American teachers are widely assessed on the basis of "value added measures," a statistical tool for analyzing the outcomes of their teaching. But as Jerry Genovese points out, this is statistically completely bankrupt -- unless you randomize your samples, you get no insight into the quality of the teaching. I asked my father, Gord Doctorow -- a mathematician, math teacher, and professor of education -- what he thought of Genovese's piece, and he sent me some great material, which you'll find after the jump.
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If you're the type of person who really needs some good visuals to make a concept stick in your head, this series of YouTube videos made by the British Psychological Society Media Centre will help you remember the meanings behind statistical concepts like "correlation", "frequency distributions", and "sampling error". There are four videos in the series so far, and they do a great job of painting pictures around abstract ideas. Bonus: Soothing music.
Errata Security CEO Rob Graham has published a blog-post speculating that ninety percent of the traffic on the Tor anonymized network can be broken by the NSA. That's because the majority of Tor users are still on the an old version of the software, 2.3, which uses 1024 RSA/DH keys -- and at keylengths of 1024 RSA/DH crypto can be broken in a matter of hours using custom chips fabbed at an estimated cost of $1B. It seems likely that the NSA has spent the necessary sum and sourced these chips (likely from IBM).
This isn't the same as being able to decrypt all of Tor in realtime, but it does suggest that the NSA could selectively decrypt its stored archives of Tor traffic.
However, the new version of Tor, 2.4, uses elliptical curve Diffie-Hellman ciphers, which are probably beyond the NSA's reach.
Graham faults the Tor Project for the poor uptake of its new version, though as an Ars Technica commenter points out, popular GNU/Linux distributions like Debian and its derivative Ubuntu are also to blame, since they only distribute the older, weaker version. In either event, this is a wake-up call that will likely spur both the Tor Project and the major distros to push the update.
Yesterday's revelations about the NSA's ability to decrypt 'secure' communications were taken by many to mean that the NSA had made fundamental mathematical or computing breakthroughs that allowed it to decrypt securely enciphered messages. But it's pretty clear that's not what's going on.
You may have heard speculation that the NSA has secretly broken the strong cryptographic systems used to keep data secret -- after all, why collect all that scrambled data if they can't unscramble it? But Bruce Schneier argues (convincingly) that this is so impossible as to be fanciful. So why have they done this? My guess is that they're counting on flaws being revealed in the cryptographic implementations in the field (or maybe they've discovered such flaws and are keeping them secret). Or they're hoping for a big breakthrough in the future (quantum computing, anyone?).
About 200 million people go to U.S. beaches each year. About 36 of those hundreds of millions are attacked by sharks. Most of them survive. In contrast, more than 30,000 of those millions of beach-goers are to be rescued from surfing accidents. And many of those humans each year die, or must be rescued, from drowning incidents in which no other creature is to blame.
So, will we see Human Week, or Human-nado mockumentaries any time soon?
At VW's request, English court censors Usenix Security presentation on keyless entry systems for luxury cars
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Flavio Garcia, a security researcher from the University of Birmingham has been ordered not to deliver an important paper at the Usenix Security conference by an English court. Garcia, along with colleagues from a Dutch university, had authored a paper showing the security failings of the keyless entry systems used by a variety of luxury cars. Volkswagon asked an English court for an injunction censoring his work -- which demonstrated their incompetence and the risk they'd exposed their customers to -- and Mr Justice Birss agreed.
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We've featured doodling, fast-talking YouTube mathematician Vi Hart a lot here, but her latest, a 30-minute extended mix, is absolutely remarkable, even by her high standards. For 30 glorious minutes, Ms Hart explores the nature of randomness and pattern, using Stravinsky's 12-tone music as a starting-point and rocketing through constellations, the nature of reality, Borges's library, and more. On the way, she ends up with a good working definition of creativity, and explores the dilemma of structure versus creation. Brava, Ms Hart, you have outdone yourself! Plus, I like your copyright jokes.
Want to play a game of Tic-Tac-Toe that's genuinely challenging and hard? Try "Ultimate Tic-Tac-Toe," in which each square is made up of another, smaller Tic-Tac-Toe board, and to win the square you have to win its mini-game. Ben Orlin says he discovered the game on a mathematicians' picnic, and he explains a wrinkle on the rules:
You don’t get to pick which of the nine boards to play on. That’s determined by your opponent’s previous move. Whichever square he picks, that’s the board you must play in next. (And whichever square you pick will determine which board he plays on next.)...
This lends the game a strategic element. You can’t just focus on the little board. You’ve got to consider where your move will send your opponent, and where his next move will send you, and so on.
The resulting scenarios look bizarre. Players seem to move randomly, missing easy two- and three-in-a-rows. But there’s a method to the madness – they’re thinking ahead to future moves, wary of setting up their opponent on prime real estate. It is, in short, vastly more interesting than regular tic-tac-toe.
The correct answer is that Brian and Angela just need to break up, already.
From Thanks, Textbooks — a fantastic Tumblr of supremely weird and hilarious textbook examples and questions.
This fantastic video by Vi Hart shows you what the math of music looks like in a visual representation — or, should that be "what visual frieze patterns sound like when turned into music"?
Frieze patterns are symmetrical repeating patterns that show up in architecture, art, and even our model of DNA. According to Hart, this video is:
A visual and musical expression of mathematical symmetry groups. The transformations done to the video are equivalent to the transformations done to the notes.
Very cool to watch! Here's the video link.
Thanks, Peter Newbury!
Turns out, math fans dig set theory for almost exactly the same reason that some Christian fundamentalists absolutely hate it — all that messy uncertainty, which is either an affront to the idea of intelligent design or really, really sexy and fascinating, depending on your outlook.
At Nautilus, which is currently hosting an entire issue on topic of uncertainty, math professor Ayalur Krishnan writes about an idea in set theory that he calls "The Deepest Uncertainty". This is the Continuum Hypothesis — an idea that, paradoxically, can be proven to be unprovable and proven to be something you can't disprove. (And, with that, I've just typed the word "proven" so many times that it has lost all meaning in my brain.)
The uncertainty surrounding the Continuum Hypothesis is unique and important because it is nested deep within the structure of mathematics itself. This raises profound issues concerning the philosophy of science and the axiomatic method. Mathematics has been shown to be “unreasonably effective” in describing the universe. So it is natural to wonder whether the uncertainties inherent to mathematics translate into inherent uncertainties about the way the universe functions. Is there a fundamental capriciousness to the basic laws of the universe? Is it possible that there are different universes where mathematical facts are rendered differently? Until the Continuum Hypothesis is resolved, one might be tempted to conclude that there are.
Read the full story, which explains what set theory and the Continuum Hypothesis actually are. I could that here, but then this link would end up being as long as the story it's trying to link you to. Ahhhh, set theory.
Here's a mesmerizing gallery of "Fabrege Fractals" created by Tom Beddard, whose site also features a 2011 video of Fabrege-inspired fractal landscapes that must be seen to be believed. They're all made with Fractal Lab, a WebGL-based renderer Beddard created.
Yitang Zhang is a largely unknown mathematician who has struggled to find an academic job after he got his PhD, working at a Subway sandwich shop before getting a gig as a lecturer at the University of New Hampshire. He's just had a paper accepted for publication in Annals of Mathematics, which appears to make a breakthrough towards proving one of mathematics' oldest, most difficult, and most significant conjectures, concerning "twin" prime numbers. According to the Simons Science News article, Zhang is shy, but is a very good, clear writer and lecturer.
For hundreds of years, mathematicians have speculated that there are infinitely many twin prime pairs. In 1849, French mathematician Alphonse de Polignac extended this conjecture to the idea that there should be infinitely many prime pairs for any possible finite gap, not just 2.
Since that time, the intrinsic appeal of these conjectures has given them the status of a mathematical holy grail, even though they have no known applications. But despite many efforts at proving them, mathematicians weren’t able to rule out the possibility that the gaps between primes grow and grow, eventually exceeding any particular bound.
Now Zhang has broken through this barrier. His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. No matter how far you go into the deserts of the truly gargantuan prime numbers — no matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million.
The result is “astounding,” said Daniel Goldston, a number theorist at San Jose State University. “It’s one of those problems you weren’t sure people would ever be able to solve.”
Unknown Mathematician Proves Elusive Property of Prime Numbers [Erica Klarreich/Wired/Simons Science News]
(Photo: University of New Hampshire)