October 2014 - mathfun@mailman.xmission.com

Re: [math-fun] Why Tough Teachers Get Good Results
by Henry Baker 24 Oct '14

24 Oct '14

I find that _writing code_ for a rote procedure helps me understand things. Then applying various kinds of standard computer optimizations to see what happens. The computer is merciless at reminding you not to divide by zero, etc. If you also have access to Maxima/Mathematica/Maple, etc., with good plotting capabilities, then you can learn even more. I think that math education could be vastly improved by incorporating a lot more computer computations into early math. This would replace most of the mindless repetition -- e.g., 4x4 matrices. At 10:45 AM 10/22/2014, Andy Latto wrote: >The thing I disagree with most strongly is the praise for rote >learning. I would have learned so much more mathematics if I wasn't >required to spend hundreds (thousands?) of hours in rote repetition of >things I already knew perfectly well how to do. And not just at the >multiplication table level! What possible benefit was their, in an >honors-level linear algebra class at a good university, of wasting my >time making me do Graham-Shmidt orthogonalization of 4x4 matrices by >hand?

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[math-fun] 3 mass ratios
by Simon Plouffe 24 Oct '14

24 Oct '14

Hello, I was digging (possible) expressions for mass ratios of fundamental particles and found 3 which I find amusing. The first one is for Mn/Mp, the neutron/proton ratio which is 1.00137841917 (45) (codata 2010). the expression is 8/27 ( 5/cos(Pi/15) - sqrt(3)). = 1.0013784197796... The other one is for Mp/Me, proton/electron ratio which is 1836.15267261 (75) 1 5 1 1/5 -------- + 6 Pi + 1/5 -------- = 1836.1526799668615... cosh(Pi) sinh(Pi) or lprinted : 1/5/cosh(Pi)+6*Pi^5+1/5/sinh(Pi); very symmetric , it reminds me of the value that R. Feynmann gave in the 60's of 6*Pi^5 but a bit far from the real value. A simple analysis of the error would reject that expression, it is 9 times the standard error. But, ? , something is missing perhaps, another symmetric term (or 2) ? Another one which does respect the error term would be -((2000 (-12 + Sin[\[Pi]/60]))/(10 + 3 Cos[\[Pi]/60])) = 1838.683664899565... very close to the real value of 1838.6836605(11) also in the CODATA 2010 values. In this case : error term is ok but the complexity, too high ? These are the only ones among millions of expressions found using the table of 13.155 billion entries. NOTE : these are just guesses, they have no pretension of exactness or physical reality, they are simply the best and simplest expressions that satisfies the error term and/or the complexity so far. Unless someone have another educated guess. ? Best regards, Simon Plouffe

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[math-fun] Kerr metric
by Warren D Smith 24 Oct '14

24 Oct '14

As far as I know, an important still-open problem is to find a way to glue the Kerr vacuum metric as the exterior, onto the metric of some realistic hunk of rotating matter as the interior, to get combined metric representing a rotating mass and its gravity field. Seems quite embarrassing that nobody has been able to do that. How hard can it be? (But there are many successful ways to do this in the non-rotating case.)

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[math-fun] Kerr Black Holes
by Henry Baker 23 Oct '14

23 Oct '14

I recently attended a lecture by a Caltech professor on the subject of Black Holes. With a large % of non-technical people in the audience, she was trying to avoid getting extremely technical, but may have confused people even more. One question was "what is the size of a black hole of 2 million solar masses?" I found a web site that performs this exact calculation for _non-spinning_ black holes, and the answer seems to be that the radius of the event horizon is ~0.08 AU -- i.e., 8% of the distance from the Sun to the Earth. The next question is what is the size of the fastest _spinning_ black hole with a mass of 2 million solar masses? I wasn't able to find a calculator for this, but I did find a ppt slide that seemed to indicate that an extreme Kerr black hole had 1/2 the Schwarzchild radius of a non-spinning black hole; i.e., 0.04AU for my 2 million solar mass example. Apparently, this counterintuitive result comes from the fact that a large proportion of the spinning black hole mass is in the form of rotational energy. Now a spinning black hole has an ellipsoidal "ergosphere" that touches the Schwarzchild radius at its poles, but is larger (?) at the equator (?) What exactly is the shape of this ellipsoid for the fastest spinning black hole? To be precise, what is the numerical value of the eccentricity of the ellipse? The "singularity" for a spinning black hole is no longer a point, but a _ring_. What formula gives me the _radius_ of this ring for a maximally spinning Kerr black hole ?

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[math-fun] Kerr Newman black hole, surface area
by Warren D Smith 23 Oct '14

23 Oct '14

there is a simple exact formula for the SURFACE AREA of the horizon of a Kerr-Newman black hole, which if I copied it right like a monkey, is: Area = 4*pi* ( 2*M^2 - Q^2 +- 2*M*sqrt(M^2-J^2-Q^2) ) where +- for the two horizons (it has two), M=mass, Q=charge, J=specific angular momentum, everything in Planck units. Also, EQ 11.1.13 and 14 gives "radii" of these horizons http://www.modernrelativitysite.com/chap11.htm but the meaning of that may not be what you think. Area A is the same in all coordinate systems, and A = 4*pi*(r^2 + J^2) so that these r values also are invariant. Their EQ 11.1.17 also mentions how to incorporate a nonzero cosmical constant as yet another parameter besides M,J,Q. "The area of a 1-solar mass black hole is about the same as the municipal area of Atlanta." -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Hollywood renders GR visible
by Henry Baker 23 Oct '14

23 Oct '14

FYI -- A Hollywood movie that really, really tries hard to get a correct visualization of general relativity... http://www.wired.com/2014/10/astrophysics-interstellar-black-hole/ Wrinkles in Spacetime: The Warped Astrophysics of "Interstellar" by Adam Rogers Kip Thorne looks into the black hole he helped create and thinks, ÂWhy, of course. That's what it would do.Â This particular black hole is a simulation of unprecedented accuracy. It appears to spin at nearly the speed of light, dragging bits of the universe along with it. (That's gravity for you; relativity is superweird.) In theory it was once a star, but instead of fading or exploding, it collapsed like a failed soufflÃ© into a tiny point of inescapable singularity. A glowing ring orbiting the spheroidal maelstrom seems to curve over the top and below the bottom simultaneously. All this is only natural, because weird things happen near black holes. For example, their gravity is so strong that they bend the fabric of the universe. Einstein explained this: The more massive something is, the more gravity it produces. Objects like stars and black holes do this so powerfully that they actually bend light and pull space and time with it. And it gets weirder: If you were closer to a black hole than I was, our perceptions of space and time would diverge. Relatively speaking, time would seem to be going faster for me. What does Thorne see in there? He's an astrophysicist; his math guided the creation of this mesmerizing visual effect, the most accurate simulation ever of what a black hole would look like. It's the product of a year of work by 30 people and thousands of computers. And alongside a small galaxy of Hollywood starsÂ-Matthew McConaughey, Anne Hathaway, Jessica Chastain, Bill Irwin, Casey Affleck, John LithgowÂ-the simulation plays a central role in Interstellar, the prestige space travel epic directed by Christopher Nolan opening November 7. Thorne sees truth. Nolan, the consummate image maker, sees beauty. Black holes, even fictional ones, can warp perception. Thorne Isn't your average astrophysicist. Sure, he's a famous theorist, but even before his retirement from Caltech in 2009 he was deeply interested in explaining the heady ideas of relativity to the general public. Just before his retirement, Thorne and film producer Lynda Obst, whom he'd known since Carl Sagan set them up on a blind date three decades earlier, were playing around with an idea for a movie that would involve the mysterious properties of black holes and wormholes. Before long, Steven Spielberg signed on to direct; screenwriter Jonathan ÂJonahÂ Nolan wrote a script. Eventually Spielberg dropped out; Jonathan's brother ChrisÂ-known for directing mind-bendy movies like Memento and Inception (plus Batman) dropped in. And while Chris Nolan was rewriting his brother's script, he wanted to get a handle on the science at the heart of his story. So he started meeting with Thorne. Over the course of a couple months in early 2013, Thorne and Nolan delved into what the physicist calls Âthe warped side of the universeÂÂ-curved spacetime, holes in the fabric of reality, how gravity bends light. ÂThe story is now essentially all Chris and Jonah's,Â Thorne says. ÂBut the spirit of it, the goal of having a movie in which science is embedded in the fabric from the beginningÂ-and it's great scienceÂ-that was preserved.Â The story the filmmakers came up with is set in a dystopian near future when crops have failed and humanity is on the verge of extinction. A former astronaut (McConaughey) gets recruited for one last flight, a desperate attempt to reach other star systems where humans can once again thrive. And therein lies a problem. See, other stars are really far away. Reaching even the nearest ones would take decades at speeds we humans have no idea how to attain. Back in 1983, when Sagan needed a plausible solution to this problem for the story that would become the movie Contact, Thorne suggested the wormhole, a hypothetical tear in the universe connecting two distant points via dimensions beyond the four we experience as space and time. A wormhole was a natural choice for Interstellar too. As Thorne talked about the movie with Nolan, their discussions about the physical properties of wormholes led to an inevitable question for a filmmaker: How do you actually show one onscreen? That's not the only headache inducing bit of physics that the film's special effects team had to grapple with. Nolan's story relied on time dilation: time passing at different rates for different characters. To make this scientifically plausible, Thorne told him, he'd need a massive black holeÂ-in the movie it's called GargantuaÂ-spinning at nearly the speed of light. As a filmmaker, Nolan had no idea how to make something like that look realistic. But he had an idea how to make it happen. ÂChris called me and said he wanted to send a guy over to my house to talk to me about the visual effects,Â Thorne says. ÂI said, ÂSure, send him over.ÂÂ It wasn't long before Paul Franklin showed up on Thorne's doorstep. Franklin knew that his computers would do anything he told them to. That was a problem and a temptation. ÂIt's very easy to fall into the trap of breaking the rules of reality,Â says Franklin, a senior supervisor of Academy Award-winning effects house Double Negative. ÂAnd those rules are actually quite strict.Â So he asked Thorne to generate equations that would guide their effects software the way physics governs the real world. They started with wormholes. If light around a wormhole wouldn't behave classicallyÂ-that is, travel in a straight lineÂ-what would it do? How could that be described mathematically? Thorne sent his answers to Franklin in the form of heavily researched memos. Pages long, deeply sourced, and covered in equations, they were more like scientific journal articles than anything else. Franklin's team wrote new rendering software based on these equations and spun up a wormhole. The result was extraordinary. It was like a crystal ball reflecting the universe, a spherical hole in spacetime. ÂScience fiction always wants to dress things up, like it's never happy with the ordinary universe,Â he says. ÂWhat we were getting out of the software was compelling straight off.Â Their success with the wormhole emboldened the effects team to try the same approach with the black hole. But black holes, as the name suggests, are murder on light. Filmmakers often use a technique called ray tracing to render light and reflections in images. ÂBut ray-tracing software makes the generally reasonable assumption that light is traveling along straight paths,Â says EugÃ©nie von Tunzelmann, a CG supervisor at Double Negative. This was a whole other kind of physics. ÂWe had to write a completely new renderer,Â she says. Some individual frames took up to 100 hours to render, the computation overtaxed by the bendy bits of distortion caused by an Einsteinian effect called gravitational lensing. In the end the movie brushed up against 800 terabytes of data. ÂI thought we might cross the petabyte threshold on this one,Â von Tunzelmann says. ÂChris really wanted us to sell the idea that the black hole is spherical,Â Franklin says. ÂI said, ÂYou know, it's going to look like a disk.Â The only thing you can see is the way it warps starlight.Â Then Franklin started reading about accretion disks, agglomerations of matter that orbit some black holes. Franklin figured that he could use this ring of orbiting detritus to define the sphere. Von Tunzelmann tried a tricky demo. She generated a flat, multicolored ringÂ-a stand-in for the accretion diskÂ-and positioned it around their spinning black hole. Something very, very weird happened. ÂWe found that warping space around the black hole also warps the accretion disk,Â Franklin says. ÂSo rather than looking like Saturn's rings around a black sphere, the light creates this extraordinary halo.Â That's what led Thorne to his Âwhy, of courseÂ moment when he first saw the final effect. The Double Negative team thought it must be a bug in the renderer. But Thorne realized that they had correctly modeled a phenomenon inherent in the math he'd supplied. Still, no one knew exactly what a black hole would look like until they actually built one. Light, temporarily trapped around the black hole, produced an unexpectedly complex fingerprint pattern near the black hole's shadow. And the glowing accretion disk appeared above the black hole, below the black hole, and in front of it. ÂI never expected that,Â Thorne says. ÂEugÃ©nie just did the simulations and said, ÂHey, this is what I got.Â It was just amazing.Â In the end, Nolan got elegant images that advance the story. Thorne got a movie that teaches a mass audience some real, accurate science. But he also got something he didn't expect: a scientific discovery. ÂThis is our observational data,Â he says of the movie's visualizations. ÂThat's the way nature behaves. Period.Â Thorne says he can get at least two published articles out of it. When Thorne discusses the astrophysics that he likes bestÂ-colliding black holes, space dragged into motion by a whirling star, time warpsÂ-he uses a lot of analogies. He talks about two tornadoes running into each other or rays of light cast about like straw in the wind. But metaphors can be deceptive; they can make people think they understand something when they only understand what it is like. But Thorne's haloed, spinning black hole and galaxy-spanning wormhole are not just metaphors. Most Interstellar viewers will see these imagesÂ-the wormhole, the black hole, the weird lightÂ-and think, ÂWhoa. That's beautiful.Â Thorne looks at them and thinks, ÂWhoa. That's true.Â And from a certain perspective, that's beautiful too.

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Re: [math-fun] Kerr Black Holes
by Henry Baker 23 Oct '14

23 Oct '14

I found this document to be extremely helpful in explaining space-time frame-dragging & other spinning black hole phenomena: http://www.eftaylor.com/pub/SpinNEW.pdf In Newtonian space, the Earth can revolve about the Sun at any inclination to the Sun's equator without "feeling" the Sun's own rotation. In GR, this isn't possible, as the Sun's rotation "drags the space-time frame" about itself. Consider "dropping" a test mass from infinity (_zero angular momentum_ about the Sun) into the Sun. In Newtonian space, the test mass would go all the way to the center of the Sun. In GR, this test mass would be deflected slightly from the center of the Sun by the "frame-dragging" of the Sun's rotation. Here's the cool part: the test mass would _still have zero angular momentum_ in the dragged space-time frame ! Thus, a test mass dropped towards a spinning black hole would end up rotating about the black hole center, but this rotation would be that of the space-time frame itself, as the test mass would still have zero angular momentum! At 09:42 AM 10/23/2014, Fred Lunnon wrote: > See > http://en.wikipedia.org/wiki/Ergosphere ><< The equatorial (maximum) radius of an ergosphere corresponds >to the Schwarzschild radius of a non-rotating black hole; the polar (minimum) >radius can be as little as half the Schwarzschild radius if the black hole is >rotating maximally. >> > > This article later observes gnomically ><< Within the ergosphere, spacetime is dragged along in the direction >of the rotation of the black hole at a speed greater than the local >speed of light in relation to the rest of the universe. >> >I sort of understand what this is getting at; but what precisely does it mean? > >WFL > >On 10/23/14, Henry Baker <hbaker1(a)pipeline.com> wrote: >> I recently attended a lecture by a Caltech professor on the subject of Black >> Holes. >> >> With a large % of non-technical people in the audience, she was trying to >> avoid getting extremely technical, but may have confused people even more. >> >> One question was "what is the size of a black hole of 2 million solar >> masses?" >> >> I found a web site that performs this exact calculation for _non-spinning_ >> black holes, and the answer seems to be that the radius of the event horizon >> is ~0.08 AU -- i.e., 8% of the distance from the Sun to the Earth. >> >> The next question is what is the size of the fastest _spinning_ black hole >> with a mass of 2 million solar masses? >> >> I wasn't able to find a calculator for this, but I did find a ppt slide that >> seemed to indicate that an extreme Kerr black hole had 1/2 the Schwarzchild >> radius of a non-spinning black hole; i.e., 0.04AU for my 2 million solar >> mass example. Apparently, this counterintuitive result comes from the fact >> that a large proportion of the spinning black hole mass is in the form of >> rotational energy. >> >> Now a spinning black hole has an ellipsoidal "ergosphere" that touches the >> Schwarzchild radius at its poles, but is larger (?) at the equator (?) >> >> What exactly is the shape of this ellipsoid for the fastest spinning black >> hole? To be precise, what is the numerical value of the eccentricity of the >> ellipse? >> >> The "singularity" for a spinning black hole is no longer a point, but a >> _ring_. What formula gives me the _radius_ of this ring for a maximally >> spinning Kerr black hole ?

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[math-fun] Fwd: geography geometry
by Bill Gosper 23 Oct '14

23 Oct '14

One of the Stanford Gardner (let's face it) Celebration talks was geography trick questions, several of which are in http://boards.straightdope.com/sdmb/showthread.php?t=233928 (e.g., the border of Colorado is a (spherical) n-gon, where n = ___.) This page also has *------------Paul>Name the three states that have parts of a perfect circle in their borderlines. If you want the answer, you have to tell me how to do a spoilers box.* Gest>Admit it; there *is* no third and you only put it in so someone would tell you how to do a spoiler box! Well it worked. Just enclose the *spoilt* text in vB code spoiler tags as you would any other kind of text. ------------- To an off-list discussion, NeilB added BG>Real, honest to God legally defined circles. NB>...with a radius less than a thousand miles. Otherwise, every latitude line would count as a circular arc whose center is at the North (or South) pole. --Neil (And longitudes are circles concentric with Earth itself. --rwg Well, not really. Is the Earth legally defined as a sphere anywhere?) Neil then ferreted out *five* states with legitimate circular arcs! Colorado spoiler. n=π⁴/ζ(4)/15 . --rwg

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[math-fun] Why Tough Teachers Get Good Results
by Henry Baker 22 Oct '14

22 Oct '14

FYI -- I suspect that many on this list would agree with these conclusions w.r.t. math education. http://online.wsj.com/articles/SB10001424052702304213904579095303368899132 Why Tough Teachers Get Good Results By Joanne Lipman Sept. 27, 2013 7:17 p.m. ET I had a teacher once who called his students "idiots" when they screwed up. He was our orchestra conductor, a fierce Ukrainian immigrant named Jerry Kupchynsky, and when someone played out of tune, he would stop the entire group to yell, "Who eez deaf in first violins!?" He made us rehearse until our fingers almost bled. He corrected our wayward hands and arms by poking at us with a pencil. Today, he'd be fired. But when he died a few years ago, he was celebrated: Forty years' worth of former students and colleagues flew back to my New Jersey hometown from every corner of the country, old instruments in tow, to play a concert in his memory. I was among them, toting my long-neglected viola. When the curtain rose on our concert that day, we had formed a symphony orchestra the size of the New York Philharmonic. I was stunned by the outpouring for the gruff old teacher we knew as Mr. K. But I was equally struck by the success of his former students. Some were musicians, but most had distinguished themselves in other fields, like law, academia and medicine. Research tells us that there is a positive correlation between music education and academic achievement. But that alone didn't explain the belated surge of gratitude for a teacher who basically tortured us through adolescence. We're in the midst of a national wave of self-recrimination over the U.S. education system. Every day there is hand-wringing over our students falling behind the rest of the world. Fifteen-year-olds in the U.S. trail students in 12 other nations in science and 17 in math, bested by their counterparts not just in Asia but in Finland, Estonia and the Netherlands, too. An entire industry of books and consultants has grown up that capitalizes on our collective fear that American education is inadequate and asks what American educators are doing wrong. I would ask a different question. What did Mr. K do right? What can we learn from a teacher whose methods fly in the face of everything we think we know about education today, but who was undeniably effective? As it turns out, quite a lot. Comparing Mr. K's methods with the latest findings in fields from music to math to medicine leads to a single, startling conclusion: It's time to revive old-fashioned education. Not just traditional but old-fashioned in the sense that so many of us knew as kids, with strict discipline and unyielding demands. Because here's the thing: It works. Now I'm not calling for abuse; I'd be the first to complain if a teacher called my kids names. But the latest evidence backs up my modest proposal. Studies have now shown, among other things, the benefits of moderate childhood stress; how praise kills kids' self-esteem; and why grit is a better predictor of success than SAT scores. All of which flies in the face of the kinder, gentler philosophy that has dominated American education over the past few decades. The conventional wisdom holds that teachers are supposed to tease knowledge out of students, rather than pound it into their heads. Projects and collaborative learning are applauded; traditional methods like lecturing and memorizationÂ-derided as "drill and kill"Â-are frowned upon, dismissed as a surefire way to suck young minds dry of creativity and motivation. But the conventional wisdom is wrong. And the following eight principlesÂ-a manifesto if you will, a battle cry inspired by my old teacher and buttressed by new researchÂ-explain why. 1. A little pain is good for you. Psychologist K. Anders Ericsson gained fame for his research showing that true expertise requires about 10,000 hours of practice, a notion popularized by Malcolm Gladwell in his book "Outliers." But an often-overlooked finding from the same study is equally important: True expertise requires teachers who give "constructive, even painful, feedback," as Dr. Ericsson put it in a 2007 Harvard Business Review article. He assessed research on top performers in fields ranging from violin performance to surgery to computer programming to chess. And he found that all of them "deliberately picked unsentimental coaches who would challenge them and drive them to higher levels of performance." 2. Drill, baby, drill. Rote learning, long discredited, is now recognized as one reason that children whose families come from India (where memorization is still prized) are creaming their peers in the National Spelling Bee Championship. This cultural difference also helps to explain why students in China (and Chinese families in the U.S.) are better at math. Meanwhile, American students struggle with complex math problems because, as research makes abundantly clear, they lack fluency in basic addition and subtractionÂ-and few of them were made to memorize their times tables. William Klemm of Texas A&M University argues that the U.S. needs to reverse the bias against memorization. Even the U.S. Department of Education raised alarm bells, chastising American schools in a 2008 report that bemoaned the lack of math fluency (a notion it mentioned no fewer than 17 times). It concluded that schools need to embrace the dreaded "drill and practice." 3. Failure is an option. Kids who understand that failure is a necessary aspect of learning actually perform better. In a 2012 study, 111 French sixth-graders were given anagram problems that were too difficult for them to solve. One group was then told that failure and trying again are part of the learning process. On subsequent tests, those children consistently outperformed their peers. The fear, of course is that failure will traumatize our kids, sapping them of self-esteem. Wrong again. In a 2006 study, a Bowling Green State University graduate student followed 31 Ohio band students who were required to audition for placement and found that even students who placed lowest "did not decrease in their motivation and self-esteem in the long term." The study concluded that educators need "not be as concerned about the negative effects" of picking winners and losers. 4. Strict is better than nice. What makes a teacher successful? To find out, starting in 2005 a team of researchers led by Claremont Graduate University education professor Mary Poplin spent five years observing 31 of the most highly effective teachers (measured by student test scores) in the worst schools of Los Angeles, in neighborhoods like South Central and Watts. Their No. 1 finding: "They were strict," she says. "None of us expected that." The researchers had assumed that the most effective teachers would lead students to knowledge through collaborative learning and discussion. Instead, they found disciplinarians who relied on traditional methods of explicit instruction, like lectures. "The core belief of these teachers was, 'Every student in my room is underperforming based on their potential, and it's my job to do something about itÂ-and I can do something about it,'" says Prof. Poplin. She reported her findings in a lengthy academic paper. But she says that a fourth-grader summarized her conclusions much more succinctly this way: "When I was in first grade and second grade and third grade, when I cried my teachers coddled me. When I got to Mrs. T's room, she told me to suck it up and get to work. I think she's right. I need to work harder." 5. Creativity can be learned. The rap on traditional education is that it kills children's' creativity. But Temple University psychology professor Robert W. Weisberg's research suggests just the opposite. Prof. Weisberg has studied creative geniuses including Thomas Edison, Frank Lloyd Wright and PicassoÂ-and has concluded that there is no such thing as a born genius. Most creative giants work ferociously hard and, through a series of incremental steps, achieve things that appear (to the outside world) like epiphanies and breakthroughs. Prof. Weisberg analyzed Picasso's 1937 masterpiece Guernica, for instance, which was painted after the Spanish city was bombed by the Germans. The painting is considered a fresh and original concept, but Prof. Weisberg found instead that it was closely related to several of Picasso's earlier works and drew upon his study of paintings by Goya and then-prevalent Communist Party imagery. The bottom line, Prof. Weisberg told me, is that creativity goes back in many ways to the basics. "You have to immerse yourself in a discipline before you create in that discipline. It is built on a foundation of learning the discipline, which is what your music teacher was requiring of you." 6. Grit trumps talent. In recent years, University of Pennsylvania psychology professor Angela Duckworth has studied spelling bee champs, Ivy League undergrads and cadets at the U.S. Military Academy in West Point, N.Y.Â-all together, over 2,800 subjects. In all of them, she found that gritÂ-defined as passion and perseverance for long-term goalsÂ-is the best predictor of success. In fact, grit is usually unrelated or even negatively correlated with talent. Prof. Duckworth, who started her career as a public school math teacher and just won a 2013 MacArthur "genius grant," developed a "Grit Scale" that asks people to rate themselves on a dozen statements, like "I finish whatever I begin" and "I become interested in new pursuits every few months." When she applied the scale to incoming West Point cadets, she found that those who scored higher were less likely to drop out of the school's notoriously brutal summer boot camp known as "Beast Barracks." West Point's own measureÂ-an index that includes SAT scores, class rank, leadership and physical aptitudeÂ-wasn't able to predict retention. Prof. Duckworth believes that grit can be taught. One surprisingly simple factor, she says, is optimismÂ-the belief among both teachers and students that they have the ability to change and thus to improve. In a 2009 study of newly minted teachers, she rated each for optimism (as measured by a questionnaire) before the school year began. At the end of the year, the students whose teachers were optimists had made greater academic gains. 7. Praise makes you weakÂ My old teacher Mr. K seldom praised us. His highest compliment was "not bad." It turns out he was onto something. Stanford psychology professor Carol Dweck has found that 10-year-olds praised for being "smart" became less confident. But kids told that they were "hard workers" became more confident and better performers. "The whole point of intelligence praise is to boost confidence and motivation, but both were gone in a flash," wrote Prof. Dweck in a 2007 article in the journal Educational Leadership. "If success meant they were smart, then struggling meant they were not." 8.Âwhile stress makes you strong. A 2011 University at Buffalo study found that a moderate amount of stress in childhood promotes resilience. Psychology professor Mark D. Seery gave healthy undergraduates a stress assessment based on their exposure to 37 different kinds of significant negative events, such as death or illness of a family member. Then he plunged their hands into ice water. The students who had experienced a moderate number of stressful events actually felt less pain than those who had experienced no stress at all. "Having this history of dealing with these negative things leads people to be more likely to have a propensity for general resilience," Prof. Seery told me. "They are better equipped to deal with even mundane, everyday stressors." Prof. Seery's findings build on research by University of Nebraska psychologist Richard Dienstbier, who pioneered the concept of "toughness"Â-the idea that dealing with even routine stresses makes you stronger. How would you define routine stresses? "Mundane things, like having a hardass kind of teacher," Prof. Seery says. My tough old teacher Mr. K could have written the book on any one of these principles. Admittedly, individually, these are forbidding precepts: cold, unyielding, and kind of scary. But collectively, they convey something very different: confidence. At their core is the belief, the faith really, in students' ability to do better. There is something to be said about a teacher who is demanding and tough not because he thinks students will never learn but because he is so absolutely certain that they will. Decades later, Mr. K's former students finally figured it out, too. "He taught us discipline," explained a violinist who went on to become an Ivy League-trained doctor. "Self-motivation," added a tech executive who once played the cello. "Resilience," said a professional cellist. "He taught us how to failÂ-and how to pick ourselves up again." Clearly, Mr. K's methods aren't for everyone. But you can't argue with his results. And that's a lesson we can all learn from. Ms. Lipman is co-author, with Melanie Kupchynsky, of "Strings Attached: One Tough Teacher and the Gift of Great Expectations," to be published by Hyperion on Oct. 1. She is a former deputy managing editor of The Wall Street Journal and former editor-in-chief of CondÃ© Nast Portfolio.

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Re: [math-fun] Someone needs to take a photo of the Riemann hypothesis
by Henry Baker 22 Oct '14

22 Oct '14

My browser complains that this site has a bad certificate, and hence is most likely a source of malware. At 06:59 PM 10/21/2014, Thane Plambeck wrote: >https://photomath.net/ > >-- >Thane Plambeck >tplambeck(a)gmail.com >http://counterwave.com/

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