Power in Numbers: The Rebel Women of Mathematics

By Talithia Williams

From rocket scientists to code breakers, “fascinating stories” of women who overcame obstacles, shattered stereotypes, and pursued their passion for math (Notices of the American Mathematical Society).

With more than 200 photos and original interviews with several of the amazing women covered, Power in Numbers: The Rebel Women of Mathematics is a full-color volume that puts a spotlight on the influence of women on the development of mathematics over the last two millennia. Each biography reveals the life of a different female mathematician, from her childhood and early influences to the challenges she faced and the great achievements she made in spite of them. Learn how:

• After her father terminated her math lessons, Sofia Kovalevskaya snuck algebra books into her bed to read at night
• Emmy Noether became an invaluable resource to Albert Einstein while she was in the Navy
• Native American rocket scientist Mary Golda Ross developed designs for fighter jets and missiles in a top-secret unit
• Katherine Johnson’s life-or-death calculations at NASA meant that astronauts such as Alan Shepard and John Glenn made it home alive
• Shakuntala Devi multiplied massive numbers in her head so her family could eat at night
• Pamela Harris proved her school counselors wrong when they told her she would only succeed as a bilinguial secretary
• Carla Cotwright-Williams began her life in the dangerous streets of South-Central Los Angeles before skyrocketing to a powerful career with the Department of Defense in Washington, DC

These women are a diverse group, but their stories have one thing in common: At some point on their journeys, someone believed in them—and made them think the impossible was perhaps not so impossible.

“A quick read . . . full of dramatic stories and eye-catching illustrations.” —MAA Reviews

“I found myself marveling at the personal anecdotes and quotes throughout the book.” —Notices of the American Mathematical Society

• Language: English
• Publisher: Open Road Integrated Media
• Release date: Apr 10, 2018
• ISBN: 9780760360286

Book preview

Power In Numbers: The Rebel Women of Mathematics

POWER IN NUMBERS

THE REBEL WOMEN OF MATHEMATICS

TALITHIA WILLIAMS, PhD

CONTENTS

Introduction

PART I:The Pioneers

WANG ZHENYI

SOPHIE GERMAIN

SOFIA KOVALEVSKAYA

WINIFRED EDGERTON MERRILL

EMMY NOETHER

SPOTLIGHT:Breaking the PhD Color Barrier

PART II:From Code Breaking to Rocket Science

GRACE HOPPER

MARY GOLDA ROSS

DOROTHY VAUGHAN

KATHERINE G. JOHNSON

MARY WINSTON JACKSON

SPOTLIGHT:From Circus Performer to Human Computer

ANNIE EASLEY

MARGARET HAMILTON

PART III:Modern Math Mavens

SYLVIA T. BOZEMAN

FERN Y. HUNT

MARIA KLAWE

AMI RADUNSKAYA

INGRID DAUBECHIES

SPOTLIGHT:Mathematical Crochet

TATIANA TORO

KAREN E. SMITH

GIGLIOLA STAFFILANI

ERICA N. WALKER

TRACHETTE JACKSON

CARLA COTWRIGHT-WILLIAMS

EUGENIA CHENG

MARYAM MIRZAKHANI

CHELSEA WALTON

PAMELA E. HARRIS

Notes & Further Reading

Acknowledgments

Image Credits

Index

INTRODUCTION

Williams on the campus of Harvey Mudd College.

Sometimes it’s hard for people to hide their shock when I tell them I’m a mathematician. I get it. After all, I didn’t know any women with PhDs in mathematics growing up in Columbus, Georgia. Those same people might ask me how I got here. I suppose for me it started in high school. I qualified for five Advanced Placement (AP) classes and, while I would like to say that the reasons I signed up for those AP courses were honorable (I wanted a rigorous academic environment or I wanted to be challenged with college-level material), the reality was, as soon as the teacher told me that making a B in AP was like getting an A in the standard curriculum, I was sold.

It was when I was taking AP Calculus that marked the first moment I considered making math my life. In a high school with well over two thousand students, there was only one AP Calculus class, with about twenty-five kids. Of those twenty-five students, only four of us were African American. My teacher, Mr. Dorman, was in his early fifties at the time. He would invite us up to the chalkboard to work out problems for extra credit. I always went up there because I wanted that extra credit. I wasn’t a star student, but I was certainly motivated for a little extra credit. Who wouldn’t be?

One day after class, Mr. Dorman pulled me aside and told me he thought I was talented in math, and that I should think about majoring in it when I went to college. As a seventeen-year-old kid, I was shocked to hear him say that. I mean, sure, my mom and dad told me I was smart, but they’re my parents; I have their genes. But Mr. Dorman was different. He was a teacher, and his encouragement would forever change my life trajectory. Actually, it was the first time an older white man affirmed my intellectual ability. Even though I never saw myself as a mathematician, he saw me as one. The conversation changed me. It changed my life.

So, you know what I did? I went to college… and I majored in math. The funniest part of the story is that at my ten-year high-school reunion, I discovered that Mr. Dorman had said the same thing to many of his students, but by then, the damage had already been done! And I don’t regret my choice for a second. It just goes to show you that the power of a teacher who believes in you can be transformational. They can act as a mirror for potential, even if you can’t yet see it in yourself.

Williams presenting her popular TED talk, Own Your Body’s Data, at the 2017 Tableau Conference.

As an undergraduate student, I spent three summers working at NASA’s Jet Propulsion Laboratory (JPL). At JPL I was assigned to a research team where Dr. Lonne Lane was the advisor. Lonne was ridiculously brilliant, and he insisted that I call him by his first name. Now being raised in the South, I was taught never to call adults by their first name. Some social title needed to come before his name—Mr., Dr., Officer, Sir, Rev, Deacon—as a sign of respect. But he wouldn’t let me do it. In fact, everyone at JPL was on a first-name basis. He brought me into his research team and made me feel like I was an asset, as if my thoughts and opinions mattered. Here I was, this kid from Georgia, working side by side with rocket scientists. But despite the camaraderie, I still couldn’t help but feel a bit out of place. That is, until I met Dr. Claudia Alexander.

Claudia was my mentor for those three summers. She was everything that I dreamed I could become. She was beautiful and smart, and had a PhD in space physics from the University of Michigan, and worked at NASA. I remember wanting to get beautiful blonde highlights just like hers. The better I got to know her, the more I became hooked on mathematics. You see, she represented the dream of what I could one day become. I didn’t know it then, but during our summers together at NASA, she had planted a seed in me, and it was, thankfully, a lot more rewarding than blonde highlights!

I graduated from Spelman College as a math major with a minor in physics, before moving on to Howard University for a master’s degree in math. At Rice University, where I received my PhD in statistics, I was not only the only female in my class, but also the only African American. I met my husband while at Rice—he was in the Computational and Applied Mathematics department—and he approached me with a line like no other: I wish I was your derivative, because then I would be tangent to your curves. Yeah, who wouldn’t pass that up? He had me at derivative.

A few years later, I landed a tenure-track position at Harvey Mudd College and have worked to make the dream of broadening participation in the mathematical sciences a reality, much like Mr. Dorman, Claudia Alexander, Lonne Lane, and my Spelman professors have done for me. And I hope Power in Numbers takes me one step closer to that dream.

As you read through the incredible stories of the women profiled in these pages, the one thread they all have in common is that at some point on their journeys, someone believed in them; someone made them think the impossible was perhaps not so impossible. It was extremely difficult to narrow down the list of women I selected for inclusion here, but what I offer is a sampling of female mathematicians throughout history—including modern mathematicians—who have shattered stereotypes, pursued their passions, and persisted even when things got tough—even when people told them they couldn’t do it. Many were alone on their journey, but with every female who enters the field of math, it makes it easier and more achievable for the next one, and the one after that.

I invite women and men of all nationalities and backgrounds to learn about these dynamic mathematicians and scientists that have shaped our society. May their stories empower the next generation of STEM rebels to conitnue advancing mathematical theory, bringing awareness to the field, and increasing our Power in Numbers.

Talithia Williams, co-host of NOVA WONDERS.

Courtesy / © 2018 WGBH Educational Foundation.

PART 1

The Pioneers

Emmy Noether

The field of mathematics is not known for being especially friendly or appealing to women throughout history, but with the explosion of sophisticated technology in the twentieth century and beyond, many female mathematicians are making essential contributions to all kinds of human endeavors, from bioinformatics to spaceflight. The rise of universities in Ancient Greece laid the groundwork for modern university education and eventually for the breakthroughs of today, with teachers presenting subjects like philosophy and rhetoric alongside mathematics and astronomy. One such professor was Theon of Alexandria (ca. 335–ca. 405), who taught mathematics at the local university and wrote commentaries on some of the greatest scientific works of Ancient Greece, including Euclid’s Elements and Ptolemy’s Almagest . Theon passed his passion for learning to his daughter Hypatia (d. 415), who helped him write those commentaries and whose reputation would surpass his own.

Hypatia studied philosophy, astronomy, and mathematics in Athens, Greece, before she became head of the Neoplatonist school in Alexandria around the turn of the fifth century. She was a particularly skilled speaker, and many people from other cities flocked to the intellectual center of Egypt to hear her lectures. Within mathematics, she is best known for her work on Apollonius’s treatise on conic sections, which divided cones into different parts using flat geometric planes and introduced the concepts of the parabola, hyperbola, and ellipse.

Like too many women in this book, Hypatia paid a heavy price for her pursuit of knowledge, though her story is the most gruesome. As recounted in the 2009 film Agora (and countless plays and works of fiction throughout history), the Bishop of Alexandria spread virulent rumors about her, and one day in 415, she was attacked by a Christian mob, stripped, stabbed to death with broken pottery, and dragged through the streets.

When a double cone is intersected by a plane, the parabola (left), ellipse (center), and hyperbola (right) are created.

An 1865 engraving depicting the death of Hypatia.

It is almost certain that other female mathematicians quietly carried on her legacy through the European Medieval period, the Chinese Song Dynasty, the Islamic Golden Age, and the Renaissance, but there is little record of female achievements in mathematics prior to the publication of French mathematician and philosopher René Descartes’s monumental treatise Discourse on the Method—original source of the famous expression cogito ergo sum (I think, therefore I am)—in 1637. Just four years later, a French woman of obscure origins named Marie Crous published a study on the decimal numerical system, which introduced the decimal point that is central to our modern notation. A hundred years after that, Frenchwoman Émilie du Châtelet (1706–1749) published Institutions de Physique (Foundations of Physics), which explained and analyzed the cutting-edge mathematical ideas of Gottfried Leibniz, who later became co-discoverer of calculus, with Sir Isaac Newton. Naturally rumors spread that Châtelet was merely rehashing the lessons of her tutor Samuel Koenig, but in the end several prominent scientists came to her defense, and she was elected to the Academy of Sciences of the Institute of Bologna in 1746. Two years later, the Italian mathematician Maria Gaetana Agnesi (1718–1799) published one of the first and most complete textbooks on finite and infinitesimal analysis: Instituzioni Analitiche (Analytical Institutions). The book was widely translated, and in 1750 Pope Benedict XIV appointed her to the chair of mathematics and natural philosophy at the University of Bologna, making her the first woman in history ever to be appointed to a mathematics university professorship.

Jumping ahead to 1890, we arrive in Cambridge, England, where Philippa Fawcett (1868–1948) has just become the first woman to earn a top score on the annual Mathematical Tripos exam (beating out all her male classmates), and students Isabel Maddison (1869–1950) and Grace Chisholm (1868–1944) are studying calculus and linear algebra at Girton College. Two years later, Maddison and Chisholm passed the Tripos exam, earning the equivalent of first-class degrees in mathematics from Cambridge. (Women were not granted formal degrees at the time.) They also challenged each other to complete the mathematics exam for the Final Honors School at Oxford, and this time Chisholm outperformed all other entrants.

Detail of an eighteenth-century painting of Du Châtelet at her desk.

THE WITCH OF AGNESI

In mathematics, Agnesi was best known for her illustration of the Witch of Agnesi, a type of versed sine curve originally studied by French mathematician Pierre de Fermat. Her textbook used the term versiera (from the Latin vertere—to turn), but when it was translated into English, versiera was taken to be an abbreviation for avversiera, the Italian word for wife of the devil. The name stuck.

The curve is commonly expressed by the Cartesian equation y = 8a³/(x² + 4a²), where a is equal to the radius of the circle in the following illustration.

Cambridge University’s Girton College in the 1890s. Mathematicians Grace Chisolm and Isabel Maddison were attending this all-women’s college when they outperformed most male students on the 1892 Mathematical Tripos exam.

Despite beating out male students at two of the world’s most prestigious universities, women were not allowed to matriculate in graduate programs in England, so both Chisholm and Maddison ended up at the University of Göttingen in Germany, studying with group-theory pioneer Felix Klein. At this global hub of mathematics research—where Sofia Kovalevskaya earned her doctorate in 1874 (see here) and where the eminent mathematician Emmy Noether later helped establish the mathematics behind Einstein’s theory of relativity (see here)—Grace Chisholm graduated magna cum laude to become only the second woman to be granted a PhD in Germany with her 1895 thesis Algebraic Groups of Spherical Trigonometry. Her friend Mary Winston Newson (1869–1959), an American mathematician who arrived in Göttingen at the same time to study with Felix Klein, became the first American woman to earn a PhD in mathematics from a European university in 1897. As for Maddison, she specialized in differential equations and was awarded her PhD in 1896 from Bryn Mawr College in Pennsylvania. Fawcett’s achievement led to her professorship at Cambridge’s Newham College and inspired an anonymous poem:

Hail the triumph of the corset

Hail the fair Philippa Fawcett

Victress in the fray

Crown her queen of Hydrostatics

And the other Mathematics

Wreathe her brow with bay.

If you entertain objections

To such things as conic sections

Put them out of sight

Rather sing of the essential

Beauty of the Differential

Calculus tonight.

Worthy of our approbation

She who works out an equation

By whatever ruse

Brighter than the Rose of Sharon

Are the beauties of the square on

The hypotenuse.

Curve and angle let her con and

Parallelopipedon and

Parallelogram

Few can equal, none can beat her

At eliminating theta

By the river Cam.

May she increase in knowledge daily

Till the great Professor Cayley

Owns himself surpassed

Till the great Professor Salmon

Votes his own achievements gammon

And admires aghast.¹

In the intervening period between Agnesi’s 1750 Bologna professorship appointment and Fawcett’s triumph at the 1890 Cambridge Mathematical Tripos exam, several outstanding women who made pioneering advances in the field of mathematics were born. They come from China, Russia, France, Germany, and the United States, forming a global coalition of female mathematicians that has paved the way for countless women in STEM.

Tripos top scorer Philippa Fawcett in her room at Newnham College, 1891.

WANG ZHENYI

1768 – 1797

Qing Dynasty Astronomer and Mathematician Who Explained Both Lunar and Solar Eclipses

There were times that I had to put down the pen and sighed. But I love the subject, I do not give up.1

—Wang Zhenyi

Wang Zhenyi accomplished so much in such a short time on Earth—a mere twenty-nine years—it is fitting that she is remembered beyond it. There is a crater named for her on Venus, a nod to her expertise in astronomy. She is renowned