Mathematics in the Eye of the Beholder

If you’re like me, you might get as excited about the intricate patterns in a museum’s parquet floor as in the art hanging on the wall. I love seeing the world through a mathematical lens and celebrating the patterns built into everyday life. We’ve shared image-based math blogs here in the past: complex functions, math gifs, and beautiful geometric designs. Today I’d like to share some of my favorite math photo blogs.

A parquet floor in the palace at Versailles. I guess the stuff on the tables and walls was nice too. Image: Evelyn Lamb

The Mathematical Tourist is a daily blog of math-adjacent photographs by math writer Ivars Peterson. He shares photos of architecture, art with mathematical themes, and occasional not-so-mathy stuff.

I also enjoy Turismo Matemático, a Spanish (both in language and in country of origin, I believe) math picture blog. They tend to share examples of mathematics in art and architecture, such as the many mathematical details of the Sagrada Familia cathedral in Barcelona and the many paintings containing compasses and astrolabes. When I was living in Paris earlier this year, I was excited to see posts about things I had seen, particularly the climate change equations in the Gare du Nord train station. Turismo Matemático has a short article to accompany each photo, too. It’s in Spanish, so an added bonus is that you can learn a few Spanish words or practice your Spanish while you enjoy some math pictures. (Or take your chances with your browser’s translation if your Spanish isn’t quite up to it.)

The MAA used to maintain a page of Found Math images updated every week. It seems to have gone dormant in 2015, but there’s a lot of good stuff in the archives on their website and on their Flickr page.

World Tessellation Day in June (the 17th, to coincide with M.C. Escher’s birthday) was a fun celebration of tiling patterns in buildings and on sidewalks. I wrote about the holiday, invented by Emily Grosvenor, here last year and on my blog Roots of Unity this year. The Twitter hashtag #worldtessellationday was a lot of fun to watch that day. I learned about a church in Spain that features a marble Penrose tiling (article in Spanish). It’s enough to make you want to plan a trip to Mahón!

Do you like to take mathy pictures? Share them on Twitter or Instagram using the hashtag #mathphoto or #foundmath.

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Math In The Capital

Visiting with Catherine Paolucci at Senator Franken’s office, where this well known friend of science displays his pride in Minnesota higher ed.

A few weeks ago I had occasion to visit the capital — and the Capitol — and as is the custom in Washington D.C., I had coffee with interesting people. We’ve blogged quite a bit here about the expressionless face emoji state of science and truth in our current world, but I was happy to catch up with the people on the ground kicking butt in the name of the AMS.

Karen Saxe, who is the director of the Washington D.C. office of the AMS, keeps us all in the loop with her blog Capital Currents. Saxe began her position in the D.C. office in January, and her role in that office is to be a link between the mathematicians who are part of the AMS and the lawmakers and legislators of Capitol Hill. She blogs about current legislation of interest to mathematicians, most recently about a bill promoting diversity in STEM fields. Also, particularly relevant to anyone who is seeking NSF funding, Saxe has blogged about the congressional budget approval process and what that means for us.

Saxe isn’t the only AMS insider with access to the legislative branch, Catherine Paolucci is this year’s AMS Congressional Fellow. Paolucci, who is also an assistant professor at SUNY New Paltz, is working in Senator Al Franken’s office and is in a unique position to work with the AMS and its members while also advising and informing policy decisions. Together with Saxe, she creates a pipeline for communicating AMS priorities to Congress and the AMS Committee on Science Policy.

As a fellow, Paolucci also helps AMS members organize successful hill visits and initiate successful grassroots efforts. It is important to have mathematicians in the political space, she says “strength and power comes from people on the ground.” Paolucci also stresses “the power of state level advocacy and importance in constituents engaging with their own senators.”

So, you know how people are always on you to call your congresspeople and tell them what you want?

It’s really important that you do that.

One thing that Paolucci has been interested in is math as a tool for social justice. She’s been studying the role that algebra plays in defining life trajectory and career paths, and she sees the importance of supporting after school programs to make sure students have access to this vital tool. She is surprised to find that many folks in D.C. don’t even realize how important math is, and often its a matter of packing mathematical ideas in a way that makes them resonate with people. Of course we know math is important, but her job as a fellow is to make sure the legislators really know what can (and should!) be done. Believe it or not, very few legislators have a background in math or science.

If you’re interested in applying to be a Congressional Fellow, applications are due each February. In the meantime, don’t forget to call your senators and representatives to remind them how much funding for math and science means to you.

I also had coffee with Senator Bob Casey while I was in the capital. He also told me to keep calling him to tell him what was important to me, and — this part was really exciting to me and should be to you too — he told me to bring my students to D.C. and his staffers would arrange a tour of the Capitol and the White House. So get your Math Club, AMS Grad Student Chapter or AWM Student Chapter pumped up about math and public policy, pack them in a bus, and tell your senator you’re on your way!

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Teaching Math to Incarcerated Students

Last month, Beth Malmskog wrote a post for the AMS blog PhD Plus Epsilon about teaching mathematics at a nearby prison. Malmskog is a math professor at Villanova, and in the post she writes about a course she and her colleague Katie Haymaker taught in the spring at Graterford State Correctional Institution. Malmskog had previously written about giving a one-off lecture at Graterford and the Math Circle she and Haymaker started there the next year. Despite the difficulties of teaching with limited supplies and access to technology, Malmskog describes the course as a great experience.

I had more fun teaching this class than I have ever had teaching. The most striking thing about the course was the amount of energy in the classroom throughout the semester.  The students were engaged and game, willing to dive in to any discussion, to speak up with questions, comments, and occasional complaints, and to try activities for themselves.  Every day when I walked out of class, I felt that I had actually connected with the students.  Along with this gameness, most of the students were fairly mature and serious about learning, while still being ready to make jokes and speak up in class.  I wished I could have brought my on-campus students, as a demonstration of what a classroom can be like.  I love working with my on-campus students, but I feel that self-consciousness and expectations of what a college classroom “should” be can really limit their experience.  What could college be like if students really engaged every minute of class time and saw class as a dialogue?  I have tried to create this classroom atmosphere in many classes, with varying degrees of success.  At Graterford, this atmosphere just happened on its own.

Francis Su spoke about one of his incarcerated students in his MAA Retiring Presidential Address, Mathematics for Human Flourishing, posted at his blog The Mathematical Yawp. I know a few of my friends have taught math in prisons as well. But before reading Malmskog’s posts I hadn’t really thought about how that worked.

It turns out there’s no one way to do it. The Villanova Graterford program is somewhat unusual in that students can earn a degree with in-person classes and professors get the same credit for teaching at Graterford as they do on campus at Villanova. Other schools and states have different programs, including correspondence classes and non-credit classes. The Prison Studies Project website has information about many, though not all, programs around the country. (Thanks to Annie Raymond of the University of Washington for pointing me to the Prison Studies Project.) A blog post from the U.S. Department of Education has information about the Second Chance Pell Program that gives grants to incarcerated students through 67 schools around the country.

Another side to the intersection of mathematicians, education, and prison is the fact that the U.S. has the largest incarcerated population in the world and an incarceration rate far higher than most other countries. The system is rife with racism and inequality. If you’re interested in getting involved in studying and working to fix some of the problems with our criminal justice system, Phil Goff writes on Cathy O’Neil’s blog that Justice Needs Nerds.

Posted in Issues in Higher Education, Math Education | Tagged | 1 Comment

Math Puzzles To Pass The Time

Apparently nothing spoils a mountain vista quite as well as a story about a man with a fox, a goose, and a bag of beans.

There are two types of people in this world: those that can only lie, and those that can only tell the truth.

You might recognize that as the opening clause of so many knights and knaves problems. These are classic logic puzzles that I love to use to torture my siblings while we go hiking every summer. Which brings me to my real point. There are two types of people in this world: those that love math riddles, and those that despise them.

Sadly, my siblings fall into the second camp, but I’m hoping that you fall into the first.

What makes math puzzles so enticing are their pleasant blend of elementary math and logic. There are ones that require some knowledge of math, like maybe some knowledge of geometry or a clever use of arithmetic. And there are others that just require some logic. I always love watching mathematicians solve both types.

I recently had a visitor ask me the Cake Icing Problem. You have an iced birthday cake, you cut a piece of size Θ (that’s the center angle of the piece, the way you would typically cut a cake), flip it upside-down and place it back in the cake. If you continue on in one direction, cutting and flipping pieces of size Θ, will all the icing ever be back on top? Will it ever all be on the bottom? For certain values of Θ the answer is obvious, but can you say something general? Project Euler also poses a slightly trickier modification on the birthday cake puzzle.

A great treasure trove of puzzles is Varsity Math, a series cohosted by the Wall Street Journal Blogs and the National Museum of Math. A new couple of problems show up each week with solutions the following week. Last week there was a particularly fun one about areas of squares inscribed in squares.

Slightly less obvious in their mathematical nature, but no less fun, Popular Mechanics also hosts a Riddle of the Week. This week’s problem was the problem of 7 lit candles arranged in a circle. If you blow on one candle it changes the status of the two neighboring candles (that is lit with become unlit and unlit will become lit). What is the minimum number of moves to extinguish every candle. The solution is quite simple, and doesn’t really require any advanced knowledge of math but it’s a nice one to think about.

Alex Bellos, who posts his Monday Puzzle every two weeks, also wrote about a recent popular internet meme, the math problem for a 5 year old that’s been stumping the web. Bellos writes that these problems are sometimes interesting, but often totally misstated and impossible to solve. He walks us through one particular viral problem from earlier this month.

As you go forth and enjoy these puzzles, one word of advice: If you’re hiking up a mountain with a mixed group and you feel compelled to ask that cake icing problem, I recommend that you stay far back from the edge.

Posted in Recreational Mathematics | Tagged , , , | 3 Comments

Twitter, but for Math, with Toots

Hooray, it’s yet another social network for you to join! I’m skeptical about new social media, but I’ve been seeing enough posts about that I finally caved and got an account. Mathstodon is the math(s) “instance” of mastodon, a new open-source, decentralized microblogging social network. It’s a lot like the Twitter with a few differences for the user: no ads, messages can be 500 rather than 140 characters, and instead of tweets they’re called toots. Har!

Mathstodon, the new social network where you can toot a proof.
Credit: Christian Lawson-Perfect

The expanded character limit on mastodon inspired Christian Lawson-Perfect (@christianp) and Colin Wright (@ColinTheMathmo) to set up mathstodon. The sadly dormant @proofinatweet Twitter account notwithstanding, it’s pretty hard to fit mathematical ideas in 140 characters. The fact that mastodon is open-source also made it possible to add LaTeX rendering to mathstodon, and there was much rejoicing. (Though also a little bit of consternation; it doesn’t always render quickly or completely for me. It could be my browser.)

In a post for the Aperiodical introducing mathstodon, Lawson-Perfect challenges us to luxuriate in those 500 characters and write some proofs in a toot. A search for the hashtag #proofinatoot reveals some fun ones, but I’d love to read more. Mathstodon also introduced me to Lawson-Perfect’s blog Interesting Esoterica, a collection of fun or strange papers he has collected. Topics range from developing a mathematical model for bobbin lace to non-intersecting circles in the plane.

Brent Yorgey, a mathematician at Hendrix College who writes the blog The Math Less Traveled, has written a couple posts about using mathstodon. One is his proof in a toot, and one is about a fun puzzle: What’s the 99th digit to the right of the decimal point in the decimal expansion of (1 + √2)500?

So far I’ve only highlighted the work of men in this post. When I scrolled through mathstodon users, I only found about four usernames that appear to belong to women. (I did not dig into it extensively, so take that number with a grain of salt.) Without diving into the gender politics of mathematics social media sites, I’d like to blandly and generically encourage mathstodon users to make it a site that is welcoming and respectful to everyone. I find Twitter, or at least the mathy corner of Twitter I inhabit, to be a good place to talk about math without being harassed or feeling like I constantly have to prove my credentials. I hope that transfers to mathstodon.

Unfortunately, as far as I can tell, there is no way to lurk on mathstodon without setting up an account, but if you’re ready to take the plunge, have at it. As usual on social media, I’m @evelynjlamb. Feel free to send me a toot!

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Best of the Bots

A slew of paint colors named by a neural network, including such gems as “turdly” and “rose hork,” made it big last week, with mentions in Ars Technica, The AV Club, and even The Atlantic. But for the story straight from the source, check out, the wacky neural network blog of optics researcher Janelle Shane. Her original post about paint colors went viral, but I got even more out of part 2, in which she writes about how she tweaked the algorithms to get better results. (Though “copper panty” is a marginal improvement at best.)

“May Picture” by Paul Klee, who as far as I know did not buy his paint from a neural network. Public domain, via the Metropolitan Museum of Art.

My introduction to the machine-generated text genre was the now-dormant King James Programming tumblr, featuring lines from a Markov chain trained on the King James Bible and some computer programming texts. Its most recent contribution was “37:29 The righteous shall inherit the land, and leave it for an inheritance unto the children of Gad according to the number of steps that is linear in b.” Timeless wisdom, to be sure. 

Neural networks and other machine learning processes are hot right now. Google’s AlphaGo, which uses neural networks to decide which moves to play, is now consistently beating the best Go player in the world (read more about that at The Math Less Traveled by Brent Yorgey), so we humans have lost the edge in basically the last game we were still comparatively good at. I think it’s only prudent for us to keep an eye on what our robot overlords have in store for us as they take over more and more formerly-human tasks.

If the paint colors, recipes, Irish tune names, and pickup lines at Lewis and Quark aren’t enough for you, there’s a lot more algorithmic creativity to choose from. At, you can click until you find a recipe that actually sounds like food. (Let me know if you try grilled coffee, with its ingredients of milk, coffee, mayonnaise, and lambchops.) High noon GMT has something even better than Irish tune names: entire Irish tunes generated by computers. I’ve really enjoyed seeing experiments involving word2vec, which embeds words as vectors in 300-dimensional space. I first remember learning about it from Jordan Ellenberg’s blog, and just the other day a friend pointed me to a word2vec reinterpretation of Genesis I using only words that begin with “a.”

For bite-sized chunks of machine learning, check out this list of bots I follow on Twitter. I’m using “bot” as sort of a catch-all term there for a lot of different kinds of computer-generated tweets. Census Americans tweets census data of randomly selected Americans. Symmetric Curves tweets beautiful randomly-generated curves with radial symmetry. Picdescbot tweets the image descriptions it comes up with for random pictures from Wikimedia Commons (with varying degrees of success). All of the bots I follow inject a dose of randomness and usually some levity into my day. So I’d like to thank the bots for amusing me so much as they work toward world domination or their own line of Sherman Williams paints — whichever comes first.

Posted in Data Science, Mathematics and Computing | Tagged , , , , , | 2 Comments

Black Hat, White Hat

Did you know that hackers are made out of floating binary masses?

Meanwhile, over in computer science…several days ago WannaCry almost brought the world to its knees until an anonymous tech blogger, MalwareTech, brought it to a screeching halt by activating a hiding-in-plain-sight kill-switch. MalwareTech blogged about the wild 12 hour epic (which, by the way, happened during his vacation) and it makes me so gleeful to read about. It’s not math, per se, but I feel like cybersecurity is sort of like a favorite cousin of cryptography. One who visits all the time and is almost like a sister. Also, who doesn’t love a good story about a blogger saving the world? In that spirit, I wanted to take today to give a quick and dirty explanation of what our dear blogger-turned-hero did, why it worked, and what we might ought to brace ourselves for.

Here is more or less what happened, in the most jargon free way that I can bring it to you. Our heroic white hat blogger got his hands of a copy of WannaCry and was playing around with it in a quarantined operating system environment called a sandbox. This is typical when programmers are dealing with malware or other suspicion software of unknown origin; it’s a way to run analyses on the potentially dangerous program without making their own device vulnerable.

He noticed that something in the WannaCry code was directing his computer to check whether a certain website was live. He also noticed that this domain name was unregistered, so he registered it, which apparently is very common practice for folks in the cybersecurity biz. The result, was that now infected machines were still running this check to see if the website URL (which was actually hardcoded into WannaCry) was live, and when they found out it was live, the ransomware just shut down. Simply registering the domain name acted as a kill-switch for the world’s biggest ransomware attack.

There seem to be two prevailing theories as to why the original programmers wrote this domain querying piece into their code. The first, and perhaps too simple, explanation is that they left it there as a kill-switch for themselves. If things got out of hand they knew they could just register the domain name and shut it down.

The idea that seems to have slightly more traction, is that the querying of an unregistered domain name was a rudimentary method for the ransomware to identify if it was living in a sandbox environment, which would typically return all URLs (registered or otherwise) as alive. The original developers would use this as an anti-analysis measure, since the only way to analyze the ransomware would be in a sandbox (unless you like to live dangerously). The important technical detail here is that an operating system in a sandbox sees all websites as live, even the unregistered ones. So pinging a live website delivers the message to shut down the ransomware. But by registering the domain name, everyone who queries the website sees it as live, meaning WannaCry always thinks its living in a sandbox environment, and therefore always shuts itself down.

MalwareTech for the win!

But it seems the beast hasn’t been slain entirely. As of last week, bad-guy hackers — who seem to be acting purely for their own amusement — have dispatched a botnet army of hijacked cameras, modems, and other internet of things devices to continually attack the newly registered domain name and flood it with traffic whose origins can be very hard to track. The idea behind this is that if enough of these zombie devices are blocking the “door” to the domain then other devices will get turned away. Kind of like a Walmart on Black Friday. This is called a Distributed Denial of Service (or DDoS) attack, and given aggressive enough traffic it could result in forcing the new domain offline and effectively flipping the kill-switch back to the “off” position. There’s also talk of the possibility that, much like an actual virus, there are mutated strains of the WannaCry virus with different kill-switch domain names coded into them.

This means WannaCry still has potential to mess your life up pretty dramatically. So take the appropriate precautions. Check if your operating systems is recommending patches. And if you are still using Windows XP, brace yourself.

Also, because every situation involving near destruction of the world could use some extra levity, Quartz has generated a great roundup of all the absurd photos that accompany news stories about WannaCry. Much like my binary people above, they make no sense at all.

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What Are You Going to Do with That?

That feeling when you just can’t decide what mathematical career is right for you. Image: “A Rose” by Thomas Anshutz. Public domain, via the Metropolitan Museum of Art.

For people in graduate school for math, the question, “What are you going to do with that?” often seems to have a clear, easy answer: “I’m going to be a math professor.” In grad school, our role models are the professors in our departments and others we meet at research conferences, who all went through graduate school, probably did a postdoc or two, and found a tenure-track job at a university. We’ve heard whispered stories of the students who left “for industry,” but industry can seem like kind of a mysterious black hole or a career of last resort.

It’s fine to have a goal of becoming a math professor. But many of us find during graduate school that our desires for our lives don’t fit well with the academic model. Beyond that, the job market is tough. Not everyone who wants a postdoc or tenure-track job can have one. Thinking of a tenured job in a math department as the only true mathematical success story is a corrosive myth. I know it took me a long time to throw myself fully into my career as a math and science writer even though a university position wasn’t making the most of my creativity and skills because part of me was holding on to the idea that if I did something other than get a tenure-track job at a university, I would be throwing away my PhD.

I believe and hope that the idea that success must take the form of university jobs is dying. I know many math professors who want to make sure their students understand the careers available to mathematicians and how to get and thrive in those jobs. I am on the record in favor of encouraging people to go to graduate school in spite of the tough academic job prospects, and I believe that another side of that coin is that we need to see non-academic jobs as success stories, not failures, and know how to prepare students for those as well.

Fortunately, there are a lot of resources out there for mathematicians and students who are interested in non-academic employment. One I found recently is the BIG Math Network, which helps connect academic mathematicians with business, industry, and government (hence BIG). The BIG Math Network blog has information about internships and other opportunities for people who want to know more about what goes on outside of the ivory tower as well as guest posts from mathematicians who work in business, industry, and government, like this one from my grad school buddy Peter Horn about his career switch from a tenure-track job in academia to being a data scientist at a research and development nonprofit.

I’ve seen a few other posts around the blogosphere about leaving math academia for other jobs. Jesse Johnson has written a few posts (1, 2, 3) for the Low Dimensional Topology blog about his move from Oklahoma State to Google. Yen Duong of Baking and Math has written about how she decided to leave academia and how she’s started her job search. Sarah Rich recently wrote about moving from math to data science on Jordan Ellenberg’s blog Quomodocumque. And earlier this month Cathy O’Neil had a guest post on her blog Mathbabe from Phil Goff called Justice Needs Nerds about doing data science to analyze biases and brutality in policing and hopefully work to mitigate them.

If you’ve got a math degree but aren’t quite sure what you want to do with it, or if you’re a professor who wants to help your students know what’s out there and how they can prepare, I hope you’ll find some useful information about the many different mathematical careers that exist. Let me know about other related resources in the comments!

Posted in Data Science, Math Education | Tagged , , , | 5 Comments

Cook’s Take on Benford

The leading digits of the heights of the world’s tallest buildings satisfy Benford’s Law.

Lately, I’ve been having fun reading John D Cook’s Blog. Cook is an applied mathematics consultant who blogs and tweets up a storm about all sorts of topics mathematical, statistical, computational, and scientific. He maintains 18 daily tip Twitter feeds giving daily facts about…well, everything…and one personal feed.

A leading digit 1 is expected to appear a whopping 30% of the time.

But what I like most are his mathematical blog posts. He writes short easy to digest posts about reasonably accessible topics in math, often with a computational bent, and I always walk away feeling like I learned something. This past week he revisited the topic of Benford’s law, which is this totally weird and strange thing that I’ve always wanted to understand more about. Benford’s law says that in many naturally occurring data sets the leading digit is more likely to be small. If the leading digit d from the set {1,…,9} were distributed uniformly you would expect each digit to show up about 11.1% of the time. But in reality, the leading digit is more often distributed according to the chart on the right. Cook can fill you in on some of the more precise formulations of Benford’s Law.

In his blog, Cook describes how the leading digits of factorials satisfy Bedford’s Law, and even gives some tips on how you can use Python to compute leading digits up to 500!! (One of those exclamations is a factorial, the other one is for my excitement.) He also show that the collection of SciPy constants follow Benford’s Law, which Cook explains and computes using Python. Cook blogged about how samples from the Weibull distribution satisfy Benford’s Law, and most recently he even showed that the iterates of the Collatz conjecture seem to follow Benford’s Law.

And you know a party is getting good when the Collatz Conjecture shows up.

These posts just give a small flavor of Cook’s writing. I also really enjoyed his recent posts on harmonic numbers and golden angles (largely because it prompted me to check out the work of the visual artist John Edmark), the lesser known cousin of the golden ratio.

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Growing Up Gifted

It seems that Hollywood can’t get enough of mathematicians. Most recently, Gifted hit theaters. It’s the story of the mathematically gifted seven-year-old Mary who is living with her uncle in Florida. We follow Mary’s struggle adjusting to a typical public school classroom, while the conflicting desires of the adults in the film — her uncle, teacher, grandmother and neighbor — to allow her a normal childhood while making sure to nourish her talents, play out around her. As the film progresses we learn that Mary’s mother was a genius herself, who died while on the cusp of proving the existence and smoothness of the Navier-Stokes equations, one of the as-yet outstanding Millennium Problems.

The movie didn’t actually involve all that much math, save for occasional references to differential equations and some teary-eyed discussion of the problem that got away. But it did capture something charming and lovely about the sometimes non-trivial dynamics of teaching exceptionally gifted children and the captivating allure of mathematics.

I stumbled upon a blog written by several educators and researchers at Duke University’s Talent Identification Program, who are not necessarily experts in mathematics, who write about whether its depiction of giftedness in the classroom was accurate and well-handled. They bring up several good points, including how different the landscape can be for a student depending on whether their parents and educators are completely aware of all of the resources available to them. They also bring up an important point that I think the movie very conspicuously missed: being mathematically gifted and being social are not necessarily in opposition to one another.

The movie concluded with a cameo from mathematician, math blogger, and recent Erdős-Bacon number-haver Jordan Ellenberg, who consulted on mathematics in the film.

In 2014, Ellenberg wrote an essay for the Wall Street Journal, The Wrong Way to Treat Child Geniuses, (sorry about the paywall) about the disproportionate and sometimes wrongheaded way that society thinks about genius in children. Ellenberg cites a Vanderbilt University study that tracked the achievements of a cohort of children identified as gifted at an early age. He was part of this cohort, a fact he discusses in a recent interview with math blogger Anthony Bonato. Ellenberg and his cohort do have a disproportionate amount of success, especially as success is defined in the academic realm, but he points out, “most child prodigies are highly successful—but most highly successful people weren’t child prodigies.” The cult of genius, he claims, might do more to scare otherwise top-notch people away from math and science than it does to foster the geniuses.

The idea that math is an area strictly reserved for super geniuses is generally speaking, a very bad one. Evelyn Lamb wrote about some of the specific problems in the genius myth as it corresponds to the retention of women in STEM fields. Lamb also wrote about how the media contributes to this stereotype.

Fields Medalist and math blogger Terry Tao, who also consulted on the film, has written about the short-sightedness of over-hyping giftedness when it comes to mathematics. Tao writes, “I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of ‘geniuses’.” Tao has also written about strategies for gifted education, and points readers to several articles about his experience growing up gifted.

These are all interesting points, and as a mathematician and educator I would strongly recommend watching this movie, if only as a well-scored and reasonably entertaining springboard to launch into all of the rich ideas surrounding giftedness, the cult of genius, and the strange otherness of mathematics.

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