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On Media Coverage of Math
Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
"The news should start with mathematics, then poetry, and move down from there," from The Humans, by Matt Haig.
See also: The AMS Blog on Math Blogs: Two mathematicians tour the mathematical blogosphere. Editors Evelyn Lamb and Brie Finegold, both PhD mathematicians, blog on blogs that have posts related to mathematics research, applied mathematics, mathematicians, math in the news, mathematics education, math and the arts, and more. Recent posts: "Highly Unlikely Triangles and Other Beaded Mathematics" and "Blogging in Math History Class."
In this article, Toby Walsh, Research Group Leader in the Optimization Research Group at NICTA, a research center in Australia, explains why the wildly popular game, Candy Crush, belongs to the class of NP problems. Walsh begins by introducing the reader to the different classes of problems, as well as the question of whether P = NP. Then, noting that the problem of finding a solution to a logical formula belongs to the class NP, Walsh shows how a logic puzzle can be reduced to a Candy Crush problem by "building" an electric circuit in a Candy Crush game, with candies representing wires, switches, and logic gates. "Expressed in terms of these electrical logic circuits," Walsh writes, "the puzzle in playing Candy Crush is deciding which switches to set so that the logic gates fire appropriately and the output bit is set to true." Walsh then describes how to do the reverse, i.e., reducing a Candy Crush game to satisfying a logical formula.
See "Candy Crush's Puzzling Mathematics," by Toby Walsh. American Scientist, November-December 2014, pages 430-433.
--- Claudia Clark
The Chronicle of Higher Education published a special report on black men in science, technology, engineering, and mathematics (STEM). In an article about four black men in STEM, two are in the mathematical sciences. Read the inspiring stories of Karl Walker, assistant professor of math and computer science, University of Arkansas at Pine Bluff, and Ryan Charles Hynd, assistant professor of mathematics, University of Pennsylvania.
--- Allyn Jackson
And now a free app that allows people to aim their smart phones at a math problem and not only will the app give them the answer, but also it will show the steps to arrive at that answer! The company that created the app, MicroBlink, claims that its goal is not to allow students to "cheat," but rather to help students who don't have access to tutors or can't get individualized instruction. The app was written about by many and performed pretty well in a promotional video but, at least according to the review "Can you really rely on an app to do your maths homework?", point-and-solve technology hasn't arrived quite yet.
See "This app will help kids cheat on math tests," by Sonali Kohli. Quartz, 21 October 2014.
--- Mike Breen
Photos (left to right) National Museum of Mathematics Co-Executive Director Glen Whitney explains an exhibit to trustee Art Steinmetz; event organizers Whitney, Co-Executive Director Cindy Lawrence, and Chief of Design Tim Nissen; Steven Strogatz (L) and Alan Alda (R) discuss the butterfly effect; courtesy of the National Museum of Mathematics.
Quantitative hedge fund owners may be notoriously secretive about strategy and earnings, but the high-powered, high-earning brains recently came together for a lavish--and, according to this article's title, geeky--affair: the Chaos Ball, to support the National Museum of Mathematics (MoMath) in Manhattan. Ball guests had the opportunity to create a fractal made of lights attached to a blackboard, play with 3-D printed geometric trinkets, and be mesmerized by metronomes and square-wheeled tricycles. Where other charity balls include performances by rock stars, the Chaos Ball had actor Alan Alda talking about chaos theory with mathematician Steven Strogatz. Where other balls have pop-culture celebrities, the Chaos Ball had a string theorist from Columbia University and the CEO of Wolfram Research. The Ball raised $830,000 for MoMath.
See "Billionaires and Mathematicians Crack Jokes at the Geekiest Event of the Season," by Bradley Hope. The Wall Street Journal, 19 October 2014.
--- Lisa DeKeukelaere
"If you are planning to post a present this Christmas, it is advisable to be good at maths, or at least go shopping with a tape measure and a pair of weighing scales," this article says. The Royal Mail has changed its prices for various types of parcels depending on their dimensions and weight. A 16-page booklet was needed to explain the changes. In Britain one can also send mail through private carriers, but their price configurations can be just as complicated. Adding to the confusion, some carriers specify volume instead of dimensions. One, MyHermes, uses something called "volumetric area." Packages sent must be under 225cm of volumetric area. The article quotes a MyHermes spokesperson as saying: "To work out the volumetric area, if you add the two shortest dimensions of the parcel and multiply them by two, add the length, the total calculation needs to be under 225cm." Instead of taxing one's brain with such arcana, the article recommends using a web site that does calculations and price comparisons automatically.
See "Why you need to be a maths genius to post a parcel," by Brian Milligan. BBC Business News, 10 October 2014.
--- Allyn Jackson
What makes the sequence 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1,… so cool? In his blog for The Guardian, Alex Bellos explains that along with its crazy mathematical properties, it also has the distinction of being the second entry in the Online Encyclopedia of Integer Sequences (OEIS). From the beloved Fibonacci sequence, to the more obscure Kolakoski sequence, the OEIS is a database of hundreds of thousands of integer sequences. It's a tremendous technical tool for mathematics researchers, but also a cool resource for the casually number-curious.
The OIES was created by Neil Sloane (left) when he was a graduate student at Cornell University in the 1960s. He was working with one particularly obscure sequence of integers, and it occurred to him that it would be handy to have a record of every integer sequence in the world. It started as a stack of 3 x 5 index cards on his desk, after a few decades became a book with 5,000 sequences, and eventually in 1996 a website with 10,000 sequences. Since then, the website has started crowdsourcing à la Wikipedia, and it now gathers about 15,000 new sequences each year.
The OEIS was honored at a conference at the Center for Discrete Mathematics & Theoretical Computer Science (DIMACS) at Rutgers University recently, coinciding with the encyclopedia's 50th anniversary, and founder Neil Sloane's 75th birthday--a twofold celebration! Recently, the OEIS and the work of Sloane also got a nod in Wordplay, The New York Times's blog on crossword puzzles.
But wait, I still haven't told the mathematical properties that make that sequence so cool. You can see that 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1,… is kind of boring, just 1's and 2's, so the numbers themselves aren't all that remarkable. But notice that they always appear in runs of 1 or 2. So if we count the numbers of 1's and 2's and make a sequence out of that, we get 1,2,2,1,1,2,1,2,2,1,… --the original sequence! Pretty neat. There is only one other sequence that does this, and you get it by just removing the leading 1 from the sequence above.
See "Neil Sloane: the man who loved only integer sequences," by Alex Bello. Alex's Adventures in Numberland--The Guardian, 7 October 2014.
--- Anna Haensch (Posted 10/20/14)
In this article, writer Jessica Lahey describes her conversations with Cornell University mathematics professor Steve Strogatz about a course for liberal arts majors that he is teaching. For Strogatz, the key to turning around many of these students' typically negative attitudes toward mathematics is to change the way the subject is taught. To that end, he is using the DAoM curriculum--Discovering the Art of Mathematics: Mathematical Inquiry in the Liberal Arts--developed at Westfield State College by Dr. Julian Fleron and three colleagues. "The DAoM approach," explains Lahey, "is rooted in inquiry-based learning: It focuses on student-led investigations into problems, experiments, and prompts…[It] aims to intellectually stimulate students, to provide cognitive gains, and get students engaged with math rather than passively listening to a teacher." After the first week of class, Strogatz reported to Lahey that it was going well: he described how students "shouted him down" when he tried to give them a hint that would help them solve a puzzle they had already worked on for 30 minutes. "They were having a true mathematical moment," Strogatz said. "That is, they were deeply engaged with a puzzle that made sense to them, and they were enjoying the struggle…Over the weekend I started to get emails from some of them, expressing the excitement they felt when they solved it." (Photo courtesy of Steven Strogatz.)
Strogatz explains the class activity in the photo: This was taken during an activity using dance to make group theory--the math of symmetry--come alive. Specifically, the students are exploring what happens when you combine two transformations to make a third. In the activity shown here, the student on the far right strikes a pose. The student in the middle transforms that pose by (in this case) rotating it 180 degrees about a vertical axis. The student on the left then applies a different transformation (in this case, a mirror reflection in a vertical plane between the left and middle student).
The question they're exploring is: how is the pose of the student on the left related to that of the student on the right? More abstractly, what symmetric transformation do you get by combining a 180 degree rotation with a mirror reflection? As they discovered, the answer turns out to be a "glide reflection"--a reflection through a mirror plane down the midline of the left student's body, along with a translation (a glide) down the line they are all standing on. These kinds of investigations are described on pp.10-11 of the free book, Discovering the Art of Mathematics: Dance.
--- Claudia Clark
Demographers use a variety of means to study the structures of populations--human, animal, and otherwise--and their evolution through birth, death, migration, and aging. Like mathematicians in every discipline, mathematical demographers model their objects of study--in this case populations--and derive relationships that make the work of their experimental collaborators easier, or even feasible. Carey's equality is a relationship describing the age structure of a so-called stationary population, one in which birth and death rates are equal. In a stationary population, one could say the number of individuals of age zero is the same as the number of individuals whose remaining lifespan is zero. Carey's equality, remarkably, suggests that this is true for all ages: the number of individuals who will die in a given time span is the same as the number of individuals who have already lived exactly that long. This equality was first put forward by entomologist James Carey in 2004 while he was searching for a way to estimate the age distributions of wild populations of Mediterranean fruit flies. The existing techniques for estimating the ages of so-called medflies--such as examining captured individuals for mechanical, chemical, or genetic markers of aging--were proving woefully inadequate. Carey began investigating how the lifespan distribution of a captive population might be used to estimate the age structure of the corresponding wild population. A simple life table model suggested that the lifespan distribution was not just an estimate of the age distribution--it was identical. The result turned out to be true for continuous and nonstationary populations ("Demographic window to aging in the wild: constructing life tables and estimating survival functions from marked individuals of unknown age," by Hans-Georg Müller, Jane-Ling Wang, James R. Carey, Edward P. Caswell-Chen, Carl Chen, Nikos Papadopoulos and Fang Yao, Aging Cell, Volume 3, Issue 3, June 2004).
Carey's equality got its name and its (paragraph-long) proof in a 2009 paper by mathematical demographer James Vaupel ("Life lived and life left: Carey's inequality," by James Vaupel, Demographic Research, January 2009). Recently, a new proof--and a new insight--came to light, when Carey met mathematical modeler and applied mathematician Arni Rao at Ohio State University's Mathematical Biosciences Institute. As Carey described his result and illustrated it with several graphical examples, Rao saw another equality emerge before his eyes: for a stationary population of individuals captured at a random point in their lifespans, the distribution of pre-capture and post-capture lifespans would be identical. This, combined with the set theory Rao had already used to study how the aging dynamics of subpopulations contribute to stability within a population ("Population stability and momentum," Arni S.R. Srinivasa Rao, Notices of the AMS, October 2014), gave rise to a new proof of Carey's equality, to appear in the Journal of Mathematical Biology. "Understanding age structure in these insect populations is a huge deal worldwide," said Dr. Carey in a recent Science Codex post. "It's the older mosquitoes that vector the West Nile fever, malaria, yellow fever, and so forth."
(Image: When individuals from a population are captured at a random time during their lifespan (top, inset), the ordered distribution of the times they spend in captivity matches the distribution of their ages (top), as well as the ordered distribution of the times they spend out of captivity (bottom). Image courtesy James R Carey.)
See "New theorem determines the age distribution of populations from fruit flies to humans," Science Codex, 6 October 2014.
--- Ben Pittman-Polletta (posted 10/28/14)
Mathematician George Lusztig (left), an MIT professor who recently visited Hong Kong to receive the Shaw Prize for his work weaving together geometry and algebra, explains that he does math for the beauty of it, rather than for the prospect of real-world applications. He notes that understanding the beauty requires years of study, which is why many people never see it. Recalling that he spent hours each day solving math problems during his childhood in Romania, Lusztig says that doing math had the benefits of minimizing his exposure to politics and allowing him to be judged on his merits. He opines that success in mathematics requires a "good mind" and good luck, and he notes that he has a single-minded focus when trying to solve problems. The annual Shaw Prize honors recipients with $1 million for achievements in the categories of astronomy, medicine and life sciences, and mathematics.
See "When problems equate to happiness," by Raquel Carvalho. South China Morning Post, 30 September 2014. (Photo: Massachusetts Institute of Technology.)
--- Lisa DeKeukelaere
Thomas Bayes was a minister's son who became a mathematician and theologian in the first half of the 18th century. He was elected as a Fellow of the Royal Society, probably on the strength of his book defending the foundations of Newton's calculus, and never published the work for which he is most known today - a specific form of Bayes' rule, shepherded to publication by his friend Richard Price, and generalized by Laplace (Wikipedia: "Bayesian probability"). Bayes' rule says, simply, that the probability of a hypothesis given certain evidence, is proportional to both how likely the evidence is given the hypothesis, and the probability of the hypothesis in the first place. In the past 40 years, this idea, fertilized by the widespread availability of tremendous computing power, has flowered into a statistical approach Bayes might not have recognized, yielding fruit for countless scientific disciplines.
The key to so-called Bayesian statistics is that second term--the probability of the hypothesis of interest. This term plays no role in so-called frequentist statistics, the competing approach to statistical testing which developed in the 18th century and has been widely embraced by science. In frequentist statistical tests, the important thing is the likelihood of the evidence given a certain "null" hypothesis. If the evidence is highly unlikely, the null hypothesis is rejected. In Bayesian statistics, not only the likelihood of the evidence, but also the likelihood of the hypothesis in the absence of the evidence--its so-called prior probability, computed from past evidence or first principles--are taken into account. This allows for more robust calculation of the likelihood of the hypothesis given the current evidence, and for the iterative refinement of that calculation as more experiments are conducted. This Bayesian approach fits with our intuitive reasoning about events and their causes. Say, for instance, that you wake one morning from uneasy dreams to find yourself presented with a photo of a human-sized cockroach occupying your bed, next to a digital alarm clock reading the current date and time. Is it likely that you have transformed into a giant insect? The evidence--a time-stamped photo of a huge cockroach in your bed where you should be--is extremely unlikely if you remain human, suggesting that perhaps you are now, in fact, a horrible vermin. But thanks to the very low probability of such overnight transformations, you may decide that the likelihood that you are still human is quite high, unless you are presented with further evidence of a metamorphosis.
Statisticians like Columbia's Andrew Gelman have argued that a Bayesian approach may be the key to ameliorating a crisis of unreproducible results in fields as varied as oncology ("Drug development: raise standards for preclinical cancer research," by C. Glenn Bagley & Lee M. Ellis), neuroscience ("Power failure: why small sample size undermines the reliability of neuroscience," by Katherine S. Button et al.), and psychology ("False-positive psychology: undisclosed flexibility in data collection and analysis allows presenting anything as significant"). Low statistical power (which makes "statistically significant" results less likely to reflect true effects), a high accepted rate of false negatives (typically one in twenty, which pales in comparison to the vast number of scientific papers being published and the vast number of ways most scientists analyze their data prior to publication), and a bias towards publishing unexpected and counterintuitive findings all play a role in the preponderance of spurious results. The calculation of prior probabilities promotes the utilization of prior research and logic when evaluating hypotheses. For instance, Gelman re-analyzed data suggesting that ovulating single women were more likely to vote for Obama than their non-ovulating peers using a Bayesian framework. When data suggesting that people rarely change their voting preference over an election cycle were incorporated, the significance of the results vanished.
While Bayesian statistics can be misused ("Posterior-hacking: selective reporting invalidates Bayesian results also," by Uri Simonsohn), and properly executed frequentist statistics can solve many of the problems that cause unreproducibility, Bayesian methods clearly have an edge when it comes to guided, iterative hypothesis selection. For instance, astronomers have used Bayesian statistics to narrow down the age of the universe, and to deduce the properties of distant planets, even while the relevant evidence depends on many underlying, unknown parameters. Bayesian analysis also plays a prominent role in the Coast Guard's search and rescue operations ("Wikipedia: Search and Rescue Optimal Planning System"), and many researchers suggest that the brain is essentially an engine for Bayesian inference ("Wikipedia: Bayesian approaches to brain function"). In the absence of evidence, my prior suggests that Thomas Bayes, if he understood, would be thrilled.
(Image: Search and Rescue Optimal Planning System (SAROPS) probability grid.)
See "The Odds, Continually Updated," by F.D. Flam, New York Times, 29 September 2014.
--- Ben Pittman-Polletta (Posted 10/9/14)
There is that moment twice each year when the hearts of all untenured faculty momentary stop: “Student evaluations are now available online.” A recent article in The Chronicle of Higher Education examines what makes these evaluations so heart-stoppingly terrible, and so deeply ineffective.
The dreaded evaluations are typically a combination of Likert scale and free-response questions. The Likert scale responses are tabulated, averaged, compared across the faculty, and then used to decide the high-stakes outcomes of promotion and tenure. Being numbers, the Likert scale responses give a false sense of security, and suggest objectivity, even if they are anything but. It doesn’t take a PhD in statistics to realize that getting a whole bunch of 1’s and whole bunch of 9’s is not the same thing as getting a whole bunch of 5’s. One is not necessarily better or worse, but they are undeniably different, and the current evaluation structure sweeps this difference under the rug.
These sorts of pitfalls are discussed in a research paper by Philip B. Stark, a professor of statistics at Berkeley, and Richard Freishtat, a senior consultant at Berkeley’s Center for Teaching and Learning. The co-authors address the current model of student evaluations as a high-stakes popularity contest from a statistical perspective.
Beyond the fallacy of objectivity in the outcomes, they argue that the questions themselves are often too open-ended. The article notes that one common question asks “was the course valuable?” This is a broad question, that perhaps a freshman student in a large section of calculus doesn’t really have the expertise to answer. A better question, the article argues, would be “Could you hear the instructor during lectures?” or “Did you leave more or less enthusiastic about the subject matter?” These are still subjective questions, but ones that any participating member of the class could answer with authority.
In closing, this particular untenured faculty feels inclined to point out that if the numerical portion is bad, the free-response part is even worse. I have gotten comments ranging from the benign to the bizarre: “Good class,” “Everybody hates this professor,” “This professor is fantastic, she should be given tenure right now,” “I like your clothes, where do you shop?” Point being, they are all over the place and really have nothing to do with the way I plan my syllabus, execute my lectures, evaluate my students, and in short, pass on a knowledge of calculus.
See "Scholars Take Aim at Student Evaluations’ ‘Air of Objectivity’," by Dan Berrett. The Chronicle of Higher Education, 18 September 2014.
--- Anna Haensch (Posted 10/16/14)
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