Friday, December 31, 2010
Wednesday, December 29, 2010
Education (General)
Previous posts :
Brooklyn Robots Teach Math, Science and Engineering at First National Science Expo
Teach For America's 'Highly Qualified' Is Highly Misleading
S. Korean City Pilots Robot English Teachers
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Teach For America's 'Highly Qualified' Is Highly Misleading
Monday, December 27, 2010
The Story of Maths
BBC The Story of Maths Ep1: The Language of the Universe
Series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
After showing how fundamental mathematics is to our lives, du Sautoy explores the mathematics of ancient Egypt, Mesopotamia and Greece.
In Egypt, he uncovers use of a decimal system based on ten fingers of the hand, while in former Mesopotamia he discovers that the way we tell the time today is based on the Babylonian Base 60 number system.
In Greece, he looks at the contributions of some of the giants of mathematics including Plato, Euclid, Archimedes and Pythagoras, who is credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today.
BBC The Story of Maths Ep2: The Genius of the East
Four-part series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
When ancient Greece fell into decline, mathematical progress stagnated as Europe entered the Dark Ages, but in the East mathematics reached new heights.
Du Sautoy visits China and explores how maths helped build imperial China and was at the heart of such amazing feats of engineering as the Great Wall.
In India, he discovers how the symbol for the number zero was invented and Indian mathematicians' understanding of the new concepts of infinity and negative numbers.
In the Middle East, he looks at the invention of the new language of algebra and the spread of Eastern knowledge to the West through mathematicians such as Leonardo Fibonacci, creator of the Fibonacci Sequence.
BBC The Story of Maths Ep3: The Frontiers of Space
Four-part series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
By the 17th century, Europe had taken over from the Middle East as the world's powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics to describe objects in motion.
In this programme, Marcus du Sautoy explores the work of René Descartes and Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He also examines Isaac Newton's development of the calculus, and goes in search of Leonard Euler, the father of topology or 'bendy geometry' and Carl Friedrich Gauss, who, at the age of 24, was responsible for inventing a new way of handling equations: modular arithmetic.
Series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
After showing how fundamental mathematics is to our lives, du Sautoy explores the mathematics of ancient Egypt, Mesopotamia and Greece.
In Egypt, he uncovers use of a decimal system based on ten fingers of the hand, while in former Mesopotamia he discovers that the way we tell the time today is based on the Babylonian Base 60 number system.
In Greece, he looks at the contributions of some of the giants of mathematics including Plato, Euclid, Archimedes and Pythagoras, who is credited with beginning the transformation of mathematics from a tool for counting into the analytical subject we know today.
BBC The Story of Maths Ep2: The Genius of the East
Four-part series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
When ancient Greece fell into decline, mathematical progress stagnated as Europe entered the Dark Ages, but in the East mathematics reached new heights.
Du Sautoy visits China and explores how maths helped build imperial China and was at the heart of such amazing feats of engineering as the Great Wall.
In India, he discovers how the symbol for the number zero was invented and Indian mathematicians' understanding of the new concepts of infinity and negative numbers.
In the Middle East, he looks at the invention of the new language of algebra and the spread of Eastern knowledge to the West through mathematicians such as Leonardo Fibonacci, creator of the Fibonacci Sequence.
BBC The Story of Maths Ep3: The Frontiers of Space
Four-part series about the history of mathematics, presented by Oxford professor Marcus du Sautoy.
By the 17th century, Europe had taken over from the Middle East as the world's powerhouse of mathematical ideas. Great strides had been made in understanding the geometry of objects fixed in time and space. The race was now on to discover the mathematics to describe objects in motion.
In this programme, Marcus du Sautoy explores the work of René Descartes and Pierre Fermat, whose famous Last Theorem would puzzle mathematicians for more than 350 years. He also examines Isaac Newton's development of the calculus, and goes in search of Leonard Euler, the father of topology or 'bendy geometry' and Carl Friedrich Gauss, who, at the age of 24, was responsible for inventing a new way of handling equations: modular arithmetic.
Maths in the news #3
Previous post
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The mathematics of cities
Bethel boy can solve complex math equations in his head
Promoting Maths
Online math program pushing Union students
First in Math lets students participate within class, school, state and nation.
Union County Public Schools is developing a new generation of math thinkers through an online program that offers students an opportunity to practice math basics and advance to more complex problems and computations.
The program, called First in Math, was created by Robert Sun.
http://www.charlotteobserver.com/2010/12/26/1928684/online-math-program-pushing-students.htmlFind x; x + 1 = a^2 and x/2 + 1 = b^2
x + 1 = a^2 and x/2 + 1 = b^2 <=> a^2 - 2b^2 = -1
Compare the different results:
A008844: Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some x.
1, 25, 841, 28561, 970225, 32959081, 1119638521, 38034750625, 1292061882721, 43892069261881, 1491038293021225, 50651409893459761, 1720656898084610641, 58451683124983302025, 1985636569351347658201
Also,
a^2 * b^2 = (x + 1)(x/2 + 1)
A008845: n+1 and n/2+1 are squares.
0, 48, 1680, 57120, 1940448, 65918160, 2239277040, 76069501248, 2584123765440, 87784138523760, 2982076586042448, 101302819786919520, 3441313796169221280, 116903366249966604048, 3971273138702695316400
1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319, 54608393, 318281039, 1855077841, 10812186007, 63018038201, 367296043199, 2140758220993, 12477253282759, 72722761475561, 423859315570607, 2470433131948081
1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, 38613965, 225058681, 1311738121, 7645370045, 44560482149, 259717522849, 1513744654945, 8822750406821, 51422757785981, 299713796309065, 1746860020068409,
Compare the different results:
A008844: Squares of sequence A001653: y^2 such that x^2 - 2*y^2 = -1 for some x.
1, 25, 841, 28561, 970225, 32959081, 1119638521, 38034750625, 1292061882721, 43892069261881, 1491038293021225, 50651409893459761, 1720656898084610641, 58451683124983302025, 1985636569351347658201
Also,
a^2 * b^2 = (x + 1)(x/2 + 1)
A008845: n+1 and n/2+1 are squares.
0, 48, 1680, 57120, 1940448, 65918160, 2239277040, 76069501248, 2584123765440, 87784138523760, 2982076586042448, 101302819786919520, 3441313796169221280, 116903366249966604048, 3971273138702695316400
A002315: NSW numbers: a(n) = 6*a(n-1) - a(n-2); also a(n)^2 - 2*b(n)^2 = -1 with b(n)=A001653(n)
1, 7, 41, 239, 1393, 8119, 47321, 275807, 1607521, 9369319, 54608393, 318281039, 1855077841, 10812186007, 63018038201, 367296043199, 2140758220993, 12477253282759, 72722761475561, 423859315570607, 2470433131948081
1, 5, 29, 169, 985, 5741, 33461, 195025, 1136689, 6625109, 38613965, 225058681, 1311738121, 7645370045, 44560482149, 259717522849, 1513744654945, 8822750406821, 51422757785981, 299713796309065, 1746860020068409,
Sunday, December 26, 2010
Leonard Mlodinow : The Drunkard's Walk
Authors@Google: Leonard Mlodinow
The Drunkard’s Walk - The Mathematics of Randomness” by physicist and one-time Star Trek writer Leonard Mlodinow provides an entertaining and thought-provoking read on probability, randomness and how it affects our day to day lives.
Professor Leonard Mlodinow visits Google's Mountain View, CA headquarters to discuss his book, "The Drunkard's Walk: How Randomness Rules Our Lives". This event took place on May 22, 2008, as part of the Authors@Google series. For more information about Prof. Mlodinow, please visit http://www.its.caltech.edu/~len/
In The Drunkard's Walk, acclaimed writer and scientist Leonard Mlodinow shows us how randomness, change, and probability reveal a tremendous amount about our daily lives, and how we misunderstand the significance of everything from a casual conversation to a major financial setback. As a result, successes and failures in life are often attributed to clear and obvious cases, when in actuality they are more profoundly influenced by chance.
Leonard Mlodinow received his doctorate in physics from the University of California, Berkeley, was an Alexander von Humboldt fellow at the Max Planck Institute, and now teaches about randomness to future scientists at Caltech. Along the way he also wrote for the television series MacGyver and Star Trek: The Next Generation. His previous books include Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace, Feynman's Rainbow: A Search for Beauty in Physics and in Life, and, with Stephen Hawking, A Briefer History of Time. He lives in South Pasadena, California.
Read also my blog:
The Drunkard’s Walk - The Mathematics of Randomness” by physicist and one-time Star Trek writer Leonard Mlodinow provides an entertaining and thought-provoking read on probability, randomness and how it affects our day to day lives.
Professor Leonard Mlodinow visits Google's Mountain View, CA headquarters to discuss his book, "The Drunkard's Walk: How Randomness Rules Our Lives". This event took place on May 22, 2008, as part of the Authors@Google series. For more information about Prof. Mlodinow, please visit http://www.its.caltech.edu/~len/
In The Drunkard's Walk, acclaimed writer and scientist Leonard Mlodinow shows us how randomness, change, and probability reveal a tremendous amount about our daily lives, and how we misunderstand the significance of everything from a casual conversation to a major financial setback. As a result, successes and failures in life are often attributed to clear and obvious cases, when in actuality they are more profoundly influenced by chance.
Leonard Mlodinow received his doctorate in physics from the University of California, Berkeley, was an Alexander von Humboldt fellow at the Max Planck Institute, and now teaches about randomness to future scientists at Caltech. Along the way he also wrote for the television series MacGyver and Star Trek: The Next Generation. His previous books include Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace, Feynman's Rainbow: A Search for Beauty in Physics and in Life, and, with Stephen Hawking, A Briefer History of Time. He lives in South Pasadena, California.
Read also my blog:
Saturday, December 25, 2010
Fermat's Last Theorem: n = 4
The easiest proof for Fermat's Last Theorem is the case n = 4 (a^4 + b^4 = c^2)
See solution at
Fermat's Last Theorem
See solution at
How about the equation a^4 + b^4 + 1 = c^2
Could we still prove that there is no solution in the positive integers for this equation?
Could you provide a counter-example?Fermat's Last Theorem
Pierre de Fermat
Friday, December 24, 2010
Thursday, December 23, 2010
Seinfeld Economics
Although my field is Number theory, I enjoy Game theory.
Jerry lies to his uncle and says he can't go to dinner with him.
“Did I do a bad thing by lying to my uncle and saying I couldn't go to dinner?
Movie Script
A Kenny Rogers Roaster restaurant opens across the street from Kramer. He can't stand the red glare from Kenny's neon sign, and moves into Jerry's apartment. But he becomes hooked on Kenny's chicken, and eventually accepts the red glare in exchange for access to the chicken. When Kenny's shuts down, the lights go out, and Kramer's overall welfare falls—the benefits of the chicken outweighed the cost of the glare.
Here’s a thought experiment: You go to the local pizzeria, order yourself a calzone to go and then leave $1 in the tip jar before you leave. Except something goes horribly wrong: The employee who helped you doesn’t notice that you’ve left him a tip. Should you A) Just be happy knowing that you’ve done a good deed, or B) Tell the employee so he knows you’ve done a good deed?
Most normal people would probably shrug it off and choose the first option, and a select few might choose the latter. But if you’re George Costanza from the popular show Seinfeld, you would actually choose a third option: Stick your hand into the tip jar and try to take out the dollar so you can put it in again while the employee is looking. And in the process you would prove a valuable point: Altruism isn’t really altruism if you’re getting credit for it.
Here are more examples : http://yadayadayadaecon.com/
The Chinese Restaurant (Lying to Uncle)
Jerry lies to his uncle and says he can't go to dinner with him.
“Did I do a bad thing by lying to my uncle and saying I couldn't go to dinner?
Opportunity cost
Movie Script
The Chicken Roaster
Cost-benefit analysis, Externality
A Kenny Rogers Roaster restaurant opens across the street from Kramer. He can't stand the red glare from Kenny's neon sign, and moves into Jerry's apartment. But he becomes hooked on Kenny's chicken, and eventually accepts the red glare in exchange for access to the chicken. When Kenny's shuts down, the lights go out, and Kramer's overall welfare falls—the benefits of the chicken outweighed the cost of the glare.
Here’s a thought experiment: You go to the local pizzeria, order yourself a calzone to go and then leave $1 in the tip jar before you leave. Except something goes horribly wrong: The employee who helped you doesn’t notice that you’ve left him a tip. Should you A) Just be happy knowing that you’ve done a good deed, or B) Tell the employee so he knows you’ve done a good deed?
Most normal people would probably shrug it off and choose the first option, and a select few might choose the latter. But if you’re George Costanza from the popular show Seinfeld, you would actually choose a third option: Stick your hand into the tip jar and try to take out the dollar so you can put it in again while the employee is looking. And in the process you would prove a valuable point: Altruism isn’t really altruism if you’re getting credit for it.
Here are more examples : http://yadayadayadaecon.com/
Two economics professors who recently launched this website that analyzes dozens of Seinfeld episodes for the money lessons that viewers can learn.
Mathematical Infinity and Human Destiny
There are two approaches to mathematical infinity. It can be seen as defining limiting cases that can never be realized or as existing in some philosophical sense. These mathematical approaches parallel approaches to meaning and value that I call absolutist and evolutionary. The absolutist sees ultimate meaning as something that exists most commonly in the form of an all powerful infinite God. The evolutionary sees life and all of a creation as an ever expanding journey with no ultimate or final goal. There is only the journey. There is no destination. This video argues for an evolutionary view in our sense of meaning and values and in our mathematical understanding. There is a deep connection between the two with profound implications for the evolution of consciousness and human destiny. Learn more at
STORY OF 1
The story of the number one is the story of Western civilization. Terry Jones ("Monty Python's Flying Circus") goes on a humor-filled journey to recount the amazing tale behind the world's simplest number. Using computer graphics, "One" is brought to life, in all his various guises, in STORY OF 1
Additive Prime Number Theory
Recent Progress in Additive Prime Number Theory -- 2009 Moursund Lectures, Day 1
Additive prime number theory is the study of additive patterns in the primes.
Terence Tao is surveying some recent advances in this subject, including the results of Goldston, Pintz, and Yildirim on small gaps between primes, the results of Green on arithmetic progressions in the primes, and the results of Bourgain, Gamburd, and Sarnak for detecting almost primes in orbits.
See also: Three topics in additive prime number theory
Additive prime number theory is the study of additive patterns in the primes.
Terence Tao is surveying some recent advances in this subject, including the results of Goldston, Pintz, and Yildirim on small gaps between primes, the results of Green on arithmetic progressions in the primes, and the results of Bourgain, Gamburd, and Sarnak for detecting almost primes in orbits.
Recent progress in additive prime number theory
See also: Three topics in additive prime number theory
Wednesday, December 22, 2010
Science and Mathematics in medieval Arabic-Islamic civilization
A BBC documentary examining the great leap in scientific knowledge that took place in the Islamic world between the 8th and 14th centuries.
Isaac Newton is, as most will agree, the greatest physicist of all time.
At the very least, he is the undisputed father of modern optics, or so we are told at school where our textbooks abound with his famous experiments with lenses and prisms, his study of the nature of light and its reflection, and the refraction and decomposition of light into the colours of the rainbow.
Yet, the truth is rather greyer; and I feel it important to point out that, certainly in the field of optics, Newton himself stood on the shoulders of a giant who lived 700 years earlier.
For, without doubt, another great physicist, who is worthy of ranking up alongside Newton, is a scientist born in AD 965 in what is now Iraq who went by the name of al-Hassan Ibn al-Haytham.
Most people in the West will never have even heard of him.
As a physicist myself, I am quite in awe of this man's contribution to my field, but I was fortunate enough to have recently been given the opportunity to dig a little into his life and work through my recent filming of a three-part BBC Four series on medieval Islamic scientists.
Modern methods
Popular accounts of the history of science typically suggest that no major scientific advances took place in between the ancient Greeks and the European Renaissance.
But just because Western Europe languished in the Dark Ages, does not mean there was stagnation elsewhere. Indeed, the period between the 9th and 13th Centuries marked the Golden Age of Arabic science.
Great advances were made in mathematics, astronomy, medicine, physics, chemistry and philosophy. Among the many geniuses of that period Ibn al-Haytham stands taller than all the others.
First shown on BBC4 05/01/2009
Professor Jim Al-Khalili presents Science and Islam
Isaac Newton is, as most will agree, the greatest physicist of all time.
At the very least, he is the undisputed father of modern optics, or so we are told at school where our textbooks abound with his famous experiments with lenses and prisms, his study of the nature of light and its reflection, and the refraction and decomposition of light into the colours of the rainbow.
Yet, the truth is rather greyer; and I feel it important to point out that, certainly in the field of optics, Newton himself stood on the shoulders of a giant who lived 700 years earlier.
For, without doubt, another great physicist, who is worthy of ranking up alongside Newton, is a scientist born in AD 965 in what is now Iraq who went by the name of al-Hassan Ibn al-Haytham.
Most people in the West will never have even heard of him.
As a physicist myself, I am quite in awe of this man's contribution to my field, but I was fortunate enough to have recently been given the opportunity to dig a little into his life and work through my recent filming of a three-part BBC Four series on medieval Islamic scientists.
Modern methods
Popular accounts of the history of science typically suggest that no major scientific advances took place in between the ancient Greeks and the European Renaissance.
But just because Western Europe languished in the Dark Ages, does not mean there was stagnation elsewhere. Indeed, the period between the 9th and 13th Centuries marked the Golden Age of Arabic science.
Great advances were made in mathematics, astronomy, medicine, physics, chemistry and philosophy. Among the many geniuses of that period Ibn al-Haytham stands taller than all the others.
First shown on BBC4 05/01/2009
Professor Jim Al-Khalili presents Science and Islam
Tuesday, December 21, 2010
Sunday, December 19, 2010
Mathematics: The Most Misunderstood Subject
Find out why ...
WHY DO SO MANY PEOPLE HAVE SUCH MISCONCEPTIONS ABOUT MATHEMATICS?
WHAT IS MATHEMATICS REALLY LIKE?
Article : Why I love mathematics
WHY DO SO MANY PEOPLE HAVE SUCH MISCONCEPTIONS ABOUT MATHEMATICS?
WHAT IS MATHEMATICS REALLY LIKE?
Article : Why I love mathematics
Saturday, December 18, 2010
Of Teaching (in general)
Diana Laufenberg: How to learn? From mistakes
Diana Laufenberg shares 3 surprising things she has learned about teaching -- including a key insight about learning from mistakes.
Diana Laufenberg teaches 11th-grade American History at the Science Leadership Academy in Philadelphia.
http://www.ted.com/speakers/diana_laufenberg.html
Transcript
I have been teaching for a long time, and in doing so have acquired a body of knowledge about kids and learning that I really wish more people would understand about the potential of students. In 1931, my grandmother -- bottom left for you guys over here -- graduated from the eighth grade. She went to school to get the information because that's where the information lived. It was in the books, it was inside the teacher's head, and she needed to go there to get the information, because that's how you learned. Fast-forward a generation: this is the one room schoolhouse, Oak Grove, where my father went to a one room schoolhouse. And he again had to travel to the school to get the information from the teacher, store it in the only portable memory he has, which is inside his own head, and take it with him, because that is how information was being transported from teacher to student and then used in the world. When I was a kid, we had a set of encyclopedias at my house. It was purchased the year I was born, and it was extraordinary, because I did not have to wait to go to the library to get to the information; the information was inside my house and it was awesome. This was different than either generation had experienced before, and it changed the way I interacted with information even at just a small level. But the information was closer to me. I could get access to it.
In the time that passes between when I was a kid in high school and when I started teaching, we really see the advent of the internet. Right about the time the internet gets going as an educational tool, I take off from Wisconsin and move to Kansas, small town Kansas, where I had an opportunity to teach in a lovely, small town rural Kansas school district, where I was teaching my favorite subject, American government. My first year -- super gung ho -- going to teach American government, loved political system. Kids in the 12th grade: not exactly all that enthusiastic about the American government system. Year two: learned a few things -- had to change my tactic. And I put in front of them an authentic experience that allowed them to learn for themselves. I didn't tell them what to do, or how to do it. I posed a problem in front of them, which was to put on an election forum for their own community.
They produced fliers, they called offices, they checked schedules, they were meeting with secretaries, they produced an election forum booklet for the entire town to learn more about their candidates. They invited everyone into the school for an evening of conversation about government and politics and whether or not the streets were done well, and really had this robust experiential learning. The older teachers -- more experienced -- looked at me and went, "Oh, there she is. That's so cute. She's trying to get that done." (Laughter) "She doesn't know what she's in for." But I knew that the kids would show up. And I believed it. And I told them every week what I expected out of them. And that night, all 90 kids -- dressed appropriately, doing their job, owning it. I had to just sit and watch. It was theirs. It was experiential. It was authentic. It meant something to them. And they will step up.
From Kansas, I moved on to lovely Arizona, where I taught in Flagstaff for a number of years, this time with middle school students. Luckily I didn't have to teach them American government. Could teach them the more exciting topic of geography. Again, thrilled to learn. But what was interesting about this position I found myself in in Arizona, was I had this really extraordinarily eclectic group of kids to work with in a truly public school. And we got to have these moments where we would get these opportunities. And one opportunity was we got to go and meet Paul Rusesabagina, which is the gentleman that the movie "Hotel Rwanda" is based after. And he was going to speak at the high school next door to us. We could walk there; we didn't even have to pay for the buses. There was no expense cost. Perfect field trip.
The problem then becomes how do you take seventh- and eighth-graders to a talk about genocide and deal with the subject in a way that is responsible and respectful, and they know what to do with it. And so we chose to look at Paul Rusesabagina as an example of a gentleman who singularly used his life to do something positive. I then challenged the kids to identify someone in their own life, or in their own story, or in their own world, that they could identify that had done a similar thing. I asked them to produce a little movie about it. It's the first time we'd done this. Nobody really knew how to make these little movies on the computer. But they were into it. And I asked them to put their own voice over it. It was the most awesome moment of revelation that when you ask kids to use their own voice and ask them to speak for themselves, what they're willing to share. The last question of the assignment is: how do you plan to use your life to positively impact other people? The things that kids will say when you ask them and take the time to listen is extraordinary.
Fast-forward to Pennsylvania, where I find myself today. I teach at the Science Leadership Academy, which is a partnership school between the Franklin Institute and the school district of Philadelphia. We are a nine through 12 public school, but we do school quite differently. I moved there primarily to be part of a learning environment that validated the way that I knew that kids learned, and that really wanted to investigate what was possible when you are willing to let go of some of the paradigms of the past, of information scarcity when my grandmother was in school and when my father was in school and even when I was in school, and to a moment when we have information surplus. So what do you do when the information is all around you? Why do you have kids come to school if they no longer have to come there to get the information?
In Philadelphia we have a one-to-one laptop program, so the kids are bringing laptops with them everyday, taking them home, getting access to information. And here's the thing that you need to get comfortable with when you've given the tool to acquire information to students, is that you have to be comfortable with this idea of allowing kids to fail as part of the learning process. We deal right now in the educational landscape with an infatuation with the culture of one right answer that can be properly bubbled on the average multiple choice test, and I am here to share with you, it is not learning. That is the absolute wrong thing to ask, to tell kids to never be wrong. To ask them to always have the right answer doesn't allow them to learn. So we did this project, and this is one of the artifacts of the project. I almost never show them off because of the issue of the idea of failure.
My students produced these info-graphics as a result of a unit that we decided to do at the end of the year responding to the oil spill. I asked them to take the examples that we were seeing of the info-graphics that existed in a lot of mass media, and take a look at what were the interesting components of it, and produce one for themselves from a different man-made disaster from American history. And they had certain criteria to do it. They were a little uncomfortable with it, because we'd never done this before, and they didn't know exactly how to do it. They can talk -- they're very smooth, and they can write very, very well, but asking them to communicate ideas in a different way was a little uncomfortable for them. But I gave them the room to just do the thing. Go create. Go figure it out. Let's see what we can do. And the student that persistently turns out the best visual product did not disappoint. This was done in like two or three days. And this is the work of the student that consistently did it.
And when I sat the students down, I said, "Who's got the best one?" And they immediately went, "There it is." Didn't read anything. "There it is." And I said, "Well what makes it great?" And they're like, "Oh, the design's good, and he's using good color. And there's some ... " And they went through all that we processed out loud. And I said, "Go read it." And they're like, "Oh, that one wasn't so awesome." And then we went to another one -- it didn't have great visuals, but it had great information -- and spent an hour talking about the learning process, because it wasn't about whether or not it was perfect, or whether or not it was what I could create; it asked them to create for themselves. And it allowed them to fail, process, learn from. And when we do another round of this in my class this year, they will do better this time. Because learning has to include an amount of failure, because failure is instructional in the process.
There are a million pictures that I could click through here, and had to choose carefully -- this is one of my favorites -- of students learning, of what learning can look like in a landscape where we let go of the idea that kids have to come to school to get the information, but instead, ask them what they can do with it. Ask them really interesting questions. They will not disappoint. Ask them to go to places, to see things for themselves, to actually experience the learning, to play, to inquire. This is one of my favorite photos, because this was taken on Tuesday, when I asked the students to go to the polls. This is Robbie, and this was his first day of voting, and he wanted to share that with everybody and do that. But this is learning too, because we asked them to go out into real spaces.
The main point is that, if we continue to look at education as if it's about coming to school to get the information and not about experiential learning, empowering student voice and embracing failure, we're missing the mark. And everything that everybody is talking about today isn't possible if we keep having an educational system that does not value these qualities, because we won't get there with a standardized test, and we won't get there with a culture of one right answer. We know how to do this better, and it's time to do better.
(Applause)
Diana Laufenberg shares 3 surprising things she has learned about teaching -- including a key insight about learning from mistakes.
Diana Laufenberg teaches 11th-grade American History at the Science Leadership Academy in Philadelphia.
http://www.ted.com/speakers/diana_laufenberg.html
Transcript
I have been teaching for a long time, and in doing so have acquired a body of knowledge about kids and learning that I really wish more people would understand about the potential of students. In 1931, my grandmother -- bottom left for you guys over here -- graduated from the eighth grade. She went to school to get the information because that's where the information lived. It was in the books, it was inside the teacher's head, and she needed to go there to get the information, because that's how you learned. Fast-forward a generation: this is the one room schoolhouse, Oak Grove, where my father went to a one room schoolhouse. And he again had to travel to the school to get the information from the teacher, store it in the only portable memory he has, which is inside his own head, and take it with him, because that is how information was being transported from teacher to student and then used in the world. When I was a kid, we had a set of encyclopedias at my house. It was purchased the year I was born, and it was extraordinary, because I did not have to wait to go to the library to get to the information; the information was inside my house and it was awesome. This was different than either generation had experienced before, and it changed the way I interacted with information even at just a small level. But the information was closer to me. I could get access to it.
In the time that passes between when I was a kid in high school and when I started teaching, we really see the advent of the internet. Right about the time the internet gets going as an educational tool, I take off from Wisconsin and move to Kansas, small town Kansas, where I had an opportunity to teach in a lovely, small town rural Kansas school district, where I was teaching my favorite subject, American government. My first year -- super gung ho -- going to teach American government, loved political system. Kids in the 12th grade: not exactly all that enthusiastic about the American government system. Year two: learned a few things -- had to change my tactic. And I put in front of them an authentic experience that allowed them to learn for themselves. I didn't tell them what to do, or how to do it. I posed a problem in front of them, which was to put on an election forum for their own community.
They produced fliers, they called offices, they checked schedules, they were meeting with secretaries, they produced an election forum booklet for the entire town to learn more about their candidates. They invited everyone into the school for an evening of conversation about government and politics and whether or not the streets were done well, and really had this robust experiential learning. The older teachers -- more experienced -- looked at me and went, "Oh, there she is. That's so cute. She's trying to get that done." (Laughter) "She doesn't know what she's in for." But I knew that the kids would show up. And I believed it. And I told them every week what I expected out of them. And that night, all 90 kids -- dressed appropriately, doing their job, owning it. I had to just sit and watch. It was theirs. It was experiential. It was authentic. It meant something to them. And they will step up.
From Kansas, I moved on to lovely Arizona, where I taught in Flagstaff for a number of years, this time with middle school students. Luckily I didn't have to teach them American government. Could teach them the more exciting topic of geography. Again, thrilled to learn. But what was interesting about this position I found myself in in Arizona, was I had this really extraordinarily eclectic group of kids to work with in a truly public school. And we got to have these moments where we would get these opportunities. And one opportunity was we got to go and meet Paul Rusesabagina, which is the gentleman that the movie "Hotel Rwanda" is based after. And he was going to speak at the high school next door to us. We could walk there; we didn't even have to pay for the buses. There was no expense cost. Perfect field trip.
The problem then becomes how do you take seventh- and eighth-graders to a talk about genocide and deal with the subject in a way that is responsible and respectful, and they know what to do with it. And so we chose to look at Paul Rusesabagina as an example of a gentleman who singularly used his life to do something positive. I then challenged the kids to identify someone in their own life, or in their own story, or in their own world, that they could identify that had done a similar thing. I asked them to produce a little movie about it. It's the first time we'd done this. Nobody really knew how to make these little movies on the computer. But they were into it. And I asked them to put their own voice over it. It was the most awesome moment of revelation that when you ask kids to use their own voice and ask them to speak for themselves, what they're willing to share. The last question of the assignment is: how do you plan to use your life to positively impact other people? The things that kids will say when you ask them and take the time to listen is extraordinary.
Fast-forward to Pennsylvania, where I find myself today. I teach at the Science Leadership Academy, which is a partnership school between the Franklin Institute and the school district of Philadelphia. We are a nine through 12 public school, but we do school quite differently. I moved there primarily to be part of a learning environment that validated the way that I knew that kids learned, and that really wanted to investigate what was possible when you are willing to let go of some of the paradigms of the past, of information scarcity when my grandmother was in school and when my father was in school and even when I was in school, and to a moment when we have information surplus. So what do you do when the information is all around you? Why do you have kids come to school if they no longer have to come there to get the information?
In Philadelphia we have a one-to-one laptop program, so the kids are bringing laptops with them everyday, taking them home, getting access to information. And here's the thing that you need to get comfortable with when you've given the tool to acquire information to students, is that you have to be comfortable with this idea of allowing kids to fail as part of the learning process. We deal right now in the educational landscape with an infatuation with the culture of one right answer that can be properly bubbled on the average multiple choice test, and I am here to share with you, it is not learning. That is the absolute wrong thing to ask, to tell kids to never be wrong. To ask them to always have the right answer doesn't allow them to learn. So we did this project, and this is one of the artifacts of the project. I almost never show them off because of the issue of the idea of failure.
My students produced these info-graphics as a result of a unit that we decided to do at the end of the year responding to the oil spill. I asked them to take the examples that we were seeing of the info-graphics that existed in a lot of mass media, and take a look at what were the interesting components of it, and produce one for themselves from a different man-made disaster from American history. And they had certain criteria to do it. They were a little uncomfortable with it, because we'd never done this before, and they didn't know exactly how to do it. They can talk -- they're very smooth, and they can write very, very well, but asking them to communicate ideas in a different way was a little uncomfortable for them. But I gave them the room to just do the thing. Go create. Go figure it out. Let's see what we can do. And the student that persistently turns out the best visual product did not disappoint. This was done in like two or three days. And this is the work of the student that consistently did it.
And when I sat the students down, I said, "Who's got the best one?" And they immediately went, "There it is." Didn't read anything. "There it is." And I said, "Well what makes it great?" And they're like, "Oh, the design's good, and he's using good color. And there's some ... " And they went through all that we processed out loud. And I said, "Go read it." And they're like, "Oh, that one wasn't so awesome." And then we went to another one -- it didn't have great visuals, but it had great information -- and spent an hour talking about the learning process, because it wasn't about whether or not it was perfect, or whether or not it was what I could create; it asked them to create for themselves. And it allowed them to fail, process, learn from. And when we do another round of this in my class this year, they will do better this time. Because learning has to include an amount of failure, because failure is instructional in the process.
There are a million pictures that I could click through here, and had to choose carefully -- this is one of my favorites -- of students learning, of what learning can look like in a landscape where we let go of the idea that kids have to come to school to get the information, but instead, ask them what they can do with it. Ask them really interesting questions. They will not disappoint. Ask them to go to places, to see things for themselves, to actually experience the learning, to play, to inquire. This is one of my favorite photos, because this was taken on Tuesday, when I asked the students to go to the polls. This is Robbie, and this was his first day of voting, and he wanted to share that with everybody and do that. But this is learning too, because we asked them to go out into real spaces.
The main point is that, if we continue to look at education as if it's about coming to school to get the information and not about experiential learning, empowering student voice and embracing failure, we're missing the mark. And everything that everybody is talking about today isn't possible if we keep having an educational system that does not value these qualities, because we won't get there with a standardized test, and we won't get there with a culture of one right answer. We know how to do this better, and it's time to do better.
(Applause)
Of Teaching Maths
Creativity in Mathematics, pt. 1 of 3, Inquiry-Based Learning (IBL) and the Moore Method
Creativity in Mathematics explores the world of Inquiry-Based Learning and seeks to identify the reasons behind its celebrated success. More than twenty-five influential teachers, top researchers, inventors, and leaders of industry attest to the life changing rewards that began for them in a classroom taught by IBL and the Moore Method. A wealth of resources is available on the web at www.discovery.utexas.edu/rlm or please call 512-469-1700 for more information.
Creativity in Mathematics explores the world of Inquiry-Based Learning and seeks to identify the reasons behind its celebrated success. More than twenty-five influential teachers, top researchers, inventors, and leaders of industry attest to the life changing rewards that began for them in a classroom taught by IBL and the Moore Method. A wealth of resources is available on the web at www.discovery.utexas.edu/rlm or please call 512-469-1700 for more information.
Geometry - interactive proofs with animations
The Menelaus Theorem
Einstein wrote, "Although the first proof is somewhat simpler, it is not satisfying. For it uses an auxiliary line which has nothing to do with the content of the proposition to be proved, and the proof favors, for no reason, the vertex A, although the proposition is symmetrical in relation to A, B, and C. The second proof, however, is symmetrical, and can be read off directly from the figure."
Search for Mathematics Articles
http://www.math.hmc.edu/journals/journalsearch2/
Probability or statistics
Frequently Asked Questions in Mathematics
Probability or statistics
How many ways are there to lace a shoe?
A shoe with two rows of six eyelets offers 43,200 different paths for a shoelace to pass through every eyelet
My Selection of Math Quotes
My quote: Mathematics is everything, the rest is just details!
A good equation should be an economical compression of truth without a symbol out of place.
Dr. Graham Farmelo
Dr. Farmelo looks for attributes like universality, simplicity, inevitability, an elemental power and ''granitic logic'' of the relationships portrayed by those symbols.
It is more important to have beauty in one's equations than to have them fit experiment.
Paul Dirac
Blaise Pascal
"This is often the way it is in physics. Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough. It is always hard to realize that these numbers and equations we play with at our desks have something to do with the real world."
Dr. Steven Weinberg
"The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful."
Henri Poincare
Wednesday, December 15, 2010
A Small Challenge
Pick any 10 numbers between 1 and 100.
Prove it!
There will always be two subsets of these 10 numbers whose sums are equal.
Thus, for example, if you were to choose 51, 11, 81, 68, 73, 87, 23, 29, 25, 94,
you would soon observe that
25 + 51 + 29 = 94 + 11.
The claim is that this works for every 10 numbers you choose.
Likewise, if you were to pick 20 whole numbers between 1 and 50,000,
you would always find two subsets of these 20 numbers whose sums were equal.
Search for semiprimes
32^2 = 1024 ..... 49^2 = 2401 ..... 66^2 = 4356 ..... 83^2 = 6889
33^2 = 1089 ..... 50^2 = 2500 ..... 67^2 = 4489 ..... 84^2 = 7056
34^2 = 1156 ..... 51^2 = 2601 ..... 68^2 = 4624 ..... 85^2 = 7225
35^2 = 1225 ..... 52^2 = 2704 ..... 69^2 = 4761 ..... 86^2 = 7396
36^2 = 1296 ..... 53^2 = 2809 ..... 70^2 = 4900 ..... 87^2 = 7569
37^2 = 1369 ..... 54^2 = 2916 ..... 71^2 = 5041 ..... 88^2 = 7744
38^2 = 1444 ..... 55^2 = 3025 ..... 72^2 = 5184 ..... 89^2 = 7921
39^2 = 1521 ..... 56^2 = 3136 ..... 73^2 = 5329 ..... 90^2 = 8100
40^2 = 1600 ..... 57^2 = 3249 ..... 74^2 = 5476 ..... 91^2 = 8281
41^2 = 1681 ..... 58^2 = 3364 ..... 75^2 = 5625 ..... 92^2 = 8464
42^2 = 1764 ..... 59^2 = 3481 ..... 76^2 = 5776 ..... 93^2 = 8649
43^2 = 1849 ..... 60^2 = 3600 ..... 77^2 = 5929 ..... 94^2 = 8836
44^2 = 1936 ..... 61^2 = 3721 ..... 78^2 = 6084 ..... 95^2 = 9025
45^2 = 2025 ..... 62^2 = 3844 ..... 79^2 = 6241 ..... 96^2 = 9216
46^2 = 2116 ..... 63^2 = 3969 ..... 80^2 = 6400 ..... 97^2 = 9409
47^2 = 2209 ..... 64^2 = 4096 ..... 81^2 = 6561 ..... 98^2 = 9604
48^2 = 2304 ..... 65^2 = 4225 ..... 82^2 = 6724 ..... 99^2 = 9801
I take a number x that is to be squared, then reverse the result.
Then add the number to its reversal.
The result may or may not be a semiprime.
Then I ask, for which number x, I get a result that is a semiprime.
The Mathematics of Christmas
A by-the-numbers guide to Santa, sleigh bells and partridges in a pear tree.
Dec. 18, 2000: Why Pine Trees Are Pointy
Origins of the Christmas tree tradition and the evergreen's evolutionary shape
Dec. 25, 2004: Mapping Santa's Path
Turning Santa's Christmas Eve journey into a mathematical equation.
Dec. 18, 2000: Why Pine Trees Are Pointy
Origins of the Christmas tree tradition and the evergreen's evolutionary shape
Dec. 25, 2004: Mapping Santa's Path
Turning Santa's Christmas Eve journey into a mathematical equation.
Thursday, December 9, 2010
Another kind of Magic Square
We all know this type of Magic Square ....
For normal magic squares of order n = 3, 4, 5, ..., the magic constants are: n(n^2+1)/2
8 1 6
3 5 7
4 9 2
where the numbers are arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number
8 + 1 + 6 = 8 + 3 + 4 = 8 + 5 + 2 = 15
For normal magic squares of order n = 3, 4, 5, ..., the magic constants are: n(n^2+1)/2
Now, let's consider a magic square where the numbers 1 to 9 in a 3x3 array so that the numbers surrounding each number add to a multiple of that number.
2 6 5
7 3 1
9 8 4
Notice that
7 + 3 + 6 = 16 (a multiple of 2)
6 + 3 + 1 = 10 (a multiple of 5)
8 + 3 + 1 = 12 (a multiple of 4)
8 + 3 + 7 = 18 (a multiple of 9)
Similarly in the following squares
2 7 9
6 3 8
5 1 4
4 1 5
8 3 6
9 7 2
4 8 9
1 3 7
5 6 2
5 1 4
6 3 8
2 7 9
5 6 2
1 3 7
4 8 9
9 7 2
8 3 6
4 1 5
9 8 4
7 3 1
2 6 5
9-digit cubes
465^3 = 100544625 .. 474^3 = 106496424 .. 483^3 = 112678587 .. 492^3 = 119095488
466^3 = 101194696 .. 475^3 = 107171875 .. 484^3 = 113379904 .. 493^3 = 119823157
467^3 = 101847563 .. 476^3 = 107850176 .. 485^3 = 114084125 .. 494^3 = 120553784
468^3 = 102503232 .. 477^3 = 108531333 .. 486^3 = 114791256 .. 495^3 = 121287375
469^3 = 103161709 .. 478^3 = 109215352 .. 487^3 = 115501303 .. 496^3 = 122023936
470^3 = 103823000 .. 479^3 = 109902239 .. 488^3 = 116214272 .. 497^3 = 122763473
471^3 = 104487111 .. 480^3 = 110592000 .. 489^3 = 116930169 .. 498^3 = 123505992
472^3 = 105154048 .. 481^3 = 111284641 .. 490^3 = 117649000 .. 499^3 = 124251499
473^3 = 105823817 .. 482^3 = 111980168 .. 491^3 = 118370771 ...............................
500^3 = 125000000 .. 525^3 = 144703125 .. 550^3 = 166375000 .. 575^3 = 190109375
501^3 = 125751501 .. 526^3 = 145531576 .. 551^3 = 167284151 .. 576^3 = 191102976
502^3 = 126506008 .. 527^3 = 146363183 .. 552^3 = 168196608 .. 577^3 = 192100033
503^3 = 127263527 .. 528^3 = 147197952 .. 553^3 = 169112377 .. 578^3 = 193100552
504^3 = 128024064 .. 529^3 = 148035889 .. 554^3 = 170031464 .. 579^3 = 194104539
505^3 = 128787625 .. 530^3 = 148877000 .. 555^3 = 170953875 .. 580^3 = 195112000
506^3 = 129554216 .. 531^3 = 149721291 .. 556^3 = 171879616 .. 581^3 = 196122941
507^3 = 130323843 .. 532^3 = 150568768 .. 557^3 = 172808693 .. 582^3 = 197137368
508^3 = 131096512 .. 533^3 = 151419437 .. 558^3 = 173741112 .. 583^3 = 198155287
509^3 = 131872229 .. 534^3 = 152273304 .. 559^3 = 174676879 .. 584^3 = 199176704
510^3 = 132651000 .. 535^3 = 153130375 .. 560^3 = 175616000 .. 585^3 = 200201625
511^3 = 133432831 .. 536^3 = 153990656 .. 561^3 = 176558481 .. 586^3 = 201230056
512^3 = 134217728 .. 537^3 = 154854153 .. 562^3 = 177504328 .. 587^3 = 202262003
513^3 = 135005697 .. 538^3 = 155720872 .. 563^3 = 178453547 .. 588^3 = 203297472
514^3 = 135796744 .. 539^3 = 156590819 .. 564^3 = 179406144 .. 589^3 = 204336469
515^3 = 136590875 .. 540^3 = 157464000 .. 565^3 = 180362125 .. 590^3 = 205379000
516^3 = 137388096 .. 541^3 = 158340421 .. 566^3 = 181321496 .. 591^3 = 206425071
517^3 = 138188413 .. 542^3 = 159220088 .. 567^3 = 182284263 .. 592^3 = 207474688
518^3 = 138991832 .. 543^3 = 160103007 .. 568^3 = 183250432 .. 593^3 = 208527857
519^3 = 139798359 .. 544^3 = 160989184 .. 569^3 = 184220009 .. 594^3 = 209584584
520^3 = 140608000 .. 545^3 = 161878625 .. 570^3 = 185193000 .. 595^3 = 210644875
521^3 = 141420761 .. 546^3 = 162771336 .. 571^3 = 186169411 .. 596^3 = 211708736
522^3 = 142236648 .. 547^3 = 163667323 .. 572^3 = 187149248 .. 597^3 = 212776173
523^3 = 143055667 .. 548^3 = 164566592 .. 573^3 = 188132517 .. 598^3 = 213847192
524^3 = 143877824 .. 549^3 = 165469149 .. 574^3 = 189119224 .. 599^3 = 214921799
Wednesday, December 8, 2010
Tuesday, December 7, 2010
math professor Edward Frenkel: Multivariable Calculus - Lectures
The true language of love? It's math, says Berkeley professor
Beauty and truth aren't the first things that come to mind, for most people, when they think about math. Berkeley math professor Edward Frenkel is trying to change that.
He tells his classes in multivariable calculus that one of his goals is to unlock the subject's inherent beauty for them, the truth revealed by a mathematical formula.
Multivariable Calculus - Lecture 1
Maths in the news #2
Previous post
Rites of Love and Math
Rites of Love and Math - Trailer
"Rites of Love and Math"
A Film by Reine Graves and Edward Frenkel.
Written and produced by Edward Frenkel and Reine Graves.
Homage to the film "Rite of Love and Death" (a.k.a. "Yukoku" ) by Yukio Mishima.
Director of Photography: Daniel Barrau.
Associate Producer: Sycomore Films.
Supported by Fondation Sciences Mathématiques de Paris.
Cast: Edward Frenkel, Kayshonne Insixieng May.
Math Puzzles’ Oldest Ancestors Took Form on Egyptian Papyrus
Report: U.S. parents overconfident about kid's math
Singapore students among top in reading, math, science
Competitive math: Middle-schoolers quietly duke it out in Federal Way
American Teens Trail Global Peers in Math Scores
But U.S. Students Show Progress in Science
The true language of love? It's math, says Berkeley professor
Report on the film "Rites d'Amour et de Maths" by Reine Graves and Edward Frenkel on the program "En attendant minuit" on the French TV channel TPS Star, October 18, 2010.
Rites of Love and Math
Rites of Love and Math - Trailer
"Rites of Love and Math"
A Film by Reine Graves and Edward Frenkel.
Written and produced by Edward Frenkel and Reine Graves.
Homage to the film "Rite of Love and Death" (a.k.a. "Yukoku" ) by Yukio Mishima.
Director of Photography: Daniel Barrau.
Associate Producer: Sycomore Films.
Supported by Fondation Sciences Mathématiques de Paris.
Cast: Edward Frenkel, Kayshonne Insixieng May.
PISA: U.S. is mediocre in reading, math, science
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