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Peter Kronfeld

Is Algebra Necessary? - NYTimes.com - 0 views

www.nytimes.com/...is-algebra-necessary.html

education math

shared by Peter Kronfeld on 07 Aug 12 - No Cached
  • Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.”
  • It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.
  • Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”
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  • I hope that mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and music — even poetry — along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet.
  • Peter Kronfeld on 07 Aug 12
     
    A better question than "is Algebra necessary?" would be "how can we make it more relevant and compelling to students?"
Peter Kronfeld

The Singular Mind of Terry Tao - The New York Times - 0 views

www.nytimes.com/...ingular-mind-of-terry-tao.html

math learning

shared by Peter Kronfeld on 29 Jul 15 - No Cached
  • his view of mathematics has utterly changed since childhood.
  • But it turned out that the work of real mathematicians bears little resemblance to the manipulations and memorization of the math student.
  • he ancient art of mathematics, Tao has discovered, does not reward speed so much as patience, cunning and, perhaps most surprising of all, the sort of gift for collaboration and improvisation that characterizes the best jazz musicians.
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  • n class, he conveys a sense that mathematics is fun.
  • at 8 years old, Tao scored a 760 on the math portion of the SAT — but Stanley urged the couple to keep taking things slow and give their son’s emotional and social skills time to develop.
  • Tao became notorious for his nights haunting the graduate computer room to play the historical-­simulation game Civilization. (He now avoids computer games, he told me, because of what he calls a ‘‘completist streak’’ that makes it hard to stop playing.) At a local comic-book store, Tao met a circle of friends who played ‘‘Magic: The Gathering,’’ the intricate fantasy card game. This was Tao’s first real experience hanging out with people his age, but there was also an element, he admitted, of escaping the pressures of Princeton
  • Gifted children often avoid challenges at which they might not excel.
  • At Princeton, crisis came in the form of the ‘‘generals,’’ a wide-­ranging, arduous oral examination administered by three professors. While other students spent months working through problem sets and giving one another mock exams, Tao settled on his usual test-prep strategy: last-­minute cramming. ‘‘I went in and very quickly got out of my depth,’’ he said. ‘‘They were asking questions which I had no ability to answer.’’
  • The true work of the mathematician is not experienced until the later parts of graduate school, when the student is challenged to create knowledge in the form of a novel proof.
  • As a group, the people drawn to mathematics tend to value certainty and logic and a neatness of outcome, so this game becomes a special kind of torture. And yet this is what any ­would-be mathematician must summon the courage to face down: weeks, months, years on a problem that may or may not even be possible to unlock.
  • Ask mathematicians about their experience of the craft, and most will talk about an intense feeling of intellectual camaraderie. ‘‘A very central part of any mathematician’s life is this sense of connection to other minds, alive today and going back to Pythagoras,’
  • ‘Terry is what a great 21st-­century mathematician looks like,’’ Jordan Ellenberg, a mathematician at the University of Wisconsin, Madison, who has collaborated with Tao, told me. He is ‘‘part of a network, always communicating, always connecting what he is doing with what other people are doing.’’
  • Early encounters with math can be misleading. The subject seems to be about learning rules — how and when to apply ancient tricks to arrive at an answer. Four cookies remain in the cookie jar; the ball moves at 12.5 feet per second. Really, though, to be a mathematician is to experiment. Mathematical research is a fundamentally creative act.
  • Peter Kronfeld on 29 Jul 15
     
    Great insight into how math is learned, and how it should be taught
Peter Kronfeld

5-Year-Olds Can Learn Calculus - Luba Vangelova - The Atlantic - 0 views

www.theatlantic.com/...284124

education math learning play

shared by Peter Kronfeld on 13 Mar 14 - No Cached
  • But this progression actually “has nothing to do with how people think, how children grow and learn, or how mathematics is built,” says pioneering math educator and curriculum designer Maria Droujkova.
  • The current sequence is merely an entrenched historical accident that strips much of the fun out of what she describes as the “playful universe” of mathematics
  • “Calculations kids are forced to do are often so developmentally inappropriate, the experience amounts to torture,” she says. They also miss the essential point—that mathematics is fundamentally about patterns and structures, rather than “little manipulations of numbers,” as she puts it.
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  • Studies [e.g.,  this one, and many others referenced in this symposium] have shown that games or free play are efficient ways for children to learn, and they enjoy them.
  • start by creating rich and social mathematical experiences that are complex (allowing them to be taken in many different directions) yet easy (making them conducive to immediate play). Activities that fall into this quadrant: building a house with LEGO blocks, doing origami or snowflake cut-outs, or using a pretend “function box” that transforms objects (and can also be used in combination with a second machine to compose functions, or backwards to invert a function, and so on).
  • What is learned without play is qualitatively different. It helps with test taking and mundane exercises, but it does nothing for logical thinking and problem solving.
Peter Kronfeld

Sizing Up Consciousness by Its Bits - NYTimes.com - 0 views

www.nytimes.com/...21consciousness.html

consciousness informationtheory mathematics

shared by Peter Kronfeld on 21 Sep 10 - No Cached
  • Dr. Tononi’s theory is, potentially, very different. He and his colleagues are translating the poetry of our conscious experiences into the precise language of mathematics. To do so, they are adapting information theory, a branch of science originally applied to computers and telecommunications.
  • Dr. Tononi began to think of consciousness in a different way, as a particularly rich form of information. He took his inspiration from the American engineer Claude Shannon, who built a scientific theory of information in the mid-1900s. Mr. Shannon measured information in a signal by how much uncertainty it reduced.
  • Dr. Tononi and his colleagues have been expanding traditional information theory in order to analyze integrated information. It is possible, they have shown, to calculate how much integrated information there is in a network. Dr. Tononi has dubbed this quantity phi, and he has studied it in simple networks made up of just a few interconnected parts. How the parts of a network are wired together has a big effect on phi. If a network is made up of isolated parts, phi is low, because the parts cannot share information. But simply linking all the parts in every possible way does not raise phi much. “It’s either all on, or all off,” Dr. Tononi said. In effect, the network becomes one giant photodiode. Networks gain the highest phi possible if their parts are organized into separate clusters, which are then joined. “What you need are specialists who talk to each other, so they can behave as a whole,” Dr. Tononi said. He does not think it is a coincidence that the brain’s organization obeys this phi-raising principle.
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  • It is impossible, for example, to calculate phi for the human brain because its billions of neurons and trillions of connections can be arranged in so many ways. Dr. Koch and Dr. Tononi recently started a collaboration to determine phi for a much more modest nervous system, that of a worm known as Caenorhabditis elegans. Despite the fact that it has only 302 neurons in its entire body, Dr. Koch and Dr. Tononi will be able make only a rough approximation of phi, rather than a precise calculation. “The lifetime of the universe isn’t long enough for that
  • Peter Kronfeld on 21 Sep 10
     
    Measuring consciousness with mathematical concept of information theory
Peter Kronfeld

World's Subways Converging on Ideal Form | Wired Science | Wired.com - 0 views

www.wired.com/...subway-convergence

math subways urbanism

shared by Peter Kronfeld on 16 May 12 - No Cached
  • After decades of urban evolution, the world’s major subway systems appear to be converging on an ideal form. On the surface, these core-and-branch systems — evident in New York City, Tokyo, London or most any large metropolitan subway — may seem intuitively optimal. But in the absence of top-down central planning, their movement over decades toward a common mathematical space may hint at universal principles of human self-organization. Understand those principles, and one might “make urbanism a quantitative science, and understand with data and numbers the construction of a city,” said statistical physicist Marc Barthelemy of France’s National Center for Scientific Research.
  • On the surface, these core-and-branch systems — evident in New York City, Tokyo, London or most any large metropolitan subway — may seem intuitively optimal. But in the absence of top-down central planning, their movement over decades toward a common mathematical space may hint at universal principles of human self-organization.
  • With equations used to study two-dimensional spatial networks, the class of network to which subways belong, the researchers turned stations and lines to a mathematics of nodes and branches. They repeated their analyses with data from each decade of a subway system’s history, and looked for underlying trends. Patterns emerged: The core-and-branch topology, of course, and patterns more fine-grained. Roughly half the stations in any subway will be found on its outer branches rather than the core. The distance from a city’s center to its farthest terminus station is twice the diameter of the subway system’s core. This happens again and again.
  • Peter Kronfeld on 16 May 12
     
    Studying subway systems throughout the world leads to insights about urban evolution
Peter Kronfeld

Rubik's Cube Enjoys Another Turn in the Spotlight - NYTimes.com - 0 views

www.nytimes.com/...her-turn-in-the-spotlight.html

math Rubik'sCube

shared by Peter Kronfeld on 07 Aug 12 - No Cached
  • In the 38 years since the Hungarian architecture professor Erno Rubik invented his cube, it has alternately been regarded as an object of fun, art, mathematics, nostalgia and frustration
  • “You can use Rubik’s Cube to teach engineering, you can use it to teach mathematics, and you can use it to talk about the interplay between design and engineering and mathematics and creativity,”
Peter Kronfeld

Apple engineer re-creates ancient computer with Legos | Technically Incorrect - CNET News - 0 views

news.cnet.com/8301-17852_3-20025264-71.html

lego computer mathematics

shared by Peter Kronfeld on 12 Dec 10 - No Cached
  • Peter Kronfeld on 12 Dec 10
     
    Mathematical calculations performed by a Lego construction!? And based on a 2000 year old computer! Pretty cool. Wonder if he sells the construction plans.
Peter Kronfeld

Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It - Wir... - 0 views

www.wired.com/...all

math data netflix

shared by Peter Kronfeld on 10 Oct 13 - No Cached
  • This approach can even be useful for applications that are not, strictly speaking, compressed sensing problems, such as the Netflix prize.
    • Peter Kronfeld
      Peter Kronfeld on 10 Oct 13
       
      Took 2006 - 2009 to accomplish, by an "international team of statisticians, machine learning experts and computer engineers"
  • Given the enormous popularity of Netflix, even an incremental improvement in the predictive algorithm results in a substantial boost to the company’s bottom line. Recht found that he could accurately predict which movies customers might be interested in purchasing, provided he saw enough products per person. Between 25 and 100 products were sufficient to complete the matrix.
  • Across every discipline, data sets are getting bigger and more complex, whether one is dealing with medical records, genomic sequencing, neural networks in the brain, astrophysics, historical archives, or social networks. Alessandro Vespignani, a physicist at Northeastern University who specializes in harnessing the power of social networking to model disease outbreaks, stock market behavior, collective social dynamics, and election outcomes, has collected many terabytes of data from social networks such as Twitter, nearly all of it raw and unstructured. “We didn’t define the conditions of the experiments, so we don’t know what we are capturing,” he said.
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  • It wasn’t the size of the data set that was daunting; by big data standards, the size was quite manageable. It was the sheer complexity and lack of formal structure that posed a problem.
  • calculus lets you take a lot of simple models and integrate them into one big picture.” Similarly, Coifman believes that modern mathematics — notably geometry — can help identify the underlying global structure of big datasets.
  • The key to the technique’s success is a concept known as sparsity, which usually denotes an image’s complexity, or lack thereof. It’s a mathematical version of Occam’s razor: While there may be millions of possible reconstructions for a fuzzy, ill-defined image, the simplest (sparsest) version is probably the best fit. Out of this serendipitous discovery, compressed sensing was born.
  • Using compressed sensing algorithms, it is possible to sample only 100,000 of, say, 1 million pixels in an image, and still be able to reconstruct it in full resolution — provided the key elements of sparsity and grouping (or “holistic measurements”) are present. It is useful any time one encounters a large dataset in which a significant fraction of the data is missing or incomplete.
Peter Kronfeld

Big Bang to Little Swoosh - NYTimes.com - 0 views

www.nytimes.com/...big-bang-to-little-swoosh.html

math physics bigbang

shared by Peter Kronfeld on 11 Apr 14 - No Cached
  • By discovering hidden mathematical patterns and regularities in nature that we call equations of physics, we have gotten progressively better at predicting things — from tomorrow’s weather to tomorrow’s technology. The planet Neptune, the radio wave and the Higgs boson were all predicted mathematically before they were observed.
Peter Kronfeld

Shooting for the Sun - Magazine - The Atlantic - 0 views

www.theatlantic.com/...8268

science solarenergy mathematics african-american

shared by Peter Kronfeld on 23 Nov 10 - No Cached
  • JTEC was only a set of mathematical equations and the beginnings of a prototype, but Johnson had made the tantalizing claim that his device would be able to turn solar heat into electricity with twice the efficiency of a photovoltaic cell
  • Peter Kronfeld on 23 Nov 10
     
    Might interest students that don't find math relevant or engaging. What teenager doesn't like a SuperSoaker?
Peter Kronfeld

Vi Hart's Videos Bend and Stretch Math to Inspire - NYTimes.com - 0 views

www.nytimes.com/...18prof.html

math art

shared by Peter Kronfeld on 19 Jan 11 - No Cached
  • Then, in November, she posted on YouTube a video about doodling in math class, which married a distaste for the way math is taught in school with an exuberant exploration of math as art .
  • At first glance, Ms. Hart’s fascination with mathematics might seem odd and unexpected. She graduated with a degree in music, and she never took a math course in college.
  • The ensuing attention has come with job offers and an income. In one week in December, she earned $300 off the advertising revenue that YouTube shares with video creators. She is also happy that, unlike in her early efforts, which drew an audience typical of mathematics research — older and male, mostly — the biggest demographic for her new videos, at least among registered users, are teenage girls.
  • Peter Kronfeld on 19 Jan 11
     
    Great argument for math's relationship to art, against math as mere calculation drudgery. Check out the links to engaging YouTube videos.
Peter Kronfeld

Will Africa Produce the 'Next Einstein'? | WIRED - 0 views

www.wired.com/...l-africa-produce-next-einstein

math Africa teaching

shared by Peter Kronfeld on 02 Jan 15 - No Cached
  • There are three formal AIMS undertakings: a master’s degree program in Mathematical Sciences, research, and teacher training. The master’s program offers free tuition to accepted students and trains them in both general principles – problem formulation, the scientific method, communication – and cutting-edge math in subjects including computer science, biomathematics, and financial mathematics. Research will allow for international collaborations and advanced student training.
    • Peter Kronfeld
      Peter Kronfeld on 02 Jan 15
       
      Brilliant: applied math (CompSci, bio, financial) and 3 keys: problem formulation, the scientific method, and communication
  • Traditionally, classrooms were led by an authoritative teacher who disseminated information to silent students, but Zomahoun hopes to turn that paradigm on its head. “We train people who can challenge the status quo,” he explains, “not just people who learn from books, listen to lectures, and just repeat it.” Rather, he hopes to instill qualities like “critical thinking, independent thinking, and problem solving” in order to prepare students for real-world problems.
Peter Kronfeld

Haresh Lalvani's SEED54 - A Sculpture With a Twist - NYTimes.com - 0 views

www.nytimes.com/...-a-sculpture-with-a-twist.html

math art sculpture

shared by Peter Kronfeld on 07 Feb 13 - No Cached
  • “I’m interested in seeing what design principles nature uses,” Dr. Lalvani said. “Math, perhaps; maybe physics, whatever. The whole D’Arcy Thompson-type stuff.” The biologist D’Arcy Wentworth Thompson’s book “On Growth and Form,” published in the early 20th century, was a seminal work on the subject of patterns in nature. Thompson, a Scotsman, argued that growth and the structures that resulted were governed by physical principles and could often be described in mathematical terms. He saw examples throughout nature — in the spiral shell of a nautilus, the branching veins on an insect wing and the scales of a pine cone, to name just a few.
Peter Kronfeld

New math model could help preserve species - 0 views

www.jpost.com/...Article.aspx

math model biodiversity

shared by Peter Kronfeld on 14 Sep 10 - No Cached
  • Instead of relying solely on empirical studies as the basis for habitat conservation, Omri Allouche, a student at the Department of Evolution, Systematics and Ecology at the Hebrew University, has developed, under the supervision of Prof. Ronen Kadmon, a predictive mathematical model.
  • Peter Kronfeld on 14 Sep 10
     
    New math model overturns assumptions of models based only on empirical evidence.
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