Lost History: Mathematics in the Muslim World

The history of mathematics, like any type of history, contains blind spots and lost information.  So much of history mirrors the movements and conquests of armies and their rulers, and armies usually destroy indiscriminately.  In 1258 Hulegu Khan, grandson of Genghis Khan and leader of the Mongols, sacked Baghdad, destroying perhaps the greatest city in existence. 

For over 500 years, the City of Baghdad was among the greatest cities in the world.  During what is sometimes known as the “dark ages” of medieval Europe, the Islamic Empire was among the most accomplished empires in the world, with Baghdad at its center.  In “Lost History” Michael Morgan quotes from a description found from a geographer’s account, the Kitab al-buldan:

“I begin with Iraq only because it is the center of this world, the navel of the earth, and I mention Baghdad first because it is the center of Iraq, the greatest city, which has no peer in the east or the west of the world in extent, size, prosperity, abundance of water or health of climate, and because it is inhabited by all kinds of people, town-dwellers and country-dwellers.  To it they come from all countries, far and near, and people from every side have preferred Baghdad to their own homeland.  There is none more learned than their scholars, better informed than their traditionalists, more cogent than their theologians”

The loss of such a great city is just one example of the “Lost History” outlined by Michael Morgan in his book.  Either through destruction, lost translations, or ignorance, there is a long history of Islamic scholarship that is unknown in much of the world.  Many advances that were later attributed to European thinkers found their origins in Muslim lands.  In particular, there are a number of mathematical ideas and tools that were either created or put into popular use by Islamic mathematicians. 

Today, the American image of the Muslim world is rife with anger, stereotypes and violence.  Years of war and terrorism have changed how Americans view countries like Iraq and cities like Baghdad.  There are limits to how much change we can make in math classrooms, but there are small steps that teachers can take to develop cultural understanding in our students.  After reading Lost History, I created a few ways in which I could incorporate some of the mathematical contributions of the Muslim world into a classroom.

Mohamed Al-Kwarizmi, the man for whom algorithms were named, did extensive work translating Indian mathematical texts into Arabic, which allowed the ideas to spread much more quickly.  One of the ground-breaking discoveries was the concept of the number zero.  Zero became the center of the numerical world and with its unique properties, made the world of mathematics much more complex and abstract.  It allowed math to move into more academic, less literal areas, like Algebra (whose root is an Arabic word, jabr describing the process of subtracting from both sides of an equation).  As an activity in an algebra class, students will be assigned to small groups, and each group will be responsible for evaluating a unit previously studied.  Students will make the determination, with proof, whether or not we could learn that unit without the concept of zero.  If it is impossible, students will gain insight into the power of zero, and if it is possible, this activity should foster creative thinking in devising new ways to solve math problems.

Al-Kwarizmi also introduced decimal numbers to the world.  Prior to the Indic-Arabic number system, there were three ways to represent numbers:  Counting on your fingers, writing the number in script, or using roman numerals.  As an activity, students will solve basic algebraic equations using either roman numerals, or script.  Students should grapple with the idea of variables.  What happens when X means 10?  How can we represent variables when we are writing out equations using words?  If students get stuck, or would like more information and inspiration, they can watch this video:

The Magic Square was also studied and popularized by a Muslim mathematician, Thabit ibn Qurra.  A magic square is one where, in a square grid, all the numbers in a row and column all add up to the same number (See below).  This is similar to what we now call Sudoku puzzles.  Rather than completing a Sudoku, students will receive extra credit for creating a magic square.  The extra credit will scale along with the dimensions of the square (3x3 is 3 points, 5x5 is 5 points, etc.)

 

Students in calculus could examine the chess board problem. Thabit ibn Qurra also worked on this type of problem, described here:

“The man who invented the game of chess asks his ruler for a favor, the receive one grain of wheat on the first square, then double that on the next, that is two on the second square, four on the third square, eight on the fourth square and so on until all 64 squares are filled”. 

This could serve as an introduction to exponential series, where students would require the new knowledge in order to solve.  Using this model, students will build the series representing the chess board problem:

1+2+4+8+16+… is the same as 2⁰+2¹+2²+2³+… which could be used as an introduction to series notation. 

As an FYI, the number of wheat grains, when calculated is a staggering 18,446,744,073,709,551,615.

If we were to generalize the contributions of Muslim mathematicians, we could say that they moved mathematics to a more abstract area of study.  Through their use of decimal numbers, they allowed math to be written more clearly.  Through the use of exponents, they evaluated numbers that were too complicated for ‘brute force’ calculation.  Moving from the literal and observable to the abstract is a big step for students.  Through studying the methods and techniques used by thinkers who were among the first to embrace abstract mathematics, students may gain insight into their own learning.  Beyond the mathematical insight, by studying the contributions of Muslim thinkers, students may also gain a more nuanced understanding of a religious and cultural force that is often misrepresented today.

Morgan, M. H. (2007). Lost History: The enduring legacy of Muslim scientists, thinkers, and artists. Washington, D.C.: National Geographic Society.

The Calculus Wars

One definition of calculus is "an advanced branch of mathematics that deals mostly with rates of change and with finding lengths, areas, and volumes".  This entire branch of mathematics is at its core, about how things change.  It is somewhat ironic, that the two men who battled in Jason Socrates Bardi’s "The Calculus Wars", Isaac Newton and Gottfried Leibniz, failed to recognize the incremental and gradual change that allowed them to "invent" this branch of mathematics.  The idea of invention suggests a brand new idea, which may radically change how we live or how we understand the world.  By fighting and arguing over who invented calculus, Newton, Leibniz, and their contemporaries overlooked the significant contributions of dozens of other men made in the years leading up to their own work.  In their argument about calculus, the study of change, they failed to address how this change to mathematics really happened.  

“The Calculus Wars” tracks the lives and careers of the two men credited with inventing calculus, Isaac Newton and Gottfried Leibniz.  Bardi makes the case that there were two core mistakes or misunderstandings that allowed the calculus conflicts to escalate into the calculus wars.  If Newton had published his work on calculus when he wrote them, in 1666, the conflict could have been avoided entirely.  He had, in his personal notes, ample evidence of his work on “fluxions and fluents”, which make up his claim to the creation of calculus.  However, Leibniz was the first to publish his ideas of calculus in the year 1684, allowing him to claim ownership of the field.  This publication did, however, come after he had limited access to Newton's journals, which would later lead to claims of plagiarism.  The conflict over which man created calculus is more complicated than just these two events, but the timeliness of Newtown’s publishing and Leibniz’s alleged access to those unpublished ideas were the catalyst for a fairly ugly academic battle.  The conflict would warp to include nationalism, academic name-calling, and claims on both sides of stolen work.  Those initial claims of plagiarism would prove to be the spark that would lead to the rising accusations that would divide these great minds for the rest of their lives. 

These men are each undeniable geniuses, both in mathematics and elsewhere in the academic world.  Newton is a familiar name for any student who has taken physics, where he established his still famous laws of motion and gravitation.  Leibniz's talents were wide ranging and profound.  He established foundational ideas in Earth science and geology, and was a renowned scholar on politics, philosophy, and even Chinese culture and history.  The abilities of these men, in such varied fields, prove that neither discovery was an isolated achievement.  It seems unlikely that either man was intellectually dishonest about their work in calculus.  The most likely order of events saw Newton as the earliest "inventor" but not publishing, with Leibniz publishing first, marginally (if at all), influenced by Newton’s journal.  Leibniz went on to create, simplify and improve the mechanics and symbolism of calculus.  He introduced enough new elements to calculus, including the familiar elongated 'S' many recognize (and dread) from integration, to prove his own merits as a mathematician.  

Prior to the work of Newton and Leibniz, there were many mathematicians working on topics very similar to calculus.  As Bardi notes, Johannes Kepler, building on the work of earlier Greek geometry experts, used infinitely small shapes to determine the volume of larger shapes, closely approximating a method of integration.  Bonaventura Cavalieri “considered a line to be an infinity of points: an area an infinity of lines; and a solid an infinity of solids” (p. 8).  This approach reflects the dimensional relationships that are at the heart of differential calculus, specifically the power rule.  Newton and Leibniz owed a debt to René Descartes, who created analytical geometry, which “allowed the analysis of geometrical shapes through mathematical equations”.  Pierre Fermat and Blaise Pascal both created methods for finding maxima and minima, and did extensive work with tangents to curves, all of which are taught in thousands of calculus classrooms across the world.  Gilles Personne de Roberval worked with geometric shapes and volumes, as did Evangelista Torricelli.  John Wallis, Johann Hudde, Christian Huygens, Isaac Barrow, and René François de Sluse and no doubt numerous others made contributions to what we understand to be calculus, all within about 100 years of Newton and Leibniz.  

Looking at the list of mathematicians who contributed to what we know as calculus, the idea of “inventing” calculus becomes even cloudier.  Both Newton and Leibniz claimed individual credit, but both owed a debt of gratitude to a long list of scholars.  Beyond the contemporaries mentioned earlier, there is a long history, a continuum of mathematical understanding that has developed, sometimes slowly, and sometimes rapidly.  It is no accident that each man claims to have created calculus in the same time period.  Kepler, Descartes, Fermat, and many others had set the stage for calculus to emerge in the 17^th century.   It is also important to note that the conflict between Newton and Leibniz only escalated in 1713, 30 years after Leibniz published his work and almost 50 years after Newton had done his.  Both men had established impressive careers and reputations across Europe, and the conflict seemed to be more about protecting those reputations than anything else.  Both men relied upon proxies to argue on their behalf, and were fairly careful to not directly attack the other man.  This professional politeness and respect seems to undercut the idea that either man really thought the other had done anything malicious.  

The “Calculus Wars”, like most wars, was destructive and unnecessary.  Two men, Isaac Newton and Gottfried Leibniz stood out among a long list of brilliant mathematical minds in the 17^th and 18^th century.  They contributed imaginative and important work to the field of mathematics that should have been enough to establish their place in history.  Instead, an ugly feud ensued over credit for work that both most likely created independently.  If each man had the benefit of hindsight, they may see that they were part of an innovative time for mathematics, and that their own contributions were significant parts of a much larger whole.  

Descartes Dreamworld

Most of us probably know the name Rene Descartes. In addition to his contributions to the mathematical world (the Cartesian coordinate system is named after him), he is also a well known philosopher, particularly for the quote "I think, therefore, I am". In the first half of the 17th century, Rene Descartes dreamed of a world completely described and governed by mathematics. He saw the trajectory of mathematics, which was, to him and many others, the pure distillation of understanding. He wanted to apply the same logic and precision that mathematics exhibited. He believed that a world dominated by mathematics would be a triumph of understanding and knowledge.

We now sit in the 21st century. Every day, we use innumerable devices and tools that are based upon digitization and mathematics.  Since the early 1600’s, the world has experienced an explosion of mechanization, automation and computerization.  We now live in a world more similar to the world Descartes dreamed about and yet, this world driven by mathematics has not given us the salvation he predicted.  We have adopted a mathematical approach to a number of sciences and arts that would impress Descartes, but there are numerous ways in which numbers and metrics are used to confuse and mislead people.  Whether it is politically, financially or in a wide variety of other fields, numbers, modeling and mathematics in general are used for questionable ends.  Beyond the intentional misuse of mathematics, the act of digitizing and analyzing information creates the feeling of a loss of humanity. 

In order to aid students as they head into the mathematized world, I created a few activities that could be inserted into a unit that might allow students to better understand the pitfalls of the information they see throughout their lives.  I used the unit below, on Linear Equations and their Graphs as a starting point, and created 4 activities that could be used at specific points in the unit. 

 

Unit Warm-Up Activity

Students will write a journal entry reflecting on the following questions. 

• What kind of information, or data, can we put in a graph?

• Graphs and charts are good ways to organize information.  Do you agree or disagree?  Explain.

• How do people use graphs?  Do they help to understand what happened?  Do they predict what will happen next?

6.3 Applying Linear Functions Activity

In order to begin to understand the limitations and specific problems created by linear graphs, students will complete this activity to better recognize what requirements are necessary for real world examples to be accurately modeled using a linear graph. 

The students will create a real world example that can be modeled using a linear graph.  They can use examples from the book as a starting point, as seen in the graph below. 

Once students do this, select students (some selected for correct work, others for incorrect work to “find the mistake”) will share their results with the class. 

Students should work to answer the following question:  What conditions do you need to exactly meet the conditions of a linear relationship?

Students should discuss different features of the situations they created.  The discussion should be guided towards the idea that in order to be a true linear relationship, the change in the dependent variable is proportional to the change in the independent variable.   They should also consider and discuss what types of questions could they answer with the graph they've created.

The text uses the graph below to describe profits from a car wash, based on the number of cars the students wash.  What variables could we introduce that would make this an imperfect linear relationship?

 

6.7 Scatter Plots and Equations of lines

Students will again grapple with the limits of linear modeling, this time by evaluating trend lines in data. 

As a class, students will work in a computer lab to explore this idea.  First, students will research and select a set of data that includes at least 5 points of data, and no more than about 25.  This can come from anywhere, and only the most extreme cases will be disallowed. 

Using Excel, students will enter their selected data into a spreadsheet and create a scatter plot of the data.  They will then create a line of best fit, or trend line, and print out their resulting graph.

Back in class, we will discuss again their findings.  We will try to identify if any students had a true and accurate linear graph, which is unlikely but not impossible.  We will also discuss the student’s graph that is least accurately represented by their trend line.  It will be important to discuss what the trend line (using vocabulary like Slope and Intercept) really represent in each graph, and whether or not that information is valuable. 

The discussion should cover what we mean when we take an average.  We can also discuss statistical ideas like sample size and deviation if possible.  The text introduces a correlation coefficient, which will be a good measure of the accuracy of a trend line.  Additionally, the students should again think and discuss what types of questions they could answer with their data and the corresponding trend line.

As a class, we will gather the heights of all students in the class, and then average the heights.  The average should be a non-integer around 5-6 or 5-7.  I will make the point that even though everyone was accounted for in the experiment, no one in the class is actually represented by the average height.  The class will discuss what is meant by the “average person” or the “average family” (graph here), and what it means to have 3.13 family members.    

At this point, students will have seen the limitations of a modeling from a few different viewpoints.  Students should have a better understanding of how statistics can be used and misused.

Final Project

Students will discuss the ways that data can be skewed and how bias can show up in seemingly scientific and mathematically sound areas.  Students should understand that data can be “spun” in different ways graphically, which can alter the conclusions an audience draws from that graph.    

As a final project, students will research some piece of information that is represented graphically (preferably linear graphs, or a scatter plot with trend line) for which specific data is available.  Students will then, using the same data and information available, re-create the graph in a way that, in their opinion, may change the utility or possible conclusions drawn by the audience.  Students can alter the scale of the graph, the orientation and even the sample of data used, but cannot change the data itself.  Students will also explain how their choices may affect the perceptions created by their new graph.

As a conclusion to the unit, students will return to the journal entries from the beginning of the unit.  They will have the option of editing (or rewriting) their journal entry, or, responding to their “earlier selves” and correcting any mistakes or misconceptions they feel are part of the entry.  These edits and explanations should demonstrate new knowledge and insight learned from the chapter, as well as these lessons.

Note:  The idea for this post came from reading Descartes Dream: The World According to Mathematics by Philip Davis & Reuben Hersh.