On a math teacher who's running for the U.S. Senate, by Mike Breen
Amanda Curtis is a math and science teacher in Montana who is also the Democratic nominee for the U.S. Senate. A member of the state legislature, which is a part-time job, Curtis got the nomination after the incumbent, John Walsh, pulled out following charges of plagiarism in his master's thesis. Did Curtis always like math? "No, I really struggled in high school. And the first college course I took was a remedial math course. [But at Montana Tech] a professor laid out the beautiful picture that is calculus, and all of a sudden I understood it. It just clicked for me."
See "Newsmakers: Three Q's." Science, 29 August 2014, page 987.
--- Mike Breen
Math Bytes Shows Fun And Tasty New Ways To Teach Math, by Anna Haensch
When Professor Tim Chartier (left) from Davidson College wanted to get an honest opinion on his new math book, his not-so-enthusiastic-about-math sister was the obvious choice. When she not only finished it, but admitted that she actually kinda liked it, he knew he was onto something.
In this latest book, Math Bytes, Chartier explores topics in mathematics from middle school math up to college-level linear algebra using clever hands-on activities, and relatable--sometimes even delicious--tools to get his message across. One activity, which he performed live on WCCB News in Charlotte, uses approximation methods to turn a photograph into a tasty M&M mosaic.
Below is another M&M mosaic that he and his family put together for Make Magazine earlier this year.
Many of the activities, Chartier explains, were developed in a seminar that he taught for public school teachers in Charlotte. So while they are primarily geared towards middle and high school students, they are really adaptive, and can be fun for people at any level. “It has a very broad appeal,” he says, “that doesn't mean that everyone can understand all of it, but I know if this part gets a little more complicated, then you'll catch me on the other side.”
“I want people to have a positive story about math,” he says, “a lot of times people stop at algebra. But it’s like you’re at the buffet of math and only made it to the salad bar. You’ve missed all the other good stuff.”
Chartier has created a companion website to help students take their activities to the next level. For more fun math bits and bytes, follow Chartier on Twitter. @timchartier(Images courtesy of Tim Chartier.)
See "Davidson College Professor Teaches Non-Traditionally With 'Math-Bytes'," by Jennifer Miller, WCCB-TV, 28 August 2014.
--- Anna Haensch (posted 9/8/14)
A Famous Graph Makes an Appearance on a Very Small Stage, by Ben Pittman-Polletta
Imagine that you are an architect in a small, two-dimensional town--either planar or spherical--having only six buildings: three homes, and three utilities--a water plant, a gas plant, and an electric plant. You are trying to connect each home to each of the three utilities, but with a very strict aesthetic: you don't want to connect the homes serially - each home must have its own connection to each utility - and you don't want any of the connections to cross. The task you've set for yourself is the utilities problem, also known as the water, gas, and electricity problem. Go ahead and take a crack at it, I'll wait.
Welcome back. I hope you didn't spend a long time trying to draw those cables and pipes, because connecting the three houses to the three utilities without having a gas line cross a water pipe turns out to be impossible. Viewing the three houses and the three utilities as vertices of a graph, the connections between them form a complete bipartite graph, also known as the utility graph or K3,3. K3,3 is non-planar - that is, there is no embedding of this graph in a two-dimensional space of genus zero ("Why the Complete Bipartite Graph K3,3 is Not Planar", by Rod Hilton from his blog Absolutely No Machete Juggling, 29 October 2011), although it can be embedded in a torus. Not only is K3,3 nonplanar, it is in some sense one of only two nonplanar graphs. According to Kuratowski's theorem, a graph is nonplanar if and only if it contains a subgraph homeomorphic to either K3,3 or K5, the complete graph on 5 vertices.
Now imagine that you are a pregnant woman in the Congo during the '60s, looking for a medicinal tea to help you induce labor. Chances are, you'll reach for a medicinal tea that goes by the name kalata kalata, made from the plant Oldenlandia affinis. The active ingredient of kalata kalata is a peptide, named kalata B1. Kalata B1 is a ring of around 30 to 40 amino acids, interrupted at six places by the amino acid cysteine. The six cysteine residues are connected in pairs by three disulfide bonds. The six links between these cysteine residues--three disulfide bonds, and three chains of amino acids--make kalata B1 a protein incarnation of K3,3, with cysteine residues as vertices. In fact, kalata B1 is only one of a huge family of plant proteins known as cyclotides, all of which share the topology of K3,3. In these proteins, the linked cysteines are a constant, but the sections of amino acids between them are highly variable, containing different functional motifs. The cyclotides all share a remarkable rigidity and stability, thanks not only to their disulfide bonds but also to their peculiar topology, and a high level of resistance to digestion. They have potent insecticidal properties, and are being explored as a backbone for peptide drugs designed for oral administration (see "Cyclotide," Wikipedia.)
Finally, imagine you are polymer chemist Yasuyuki Tezuka. Polymers are macromolecules composed of many repeating subunits. Their behavior in aggregate--they may form materials that are tough, viscous, elastic, or combinations of all three--are dictated by their molecular properties. While many interesting things can be done with linear polymers--molecules made up of chains of subunits--you are interested in the unexplored frontier of cyclic polymers. You want to know how a plastic made of Hopf links or figure eights might behave. So, you develop a process allowing for the creation of molecules with simple but nontrivial topologies--such as a "theta" shape or an unfolded tetrahedron. Now you want to set your sights higher, to create a mathematically interesting as well as potentially useful cyclic polymer. What graph would you look to sculpt out of molecular bonds? As you've certainly guessed, Tezuka and his team set out to synthesize a tiny version of K3,3. They succeeded in part because K3,3 has an exceptionally compact 3D shape, when compared to other topological arrangements, allowing it to be isolated from these other molecules, and perhaps helping it to "achiev[e] exceptionally thermostable bioactivities" ("Constructing a Macromolecular K3,3 Graph through Electrostatic Self-Assembly and Covalent Fixation with a Dendritic Polymer Precursor" by Suzuki, et al.). Tezuka credits his graduate student Takuya Suzuki, the paper's first author, with recognizing the utility of K3,3's compactness. "It's a very nice example of Japanese craftsmanship!" he says. But they aren't finished yet. "There are many other structures that are not easy to make at the nanoscale," he says. The "Konigsberg bridge-graph" appearing in their paper suggests what Tezuka's group might look to build next.
Image: The K3,3 graph, on the far right, has the smallest volume of all configurations shown, making it the fastest molecule in size-exclusion chromatography. Image courtesy of Dr. Yasuyuki Tezuka.
See "Materials scientists, mathematicians benefit from newly crafted polymers." R&D Magazine, 26 August 2014 (from Tokyo Tech News, 19 August 2014).
--- Ben Pittman-Polletta (posted 9/4/14)
On the mathematical landscape, by Lisa DeKeukeleare
Examining Rene Thom's quote that "any mathematical pedagogy… rests on a philosophy of mathematics," columnist Ifran Muzaffar posits that while the quote may hold true for university professors, most K-12 teachers organize their instruction based on a blend of philosophies, rather than standing by a single philosophy to shape their approach to teaching. The article describes three philosophies and how each would be applied in the classroom: 1) mathematics as an objective reality to be discovered , as theorized by G.H. Hardy; 2) mathematics as a set of abstract rules and procedures to be memorized; and 3) mathematics as an iterative process of conjectures to be tested, as popularized by Karl Popper. Muzaffar recounts observing a fourth-grade teacher who adeptly used multiple techniques--without knowing about the underlying philosophical constructs--to meet the multiple demands of school mathematics.
“On the mathematical landscape,” by Irfan Muzaffar. The News on Sunday, 24 August 2014.
--- Lisa DeKeukeleare
On bicycle sharing, by Lisa De Keukelaere
As bike sharing programs spring up in cities around the world, mathematicians are tackling the problem of how to ensure a steady supply of bikes in all stations in spite of user habits that tend to leave some stations empty and others full. Researchers from Vienna to New York are fine-tuning algorithms for efficiently guiding bike-moving trucks to "rebalance" stations by predicting empty and full stations based on the season, the day of the week, and the weather. A computer scientist in Vienna explains that his algorithm resembles those that package delivery services use, and provides updates throughout the day. Bike-transfer truck drivers are still working out the kinks of applying the theories in practice, however, in particular when the algorithm advises picking up less than a full load of bikes or unloading at a station difficult to reach under certain traffic conditions.
See "Wheels when you need them," by Chelsea Wald. Science, 22 August 2014, v. 375 issue 5199, pages 862-863.
--- Lisa DeKeukeleare
Study shows practicing multiplication tables definitely worth it, by Anna Haensch
At some point when we were kids, maybe 8 or 9, we stopped counting on our fingers and answers started to just…sort of appear in our brains. As a recent article in the Detroit Free Press explains, this transition, while easier for some that for others, turns out to be a pretty good predictor of the course of a kid's mathematical life. Youngsters who make this transition easily will likely excel, and those who don't, often face severe difficulty later in life. A recent study funded by the NIH examines what exactly goes on in the grey matter during this transition. (Image courtesy of Jimmie, via Flickr Creative Commons.)
The study was carried out by Professor Vinod Menon and his team at Stanford. Menon put 28 lucky kids into a brain-scanning MRI machine and asked them to solve simple addition problems. First they gave the kids equalities, like 2+5=8, and had them press a button to indicate "'right" or "wrong" (hint: that one's wrong). Next, the kids did the same exercise, but the researched watched them face-to-face, to see if they moved their lips or used their fingers.
Then they did the whole thing again, nearly a year apart. Turns out, kids who relied more on their memory--signified by an active hippocampus--were much faster than the kids who showed heavy activity in their prefrontal and parietal regions, areas associated with counting.
The hippocampus (left, courtesy of Wikimedia Commons) is sort of like a traffic staging area. When new memories pull in, a traffic controller directs them into a more long-term parking spot for later retrieval. But for memories that come in and out often, they get used to the routine. They always go to the same parking spot and eventually don't even need the help of traffic control to get there. So for frequently accessed memories, like 2+5=7, we don't even need to rely on our hippocampus.
What does this mean for children learning simple arithmetic? Practicing multiplication tables, with the end goal of rote memorization, actually helps to shape a kid's brain. And this is particularly helpful in the long run, because kids who work too hard to understand the simple arithmetic, will often feel confused and fall behind as soon as more complicated topics are thrown into the mix.
So bust out those flashcards and fire up that hippocampus. Your future self will thank you.
See: "Brain scans show how kids' math skills grow," by Lauran Neergaard, Detroit Free Press, 19 August 2014.
--- Anna Haensch (posted 8/26/14)
Take my peer review... please, by Ben Pittman-Polletta
Being asked to review a 50-page paper can be a frightening proposition, and debugging someone else's code can be a nightmare. Imagine the horror, then, of being asked to review Thomas Hales's computer-assisted proof of the Kepler conjecture, over 300 pages long and depending on approximately 40,000 lines of custom code ("Mathematical proofs getting harder to verify," by Roxanne Khamsi, New Scientist, 19 February 2006). The reviewers charged with this task by the Annals of Mathematics spent five years vetting the proof. "After a year they came back to me and said that they were 99% sure that the proof was correct," says Hales in the above article. To eliminate this uncertainty, the reviewers continued their evaluation. "After four years they came back to me and said they were still 99% sure that the proof was correct, but this time they said were they exhausted from checking the proof." (Image: Thomas Hales talking about sphere packing at the 2010 Arnold Ross Lecture in Pittsburgh.)
The Kepler conjecture asserts that no sphere packing (i.e., arrangement of spheres in three dimensions) can be denser (i.e., have a larger ratio of sphere to empty space) than the "greengrocer's" or hexagonal lattice packing. Hales' original proof comes in six chapters, and is frankly bewildering. As far as I can tell, it involves finding the minimum value of a function of 150 variables over a set of ~50,000 sphere configurations, each of which represents some neighborhood of a compact topological space, the points of which represent sphere packings ("A Formulation of the Kepler Conjecture," by Thomas C. Hales and Samuel P. Ferguson, a chapter from The Kepler Conjecture, Springer, 2011, available for a fee). With the help of graduate student Samuel Ferguson (who seems to have disappeared from at least the internet after his graduation from the University of Michigan in 2007), Hales spent six years solving around 100,000 linear programming problems to complete his computer-assisted proof.
When Hales was met with the reasonable doubts of his reviewers, he began the FlysPecK Project -- an attempt to provide a formal proof of the Kepler conjecture -- and made the natural choice of computer-assisted peer review for a computer-assisted proof. Flyspeck consists of three parts: a classification of the so-called tame graphs, which "enumerates the combinatorial structures of potential counterexamples to the Kepler conjecture"; a "conjunction of several hundred nonlinear inequalities," which I can only assume are related to the minimization of the function described above, and which were broken into 23,000 pieces and checked in parallel on 32 cores; and a formalization of the proof, combining the above two pieces. The automated proof checkers utilize two "kernels of logic" that have themselves been rigorously checked. "This technology cuts the mathematical referees out of the verification process," says Hales. "Their opinion about the correctness of the proof no longer matters."
Whether the rest of the mathematical community is any more likely to trust automatic proof checkers than computer-assisted proofs -- not to mention the automatic theorem generators that have recently come into existence and gone into business ("Mathematical immortality? Name that theorem," by Jacob Aron, New Scientist, 3 December 2010) -- remains to be seen. In the meantime, we can take comfort in Wikipedia's list of long proofs. While a cursory glance suggests that proofs have gotten longer over the years, a second look suggests that the long proofs of the past have been made vastly shorter by advances in our collective mathematical sophistication. Perhaps the long proofs of today, even those mostly built and checked by computers ("Wikipedia-size maths proof too big for humans to check," by Jacob Aron, New Scientist, 17 February 2014), await only time and the slow accumulation of mathematical insight to be cut down to size. As for Hales, he's no longer holding his breath. "An enormous burden has been lifted from my shoulders," he says. "I suddenly feel ten years younger!"
See "Proof confirmed of 400-year-old fruit-stacking problem," by Jacob Aron. New Scientist, 12 August 2014.
--- Ben Pittman-Polletta (Posted 8/22/14)
Coverage of the 2014 Fields Medals, by Allyn Jackson. Allyn writes about some Fields firsts (below the photos and links).
"Top Math Prize Has Its First Female Winner", by Kenneth Chang. New York Times, 12 August 2014.
"Maryam Mirzakhani Named the First Female Fields Medalist", by Mary Grace Garis. Elle, 12 August 2014.
"Top Mathematics Prize Awarded to a Woman for First Time", by Alex Bellos. Time, 12 August 2014.
"Iranian woman wins maths' top prize, the Fields medal", by Dana Mackenzie. New Scientist, 12 August 2014.
"Fields Medal won by woman for first time", by Núria Radó-Trilla. Times Higher Education, 13 August 2014.
"First female winner for Fields maths medal", by Jonathan Webb. BBC News, 12 August 2014.
"These 4 People Just Won The Most Prestigious Award In Mathematics", by Andy Kiersz. Business Insider, 12 August 2014.
"Fields Medals 2014: prizes for maths work that few of us can grasp", by Alex Bellos. Guardian, 13 August 2014.
"Stanford professor becomes the first woman to win the Fields Medal - the highest honor in mathematics", by Lauren Lumsden. Daily Mail, 13 August 2014.
"After 78 Years, A First: Math Prize Celebrates Work Of A Woman", by Geoff Brumfiel. National Public Radio, 13 August 2014.
Photos: Artur Avila (left) and Manjul Bhargava (right).
"Fields-Medaille an Iranerin Maryam Mirzakhani: Das gab es noch niemals zuvor. Eine Frau hat die höchste Auszeichnung für Mathematik erhalten, die Fields-Medaille (Fields Medal to Maryam Mirzakhani: This has never happened before. A woman has received the top honor in mathematics, the Fields Medal)", by Manfred Lindinger. Frankfurter Allgemeine Zeitung, 13 August 2014.
"Stanford math professor becomes first woman to receive prestigious Fields Medal", by Catherine Garcia. The Week, 13 August 2014.
"Sanskrit, music and mathematics: Manjul Bhargava wins Fields Medal, considered Nobel Prize for maths", by Bibhu Ranjan Mishra. Business Standard, 14 August 2014.
"Curiosity key to learning mathematics", by Chung Hyun-chae. Korea Times, 20 August 2014.
"Kenilworth professor awarded 'Nobel Prize of the maths world'". Kenilworth Weekly News, 20 August 2014.
"Maryam Mirzakhani: the right woman at the right time", by Caroline Series. Times Higher Education, 21 August 2014.
Left: Martin Hairer, right: Maryam Mirzakhani.
Above are links to a sampling of the worldwide coverage of the 2014 Fields Medals, which were presented on August 13 at the International Congress of Mathematicians (ICM) in Seoul. Though often called the "Nobel Prize" of mathematics (there is no Nobel in mathematics), the Fields Medal differs from the Nobel Prize: The medal is given every four years and, instead of honoring a career-long body of work, it is presented to young (under 40 years of age) mathematicians as an encouragement to further achievements. [See a summary of an article about the Fields Medal's label as the "Nobel" of mathematics.]
Since its establishment in 1936, the Fields Medal had never gone to a woman, until this year. Naturally, most of the coverage centered on the first-ever woman Fields Medalist, Maryam Mirzakhani. The article by Caroline Series, a distinguished British mathematician, provides insights on why it took so long for the Fields Medal to be awarded to a woman. "[T]he generation of women born after the Second World War and currently reaching retirement is really the first in which aspiring mathematicians have been able to pursue their chosen career without institutional obstacles in their path," she writes. "Combine this history with the level of concentration that is needed in those precious twenties and thirties---the years in which most of us want to be building a family, the years of juggling the demands of two careers in a discipline that may require relocating anywhere in the world, perhaps with a husband, who may, or may not, consider his wife's career as important as his own. It then becomes a little clearer why it is that women have lacked the support networks, the role models and the contacts that most people need to get to the very top."
Other firsts in this crop of Fields Medals: Mirzakhani is the first Iranian Fields Medalist, Artur Avila the first Brazilian, Manjul Bhargava the first of Indian origin, and Martin Hairer the first Austrian. The International Mathematical Union, which awards the Fields Medals, works hard to nurture and support mathematical development the world over. The Time magazine story quoted IMU President Ingrid Daubechies: "At the IMU we believe that mathematical talent is spread randomly and uniformly over the Earth---it is just opportunity that is not. We hope very much that by making more opportunities available---for women, or people from developing countries---we will see more of them at the very top, not just in the rank and file."
Because Mirzakhani dominated the coverage, the other IMU honors presented at the ICM received less attention: The Nevanlinna Prize went to Subhash Khot, the Gauss Prize went to Stanley Osher, and the first-ever Leelavati Prize went to Adrian Paenza.
Don't miss the outstanding articles on the work of the Fields Medalists that appear in Quanta magazine.
--- Allyn Jackson
On fonts from puzzles, by Claudia Clark
In this article, Rosen tells the story behind a few of the fonts designed by the father-son team of Martin and Erik Demaine, an artist-in-residence and a professor in computer science, respectively, at MIT. Perhaps more well known for their work with geometric folding, the two have applied mathematics and computational geometry to design a number of fonts. The idea for the "conveyor belt" font--imagine letters formed from thumb tacks and elastic bands--occurred during a break the Demaines and a colleague were taking from working on the following question: Can a single 2-D conveyor belt be stretched around a set of wheels such that the belt is taut and touches every wheel without crossing itself? The "glass-squashing" font resulted from their interest in glass blowing: clear disks and blue glass sticks can be arranged in such a way that, when heated and pressed together horizontally, the blue glass sticks form letters. Both are called puzzle fonts because, in one form, the letters are difficult to discern. Visit their website to play with these and other fonts.
See "Father-son mathematicians fold math into fonts," by Meghan Rosen. Science News, 10 August 2014.
--- Claudia Clark
Background on the Fields Medal, by Lisa DeKeukelaere
In preparation for the mid-August announcement of the 2014 Fields Medal winner, this article (published in early August) examines the history of the Medal and the intersection between mathematics and politics. Debunking the myth that Alfred Nobel neglected to create a mathematics prize to spite a Swedish mathematician rival, the article explains that mathematics simply was not important to Nobel, and Canadian mathematician John Charles Fields created the award in 1950 to unite the divided scientific community following World War II. The Medal did not gain widespread recognition--or the "Nobel of mathematics" tag line--until the 1960s, when media outlets championed the award to help Medal recipient Stephen Smale evade censure for alleged anti-Communist activities. Math and politics continue to be intertwined, as mathematicians consider the implications of military funding and working for the NSA, but the author argues that acknowledging this overlap bolsters the meaning and promise of mathematics.
See "How Math Got Its 'Nobel'," by Michael J. Barany. The New York Times, 8 August 2014 and coverage of the 2014 Fields Medals winners, above, and in Tony's Take.
--- Lisa DeKeukelaere
On a Google Doodle saluting John Venn, by Mike Breen
It's perhaps not quite media coverage, but definitely worthy of mention. August 4 was the 180th birthday of mathematician and logician John Venn, of Venn diagram fame. Google saluted him with a very clever animated Doodle, which you can still see in the Doodle archive. The site also has an interview with the Doodle's creators as well as images, such as the one at left, that show their thought process as they developed the Doodle.
--- Mike Breen
Five reviews in The New York Times, by Mike Breen
Each week on a page called "The Shortlist," The New York Times Book Review publishes reviews related to a certain theme. In the August 3 issue, the topic was math. Jennifer Ouelllette writes short reviews of How Not to Be Wrong by Jordan Ellenberg, The Improbability Principle by David J. Hand, The Norm Chronicles by Michael Blastland and David Spiegelhalter, Infinitesimal by Amir Alexander, and The Grapes of Math by Alex Bellos. She likes them all. The Reviews page has links to more reviews of books, as well as reviews of plays and films.
See "The Shortlist: Math," by Jennifer Ouellette. The New York Times Book Review, 3 August 2014, page 30.
--- Mike Breen
Math Digest Archives
|| 2010 || 2009 ||
2008 || 2007 || 2006 || 2005 || 2004 || 2003 || 2002 || 2001 || 2000 || 1999 || 1998 || 1997 || 1996 || 1995
Click here for a list of links to web pages of publications covered in the Digest. |