solidify students' grasp of the pattern, I follow up with a series of exercises along these lines:

**Exercise.** Use the same logic as ibn Mun‘im to complete these expansions:

a) \(\binom{10}{6}=\binom{5}{5}+{\phantom{xxxxxx}}+{\phantom{xxxxxx}}+{\phantom{xxxxxx}}+{\phantom{xxxxxx}}.\)

b) \(\binom{10}{7}={\phantom{xxxxxx}}+{\phantom{xxxxxx}}+{\phantom{xxxxxx}}+{\phantom{xxxxxx}}.\)

Interpret your answer in words, in terms of making a tassel from silk threads of various colors.

c) \(\binom{8}{3}=\)

Interpret your answer in words, in terms of selecting whom, from the workers in one office, will be chosen to attend a professional development seminar.

d) Use numbers and words of your own choosing to make up a problem

like that in part (c), and provide an answer for it.

The type of verbal interpretation that I am looking for in part (b) is: “To make a tassel from 7 of the 10 colors, choose either color 7 and all 6 of the 6 colors above it, or color 8 and any 6 of the 7 colors above it, or color 9 and any 6 of the 8 colors above it, or color 10 and any 6 of the 9 colors above it.”

This exercise illustrates three more of the five points of my strategy. First, notice that I use modern binomial notation even though it did not exist in ibn Mun‘im’s time. Such “translations” help streamline the work (but take care, in making such translations, not to inadvertently misrepresent the way in which the knowledge was conceived in the original context). Second, this exercise leads students to state their results in both a numerical and a verbal form. Other exercises in the same module make use of graphical and algebraic representations, in the form of the arithmetical triangle and formulae with symbols such as \(\binom{n}{r}.\) The use of such multiple perspectives leads to a sturdier understanding by the student, appeals to different learning styles, and is also intrinsically interesting. Third, even within this single exercise, the tasks required of the student progress gradually from easier to more difficult. At first the scaffolding of blank spaces is pre-filled in part, then not pre-filled at all, and finally the scaffolding is withdrawn altogether. In addition, what is requested is only a numerical answer at first, then a verbal interpretation in terms of the by-then-familiar 10 colors of dyed silk, then in terms of a different situation altogether, and finally in terms of a situation dreamt up by the student.

Download:

- the module that introduces students to counting colored tassels as ibn Mun‘im did, Combinations and Their Sums (Elementary Statistics, Finite Mathematics), and
- the follow-up module, Binomial Coefficients and Subsets (Finite Mathematics).

In the module, "Combinations and Their Sums," ibn Mun'im’s pattern is also used to compress rather than to expand. A surprising range of problems can be simplified by this means.

For example, I ask my students to count the number of different possible sequences of wins and losses in the baseball World Series, assuming that the Detroit Tigers win the Series. Since the Series winner is the first to win 4 games, the length of the Series ranges from 4 to 7 games. In each case, the Tigers would need to win not only the final game played but any 3 of the preceding games, with the number of preceding games ranging from 3 to 6. Thus, it was easy to construct an exercise that leads students to see that the number of possibilities is:

\[\left(\begin{array}{c} 3\\3\end{array}\right)+\left(\begin{array}{c} 4\\3\end{array}\right)+\left(\begin{array}{c} 5\\3\end{array}\right)+\left(\begin{array}{c} 6\\3\end{array}\right).\]

The students are then asked to figure out how to use ibn Mun'im’s pattern to compress this into a single coefficient, namely \(\left(\begin{array}{c} 7\\4\end{array}\right)=35\) sequences. This shortcut to the answer surprises many of them, in part because most Series do not last 7 games.

As another example of compression, I ask my students to count the total number of gifts exchanged in “The Twelve Days of Christmas” according to the traditional song:

On the 1st day of Christmas my true love gave to me: A Partridge in a Pear Tree.

On the 2nd day of Christmas my true love gave to me: Two Turtle Doves, and A Partridge in a Pear Tree.

.

.

.

On the 12th day of Christmas my true love gave to me: Twelve Drummers Drumming, Eleven Pipers Piping, Ten Lords A-leaping, Nine Ladies Dancing, Eight Maids A-milking, Seven Swans A-swimming, Six Geese A-laying, Five Golden Rings, Four Calling Birds, Three French Hens, Two Turtle Doves, and A Partridge in a Pear Tree.

Here, the students must compress repeatedly and on two levels. First, they discover that on any given day, the number of gifts can be found by a combination involving a choice of two items. For instance, on Day 12 the gifts total:

\[12+11+\cdots+1= \left(\begin{array}{c} 12\\1\end{array}\right)+\left(\begin{array}{c} 11\\1\end{array}\right)+\cdots +\left(\begin{array}{c} 1\\1\end{array}\right)=\left(\begin{array}{c} 13\\2\end{array}\right).\]

Based on this, the students are able to conclude that over the 12 days, the grand total is:

\[\left(\begin{array}{c} 2\\2\end{array}\right)+\left(\begin{array}{c} 3\\2\end{array}\right)+\cdots +\left(\begin{array}{c} 13\\2\end{array}\right)=\left(\begin{array}{c} 14\\3\end{array}\right)=364\,\,{\rm gifts,}\]

with the final calculation based on the standard factorial formula. Thus, instead of needing to calculate many different numbers and then their total, the students are able to count the gifts by a single calculation involving factorials. That the work can be compressed into a single binomial coefficient is very surprising, especially since the numbers 3 and 14 bear no obvious relation to the original statement of the problem. Such examples illustrate in a dramatic way the utility of ibn Mun'im’s insight.

The students go on to compare ibn Mun'im’s arithmetical triangle with later ones from Chu Shi-Chieh (China, 1303) and Blaise Pascal (France, 1665) (on both, see Edwards 1987). Interestingly, for Pascal and others in Europe investigating combinatorics, the major stimulus was to answer questions arising from games of dice. By contrast, in the medieval Muslim world such games were rarely played because Islam disapproves of wagering and other kinds of financial speculation. Instead, combinatorial research among Muslims was driven by their interest in Arabic, which they considered sacred because it is the language of the *Qur’an.* Their goal was to count the number of ways that Arabic letters and sounds can be combined; this was, in fact, the larger project for which ibn Mun'im developed his triangle. In medieval India, combinatorics came to be of special interest to mathematicians of the Jaina religion, who studied combinations of senses, of philosophical categories, etc. (Joseph 2000: 253-255; Katz 1996: 99-101; Katz 2009: 250-252). All of these applications provide wonderful raw material for student activities. Comparing such examples (the fifth and final point of strategy mentioned above) allows us to impart to our students the lesson that mathematics takes diverse forms in different parts of the world because it is shaped by culture.

Download:

- the module that includes the exercises described on this page, Combinations and Their Sums (Elementary Statistics, Finite Mathematics), and
- the follow-up module, Binomial Coefficients and Subsets (Finite Mathematics).

In the U.S., people from non-European cultures often yearn for a public recognition that important contributions to knowledge were made in other parts of the world. This becomes evident when I ask my students to write down their reactions to the cross-cultural activities. Alice, a young woman from Beirut, proudly wrote, “Beginning with colors and threads and finishing with a combination formula is a great breakthrough in the historical past of any country.” Ahmad, a business student from Tripoli, Lebanon, decided to turn in his paper with his name written in ancient Phoenician and accompanied by 11 pages about Phoenician contributions to history, which he’d printed for me from the Internet. He went on to comment:

Arab, Indian and Chinese did not only invent some mathematical techniques, but they also invented physics, chemistry, biology, astronomy, and philosophy when Europeans were still living in the dark, and the American continent wasn’t even discovered yet. Not to mention that Phoenicians (my ancestors) discovered the alphabet and spread it all over the world. Trying to internationalize the curriculum might be a good idea, since we live in a world full of hatred, bigotry and racism, and since some Americans (with all due respect) don’t know a lot about the rest of the world.

While this student went too far in belittling European achievements, I find even such strong remarks quite understandable. They represent a natural reaction to a curriculum, and a cultural mindset, that has been dismissive of the mathematics of Arab and other non-European peoples.

At the same time, these activities also have broad appeal to non-minority students born in the U.S. Some of these students simply find the cultural connections intriguing, as with one who wrote, “I thought it was very interesting because, not only did we learn different ways of doing the problems, but we also learned where these methods originated from and how they were used.” Other U.S.-origin students consciously want greater cross-cultural and cross-national understanding. Kathy, a returning student in marketing, wrote:

In this day of global hostility as well as global economics I believe it almost a mandate that all things possible be done in order to lay the foundation for not merely acceptance but rather an

appreciationof cultures and people different from those with whom we are accustomed […] Gradually, I submersed myself into the worksheets and found them to be intensely interesting. I found enough culture description that I almost felt as though I could see the beautiful colors of silk threads used to make the tassels from which the problems were derived. All the while I was intrigued with the realization that these techniques originated from a region previously given little attention yet is now of great national interest.

Of course, it is too much to expect that historical and cross-cultural activities will appeal to all of the students enrolled in traditional mathematics courses. However, in my experience the number of adverse reactions has been tiny. Specifically, some students think that the activities represent “extra work” that isn’t strictly necessary in order for them to learn the mathematical skills needed to pass the course. Occasionally a student will even say, “I would rather just be told the formula to memorize, and forget about who discovered it or how it was used.” Unfortunately, such an attitude actually impairs a student’s ability to learn (as opposed to “memorize”) mathematics; in fact, it is a symptom of the narrow pragmatism that has nourished a Eurocentric bias in Western education

Many problems from cross-cultural sources are not as directly suited for classroom use as is the tassel problem discussed above. Teachers can opt to formulate their own examples, but they should also consider recasting the original ones.

Such recasting presented a fun challenge for me in Business Calculus. In my syllabus for that course, I now routinely include “al-Tusi’s Derivative” alongside “Newton’s Derivative.” It is eye-opening to learn that a discovery relevant to calculus was made so long ago as the year 1209, by the Persian-born mathematician Sharaf al-Din al-Tusi in Baghdad. Also surprising is that his work was stimulated by a problem from ancient Greek geometry that Archimedes had studied but had been unable to solve. The problem was how to cut a given line segment of length \(a\) into two pieces such that the square of the first piece of length \(x\), times the second piece of length \(a-x\), equals a given volume \(c\). Al-Tusi’s predecessor Muhammad ibn Isa al-Mahani (Baghdad, *c.* 850) succeeded in converting this problem into an algebraic equation, \(x^3+c=ax^2\), and in the following century that equation was solved using conic sections. But al-Tusi took things even further, wondering how large the volume \(c\) can be and still admit of a solution; in modern terms, he was trying to maximize the polynomial function \(c=ax^2-x^3\). His search led him to the discovery that the relative maximum of a cubic polynomial can be located by finding the roots of a certain quadratic polynomial associated with it—in modern terms, derived from it (Houzel 1995; Katz 2009: 290-292).

In writing the module, "Using the Derivative to Solve an Optimization Problem," I devised a story that helps make the problem concrete and contemporary (see Figure 2):

Sami is a maintenance worker at a hospital in Baghdad. He has been asked to curtain off the area around a patient bed for privacy. The bed is in a corner of one ward, so the curtain needs to shield only two of its sides: there will be a square drape shielding one long side, and a smaller rectangular drape shielding the foot of the bed. Both lengths of curtain will drape from curtain rods suspended at equal heights from the ceiling of the hospital ward. But this is where Sami faces a problem. Because of war and supply shortages in Baghdad, the only material that he has available for the curtain rods is a 6-meter wooden pole, which he can cut into the two pieces that he needs. Sami is concerned to know how large a rectangular volume he will be able to enclose with these two pieces of curtain rod.

**Figure 2.** Al-Tusi’s discovery about cubic polynomials can be recast as a problem in optimizing a volume for hospital privacy.

I ask my students to convert this problem into a cubic equation, much as al-Mahani had done, and to examine the polynomial numerically, graphically, and analytically. The module guides them through the steps that al-Tusi is supposed to have taken (Farès 1995: 220-2), which are reminiscent of the development of a difference quotient. Eventually, we find that the maximum volume of privacy is 32 cubic meters, achieved by making the square curtain 4 meters by 4 meters, and the rectangular curtain 2 meters by 4 meters. Al-Tusi’s insight, and its use in resolving Sami’s dilemma, is the first example of optimization that my students encounter in this course.

- Download the module discussed on this page, Using the Derivative to Solve an Optimization Problem (Calculus for Business and Social Sciences).
- Download another module designed for my Calculus for Business and Social Sciences course, The Rule of Double False Position.

Download the three modules discussed in this article:

- Combinations and Their Sums (Elementary Statistics, Finite Mathematics)
- Binomial Coefficients and Subsets (Finite Mathematics)
- Using the Derivative to Solve an Optimization Problem (Calculus for Business and Social Sciences)

Besides the topics addressed in these modules, other topics from Islamic mathematics that can be adapted for classroom activities or student projects include:

- double false position (
*hisāb al-khatā’ayn*), an arithmetical procedure for evaluating linearly related quantities (Chabert 1999: 83-112; Schwartz forthcoming;*cf.*Shen et al. 1999: 349-385 and Sigler 2002: 447-487).

Download modules on this topic:

The Rule of Double False Position (Calculus for Business and Social Sciences)

The Rule of Double False Position (Linear Algebra) - the use of arithmetical and algebraic techniques for the Qur’ānic division of estates and the calculation of alms taxes (Berggren 1986: 63-68; Lesser 2000: 63; Rosen 1831: 86-174)
- the use of spherical trigonometry in determining the times for Islamic holidays and the five daily prayers, as well as the
*qibla*or prayer direction facing Mecca (Berggren 1986: 157-186; Katz 2009: 306-317; Van Brummelen 2009: 170-172, 192-201) - the geometry of patterns used in the design and decoration of mosques and palaces (Niman and Norman 1978; Özdural 2000)
- combinatorial and inductive techniques that lead toward integral calculus (Katz 1995)
- discoveries in number theory resulting from the search for perfect, abundant, deficient, and amicable numbers (Rashed 1994: 277-287).

The practical needs of Islam, reflected in the above list, were among the reasons that the spread of this faith stimulated mathematical work in the Middle Ages. More fundamental was the impact of its central doctrine of *al-tawhīd,* variously translated as “unification” or “unity in multiplicity.” This doctrine encouraged a sweeping embrace of all knowledge and all people as the best way to know God (Schwartz 2001). In a multicultural society such as ours, the outlook of unity in multiplicity has great relevance to education. The work described in this article shows that in a classroom consciously based on global diversity, students not only learn more mathematics, they learn how every culture has important contributions to make to the unified stock of human knowledge.

**About the Author**

Randy K. Schwartz holds degrees in mathematics from Dartmouth College and the University of Michigan. He is a Professor of Mathematics at Schoolcraft College, a community college in Livonia, Michigan, where he has taught since 1984. At Schoolcraft, his teaching focuses on preparing students for careers in engineering, science, health care, and business. Prof. Schwartz is a member of the Commission on the History of Science and Technology in Islamic Societies (CHSTIS), and has participated in several international conferences on the history of Arab mathematics. In 2000, he was awarded the Democracy in Higher Education Prize (National Education Association) for his essay, “Unity in Multiplicity: Lessons from the Alhambra,” an argument for a multicultural approach in mathematics education.

Berggren, J. Lennart. *Episodes in the Mathematics of Medieval Islam.* New York: Springer-Verlag, 1986.

Chabert, Jean-Luc (ed.). *A History of Algorithms: From the Pebble to the Microchip.* New York: Springer-Verlag, 1999.

Edwards, A. W. F. *Pascal’s Arithmetical Triangle.* London: Charles Griffin & Company, 1987.

Farès, Nicolas. “Le Calcul du Maximum et la ‘Dérivée’ Selon Sharaf al-Din al-Tusi.” *Arabic Sciences and Philosophy: A Historical Journal* 5 (1995): 219-237.

Houzel, Christian. “Sharaf al-Din al-Tusi et le Polygône de Newton.” *Arabic Sciences and Philosophy: A Historical Journal* 5 (1995): 239-262.

Joseph, George Gheverghese. “Foundations of Eurocentrism in Mathematics.” *Race & Class* 28 (1987): 13-28.

Joseph, George Gheverghese. *The Crest of the Peacock: Non-European Roots of Mathematics,* Second edition. Princeton: Princeton University Press, 2000.

Katz, Victor J. “Ideas of Calculus in Islam and India.” *Mathematics Magazine* 68:3 (June 1995), pp. 163-74.

Katz, Victor J. “Combinatorics and Induction in Medieval Hebrew and Islamic Mathematics.” In *Vita Mathematica,* edited by Ronald Calinger, pp. 99-106. Washington, D.C.: Mathematical Association of America, 1996.

Katz, Victor J. *A History of Mathematics: An Introduction,* Third edition. Boston: Addison-Wesley, 2009.

Kline, Morris. *Mathematical Thought from Ancient to Modern Times.* New York: Oxford University Press, 1972.

Lamrabet, Driss. *Introduction à l’Histoire des Mathématiques Maghrebines.* Rabat, Morocco: Imprimerie El maârif Al Jadida, 1994.

Lesser, Lawrence M. “Reunion of Broken Parts: Experiencing Diversity in Algebra.” *Mathematics Teacher* 93 (January 2000): 62-67.

Niman, John, and Jane Norman. “Mathematics and Islamic Art.” *American Mathematical Monthly* 85 (June-July 1978): 489-490.

Özdural, Alpay. “Mathematics and Arts: Connections between Theory and Practice in the Medieval Islamic World.” *Historia Mathematica* 27 (2000): 171-201.

Rashed, Roshdi. *The Development of Arabic Mathematics: Between Arithmetic and Algebra.* Dordrecht, Netherlands: Kluwer Academic Publishers, 1994.

Rosen, Frederic (ed.). *The Algebra of Mohammed ben Musa [al-Khuwarizmi].* London: Oriental Translation Fund, 1831.

Schwartz, Randy K. “Unity in Multiplicity: Lessons from the Alhambra.” *Thought and Action* 17 (Summer 2001): 63-75. Available at http://www.nea.org/assets/img/PubThoughtAndAction/TAA_01Sum_07.pdf

Schwartz, Randy K. “Adapting the Medieval ‘Rule of Double False Position’ to the Modern Classroom.” In Amy Shell-Gellasch and Richard Jardine (eds.), *Mathematical Time Capsules.* Mathematical Association of America, forthcoming.

Shen Kangshen, John N. Crossley, and Anthony W.-C. Lun. *The Nine Chapters on the Mathematical Art: Companion and Commentary.* Oxford: Oxford University Press, 1999.

Sigler, Laurence E. (ed.). *Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation.* New York: Springer-Verlag, 2002.

Van Brummelen, Glen. *The Mathematics of the Heavens and the Earth: The Early History of Trigonometry.* Princeton: Princeton University Press, 2009.

**Additional Resources on Mathematics in the Medieval Middle East**

Abdeljaouad, Mahdi. “Quelques Éléments d’Histoire de l’Analyse Combinatoire.” *Journées Nationales 2003 de l’Association Tunisienne des Sciences Mathématiques.* Available at http://www.math.unipa.it/~grim/MahCombinatoire.pdf

Berggren, J. Lennart. “History of Mathematics in the Islamic World: The Present State of the Art.” *Middle East Studies Association Bulletin* 19 (1985): pp. 9-33.

Berggren, J. Lennart. “Mathematics and Her Sisters in Medieval Islam: A Selective Review of Work Done from 1985 to 1995.” *Historia Mathematica* 24 (1997): pp. 407-440.

Covington, Richard. "Rediscovering Arabic Science." Originally published in *Saudi Aramco World* (May/June 2007): pp. 2-16. Available at http://www.saudiaramcoworld.com/issue/200703/rediscovering.arabic.science.htm

Djebbar, Ahmed. *Enseignement et Recherche Mathématiques dans le Maghreb des XIIIème - XIVème siècles,* doctoral thesis. Orsay, France: Université de Paris-Sud (Publications Mathématiques d’Orsay no. 81-02), 1981.

Djebbar, Ahmed. *Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa.* Available at http://muslimheritage.com/topics/default.cfm?TaxonomyTypeID=12&TaxonomySubTypeID=59&TaxonomyThirdLevelID=-1&ArticleID=952

Langermann, Y. Tzvi, and Shai Simonson. “The Hebrew Mathematical Tradition.” In *Mathematics Across Cultures: A History of Non-Western Mathematics,* ed. by Helaine Selin, pp. 167-188. Dordrecht, Netherlands: Kluwer Academic Publishers, 2000.

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