Studies show most football coaches make poor decisions on fourth down. Does Bill Belichick have a secret advantage? Photo by flickr user Keith Allison

Read our other posts about the history of football, what to bring to your Super Bowl party, the innovations of television advertising and much more.

This Super Bowl Sunday, as you watch grizzled coaches pace the sideline and bark at players, feel free to play armchair quarterback—or even head coach. Despite the hours they spend scouting players, analyzing game tape and drawing up complex tactical schemes, a pair of recent scientific studies indicates that many football coaches are no better at making some in-game decisions than you or I would be.

A 2006 paper by David Romer (pdf), a University of California at Berkeley economist, started things off by looking at a choice frequently encountered by coaches on fourth down: kick a field goal or try for a touchdown? Using data from more than 700 NFL games, Romer calculated the average chance of winning generated by each choice at different positions on the field. He then compared the data to the actual choices made by NFL coaches.

The conclusion: most avoid risk to an irrational extent, often opting to kick a field goal when going for a touchdown would provide a better chance of winning. Why would coaches—with their salaries and job security determined by on-field success—depart from the best possible choice? Romer speculates:

Perhaps the decision makers are systematically imperfect maximizers. Many skills are more important to running a football team than a command of mathematical and statistical tools…thus the decision makers may want to maximize their teams’ chances of winning, but rely on experience and intuition rather than formal analysis.

Another possible interpretation: for job security, coaches may prefer closer losses, coming after seemingly safe decision-making, to blowouts. A 23-0 loss may get a coach fired faster than a 23-6 score, which gives coaches incentive to kicking meaningless field goals rather than going for touchdowns.

Soon after the Romer study, Indiana University scientist Chuck Bower and partners from the business world went one step further. Using a similar dataset of actual NFL games, they built ZEUS: a powerful computer program that can analyze in-game situations on the fly and provide high-volume data analysis to coaches in real time. Bower said:

ZEUS is a valuable addition to a coaching staff’s tools, and one that can provide that elusive edge over the competition. The ZEUS engine is powerful enough to simulate the equivalent of every game played in the history of the NFL in less than a second. ZEUS can objectively assess crucial play-calling decisions with startling accuracy.

Comparing live data from the game with the historical track record of the NFL, ZEUS can indicate the choice that leads to a better chance of winning for a number of situations: not just what to do on fourth down, but whether to accept or decline penalties, attempt onside kicks, or try for two-point conversions.

In designing ZEUS, Bowers’ team drew upon many of the principles used in building computer models for other games—such as backgammon or chess—and applied them to football. “While the physical nature of the game is very different, the situational nature is strikingly similar. A football coach is constantly making decisions with respect to multiple variables: score, field position, down, yards to a first down, etc.,” said Bowers, an expert backgammon player.

NFL head coaches are a notoriously secretive bunch when it comes to strategy, so if anyone is currently using ZEUS, we’d likely not hear about it. But ZEUS’ own analysis indicates that one coach in particular might be using the cutting-edge program: New England Patriots coach Bill Belichick, set to coach in his 5th Super Bowl on Sunday.

The evidence? Belichick is famous for his unconventional decision-making, often opting to go for an aggressive play on fourth down when most coaches would punt or kick a field goal. The New York Times “Fifth Down” blog has used ZEUS to evaluate real-world decisions on a number of occasions. And when ZEUS was used to analyze a particularly controversial fourth down call made by Belichick—at the end of a crucial 2010 game against the Indianapolis Colts, he opted to go for it on his own 28-yard line, an unusually aggressive choice—ZEUS surprised many by saying he had, statistically, made the right call. The analysis indicated that, overall, it gave him team the best chance of winning.

Of course, statistical projections are not guarantees. In that case, the decision didn’t work out, and the Patriots lost the game. But if Belichick does have ZEUS on his sideline, it might give him that much better odds of being the winning coach on Sunday.

The first Death Star from Star Wars (via Wookieepedia)

Obi-Wan: That’s no moon. It’s a space station.

That space station was the Empire’s first Death Star in Star Wars: A New Hope. Obi-Wan and company had just bounced through a debris field, the remnants of the planet Alderaan. Such an act of destruction would seem impossible to us–it seemed so to many of the movie’s characters until it happened. But perhaps not, say three students at the University of Leicester in England who last year published a study on the subject in their university’s undergraduate physics and astronomy journal.

The study’s authors start off by making some simple assumptions: The planet being fired upon doesn’t have some sort of protection, like a shield generator. And it’s about the size of Earth but solid through and through (Earth isn’t solid, but the planet’s layers would have significantly complicated the math here). They then calculate the planet’s gravitational binding energy, which is the amount of energy required to pull apart an object. Using the mass and radius of the planet, they calculate that destruction of the object would require 2.25 x 10³² joules. (One joule is equal to the amount of energy required to lift an apple one meter. 10³² joules is a lot of apples.)

The energy output of the Death Star isn’t given directly in the movie, but the space station was said to have had a “hypermatter” reactor that had the energy output of several main-sequence stars. For an example of a main-sequence star, the authors look to the Sun, which puts out 3 x 10²⁶ joules per second, and they conclude that the Death Star could “easily afford to output [the energy required for an Earth-like planet's destruction] due to to its tremendous power source.”

It would be a different story, though, if the planet scheduled for destruction had been more like Jupiter than Earth. The gravitational binding energy of Jupiter is 1,000 times that of the Earth-like planet in the study. “To destroy a planet like Jupiter [the space station] would probably have to divert all remaining power from all essential systems and life support, which is not necessarily possible.”

Of course, that assumes that the Emperor wouldn’t be willing to sacrifice a space station full of people to wipe out his enemies. And considering that he was just fine with wiping out whole planets, I’m not sure I’d take that bet.

Sofia Kovalevskaya, Emmy Noether and Ada Lovelace are just three of the many famous female mathematicians you should know. Images courtesy of Wikicommons

If you haven’t yet read my story “Ten Historic Female Scientists You Should Know,” please check it out. It’s not a complete list, I know, but that’s what happens when you can pick only ten women to highlight—you start making arbitrary decisions (no living scientists, no mathematicians) and interesting stories get left out. To make up a bit for that, and in honor of Ada Lovelace Day, here are five more brilliant and dedicated women I left off the list:

Hypatia (ca. 350 or 370 – 415 or 416)

No one can know who was the first female mathematician, but Hypatia was certainly one of the earliest. She was the daughter of Theon, the last known member of the famed library of Alexandria, and followed his footsteps in the study of math and astronomy. She collaborated with her father on commentaries of classical mathematical works, translating them and incorporating explanatory notes, as well as creating commentaries of her own and teaching a succession of students from her home. Hypatia was also a philosopher, a follower of Neoplatonism, a belief system in which everything emanates from the One, and crowds listened to her public lectures about Plato and Aristotle. Her popularity was her downfall, however. She became a convenient scapegoat in a political battle between her friend Orestes, the governor of Alexandria, and the city’s archbishop, Cyril, and was killed by a mob of Christian zealots.

Sophie Germain (1776 – 1831)

When Paris exploded with revolution, young Sophie Germain retreated to her father’s study and began reading. After learning about the death of Archimedes, she began a lifelong study of mathematics and geometry, even teaching herself Latin and Greek so that she could read classic works. Unable to study at the École Polytechnique because she was female, Germain obtained lecture notes and submitted papers to Joseph Lagrange, a faculty member, under a false name. When he learned she was a woman, he became a mentor and Germain soon began corresponding with other prominent mathematicians at the time. Her work was hampered by her lack of formal training and access to resources that male mathematicians had at the time. But she became the first woman to win a prize from the French Academy of Sciences, for work on a theory of elasticity, and her proof of Fermat’s Last Theorem, though unsuccessful, was used as a foundation for work on the subject well into the twentieth century.

Ada Lovelace (1815 – 1852)

Augusta Ada Byron (later Countess of Lovelace) never knew her father, the poet Lord Byron, who left England due to a scandal shortly after her birth. Her overprotective mother, wanting to daughter to grown up as unemotional—and unlike her father—as possible, encouraged her study of science and mathematics. As an adult, Lovelace began to correspond with the inventor and mathematician Charles Babbage, who asked her to translate an Italian mathematician’s memoir analyzing his Analytical Engine (a machine that would perform simple mathematical calculations and be programmed with punchcards and is considered one of the first computers). Lovelace went beyond completing a simple translation, however, and wrote her own set of notes about the machine and even included a method for calculating a sequence of Bernoulli numbers; this is now acknowledged as the world’s first computer program.

Sofia Kovalevskaya (1850 – 1891)

Because Russian women could not attend university, Sofia Vasilyevna contracted a marriage with a young paleontologist, Vladimir Kovalevsky, and they moved to Germany. There she could not attend university lectures, but she was tutored privately and eventually received a doctorate after writing treatises on partial differential equations, Abelian integrals and Saturn’s rings. Following her husband’s death, Kovalevskaya was appointed lecturer in mathematics at the University of Stockholm and later became the first woman in that region of Europe to receive a full professorship. She continued to make great strides in mathematics, winning the Prix Bordin from the French Academy of Sciences in 1888 for an essay on the rotation of a solid body as well as a prize from the Swedish Academy of Sciences the next year.

Emmy Noether (1882 – 1935)

In 1935, Albert Einstein wrote a letter to the New York Times, lauding the recently deceased Emmy Noether as “the most significant creative mathematical genius thus far produced since the higher education of women began.” Noether had overcome many hurdles before she could collaborate with the famed physicist. She grew up in Germany and had her mathematics education delayed because of rules against women matriculating at universities. After she received her PhD, for a dissertation on a branch of abstract algebra, she was unable to obtain a university position for many years, eventually receiving the title of “unofficial associate professor” at the University of Göttingen, only to lose that in 1933 because she was Jewish. And so she moved to America and became a lecturer and researcher at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. There she developed many of the mathematical foundations for Einstein’s general theory of relativity and made significant advances in the field of algebra.

A chocolate peanut butter pie for Pi Day (photo, and pie, by Sarah Zielinski)

Today is March 14, or 3.14, the day we celebrate the mathematical constant pi (π). Pi, the ratio of a circle’s circumference to its diameter, is an irrational number, meaning that it can’t be expressed as a simple fraction of two integers. It is also a transcendental number, which means it is not algebraic. The celebrated 3.14 is just the beginning of pi—it continues into infinity, and that may be one of the reasons people find it so fascinating. So in honor of Pi Day, here are some suggestions for how to celebrate:

1 ) Read about the long history of pi.

2 ) Memorize as many digits of pi as you can (here’s pi to a million digits). A Japanese man in 2005 memorized pi to 83,431 digits.

3 ) See how far you can calculate pi. Computer programmer Fabrice Bellard calculated pi to 2.7 trillion digits using his home computer.

4 ) Watch the movie Pi, a 1998 thriller about a paranoid mathematician.

5 ) Make a pi-themed pie (I went for chocolate peanut butter pie, but any flavor is appropriate).

6 ) Celebrate with music: Learn the song, “Pi, Pi, Mathematical Pi,” set to the tune of “American Pie”; listen to the Pi Rap; or sing Pi Day carols.

Whoever builds a better light bulb could win up to $10 million (photo courtesy of flickr user djaquay)

Last week, a neurologist at Beth Israel Deaconess Medical Center in Boston won $1 million from Prize4Life for his discovery of a reliable way to monitor progression of amyotrophic lateral sclerosis (ALS), also known as Lou Gehrig’s disease. Prize4Life, which also has an ongoing competition for development of a treatment for ALS, is just one of several organizations that are trying to promote the development of solutions to sometimes longstanding problems with crowdsourcing and the lure of a big prize. Here are some ways you can pad that bank account—if you’re smart enough:

Millennium Prize Problems: In 2000, the Clay Mathematics Institute announced that they would award $1 million to anyone who solved one of seven math problems: the Birch and Swinnerton-Dyer Conjecture, the Hodge Conjecture, the Navier-Stokes Equation, the P vs. NP problem, the Poincaré Conjecture, the Yang-Mills and Mass Gap and the Riemann Hypothesis, which has been on mathematicians’ wish list since 1900. Russian mathematician Grigoriy Perelman received a Millennium Prize last year for resolving the Poincaré Conjecture, the only problem on the list solved so far, but he turned down the prize money.

NASA Centennial Challenges: The space agency has already given away millions in competitions, including competitions over the development of better space suit gloves and reusable rocket-powered vehicles. Current challenges range from the development of super-efficient, “green” aircraft to the demonstration of a solar-powered rover that can operate at night. NASA’s budget plans include $10 million per year for future competitions.

Bright Tomorrow Lighting Prize (aka, L-Prize): If you can build a better light bulb, you might be able to win this Department of Energy-sponsored competition, which was developed to spur manufacturers to create high-quality, high-efficiency products. At stake are a $5 million and a $10 million prize along with promises from electric companies across the country to promote the winning designs.

Life Grand Challenges: Life Technologies, a biotech company, has announced the first four of seven $1 million challenges intended “to accelerate innovative solutions to technical life science challenges.” They’re starting with challenges to increase the capacity, speed and accuracy of DNA sequencing and to sequence the genome from a single cancer cell.

X Prizes: There are three current competitions: The Wendy Schmidt Oil Cleanup X Challenge will give $1.4 million to the team that demonstrates the best way to recover oil from the surface of the sea. The Archon X Prize for Genomics will award $10 million to the person who develops a method to sequence 100 human genomes in 10 days at a cost of less than $10,000 per genome. And the Google Lunar X Prize will give a jackpot $30 million to whoever manages to send a robot to the moon, have it travel 500 meters and then send video back to Earth.

A diagram showing partitions for the integers 1 through 8 (via wikimedia commons)

You’re familiar with partition numbers, even if you don’t recognize the term; even kindergartners know them. The partition of a number is all the ways that you can use integers to add up to that number. Start with 2. There is only one way to get there: 1 + 1. The number 3 has 2 partitions: 2 + 1 and 1 + 1 + 1. Four has 5 partitions: 3 + 1, 2 + 2, 2 + 1 + 1 and 1 + 1 + 1 + 1. And so forth. But partition numbers get unwieldy pretty quickly. By the time you get to 100, there are more than 190,000,000 partitions. We’re well beyond elementary school math.

Mathematicians have been searching for the past couple of centuries for an easy way to calculate partition values. In the 18th century, Leonhard Euler developed a method that worked for the first 200 partition numbers. Solutions proposed in the early 20th century for larger partition numbers proved to be inexact or impossible to use. And the search continued.

The most recent mathematician to tackle the problem was Ken Ono at Emory University, who had a eureka moment while on a walk through the north Georgia woods with his post-doc Zach Kent. “We were standing on some huge rocks, where we could see out over this valley and hear the falls, when we realized partition numbers are fractal,” Ono says. “We both just started laughing.”

Fractals are a kind of geometric shape that looks incredibly complex but is actually composed of repeating patterns. Fractals are common in nature—snowflakes, broccoli, blood vessels—and as a mathematical concept they’ve been hauled into use for everything from seismology to music.

Ono and his team realized that these repeating patterns can also be found in partition numbers. “The sequences are all eventually periodic, and they repeat themselves over and over at precise intervals,” Ono says. That realization led them to an equation (all math leads to equations, it sometimes seems) that lets them calculate the number of partitions for any number.

The results of their studies will soon be published; a more detailed analysis is available at The Language of Bad Physics.

On the episode I watched, Chris Budd, a mathematician at the University of Bath, presented his theory, “there is nothing random about chaos.” I’m still floored that such a thing could exist on TV (and wish they would show it here in the U.S.). Here’s how the segment went:

The current fuel economy sticker found on U.S. vehicles (credit: EPA)

Today, if you go to buy a new car, you’ll find a sticker like the one on the right giving you a bunch of data on fuel economy: the miles per gallon you’ll get on the highway and in the city and the estimated annual fuel cost (based on 15,000 miles driven over a year and gas costing $2.80 per gallon). You’ll also see a little diagram that rates and compares that vehicle’s fuel economy with others in its class.

The EPA has now proposed changing the fuel economy sticker, this time adding information about how much greenhouse gases and other pollutants are emitted by the vehicle and how that compares to others in its class. They also propose adding another measure of gas mileage, this time presenting the number in terms of gallons of gas used per 100 miles.

It’s the last bit of information that is potentially the most useful. That’s because measuring fuel economy in MPG is rather misleading. Jennifer Ouelette explains in The Calculus Diaries:

One of the newly proposed labels for fuel economy (credit: EPA)

Why doesn’t everyone ditch their current gas-guzzling cars for a Prius or similar hybrid? The answer might surprise you. It turns out that many of us assume that saving gas (and therefore money) corresponds linearly with miles per gallon. But according to a June 20, 2008, article in Science by Richard Larrick and Jack Soll at Duke University’s Fuqua School of Business, the gas used per mile is actually inversely proportional to miles per gallon. They call this the MPG illusion.

If you do the math, this becomes immediately obvious. A car that gets 10 MPG uses 10 gallons every 100 miles. A car that gets 20 MPG uses 5 gallons per 100 miles. An MPG of 30 equals 3.3 gallons per 100 miles. And 40 MPG is only 2.5 gallons per 100 miles. Each improvement in 10 MPG does not result in the same improvement in gallons per 100 miles. And it’s that number that matters in terms of money saved.

That’s why the MPG illusion can also been seen when looking at estimated annual fuel costs, which is probably the easiest number to relate to on the sticker. We all understand money coming out of our pockets. But it’s good to see the addition of the new numbers, to move people away from thinking of their vehicle in terms of MPG. And perhaps in a few years, after everyone is familiar with calculating fuel economy in this new way, we could scrap MPG all together. It’s an easy way to keep track of your car’s health and your driving habits when you own the car, but, as we can see from the math, it’s not very useful when buying one.

The Calculus Diaries, by Jennifer Ouellette

It doesn’t have to be that way, though, as Jennifer Ouellette demonstrates in her new book The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse. There are plenty of opportunities in the world around us to find interesting examples of math, and especially calculus. Ouellette explains how to use calculus to analyze your odds of winning at craps and why your best option is simply not to play. She examines the Thermodynamics Diet, in which you can use calculus (or at least your own judgment) to optimize your diet and exercise regime so that you burn more calories than you consume. She links cholera, the black plague and zombies. (Okay, I’ll admit that last one falls into the fictional category that troubled me so much in school. But she links it to disease epidemiology. And beside, zombies are way more fun than Houdini tricks, at least in my world.)

The book has plenty of math and science history, and plain history itself—William the Conqueror makes an appearance—along with references to pop culture (the Mythbusters) and literature (the Aeneid). There’s a trove of material here for math teachers hoping to catch the attention of non-math students. Historical problems in math and physics show up regularly as do more recent analyses by modern scientists (there’s an actual study that goes with the zombie discussion).

The appendix includes many of the equations and graphs discussed in the text. However, I found that inadequate as I read through the book. What I really wanted was a workbook that would guide me along through the problems and scenarios that Ouellette posed in her writing. But that’s what surprised me: the book made me want to do the math, to work through the equations with a pencil and calculator, to graph out the curves and see for myself how all these things fit together.

I’m not sure I would have pursued math any more than I did if the teachers had made it this interesting in class. But perhaps I wouldn’t have slept through quite so many hours of it.

The exhibit is part of the Hyperbolic Crochet Coral Reef, a project started by two sisters in Los Angeles who run the Institute for Figuring, an organization that educates people about math and science. In about 2003, they started making models of hyperbolic space, a kind of space with surfaces that look undulating and ruffly, like a leaf of ornamental kale or a piece of kelp. The discovery of hyperbolic geometry in the early 19th century revolutionized how mathematicians thought about space; it launched the study of non-Euclidean geometry, the kind of math that underlies general relativity. Many cosmologists think the universe’s shape may be best described using hyperbolic geometry.

The writer says that hyberbolic crochet (her works, above) can be "kind of addictive." (Photo courtesy of Helen Fields)

It’s a tricky concept to visualize – unless, it turns out, you use crochet. After a few years, the sisters started varying the patterns in their crocheted work, and the pieces of frilly mathematical space piled up until, one day, they noticed it looked like a coral reef. A project was born; with contributions from volunteer crafters, the reef has been displayed in museums in London, Dublin, New York, San Francisco and others. Now it’s headed for the Smithsonian.

So the other night, I went on an adventure in math, crochet and coral. About three dozen women turned up at the Yarn Spot, a store in Wheaton, Maryland. (The all-female crowd wasn’t unusual; the vast majority of the coral pieces have been made by women.) The Yarn Spot is one of 10 yarn stores in the D.C. area that are hosting workshops and crochet-along parties for the Smithsonian Community Reef.

Jennifer Lindsay, the program coordinator, talked about the history of the project, passed around sample pieces and explained how to crochet hyperbolic planes, pseudospheres and other shapes. Then she set us loose to crochet. People who needed to borrow a crochet hook or some yarn dug through one of Lindsay’s bins. Experienced crocheters crowded her to ask questions, while store owner Victoria Rothenberg took the beginners aside to teach them how to wield a crochet hook. A lot were knitters who are perfectly capable with two needles but flummoxed by the single hook of crochet (crocheting is, by the way, much easier).

The coral reef has moved away from the strict requirements of modeling hyperbolic space; crafters are encouraged to experiment with varying the shape, increasing stitches (which widens the fabric and makes it ruffle like a hyperbolic plane) as often as they want to, for example. This is just the way nature works, says Margaret Wertheim, one of the sisters behind the Institute for Figuring. “All these frilly and crenulated structures on the coral reef—sponges, nudibranchs—those are all basically imperfect hyperbolic variants.” Of course, the animal isn’t counting stitches, but it is varying its growth. “They have it in their DNA to grow like this, but it’s affected by their immediate environmental conditions.”

And you don’t have to stick to hyperbolic shapes; they will take crochet models of anything that sits on a coral reef, like clams and anemones. Heck, you don’t even have to stick to crochet. Knitting is welcome, too, as long as the product looks reef-like. Knitting is welcome, too, as long as knitters make hyperbolic shapes or combine knitting (or other fiber techniques like felting, tatting, embroidery, etc.) with crochet. Anyone can mail in pieces by August 30; the deadline for dropping pieces off at a local yarn store or at the museum hasn’t been set yet. The reef will be on display in the Ocean Hall from October 16, 2010 to April 24, 2011.

For more information on the Hyperbolic Crochet Coral Reef at NMNH, including how to contribute: http://www.mnh.si.edu/exhibits/hreef/index.html

Guest blogger, and knitter, Helen Fields has written about snakeheads and dinosaurs for the magazine and can be found at Hey Helen.

How much of $100 in taxes goes to science? (courtesy of flickr user gtorelly)

Since I did not want to slog my way through federal budgets and appropriations, I took a shortcut and consulted a very nice interactive graphic from the New York Times. In Fiscal Year 2010, the federal government will spend a total of $3.60 trillion. If I paid $100 in taxes and it got divided proportionally among all programs, where would it go?

Well, about $20 would go to social security and $13 to medicare. Another $20 for national defense. And $5 for interest on the national debt.

Science is a bit difficult to figure out as it’s not a separate category. And to make matters worse, there are government agencies—like the National Oceanic and Atmospheric Administration and the U.S. Geological Survey—that perform scientific research but don’t break those dollars out, at least in this graphic. But adding up all the various science and research programs and agencies, I come up with about $72 billion spent on science in FY2010, and that’s probably on the generous side of things. So how much of my $100 goes to pay for that science? Just $2.

Will these girls learn to fear math from their teacher? (courtesy of flickr user woodleywonderworks)

We know that girls can do math, and be very good at it. But a new study published this week in PNAS shows that some girls in elementary school aren’t learning just how to add one plus one—they are learning that girls should be scared of those numbers. Just like their teachers.

University of Chicago researchers assessed the math anxiety of 17 first- and second-grade teachers in a large Midwestern urban school district. (When someone has math anxiety, they can master mathematical concepts but tend to avoid the subject and perform more poorly than their abilities allow.) They also assessed the math performance levels of the teachers’ students at the beginning and end of the school year as well as whether or not the students believed the stereotype that girls do better in reading and boys do better in math.

The researchers discovered that in classes with teachers that have math anxiety, math achievement at the end of the school year was worse for girls but not for boys. Girl with such teachers were also more likely to endorse the stereotype that boys are better in math and girls are better in reading. What’s going on? The teachers in question were not worse at teaching math, the scientists say, but they were somehow passing on the idea to the young girls in their classrooms that math is scary. The researchers write:

We speculate that having a highly math-anxious female teacher pushes girls to conform to the stereotype that they are not as good as boys at math, which, in turn, affects girls’ math achievement. If so, it follows that girls who confirm traditional gender ability beliefs at the end of the school year should have lower math achievement than girls who do not and than boys more generally. This is exactly what we found. …

In addition, children do not blindly imitate adults of the same gender. Instead, they model behaviors they believe to be gender-typical and appropriate. Thus, it may be that first- and second-grade girls are more likely to be influenced by their teachers’ anxieties than their male classmates, because most early elementary school teachers are female and the high levels of math anxiety in this teacher population confirm a societal stereotype about girls’ math ability.

The problem really starts in college, where the elementary education major requires little math. This appears to be attracting math-phobes, and, not shockingly, there is a higher incidence of math anxiety among elementary education majors than individuals in any other college major. So our college education system is churning out a disturbing number of role models for little girls who find math harder and scarier than Barbie ever did. And they’re teaching their charges—if unintentionally—to follow their lead.

What should be done? My impulse is to say that we should raise the math requirements for elementary education majors beyond basic algebra and geometry and weed out some of the math-phobes. And if you’re thinking of becoming an elementary school teacher and are scared of math, perhaps you should find another profession.

Bruegel's drawing The Seven Deadly Sins or the Seven Vices - Gluttony (via wikimedia commons)

Pieter Bruegel the Elder was a 16th-century painter from the Netherlands known for his landscape paintings populated by peasants (though you may also be familiar with his version of the Tower of Babel). He also produced dozens of drawings and prints. In the early 1990s, though, several Alpine drawings attributed to Bruegel were identified as fakes when it was discovered they were drawn on paper made after 1569, when the artist died.

Although the identification of the drawings as imitations might have been distressing for the owners of the works, it provided a group of computer scientists with an ideal test case for the development of a statistical method for spotting fake art. Their latest paper appears in this week’s PNAS.

The scientists used a method called “sparse coding” that breaks down an artist’s works into tiny, random pieces that, when recombined, can recreate the original works but not a piece done by another hand. BBC News explains:

The method works by dividing digital versions of all of an artist’s confirmed works into 144 squares – 12 columns of 12 rows each.

Then a set of “basis functions” is constructed – initially a set of random shapes and forms in black and white.

A computer then modifies them until, for any given cut-down piece of the artist’s work, some subset of the basis functions can be combined in some proportion to recreate the piece.

The basis functions are refined further to ensure that the smallest possible number of them is required to generate any given piece – they are the “sparsest” set of functions that reproduces the artist’s work.

This method easily picked out the fake Bruegels from the real ones and did so more easily and accurately than other approaches used to find imitations. “These digital techniques can assist art historians in making judgments and may provide detailed information about subtleties inherent to a particular artist’s style that are not immediately observable,” the scientists write.

Eternal Sunshine of the Spotless Mind, courtesy of Focus Features

The last decade has been a pretty good one for science in the movies (though there are exceptions, as we’ll see tomorrow). Here are 10 movies we enjoyed:

• A Beautiful Mind (2001): This is the nearly-true story of John Nash, the mathematician who won a Nobel Memorial Prize in Economics for his work in game theory but later struggled with paranoid schizophrenia. The film won four Academy Awards, including Best Picture.
• Eternal Sunshine of the Spotless Mind (2004): Jim Carrey erases Kate Winslet from his brain. It may seem like crazy science fiction, but scientists know how to do it in mice, and this week New York University researchers claimed that they have figured out how to rewrite fear memories.
• Primer (2004): This $7,000 film about time travel was praised for its attempt to portray scientific discovery—even if it’s outlandish and impossible—in a realistic and down-to-earth manner.
• March of the Penguins (2005): We can forgive the anthropomorphization of Antarctic emperor penguins in this French documentary because not only was the movie beautiful and charming, but it also got thousands of people, especially children, interested in nature. The film won the 2005 Academy Award for Best Documentary.
• An Inconvenient Truth (2006): The documentary about Al Gore’s slideshow woke up the United States to the issue of climate change. (And before the skeptics start arguing with us: Gore got most of the science right.) The movie won an Academy Award, Gore got a Nobel Prize and it looks like the country might be on its way finally to tackling the problem.
• Flock of Dodos (2006): Marine biologist-turned-filmaker Randy Olson explores the evolution-intelligent design debate, smacking down the proponents of creationism and intelligent design and chiding scientists for losing the message war.
• Idiocracy (2006): Two modern-day people have their bodies put into stasis by the military—which forgets about the experiment—and wake up 500 years in the future to find the human race has devolved. It’s crass comedy but one of the best examples of human evolution to be portrayed in a movie.
• Encounters at the End of the World (2007): This was acclaimed filmmaker Werner Herzog’s answer to March of the Penguins. While there are penguins in the movie, there are also volcanologists and physicists, maintenance workers at science stations and stunning footage of the Antarctic underwater.
• WALL-E (2008): The sweet love story of the only robot left cleaning up the Earth after humans have fled takes on the themes of environmentalism, technology and even human evolution. The film won the 2008 Academy Award for Best Animated Feature.
• Star Trek (2009): There’s this one lovely moment at the beginning of the movie where there is silence in space, a rarity in science fiction films. So the movie makers got much of the rest of the science wrong. Who cares? We really like the reinvented Star Trek universe, especially the new Spock.

What was your favorite science-y movie of the 2000s? Tell us in the comments below.

There is something we find beautiful about fractals, those curious geometric structures with repeating shapes that seem to go on for infinity (see video below). Perhaps it is because these mathematical oddities remind us of nature; river networks, ferns and Romanesco broccoli are all examples of natural fractals.

The most famous fractal is probably the Mandelbrot set, named for Benoît Mandelbrot, the French-American mathematician who coined the term “fractal” in 1975. The Mandelbrot set is a 2-dimensional object created through a mathematical equation. Mathematicians had first pondered how to turn it into a 3-dimensional object, a “Mandelbulb,” more than 20 years ago, but they didn’t figure out how to do it until recently. The result is above. Quite pretty, don’t you think?

(Hat tip: Bad Astronomy)