Derive

December 2012

Breaking Through: How One Girl Beat the Gender Gap in Math

To-Fold Sky High Make orgami objects Seven Facts on the with ease New Freedom Tower Page 14 Page 8

Star Struck UT professor on math education in K-12 Page 26

2 December 2012

About the Authors Ethan Russo has been dedicated to improving at math since he was snubbed of the title “Best Mathematician” in the third grade. He competed in MATHCOUNTS competitions in middle school to some success. In his free time Ethan is liable to read the newspaper, listen to NPR, and play his violin, as well as read both internet memes and the news. A member of the Quiz Bowl team and Math Club, he is also fond Pixar movies, of which his favorite is “Up”.

PHOTOGRAPH BY JONATHAN S.

Jonathan S. grew up surrounded by math. It was one of the most common dinner table conversations. In preschool, he came home proud that he had counted to 800 in half an hour. By fourth grade, he was learning algebra. Jonathan, now a freshman as LASA, is involved in a plethora of after school activities, ranging from Quiz Bowl to Robotics to Math Club. In his spare time, he enjoys reading and programming.

PHOTOGRAPH BY ETHAN RUSSO.

Travis Kennedy is a freshman at LASA. He enjoys origami and playing Ultimate Frisbee. Travis is odd because he didn’t got to Kealing, but rather to Saint Mary’s and Saint Austin’s. He wants to attend the University of Texas at Austin and move to Vancouver, Canada.

PHOTOGRAPH BY ETHAN RUSSO.

December 2012 3

Letter From the Editors Multiply your knowledge. Discover the story behind Derive.

D

erive is, first and foremost, not a magazine about the kind of math one would find written up in a textbook. It is about the patterns, the forms, and all that has made math one of the most continuous sources of human interest in all of history. It is about things like origami and architecture, and even things like social issues surround math and science.

We seek to make math something one shouldn’t be afraid to admit to loving. We strive to make it interesting, to make it cool, sleek, shiny and engaging.

We hope to help you find the infinite number of ways to see the math that have barely been eluding you before.

It makes us all cringe a little bit when someone laments that math is a cruel, useless tool because their only exposure to it has been through a similarly useless and cruel worksheet or exercise. The things that drew us to math are what we want to show to you.

Our magazine is divided into two parts: one on origami and one dealing with a small sample of the many ways math has come into contact with you and issues surrounding math today.

Each of us has found some beauty within the world of numbers and shapes that we know as math, and it is our wish that you do the same.

From left, Ethan Russo, Jonathan S. and Travis Kennedy—your guys at Derive.

PHOTOGRAPH BY IFY EZEKOYE.

4 December 2012

Table of Contents About the Authors

2

Letter From the Editors

3

Reel Math

7

The Sky’s the Limit

8

Neat Streets

12

Foldin’ Age

14

The Magnificent Seven

15

December 2012 5

16

The Basic Folds

17

Beginner: The Crane

19

Intermediate: The Marigold

21

Advanced: The Longhorn

23

The Origami Pig Massacre

24

A Man’s World

26

Reach for the Starbird

Come Discover Our Math and Sciences.

The Maine Arts and Sciences Academy features a world-class faculty nested in the gem of the Atlantic Seaboard. With advanced programs in physics, topology and more, MASA allows anyone to unleash their mathematical potential.

PHOTOGRAPH FROM PUBLIC DOMAIN

December 2012 7

Reel Math Math isn’t just made for paper. Here are a few examples of where it’s found its way to the movie and television screen.

Lindsay Lohan (bottom left) appears as Cady Heron along side Rachel McAdams, Lacy Chabert and Lizzie Caplan in the 2004 teen comedy “Mean Girls.” IMAGE USED UNDER FAIR USE.

C

an you recognize the limit to the left? You might think that the equation is taken right from a calculus textbook, but it actually comes straight from the 2004 film “Mean Girls”. It satires social cliques among girls by chronicling the social rise, fall and redemption of Cady Heron, played by Lindsay Lohan. In one of the last scenes of the film, Lohan’s character evaluates the limit in a sudden death showdown to win the state math championship.

Besides spawning a number of notable quips, the movie is an example of where math seeps it way into popular culture. Here are a few of our favorite moments where the two fields have collided.

“The Princess Bride” The 1987 cult-classic features an intriguing example of a branch of mathematics known as game theory. Game theory concerns the mathematics of decision making, and what decision is more important than one of life or death? The hero, Westley, disguised as the Dread Pirate Roberts, plays a game of sorts with his Sicilian foe Vizzini.

Westley presents two glasses to his enemy, one of which he says is poisoned with iocane. The game is variation on the matching pennies game from game theory, however it differs slightly, as Westley already knows the outcome of the duel, remarking at the end, “they were both poisoned. I spent the last few years building up an immunity to iocane powder.”

“Numb3rs” This CBS detective show produced by brothers Tony and Ridley Scott featured the another pair of brothers, Don and Charlie Epps, an FBI agent and a professor of mathematics, solving crimes using their respective talents. Mathematics, real and legitimate, was presented in most episodes of the series. Professional mathematics were consulted during the series, although some criticized “Numb3rs” for only using them to check jargon, not in story development.

By ETHAN RUSSO

ln(1 − x) − sin x lim x→0 1 − cos2 x This expression, used in the 2004 film “Mean Girls,” was solved by Cady Heron, played by Lindsay Lohan, at the end of the movie.

“A Beautiful Mind” A 2001 film based on the book of the same name by Sylvia Nasar and directed by Ron Howard, “A Beautiful Mind” tells the story of Nobel Prize-winning economist John Nash. The picture, which won its own award for best picture at the Academy Awards, focuses on Nash’s beginnings at Princeton University, his schizophrenia and hallucinations, and his eventual resurgence. The films concludes with Nash being awarded the 1994 Nobel Prize in Economics for his work in game theory.

“Bye Bye Nerdie” from “The Simpsons” The episode from Season 12 of the Emmy-winning animated series includes Lisa Simpson presenting the scientific reason behind the bullying of nerds during the “12 Annual Big Science Thing” in Springfield, and also Professor Frink, in an attempt to quiet down the rowdy attendees of the conference, announcing that, “pi is exactly three!” This might be simply pure fun if the state of Indiana didn’t almost legislate the erroneous fact into law.

The so-called “Indiana Pi Bill” of 1897 sought to set the value of the constant Pi exactly at three and to defy thousands of years of mathematical logic and theory. The bill, which unanimously was passed in the state house, almost passed in the senate. Opinions shifted, however, after one senator remarked they lacked any power to change mathematical truth.

The Sky’s The Limit New York City is about regain a titan among its buildings. Derive presents four startling figures about the new skyscraper.

By ETHAN RUSSO The tallest building in all of New York City is one that has not even been completed yet, as reported by Bloomberg News. In fact, it isn’t scheduled to be finished until 2013, by which time it will have surpassed nearly every other building on the globe. The skyscraper is the new One World Trade Center, also known as the Freedom Tower, built on the site of the old Six World Trade Center, which was destroyed in the September 11 attacks. The tower will feature over 3.5 million square feet of space and 71 elevators, but that’s not whole story. Here are four fascinating figures about Manhattan’s new crown jewel.

1776 The Port Authority of New Jersey and New York states that One World Trade Center will stand 1776 feet high, in reference to the year the US declared its independence. The tower would be the tallest in the Western Hemisphere, and runner-up globally only to the Burj Khalifa and Abraj Al Bait hotels.

But controversy has arisen over whether an antenna to be placed at the apex of the tower ought be included in the overall height of the building. Without it, the tower stands 1368 feet, making it the third tallest tower in One World Trade Center is shown as the tallest structure in this rendering of the of the New York City skyline post-completion of the new World Trade Center towers.

IMAGE FROM THE LOWER MANHATTAN DEVELOPMENT CORPORATION COURTESY OF SKIDMORE, OWINGS & MERRILL LLP.

ABOVE LEFT: The memorial for the victims of the September 11 attacks, known as Reflecting Absence, opened in 2011. The monument is large fountain and includes the names of all the victims of the attacks inscribed on bronze plaques. PHOTO

BY KAI BRINKER USED UNDER CREATIVE COMMONS LICENSING. ABOVE RIGHT: One World Trade Center, scheduled to be opened in 2013, will be the tallest building in all of New York and the Western Hemisphere at 1776 feet tall.

PHOTO BY BENJAMIN KRAFT USED UNDER CREATIVE COMMONS LICENSING. BELOW RIGHT: In this photo of the island of Manhattan as seen from a helicopter. One World Trade Center stands at the left of the photo. PHOTO BY DAVID JONES USED UNDER CREATIVE COMMONS LICENSING.

the Western Hemisphere, behind Chicago’s Willis and Trump Towers. Ultimately, the decision will be made by the Council on Tall Buildings and Urban Habitat. The CTBUH announced the decision would be made after the tower’s completion in 2013.

Seven While the Freedom Tower has received most of the spotlight from the media, there are actually seven other structures flanking it at the World Trade Center. According to the Lower Manhattan Development Corporation, six of those are other towers and the other is the National September 11 Memorial and Museum. Several of the towers were designed by winners of the Pritzker Prize (the “Nobel Prize” for architects). Though every structure is now expected to open by 2016 , as recently as 2010

the developers of the site were criticized for not moving the projects forward faster.

Particularly distressing to some was the fact that prior to 2010 the estimated completion date of the towers was 2037. Since then, Seven World Trade Center has opened its doors, as has the memorial portion of the memorial and museum.

104 One World Trade Center has already lined up tenants as notable as Conde Nast, a wellrespected magazine publisher. According to The New York Times, the tenants will occupy 69 of the skyscrapers 104 floors, out 74 total above-ground levels that are usable for commercial purposes. The remaining stories will be used for sky lobbies, observation decks and mechanical spaces. The first 20 floors will constitute the base of the tower.

Also of note is that the level of the window washing tracks will be designated as floor 110, in remembrance of the 110 stories of the original Twin Towers.

3.1 Billion Although only estimated to be 3.1 billion dollars in 2008, the total cost of the Freedom tower in 2012 was set at 3.8 billion dollars according to The Wall Street Journal, putting it at the largest amount put into a single building in the world. The cost was divided up among the developers, Larry Silverstein and his property group chief among them, New York state, and the Port Authority of New Jersey and New York. Silverstein put up one billion dollars toward the project. The state contributed a quarter billion dollars, while the Port Authority added another billion via the sale of bonds.

12 December 2012

Neat Streets The history of a city built like a coordinate grid and how it shaped city planning throughout America

The Plat of Zion, created by Joseph Smith, the leader of the Mormons, showed how the city of Salt Lake would be built in the shape of a coordinate grid.

PHOTOGRAPH COURTESY OF MICHAEL MALLOY.

By TRAVIS KENNEDY

J

oseph Smith knew exactly what he was going to do.

Smith, the leader of the Church of Jesus Christ of Latter-day Saints, also known as the Mormon church, had a city already mapped out. All he had to do was build it.

He and the Mormons tried to start a city in Illinois, but were kicked out. In territory after territory, they got kicked out until they came across modern-day Utah.

In present day Salt Lake City, Smith began the construction of the Salt Lake Temple and built a city according to the Plat of Zion around it.

gave room for each family to have a house and a small garden.”

A Template for the Ages Hundreds of cities and towns have used the Plat of Zion as a template for their streets.

The grid-like system of streets is a simple and organized way to start a city. The Plat of Zion has been nominated for the American Institute of City Planner’s landmark award for its influence on American cities.

Present-day Expansion in Salt Lake City has kept true to the grid like structure. Having the

“There is an urban legend that the streets were made 132 feet wide so an oxenpulled carriage could turn around without backing up.” wide streets is a benefit because it allows firefighters to have “staging room” while extinguishing fires. However, the wide, straight streets allow cars to speed away from police. Additionally, the roads’ width makes it difficult for some pedestrians cross the street.

The Plat of Zion The Plat of Zion, created by Smith, was a map that showed how the city would be layed out.

Michael Malloy, Salt Lake City planner for six years said, “There is an urban legend that the streets were made 132 feet wide so an oxen-pulled carriage could turn around without backing up.”

Each city block was made a quarter mile long, and was divided into twenty-three plots of land. Malloy said that, “the Plat of Zion

This is a map of Salt Lake City that shows how it looks like a coordinate grid. MAP © OPENSTREETMAP CONTRIBUTORS. USED WITH PERMISSION.

December 2012 13

Salt Lake City, Utah

MAP Š OPENSTREETMAP CONTRIBUTORS.

Saint George, Utah

Preston, Idaho

Colorado City, Arizona

Around the Map Salt Lake City isn’t the only town that the Plat of Zion has influenced. As Mormons from the area engaged on missions to spread their beliefs, they also spread the the design principles of the Plat of Zion. For this reason, many cities in Utah, Arizona, Colorado, Nevada , and even Idaho are grid-like. The cities denoted with blue in this map represent those directly influenced by the Plat of Zion.

14 December 2012

Foldin’ Age A look into the mathematics of a centuries old art

1

2

8

3

7

4 6

5

By TRAVIS KENNEDY AND JONATHAN S.

Welcome to a new age.

The Mathematics

Welcome to a time where a centuries-old art has found a new following in a world dominated by newfangled ideas.

Welcome to the era of origami.

Origami is built on a foundation of mathematics—from constructions to axioms, and theorems and more. A great many branches of math can impact what goes on in folds.

Evolution

Constructions

Interest in this nearly ancient Japanese art of paper folding has recently been revived with the introduction of computers and mathematical analysis and rigor. These two developments are among those that have allowed many modern advancements, the formation of new organizations and novel groups, as well as several previously unknown applications.

Put simply, constructions are accurate depictions of lines, angles and points. In origami, constructions come in handy when trying to divide a square into “x” number of parts. The way to do this varies in accordance with “x”. Robert J. Lang, an origami artist and mathematician has created Reference Finder, a computer program that finds a way to divide a square into “x” parts without using a ruler or marking the paper.

For example, in April of this year, Christophe Guberan, a student of industrial design at the Ecole Cantonale d’art de Lausanne developed a method of making creases by using a printer. Additionally, not long ago, Akira Yoshizawa developed the technique of wet-folding, making it easier to create smoother, more realistic designs. By adding water to thicker paper, wet-folding can allow one to sculpt shapes and add a layer of complexity beyond the basic geometric figures. But now the artistic angle of the discipline seems to be moving into a new, sharper world thanks to mathematics.

Theorems Two of the most famous origami theorems are Maekawa’s theorem and Kawasaki’s theorem. Each has to do with crease patterns, or the lines the folds make when you unfold a model. Kawasaki’s theorem states if there is a point with mountain and valley folds going through it, in a crease pattern, there will always be two more of one than the other.

Maekawa’s theorem says if one alternately adds and subtracts angles as one goes around a point one will get zero.

This diagram demonstrates Maekawa’s theorem. That law states that if one has a point with certain folds going through it in a crease pattern, the results of alternately adding and subtracting the angles around the point will be zero. In this diagram, each of the eight angles around the center point has a measure of 45 degrees. Alternately adding and subtracting them gives a result of zero. DIAGRAM BY TRAVIS KENNEDY.

How the Math Helps Mathematics and origami allowed Robert J. Lang to work with Lawrence Livermore National Laboratory to design a massive satellite meant to be folded and fit into a small pod for expansion in space. He used the same technology that has been used in installing small, quick-release air bags in new automobiles. Lang’s talent has even been implanted into humans. He designed a small expandable frame for patients with heart failures to keep their hearts from collapsing.

The Future More and more mathematicians and scientists have turned to origami from solutions to their problems. With origami helping the world through mathematics at this time, no one knows what’s possible years from now.

December 2012 15

The Magnificent Seven Learn the unbreakable seven laws that let you turn a sheet of paper into a work of art

The seven Huzita-Justin Axioms are the only known way to fold a piece of paper using lines and points as “landmarks� for folding. The Italian-Japanese scientist and mathematician Humiaki Huzita formulated the first of six of the axioms, while Jacques Justin stated the final member of the group.

The seven axioms constitute the foundation for any origami model and have been proven mathematically over the years as being the complete set of ways to fold paper.

Axiom 1: One can fold a line connecting two points.

Axiom 2: One can fold a line that sets one point on another point.

Axiom 3: One can fold one line onto another line.

Axiom 4: One can fold a line perpendicular to another line while passing through a point.

Axiom 5: One can fold a point onto a line while passing through a second point.

Axiom 6: One can fold one point onto one line and another point onto another line.

Axiom 7: One can fold a line perpendicular to another line while placing a point onto a third line.

16 December 2012

The Basic Folds The valley fold is the simplest of all the folds.

To do one, simply fold the paper along the indicated line, folding the farther away part towards the folder. It is called the “valley fold“ because, while in progess, the fold line forms a V shape, reminiscent of a valley. It is shown by a dashed line.

Mountain folds are very similar to valley folds. The difference lies in the direction of the fold. This time, the section of the paper is folded is away from the folder, below the remaining paper. Its name originates from the fact that, while in progress, the fold line makes an upside down V, similar to a mountain. It is show by a line that alternates between dots and dashes.

Squash folds are the most complex of those presented here. It starts with a piece of paper that already has a fold on it. The folder stick a finger inside the fold, opening it up. Then the opening is flattened, or squashed, producing the final result.

December 2012 17

Beginner: The Crane A Japanese legend promises a wish to whomever folds 1000 cranes. Here’s how to fold the origami classic.

The origami crane is a centuries old classic that remains popular today. One Japense legend states that one wish will be granted to whomever folds 1000 of the cranes. PHOTOGRAPH

BY ETHAN RUSSO.

Crease along the diagonals.

Turn over and crease in half both vertically and horizontally.

Begin the preliminary fold.

Continue the preliminary fold.

Crease to the middle.

Crease the top down where the two previous creases hit the edges.

18 December 2012

Begin a petal fold.

Fold the edge to the center.

Fold down the head.

Repeat beind.

Repeat behind.

Fold the head and tail upward.

The final product is shown.

December 2012 19

Intermediate: The Marigold It’s not too easy, nor too hard; it’s just right. Figure out how to fold the flower that would please even Goldilocks.

Even the picky Goldilocks would take pleasure in folding the marigold, a model that’s just right for intermediate level folders. PHOTOGRAPH BY TRAVIS KENNEDY.

Crease into quarters in verticals, horizontals, and the diagonals.

Fold in the quarters .

Fold in the quarters and turn over.

Crease the quarters.

Fold in two adjacent quarters and squash the 3D triangle into a square.

Repeat on the other three corners.

20 December 2012

Pull out the single layers of one of the corners to make a rectangle.

The result is shown. Repeat on the other three corners.

The result is shown.

“Round” the corners by folding the points under.

Pull out the layers to make a rectangle similar to the past steps.

The result is shown. Repeat on the other three flaps.

The result is shown.

“Round” the corners on the points again.

The finished marigold is shown.

December 2012 21

Advanced: Longhorn Show off your University of Texas at Austin pride with this model created

by Derive’s own Jonathan S. Learn how to fold the advanced longhorn.

Learn how to fold this original creation of Derive’s Jonathan S. The longhorn is the most challenging of the models presented here. PHOTOGRAPH BY ETHAN RUSSO.

Crease along the diagonals.

Crease in half vertically and diagonally.

Start the waterbomb base.

Continue the waterbomb base.

The finished waterbomb base.

Take a flap and begin to squash it across the center of the paper.

22 December 2012

Finish squashing.

Pull out from under the squashed paper.

Begin to fold over the pulled out paper.

Finish folding and repeat behind.

Reorient the paper.

Fold the top edges to the bottom.

In the process, collapse the extra paper.

Push in the baggy sides of the face.

Repeat behind.

Thin the horns.

Repeat in the other 3 locations.

The final product is shown.

December 2012 23

The Origami Pig Massacre Sometimes, people won’t take your money— even when you give it to them.

A man who goes by “BaconMoose” on YouTube went to pay a ticket in the town of Jersey Village, near Houston, Texas. He walked up to the clerk and said he was paying his 137 dollar traffic ticket in cash, due to the five percent processing fee for using a credit card.

The story of one man’s encounter with origami and the law.

He handed the clerk two donut boxes. The confused clerk asked what was in the boxes. “BaconMoose” calmly said it was the money.

The confused clerk opened the boxes to find 137 origami pigs folded from one dollar bills. The clerk called for an officer.

The officer came to the window and said he would have to unfold them all, because there was no way to store the petit porkers.

“BaconMoose” argued that the folded bills were different from crumpled bills, but to no avail.

While he was unfolding his army of origami pigs, which he said was lead by a one “Admiral Ackbacon” according the Houston Chronicle, the clerk and officer were taking pictures of the origami. The officer

By TRAVIS KENNEDY also said that he “got the joke” about pigs and donut boxes.

“BaconMoose” at last unfolded 136 pigs, saying the remaining pig was the lone survivor of the “Great Origami Pig Massacre” “BaconMoose” said it took him four to six hours to fold all of the pigs. As of November 30, his video had 1,140,284 views.

“Bacon Moose” went to the court with 137 pigs, just like the one above, to pay for a traffic ticket, but was told to unfold them by the clerk. PHOTOGRAPH BY JONATHAN S.

24 December 2012

A Man’s World Math and science have long been fields dominated by men. Here’s the story of how one LASA student broke through and beat the gender gap.

By ETHAN RUSSO

Jessica Wang loves pandas.

“It started in first or second grade when I did a project on them,” the LASA junior said, “and in middle school it sort of defined me.

I was panda-girl.” But that’s not the only the thing that defines her. Wang is also the president of LASA’s Math Club and a copresident of the Science Olympiad team.

“I’ve always just really liked math. My family just talked about it at dinner and it it just was something we talked about,” she said, recounting how she first got into numbers.

“I also liked how there was no gray area.

Everything’s the same wherever you go and there’s always a right answer. In English class, you always have to kind of guess what the author’s doing.”

She first got her start in math and science competitions during the sixth grade. “I joined MATHCOUNTS because my brother did it and I joined Sci-O, which is what we call Science Olympiad, after I took the elective.

We got to build bridges which I thought was pretty cool, so I started showing up to some practices.”

LASA Junior Jessica Wang, president of the math club, works on a set of math problems. Wang said her love of math started at an early age, and has participated in math competitions since sixth grade. PHOTOGRAPH BY ETHAN RUSSO.

It wasn’t a quick path to the top, though.

She didn’t make her school’s official team for MATHCOUNTS, a middle school math competition, in sixth grade.

Wang, however, wasn’t about to bail out on the competition quite yet. “I wasn’t that disappointed. The older kids were a lot better than me,” Wang said, “and our coach, Ms. [Nannette] Strickland, told me just to keep practicing and then I made the team in seventh grade.”

Alone in the Room However, she is also an outlier: Jessica Wang is a girl competing, and succeeding, in the fields of math and science, which are usually dominated by men. She’s broken through the so-called gender gap that plagues math and the sciences today.

“There are people who think that boys are a lot better at math and science than girls.

They’re wrong. But a lot of people also think that ther are a lot more boys in math and science competitions and such. And they’re right.”

The statistics bear Wang’s assessment out as well, especially at extremely high levels of mathematical achievement. One analysis of the gap at these exceptional levels was done by professor Glenn Ellison of MIT. Ellison measured the gender gap at high levels of skill by examining the results of the AMC, or American Mathematics Competitions, a series of contests given to about 225,000 students yearly designed to differentiate between the highest performing students.

The eventual goal of the competitions is to select a team to represent the US at the International Math Olympiad (IMO).

The IMO describes itself as the “World Championship Mathematics Competition for High School students,” and is considered by many to be the pinnacle of high school math competition.

The first steps of the test are the AMC 12 and 10, for students in grades 12 below and 10 below, respectively. The maximum score on either of the tests is a 150, and a score of 100 on the AMC 12 or a 120 on the AMC

December 2012 25

10 qualifies one for the next level, the AIME. Among those students who take the AMC 12, only 43 percent were female and of those who scored sufficiently to make the AIME, a grand total of 22 percent were girls.

The trend isn’t unique to the United States, either. In the International Math Olympiad, only 5.7 percent of participants from all countries over the years have been girls.

The US has sent teams to the event since the year 1974. In that time, it has sent 224 students, and exactly five of those students have been girls. For the first 24 years, no team even included a girl. Since then, there has been slight improvment: the ratio of boys to girls now stands at twelve to one.

Wang’s experiences have mirrored this reality ever since she started competing.

“I was almost always one of two or three girls there,” she said, “In MATHCOUNTS and in Sci-O. Even though the guys can be a little irritating sometimes, it’s not bad because you learn to stick together. I was at a math competition at Rice [University], I

think, and we [the girls] were just huddled up while there were guys everywhere else.”

What’s the Problem?

Explaining why she thinks the gender gap exists, Wang says that the problem is two pronged. “Just to me, it seems like girls are more inclined to the liberal arts, and it doesn’t really help that most of the time guys already make up most of the [math or science] club,” Wang said.

At the same time though, she does not think math does a particularly good job of selling itself to anyone with the method that is usually used to teach it in schools, either: “There’s just a lot of memorizing things, like formulas. It’s not very fun. And that’s the thing I’d really change about the way math is taught. It should be more about problem solving than just plugging things into an equation.”

She explains that girls would come to a few meetings of the math or science club, but subsequently stop attending. To girls on the

“There are people who think that boys are a lot better at math and science than girls. They’re wrong.” fence about joining or staying in one of the math or sciences activities she says, “it really does get better. It doesn’t even really bother me anymore because I’ve gotten used to it.

Just keep trying and you’ll be okay.”

To Infinity and Beyond Wang is most certain that she will be continuing to break barriers in the math and sciences. She said, “I’ll definitely go into a career in STEM [Science, Technology, Engineering and Mathematics], but I’ll probably go into in engineering or something like that. I probably won’t go into math, because you have to be really, like really, good at it to get anywhere professionally.” Strong words from a panda-loving girl in a man’s world.

This diagram shows how, since 1974, of the 224 students selected by the US for its International Mathematics Olympiad team, only 5 have been female. No team, in fact, for the first 24 years, included a girl. Since then, females have been outnumbered by males by a ratio of twelve to one. Each figure represents 5 contestants. DIAGRAM BY ETHAN RUSSO. FIGURES USED UNDER FAIR USE.

26 December 2012

Reach for the Starbird Advisors

An inside look at a man who is repairing, bit by bit, the teaching of mathematics

Birth Date Alma Mater Awards

By JONATHAN S. When growing up, Dr. Michael Starbird had an unconventional idea of fun.

“My father was a math teacher at a community college,” he said. “He would bring math puzzles to the dinner table, so that my brother and I would do these puzzles and just love ‘em. So, to us, math was something we did at dinner and enjoyed.”

This continued all through his adolescence, and led to his eventual career. “It was sort of the family business,” he said. As such, after a fruitful university education, he followed his advisor, R. H. Bing, to the University of Texas at Austin.

Unfortunately, he said, most people don’t have such an experience.

“To many people, mathematics is something that occurs in a classroom and only in a classroom, whereas to me, mathematics was what you did at home,” said Starbird. “It really was quite a different relationship of mathematics to my life. It really made it so much more fun and just... a part of life than just an isolated school kind of event,” remarked Starbird, “so in fact I always wish that students could all have that kind of experience, where they thought of school, not only mathematics but other things, as part of the way they conduct their everyday lives, rather than something in an isolated, special place.”

Transforming Teaching Starbird is known among his peers for his uncanny ability to bring the experience of playing with math to his students, as well as to the general public through his numerous books.

Students who struggle, according to Starbird, “think of mathematics as either a foreign language which they don’t really speak or as a kind of artificial enterprise that’s not associated with they way they think in real life.” In his view, though, math and the real word are intimately related.

Job Field Degree

R. H. Bing and Mary Rudin 1948 Pomona College and University of Wisconsin Mathematical Association of America Deborah and Franklin Haimo Awards for Distinguished College or University Teaching of Mathematics in 2007, Jean Holloway Award for Teaching Excellence, President’s Associates Teaching Excellence Award, Eyes of Texas Excellence Award, and many more Professor at The University of Texas at Austin Topology PhD of Mathematics

As Albert Einstein said, “the whole of science [and mathematics] is nothing more than a refinement of everyday thinking.” Starbird subscribes wholly to this philosophy. “I like for students to think that when they’re doing something in mathematics class, they should be thinking in the same kind of a way that they apply to their everyday life,” he said.

Starbird accomplishes this by not force-feeding the information to the students, as he said is done at many schools, but rather by “present[ing] the challenges that actually get the students to think for themselves.”

“I concentrate on what the students are doing, not what I’m doing, so I’m thinking to myself, ‘how can I encourage the students to have the fun of figuring out the mathematics,’” he explained.

“I set up the classroom situation where I try to maximize the amount of the time that they’re having fun, enjoying figuring things out for themselves,” Starbird said about his teaching methods. “I will just give them a bunch of problems, which are basically theorem statements to prove, or other exercises to do, but I think of them as just a bunch of fun puzzles for them to think through.”

After the students have figured out those puzzles, he “ask[s] them to present their own solutions so they have the satisfaction of having

December 2012 27

figured things out.” Those presentations, he said, are his favorite parts of the classes — seeing the students “enjoying the fun part of discovery, of finding things by virtue of thinking them through on their own. To me, that’s the fun part of doing mathematics and that the fun part of teaching mathematics is to see other people enjoy the ‘aha’ moments.”

It also makes up the core of his teaching philosophy; he focuses on allowing the students to discover a subject, not teaching it. This is because he said he wants to “change everything about a student — not just how they do at mathematics. [I want] them to be a better writer because of my class, a better member of society.”

A clear example of this is in one of his popular textbooks, “Number Theory Through Inquiry.” Very little in the book is explanation. It is made up of a sequence of definitions and theorems for the reader to prove. By working through these problems, the student gathers a

“[I want to] change everything about a student — not just how they do at mathematics. [I want] them to be a better writer because of my class, a better member of society.”

comprehensive knowledge of the subject, as well as becoming better at proof and logical reasoning.

Riveting Results All this work on Starbird’s part has not been in vain. He has received numerous teaching excellence awards, and his books have been very well received. In 2001, his “The Heart of Thinking: An Invitation to Effective Thinking” won a Robert W. Hamilton award, a prestigious prize for books written by University of Texas faculty. Even now, he continues to turn out easy-to-read and compelling publications.

Most are focused on math, but his latest book, “The 5 Elements of Effective Thinking,” is purely about logical thought.

His philosophy has also gained him a large reputation. His lectures, ranging from videos for adults to classes at UT to those for students in grade school at the local Saturday Morning Math Group, are well attended. Ironically, though, according to Starbird, numerous students tell him, “I really enjoyed that,” after a class, and then say “[But] I still hate mathematics,” not realizing that he was, in the end, teaching them mathematics. Despite this, Starbird has made significant progress in helping society acquire a more friendly, personal attitude towards math.

Michael Starbird, 64, talks about his most recent book, “The 5 Elements of Effective Thinking.“ Starbird, a professor of mathematics at the University of Texas, focused on logical thought in the book, instead of straight-up mathematics. PHOTGRAPH BY JONATHAN S.