Category Archives: Education






Home page of Nate Silver’s Burrito Bracket
Ranking system (upper left quadrant)


Google Spreadsheets used to create the competition bracket


Flickr photo archive for The Burrito Bracket (




FiveThirtyEight two months after launch, in May 2008










Is it addiction?






FiveThirtyEight on the New York Times website, 2010

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The Last Digit of Pi

[This is a rough transcript of my TEDxNYED talk, delivered on March 6, 2010, in New York City at the Collegiate School. TEDxNYED was an all-day conference “examining the role of new media and technology in shaping the future of education.” For a meta-post about the experience of giving a TED(x) talk, please read “Academic Theater (Reflections on TED & TEDxNYED).” What I actually said and did at TEDxNYED deviated from this transcript; I engaged the audience directly a couple of times, once for fun and once to get their ideas about the subject. I’ll post the video when it’s available.]

I want to tell you a story about a forgotten realm of education and knowledge. It is a cautionary tale, a parable of what happens when the world changes, when tradition is challenged.

Until relatively recently in human history, pi was the much sought-after solution to what was long called the “rectification” or “quadrature” of the circle, fancy words more easily symbolized by the diagram in this slide. How can you transform that circle into the overlaid square? One side of the square would be one-quarter of pi if the diameter of the circle is 1.

Pi was a coveted number for thousands of years, imbued with magical properties. Generations of scholars pursued it doggedly, often considering it the be-all and end-all of geometry.

This is a different pi—pi as we moderns know it:

Well, not all of it, as I’m sure you know. It’s just the first 200 or so digits. The number stretches on forever. I hope you weren’t expecting me to reveal the actual last digit of pi. Because there isn’t one. Strange, no?

Pi wasn’t always this strange. The ancient Egyptians knew better, pegging the ratio of the circumference to the diameter of a circle at 4 over 3 to the 4th power. That’s considerably more definite, and thus much more sensible.

Archimedes knew better, homing in on the value of pi between a couple of very close fractions.

If you are a biblical literalist, pi would seem to be 3, since the Bible clearly describes 30 cubits as encompassing a circle of 10 cubit diameter.

And the solutions kept coming. From ancient mathematicians and philosophers, to medieval scholars, to the Renaissance and the Enlightenment. Everyone seemed capable of finding—with enough effort—the exact value for pi. Squaring the circle was an effort of genius in an ancient science perfectly described centuries ago by Euclid.

But something changed radically in the eighteenth century, just after that book on the right by Joubert de la Rue. A few mathematicians started to take more seriously the nagging feeling that pi didn’t have a perfect solution as a magical fraction. It might not have a last digit after all. This critical number at the center of mathematics might, in fact, be irrational. One mathematician began to reconceptualize pi.

And there he is: the dapper Swiss German mathematician Johann Heinrich Lambert:

He was the son of a tailor, obviously, and was mostly self-taught in mathematics. His brilliant work in the 1760s showed that π/4 could not be a rational number—you could never exactly figure out the value of one side of that square—and thus that pi too was irrational. After Lambert, math textbooks declared the matter solved.

That’s right, problem solved…

Except….circle-squaring kept on going. The world of mathematics had changed with the discoveries of the eighteenth century but somehow the message didn’t get through to many people. John Parker, on the left, came up with my personal favorite solution: pi is precisely 20612/6561. Some circle-squarers, like James Smith on the right, mocked Lambert’s proof as the work of a dilettante.

Things then got testy between the new mathematicians and those who clung to the prior vision of pi. The record of this warfare is as informative as it is humorous. In the 1860s and 70s, James Smith took on Augustus De Morgan, a math professor in London, in a series of short pamphlets, which were the Victorian equivalent of Twitter.

But unsurprisingly, the castigations of professors of mathematics didn’t stop the circle-squarers. Their solutions kept on coming, even in the face of criticism, even after pi had been shown to be transcendental, meaning it couldn’t even be the root of some other number or equation. My favorite book from the turn of the twentieth century had this subtitle on the cover: “The great problem which has baffled the greatest philosophers and the brightest minds of ancient and modern times has now been solved by a humble American citizen of the city of Brooklyn.”

Now, it’s easy to laugh at these misguided circle squarers, especially when they’re from Brooklyn. But if you read circle-squarers seriously, and stop to think about it, they are not so different from you or me. Even in our knowing times, we all persist in doing things that others have long since abandoned as absurd or passé.

History tells us that people are, alas, not very good at seeing the new, and instead are very good at maintaining the past at all costs. This is particularly true in education: Euclid’s Elements, written over 2,000 years ago, was still a standard math textbook well into the 19th century, despite major mathematical advances.

So it’s worth pausing to think about the last digit of pi. Why did so many continue to pursue pi as it was traditionally conceived, and why did they resist the new math?

Think for a moment about the distinction between the old and the new pi. The old was perfect, simple, ordered, divine; the new, seemingly imprecise, prosaic, chaotic, human. So the story of pi is the story, and the psychology, of what happens when the complex and new tries to overtake the simple and traditional.

It’s happening all around us in the digital age. We’re replacing what has been perceived as perfect and ordered with the seemingly imprecise and chaotic.

Look at what has happened, for instance, in the last decade with Wikipedia and the angst about the fate of the traditional Encyclopedia.

Or newspapers in the face of new forms of journalism, such as blogging. A former baseball statistician, Nate Silver of, can brazenly decide to analyze elections and economy better than most newspapers? Yes indeed.

Now this audience, hip to the right side of these screens, may want to be as mean as Augustus De Morgan to those still on the left. We may want to leave modern circle-squarers behind, and undoubtedly some of them will be left behind. But for the majority who are unsettled and are caught between the old and the new, we need other methods to convince them and to change the status quo. History tells us it’s not enough to say that people are blind to the future. We have to show precisely what the weaknesses of the old are…

…and we have to show how the new works better than the old.

Knowing pi correctly to the 10th digit is enormously helpful when accurately predicting the movements of heavenly bodies; try using James Smith’s 3 1/8 when tracing the arc of a planet or moon. For some physics, knowing pi accurately to the 40th digit is critical.

Moreover, this modern pi may be strange, but its very strangeness opened up new avenues of research and thought that were just as intellectually challenging and rewarding as squaring the circle. The transcendental nature of pi led mathematicians to ponder infinite sequences of fractions and had an impact on chaos theory. In computer science, coming up with algorithms to reach a billion or trillion digits of pi as quickly as possible advanced the field. And, if you still want an unsolved problem to crack, see if you can figure out if pi is what is called a “normal number,” where the distribution of the digits 0-9 is uniform…

…or is there instead a preponderance of eights. Now that’s a tough problem, related to real issues in modern math. So there are still problems to be solved, more advanced problems. Math didn’t end with the end of the old pi—it just moved in new, more interesting directions.

But to get to that point, mathematicians had to show in a comprehensible way how the new pi created a new order.

Academic Theater (Reflections on TED & TEDxNYED)

This past weekend’s TEDxNYED event in New York took place in the theater of a school just off Broadway. I couldn’t help thinking about the symbolism of that location during the day’s proceedings. TEDx, a spinoff regional program of the billionaires-and-brains edutainment summit in California, TED, pushes speakers like me towards theatrics.

TEDxNYED was enjoyable and I greatly appreciated the opportunity to rub elbows with some digital luminaries and some very smart educators who are doing all the hard work in the trenches while I sit here in the ivory tower blogging. Whatever criticisms may be leveled, TEDxNYED was incredibly well-run and engaging. Before you read my thoughts below, you should first read the wrap-up from Dave Bill, the TEDxNYED “curator,” who gets it exactly right. I’m enormously appreciative of Dave’s hard work and the hard work of his TEDxNYED colleagues.

Back to Broadway: among other things, TEDxNYED gave me a chance to think more about the academic lecture as theater. (It also gave me a welcome chance to summon the vaudevillian genes of my New York Jewish heritage, the effectiveness of which you will be able to assess when the video is posted to the TEDx channel on YouTube in a couple of weeks.)

Take Larry Lessig, the de facto headliner of TEDxNYED. He’s clearly a first-rate legal scholar and influential activist. But after viewing him live, I realized more than ever that he’s also a rather talented performance artist, with crack comedic timing. (Here’s his talk; judge for yourself.)

We professors don’t like to admit it, but comedy and performance are important ingredients in most successful academic lectures, and can spur the pursuit of knowledge and action far better than serious monograph or article. When I was in college nearly everyone interested in history—from any era or place—took Stephen Cohen’s class on Soviet history, mostly because he was entertaining. He even had one lecture consisting entirely of jokes. Sure, it was gimmicky. But I also know several of my classmates who went into careers in diplomacy and history because of the inspiration.

Of course, academic theater can also lead to problems. TED talks are limited to 18 minutes, inevitably leading to reductionism. As I quipped in my talk on the 6,000 year history of π, “Portions have been condensed.” The humanities particularly suffer from this condensation. For instance, as hugely entertaining as Lessig’s talk was, if you watch it I’m sure you’ll pick up that it conflates, quite problematically, two kinds of conservatism: religious conservatism and libertarianism. Just because the Cato Institute can imagine a role for remixes doesn’t mean that those who attend free church potlucks can. Modern conservatism is an extraordinarily complex mix; one need only look at the tension between libertarian and evangelical views of homosexuality. Gina Bianchini, the CEO of Ning, a network of social networks, presented her work as “the joy of connecting optimists from around the world,” leaving out the fact that the history of Ning is far more interesting: it started out as an engine for making web apps, only later turning toward social networking. That’s actually a fascinating, complex business history that I would have liked to hear more about.

TED’s tagline is the catchy “Ideas Worth Spreading.” I’m an intellectual historian and appreciate the emphasis on ideas; as an educator I’m in favor of spreading knowledge. But in my later years I’ve also come to realize that while ideas are important, execution is probably more important. Lessig and Bianchini also know this—Lessig is now working on methods of more effective lobbying and Bianchini is obviously a talented CEO—and it would have helped TEDxNYED if they had explained to the audience the nitty-gritty details of making real change and progress. It doesn’t come from clever sound bites.

The TED spotlight-on-the-stage format also encourages the audience to perceive the speakers as isolated geniuses, coming out to impart wisdom. The host who introduced me credited me as being the solitary creator of several projects and works, all of which were actually broad collaborations. Again, collaboration is more complex than the format allows. Jeff Jarvis decided to blow up the format by getting up on stage with the lights on and ranting about the insanities and inanities of modern education. This was effective in a Lenny Bruce sort of way, but like Bruce, it was the exception that proved the rule that we speakers were bound to a certain form of academic theater. Inspired by Jarvis, I broke the fourth wall and interacted with the audience a couple of times during my talk, but it was perhaps a little superficial.

Regardless of these criticisms—which I give, again, entirely in recognition of the success of the event and with an eye toward improvement for next year—I enjoyed the challenge of doing a TED talk. I’m working on a much more formal Big Lecture at Cambridge University, and TEDxNYED helpfully made me think about the problems with that format as well. Indeed, I’m not blaming TED for the problems of academic theater. I actually believe the fault lies with academics themselves, who have ceded the ground of public intellectualism in the past generation or two, leaving a vacuum that TED and TEDx are happy to fill.

Hopefully—and judging by the tweets and blog posts this is true—the attendees took away more of the advantages than the disadvantages of the format, and will go on from thought to action.

[photo credit: Kevin Jarrett]

WordCamp Ed: Conference on WordPress for Education

From CHNM‘s Dave Lester, one of founders of THATCamp: The Humanities and Technology Camp, comes WordCamp Ed:

WordCamp conferences are taking the blogging community by storm as one-day events to meet fellow WordPress users in regional communities. WordCamp Ed has been organized to specifically focus on WordPress and Education. The day-long event to take place November 22, 2008, and will bring together a wide-range of institutions of higher-ed, professors, high school teachers, and students.

WordCamp Ed will be hosted at the Center for History and New Media at George Mason University, and is co-sponsored by the Center for New Designs in Learning and Scholarship at Georgetown University.

Dave and many others, including CHNM’s Jeremy Boggs, have been hacking and creating plugins for the open-source WordPress blogging platform for some time now. This seems like a great opportunity to see what others are doing and to exchange knowledge and ideas.

Digital Campus #20 – Open to Change


10 Most Popular Philosophy Syllabi














“Legal Cheating” in the Wall Street Journal

In a forthcoming article in the Chronicle of Higher Education, Roy Rosenzweig and I argue that the ubiquity of the Internet in students’ lives and advances in digital information retrieval threaten to erode multiple-choice testing, and much of standardized testing in general. A revealing article in this weekend’s Wall Street Journal shows that some schools are already ahead of the curve: “In a wireless age where kids can access the Internet’s vast store of information from their cellphones and PDAs, schools have been wrestling with how to stem the tide of high-tech cheating. Now some educators say they have the answer: Change the rules and make it legal. In doing so, they’re permitting all kinds of behavior that had been considered off-limits just a few years ago.” So which anything-goes schools are permitting this behavior, and what exactly are they doing?

The surprise is that it is actually occurring in the more rigorous and elite public and private schools, and they are allowing students to bring Internet-enabled devices into the exam room. Moreover, they are backed not by liberal education professors but by institutions such as the Bill and Melinda Gates Foundation and pragmatic observers of the information economy. As the WSJ (as well as Roy and I) point out, their argument parallels that of the introduction of calculators into mathematics education in the 1980s, eventually leading to the inclusion of these formerly taboo devices on the SATs in 1994, a move that few have since criticized. Today, if one of the main tools workers use in a digital age is the Internet, why not include it in test-taking? After all, asserts M.I.T. economist Frank Levy, it’s more important to locate and piece together information about the World Bank than to know when it was founded. “This is the way the world works,” Harvard Director of Admissions Marlyn McGrath commonsensically notes.

Of course, the bigger question, only partially addressed by the WSJ article, is how the use of these devices will change instruction in fields such as history. From elementary through high school, such instruction has often been filled with the rote memorization of dates and facts, which are easily testable (and rapidly graded) on multiple-choice forms. But we should remember that the multiple-choice test is only a century old; there have been, and there will surely be again, more instructive ways to teach and test such rich disciplines as history, literature, and philosophy.