The NSF Gets Serious And #MeToo

We Must. Image Courtesy of Molly Adams via FlickrCC

It looks like the NSF is finally getting serious about its stance on researching funding and harassment in the sciences. Two years ago in January 2016, in an official statement, the NSF threatened to pull funding from Universities that didn’t follow Title IX mandates. They warned, “NSF encourages NSF-funded researchers and students to hold colleagues accountable to the standards and conditions set forth in Title IX.” This was a good start, if a somewhat toothless threat. Let’s just say that personally holding your colleagues accountable for their actions (while admirable) is something that only seems remotely reasonable when you’re sitting in a position of relative power and privilege.

Then the last two years happened. And things got so real.

The #MeToo movement has been picking up steam across industries and math is no exception. Stories of blatant sexism and harassment in the math and tech sector have made their way into the mainstream media, and earlier this year an anonymous crowdsourced list of Sexual Harassment in The Academy was publish by Karen Kelsky who writes The Professor is In.

As of last week, the NSF has gotten more formal in their stance about harassment on their dime. In particular, Important Notice No. 144 spells out the three major changes effective in their new policy:

  1. If a PI, co-PI or other person funded by a grant is found to have harassed, this must be reported to the NSF. Then the agency has the right to take unilateral action such as suspending the grant, killing the grant, or removing people from the grant.
  2. Organizations that are funded by the NSF are expected to have clear and formal structures in place for dealing with the reporting and investigation of harassment.
  3. The Office of Diversity and Inclusion is launching a new website to make handling these sorts of things as easy and transparent as possible.

Two things that bear mentioning. The first, is that the NSF is only made aware if there is a finding of harassment after a formal investigation or if the person being investigated is put on administrative leave as a consequence of the investigation. So, due process. Also worth pointing out, it doesn’t look like the NSF is requiring any harassment-type analogue to the disclosure of current and pending support as part of their application. These policies are only relevant to individuals who already hold NSF grants. Oh to be so blessed.

In this vein, Izabella Laba who blogs as The Accidental Mathematician recently wrote a post for the men in math who are bothered by the recent revelations (and want to do better). She tackles (brilliantly in my opinion) some of the tough questions about due process and the advocacy that women so desperately need. She clarifies the difference between a friendly touch and career-derailing harassment and the historical absence of formal structures to separate and deal with the two. This is where items 2 and 3 in the new NSF guidelines are very helpful.

The NSF Office of Diversity and Inclusion also put out their own bulletin, reminding people, “if in doubt, reach out.” This would probably be a good time to brush up on your Title IX FAQs and take a moment to remind yourself what harassment looks like. And after you do that, find someone junior to you and have a conversation letting them know how seriously you take this sort of thing.

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How to Have an Excellent E Day

A portrait of Leonhard Euler, who named the mathematical constant e, painted by Jakob Emanuel Handmann.

Tomorrow, February 7, is e Day! This year is the best year to celebrate the base of the natural logarithm because, like Pi Day 2015, the year lines up along with the month and day. Hurrah! People who prefer the day/month/year date convention can save these suggestions for July 2 or wait until January 27, 2082 and get one extra digit. (That seems like a long wait for one extra digit, but you do you.)

I wrote about the excellent number e for Slate and included some possible ways to observe e Day. In addition to putting some money in a checking account to take advantage of compound interest, I humbly suggest using a constant so tied up with change and growth to do some personal growth. Think of e Day as a second chance at my favorite holiday, New Year’s Day.

I have some other suggested e Day reads for you from around the math internet. In 2010, Math Goes Pop suggested January 27 for a yearly e Day observance that privileges the day/month convention countries the way Pi Day privileges month/day users. Wired and Mathnasium have shared e Day suggestions in the past. (For those keeping score, they posted those articles in February.)

Ben Orlin’s ABC book of e does not disappoint, with ebundant elliteration and a reminder that Euler was the wearer of one of the greatest hats in the history of mathematics.

For more on the number e itself, the MacTutor math history website has a nice overview of the history of the constant from its plucky days as the unrecognized base of the natural logarithm to the proof that it is not algebraic. John D. Cook has been making pretty pictures with sums involving complex exponential functions. You can find the exponential sum of the day here. He’s also written about why the natural logarithm is the most natural and how to live with exponential growth. (Hint: it might start slower than you think.)

An equation called Euler’s identity, eπi+1=0, has been called the “most beautiful equation.” I’m on the record as feeling kind of meh about Euler’s identity, and I’m not the only one. For one, just write eπi=-1 instead of obscuring it with a weird need to have a 0 in your formula. You can put a 0 into any formula you want! Beyond that, I love Euler’s formula e=cosθ + i sinθ. Plugging in the number π for the angle θ doesn’t do much for me. But I must say I love 3Blue1Brown’s explanation of the identity in this video. (He has since expanded on the ideas in another video.)

Update, February 7: Imaginary.org has a page where you can find your birthday in e. Anthony Bonato wrote about e Day for his blog as well, describing e as the “Jan Brady of transcendental numbers.”

How will you celebrate e Day? Growth and reflection are well and good, and I’m all for busting out a little complex analysis, but after that, I’m going out for Ethiopian food.

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On Seashells, Spirals, and Solids

Fibonacci Dodecahedron, by Rafael Araujo


Recently, a friend sent me a link to the drawing Fibonacci Dodecahedron by the Venezuelan artist Rafael Araujo. I found it quite beautiful but was immediately skeptical of the words Fibonacci and dodecahedron appearing together. It’s no secret that I am somewhat of a phi detractor. And while I respect Fibonacci, and certainly crave order in the physical universe, I know that phi is not actually everywhere. But after some quick digging, I am embarrassed to admit — but apparently not too embarrassed to publish it on a blog — that I had no idea the relationship of the golden ratio to the platonic solids and I am simultaneous soothed and pleased at this revelation and the art that exists in its name.

Feeling sufficiently chastened and impressed, I reached out to Araujo to chat with the man who harnessed the power of phi so compellingly. Araujo is trained as an architect, but has been doing technical drawing using polar geometry and classical perspective (always by hand!) since he was a teen. Much of his mathematical art is build around spirals, helices, and constructions rooted in sacred geometry, and he keeps a blog on his website where he describes some of his processes for creating the technical mathematical drawings.

Wondering how to draw a perfectly twisting seashells? Wonder no more. Curious what sort of platonic solids you can build using the principles of sacred geometry? Have a look at these. Araujo also posts time-lapse videos of his creations.

Metatron’s Cube, by Rafael Araujo

I asked Araujo what his relationship was to math as he saw it in nature, in everyday life. “I’d like to think that there are a few predictable things in life, meaning that you can calculate them,” but in the end, he says “all is prone to chaos. But, being earnest, there is a lot of math all around us, and that fascinated me.”

Many artists find themselves captivated by the order (and disorder) in the world that can be charted by mathematics. Here on this blog we’ve featured the paintings of Lun-Yi London Tsai, the prints of Tilman Zitzmann, and recently the digital art of David Whyte. Sometimes it’s not obvious to me why I consider certain art to be mathematical art and others not. I’m still trying to figure out what that geometric quality is that tickles my mathematical brain. There’s something about the structure, geometry, and consistency of mathematical art that calls out to me, the viewer. What do you consider mathematical art?

Shell Detail, by Rafael Araujo

Last year Araujo published the Golden Ratio Coloring Book, which renders his beautiful geometric designs in black and white for your doodling pleasure. If you like drawing and want to know more about how me makes these geometric sketches by hand, definitely have a look at his blog or check out Araujo’s YouTube channel, where he posts videos of his images through various stages of completion. It is truly mesmerizing.

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Math with Martin

Most teachers and students in the U.S. didn’t have math class today because of the Martin Luther King, Jr. Day holiday. But when you get back to the classroom, the online math world has some suggestions of how to incorporate ideas civil rights and justice into a math or statistics class.

Jessica Hartnett, who writes a blog about not awful and boring ideas for teaching statistics, suggests looking at public opinion polling data from the 1960s addressing attitudes towards Civil Rights protesters. (She was inspired by a Washington Post article by Elahe Izadi.) Students might be interested in comparing those attitudes to current attitudes about the Black Lives Matter movement.

The Stats Medic, a site that helps provide middle- and high-school math teachers with resources and ideas for statistics lessons, has a suggested Martin Luther King, Jr. Day lesson about poverty and educational outcomes.

Annie Perkins, a math educator in Minneapolis and author of the arbitrarily close blog, has been hosting a discussion of The New Jim Crow by Michelle Alexander as part of a #MTBoS (math Twitter blogosphere) book club. The book discusses the way the war on drugs continues the country’s shameful legacy of white supremacy and racial injustice. I read it not too long ago and can attest it is eye-opening and enraging. Perkins shared a discussion she had with some students about statistics from the book, and today she is writing an extensive Twitter thread about the book. The official discussion will take place on January 20 in person for Twin Cities residents and on Twitter for the rest of us.

Though it may be too late to incorporate any of these posts into your lesson plans for tomorrow, the good news is that any day is a good day to talk about Civil Rights history, Dr. King’s legacy, and current issues of racial justice.

King himself, in addition to well-known works such as the “I Have a Dream” speech and the Letter from a Birmingham Jail, wrote about his views on education for the Morehouse College newspaper the Maroon Tiger in 1947. His words are relevant to teachers in any subject: “The function of education, therefore, is to teach one to think intensively and to think critically. But education which stops with efficiency may prove the greatest menace to society. The most dangerous criminal may be the man gifted with reason, but with no morals.” As you ponder his writing about education or any other topic, make sure to think before co-opting his words, as math educator José Luis Vilson writes at Educolor.

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Elevating The Art Of Math(s) Communication

The 2018 Joint Meetings are just around the corner, and lots of great stuff is going on in San Diego. There will be lectures on current events in math, panels and sessions highlighting inclusion in the field and on January 11th Vi Hart and Matt Parker will be presented with the 2018 JPBM Communications Awards.

Vi Hart and Matt Parker will be receiving the JPBM Communications Awards at the 2018 JMM.

Since math communication is both the medium and raison d’être of this blog, I thought it would be nice to give a timely homage to these fantastic, inspirational, and aspirational communicators.

I first learned about Vi Hart in the way back when she first made a splash with her charmingly narrated doodly-math videos. The first I ever saw was Infinity Elephants and having just rewatched it a moment ago: it’s perfection! She takes you from hum-drum stream of consciousness math class doodling to infinite series and the sizes of infinity and you almost don’t notice it happening. It’s really mesmerizing. From her more recent submissions, I really liked Smarties in which she discusses the mathematical openability of a roll of Smarties and does some geometric Smartie constructions. Arguably one of Hart’s finest works is Twelve Tones, a doodly discourse on 12-tone musical compositions that made me lol on several occasions. Hart also keeps a blog on her website where she writes about all sorts interesting stuff.

Matt Parker, who tweets @standupmaths is a mathematical comedian, and to my knowledge still the only person to sell out a many-thousand seat comedy venue by telling math jokes. He posts his own videos to his YouTube channel. Some in which he teaches math concepts, like in What does i^i=?, others wherein he and his friend Hugh play with a thing and then do some working out, and others yet where he does incredible standup comedy about math which make me lol a whole whole lot. Parker is also a frequent guest on the very popular Numberphile video series.

The humor and levity that these two are able to inject into their conversations about math is truly a thing of beauty. I am captivated by them both, and I know that many people who consider themselves no-so-mathy have been won over by Hart and Parker. So as a fellow communicator of math, my sincerest thanks to both of you for raising the bar, elevating the art, and making math seem so stinking cool.

If you’re interested in getting on the math communication bandwagon, I’d urge you to apply for the AAAS-AMS Mass Media Fellowship. You will be embedded at a top tier news media outlet where you’re get to learn the skills of the trade and share your passion for mathin’ with the whole wide world. Both Evelyn and I are alumni of the program. Applications are due on January 15th.

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Crunching the Numbers on Energy Efficiency

This fall and winter, I’ve been making some improvements around the house. I’ve gotten some new furniture, added several new houseplants, and added a handrail to the uneven front steps. The big project now is energy efficiency. 

Credit: Jonathan Cohen, Flickr. CC BY-NC 2.0

It’s a bit overwhelming to figure out which energy upgrades are the most important and most effective. It can be hard to find breakdowns of the energy, environmental, and cost impact of different choices. Companies are happy to put information about my electric bill and current home energy profile into spreadsheets and spit out energy savings numbers, but I want to see the nuts and bolts. What were their assumptions and how do different numbers or assumptions affect the output?

As I’ve tried to sort the facts from the sales pitches, I’ve found several blogs with helpful information. I’m writing about them now not just because I have energy efficiency recommendations coming out of my ears but also because it’s the beginning of the year, a good time to think about the improvements we want to make in our lives. If addressing your energy consumption is one of your goals for the coming year, these blogs have some facts, figures, and tips that might help. They’re a little less obviously mathematical than most of our recommendations, but deep down, they’re full of arithmetic, statistics, and fluid dynamics.

As I started my quest for reliable energy information, in a fog of recommendations about windows, weather stripping, and insulation, I stumbled on the blog of the Energy Institute at Haas, a joint program of the Haas School of Business at Berkeley and the UC Energy Institute. It covers a lot of different topics at the intersection of the environment and economics.

Take light bulbs. Growing up, my parents were early adopters of fluorescent/CFLs, so I’ve always defaulted to getting energy-efficient bulbs. In recent years, LEDs have supplanted CFLs as the superior energy-efficient lightbulb and may be responsible for the recent slight decline in household electricity usage in the US. When I buy a lamp, I generally buy an energy-efficient lightbulb to put in it. But the previous owners of our home didn’t have the same approach. When we moved in, there were dozens of incandescents in the fixtures. As they’ve gone out, I’ve replaced them with LED bulbs, but according to this post, I should think about throwing away those perfectly functional energy hogs and replacing them all now. Well, maybe. It turns out that in the parts of the house where I use the lights the most (and therefore switching to LEDs would do the most good), the old incandescents have already burned out and been replaced. But the post has equipped me to run the numbers for the remaining bulbs and decide whether to replace them now or later.

The most expensive decision we’re looking at right now is solar. Will the unshaded part of our roof, which faces east, generate enough electricity to make it worthwhile? What are the energy and financial risks and rewards? If we go for it, should we install it while there are still federal and state tax incentives or wait a few years, assuming technology will improve? Digging into their archives for posts about solar hasn’t exactly made the decision easier. In fact, it’s probably made it more difficult by giving me even more information to consider! As a side note, I’m glad I’m not the only one who is frustrated with the way solar companies tend to present financial savings estimates.

Although I found the blog looking for answers to my own particular questions about my own particular home, the blog has articles that questions about energy policies at a much larger scale. I was particularly interested in this post about making energy improvement programs more fair and this one about whether climate change will have an overall impact of increasing or decreasing energy consumption in the US. In other words, how does the reduced cold days vs. increased hot days cage match turn out?

After finding the Haas blog so helpful, I decided to look for more energy efficiency blogs to help me answer other lingering questions. I knew I had found a keeper when I ended up on a series about building science and the laws of thermodynamics at the Energy Vanguard blog. I also consulted the Energy Auditing Blog after our home energy audit to get information about some of the products and problems our auditor had pointed out. The blogs at Green Building Advisor are good, too, though some posts are behind a paywall. I was particularly interested in the posts at Energy Vanguard and Green Building Advisor about the global warming potential of insulation itself, as we are adding some insulation to the house and would prefer to have a net positive impact on the environment.

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Best and Worst of 2017

New Year’s Eve, that magical night in the long cold dark of winter when we reflect on the best and worst of the year gone by. Image via Flickrcc @shaireproductions.com

As this year comes careening to a screeching halt, it’s time once again for that annual tradition of the best and worst of the year…in math. And what a year it’s been! Where to begin? Let’s start with the good stuff.

The Best of 2017

The movie Hidden Figures (based on the book Hidden Figures by Margot Lee Shetterly), about role of the african american lady mathematicians in space travel, was a huge and exciting success. The movie is fantastic, and may even pass some of 538’s new and improved spins on the classic Bechdel Test.

There has also been a bunch of fun stuff going on in the computer science realm this year. An unknown blogger turned hero stopped a multinational terrorist conglomerate in the WannaCry episode, neural networks have been getting more sophisticated and we’ve started to get a better understanding of the apparent bottlenecks, and cryptocurrency and the blockchain is currently the coolest thing to blab about at holiday parties. Although, full disclosure, I’m writing this on Dec. 22 and as of this morning Bitcoin is down 30$% from its all-time high, so cryptocurrency may be full-on milkshake ducked by the time you read this.

In very (very) recent news, it looks like some doubt has been cast on the famed Navier Stokes equations. Physicists use the equations to model fluid flows but mathematicians have long been in search of a rigorous proof of (or counterexample to) the equations. As reported by Kevin Hartnett in Quanta, it looks like a group out of Princeton has shown that under certain conditions the Navier Stokes equations return rubbish. This is a compelling development. Incidentally, these equations were also a critical plot point in the mathematical blockbuster Gifted that also came out this year.

Finally, one of my personal favorites in math writing this year was Mathematics For Human Flourishing, written Francis Su on his blog The Mathematical Yawp. It was, and is, an important and thoughtful reminder as we bring this tumultuous year to a close. Which brings us to…

The Worst of 2017

The situation in Washington DC has continued to be the worst. Karen Saxe, who writes the AMS blog Capital Currents has done a great job keeping us abreast of how changes in Washington will affect mathematicians. Most recently, how the Tax Cuts and Jobs Act will affect graduate students (read: badly), and the current climate of federal funding for mathematics (read: doomed).

In that vein, the Simons Foundation updated their eligibility requirements this year for their Collaborative Travel Grant. Now it seems that only tenure stream faculty at institutions granting PhDs in mathematics are eligible to apply. This is really bad for the great many of us who work at institutions that grant many PhDs in fields *other* than math and therefore are eligible neither for Simons funding, nor for NSF Research at Undergraduate Institution funding. Let’s fix that, eh?

This year saw the passing of several notable mathematicians, including Maryam Mirzakhani, the first female Fields Medalist, and Fields Medalist Vladimir Voevodsky.

And now, as I’ve done for many years, I’ll close with an update on Mochizuki’s claimed proof of the ABC conjecture. I’ve been wrestling with whether to put this in the best or worst subheading, but here is where the hammer dropped. It looks like Mochizuki’s proof is poised to be published in the Publications of the RIMS. In a persuasively written post on his blog Persiflage, Frank Calegari argues that the ABC conjecture has (still) not been proved. The current state of things with ABC is, as Calgari puts it, “a complete disaster.” Largely, he says, because whether or not the paper is correct it hasn’t gained any clarity in the referee process and it still completely opaque, despite the fact that there was a 300-page summary published by Go Yamashita this year, saving the intrepid reader ~100 pages. Other number theorists disagree. On Facebook, Chris Rasmussen argues that the state of ABC is hardly a disaster, and in fact is no different from the verification of any mathematical breakthrough, it’s simply proceeding at a slower pace.

I’m going to close 2017 of the mind that ABC is still a conjecture. I’ll let you know where I stand next December. If you disagree with me, let me know why @extremefriday. Until then, a happy new year to you and yours. May you find clarity in all of your logical arguments, satisfaction in all of your mathematical endeavors, and grant funding raining from the sky!

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News in Numbers and Nurturing Numeracy

My attention span is a little short right now. There’s always the news, of course, but the lack of daylight around the end of the year gives me a weird combination of restlessness and lethargy. That’s why one of my favorite columns in my blog feed right now is Significant Digits from FiveThirtyEight. Every weekday, Walter Hickey rounds up a few numbers from the news. FiveThirtyEight is well-known for its data-driven reporting about sports, politics, health, and pop culture, and I highly recommend their multi-article in-depth reports. (The series on gun deaths was particularly compelling, and I’m looking forward to reading and listening to their work on gerrymandering.)

Credit: Jorge Franganillo, via Flickr.

Significant Digits boils everything down to bite-sized nuggets of numbers, often with a splash of humor. I can almost ignore the occasional sponsored content and the fact that the numbers are listed in increasing order, but he treats percentages as 100 times their value. (That is, 85% is filed under 85, not 0.85. Am I pedantic? Very well then I am pedantic.)

Numbers by themselves can do as much to obscure as to clarify. A million is a lot. But is raising or lowering the national debt by a million dollars a lot? Does a disease with a million sufferers affect a lot of people? I appreciate the fact that Significant Digits tries to contextualize the numbers and links to longer news stories so people who want to can get a more complete picture.

I think of myself as a fairly numerate news consumer. When I read numbers in the news, I try to roll them around in my head a little to put them into context. A million dollars added to the US national debt is $0.30 per person or on the order of magnitude of 0.00001% of the current national debt. I like to play with Fermi problems, as I’ve written about here before. So it was especially hard on my ego when I visited math teacher Fawn Nguyen’s website Between 2 Numbers and found out I was out of my depth trying to answer a lot of the questions. How many grains of sand does it take to cover LA to the depth of a foot? I have no idea. I guess I have some work to do to catch up to her students!

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The Blockchain Party

Image via Flicker CC, Peter Shanks.

Maybe you think bitcoin is silly, maybe you think it’s dangerous and socially irresponsible, maybe you’re a bitcoin millionaire (in which case, hi), or maybe you are desperate to join the blockchain party. Whatever your stance, it seems that you can’t escape bitcoin and the blockchain these days. And yet it feels like every time I try to understand it, I’m treated to some mumbly semi-confused version of the truth.

So in case you’re confused, here’s a little bit about the math behind bitcoin. Transactions in the bitcoin (or generally the cryptocurrency) marketplace are made very secure by a public ledger system called the blockchain. This works by using something called the elliptic curve digital signature algorithm. The blog The Mathematical Investor gives a really thorough rundown of the algorithm, but for anyone with a passing knowledge of elliptic curves, I’ll just mention that it’s related to the the modular arithmetic on the group structure of points on an elliptic curve. The point of the blockchain is that it acts as a decentralized digital public record of all bitcoin transactions and since its continually updated and there’s no master version, it’s thought to be unhackable. In theory this means it should be impossible for people to steal bitcoins, spend the same bitcoin twice, or just invent bitcoins out of thin air.

New blocks in the chain are created when miners solve certain computationally laborious but not particularly difficult hash functions. The Programster’s Blog gives an easy to digest explainer on hash functions and bitcoins. Essentially, the bounty for solving one of the hash functions is 25 bitcoins and a new block in the chain with your name on it. To solve the function you can’t do too much better than random guessing, so solving any particular one might take millions or billions of tries. This is why miners set up giant farms of CPUs cranking away at these problems all day until they hit a solution.

So basically there are two ways to get bitcoins and get yourself onto the blockchain ledger: you can mine them, or you can buy them. And mining them, in principle, is not illegal or bad any more than mining for gold or diamonds is illegal or bad. But much like diamond mining, it’s not without deeply troubling consequences.

The big problem that’s been hot in the news in the past week is the tremendous environmental cost of mining bitcoins. Mining for bitcoins uses tons of electricity – by some estimates more energy than the entire country of Ireland annually – and that is having a big global effect. Countries with cheap (or government subsidized) energy are sucking up electricity to the point of causing rolling blackouts. Some suggest that at the current rate bitcoin mining is poised to eat up more electricity than is available on planet earth by the year 2020. This is bad news for anyone interested in quitting fossil fuels and advocating for responsible energy usage.

It seems pretty clear that for any individual trying to mine bitcoins at home, there is no way you can make more money that you’re spending on electricity. This inspired one enterprising miner to rig up a mining operation in his Tesla with the goal of stealing free electricity from Tesla charging stations. One charity organization trying to raise money to help people pay bail – on its face a worthwhile charity – is advocating mining cryptocurrencies on the servers at your corporate day job to donate. You know, really stick it to the man. And the same charity recently doubled down and suggest that you use the electricity and networks at your local evil gentrifying coffee shop to mine for charitable cryptocurrency.

There has even been speculation about rich regime type North Koreans burning coal to mine bitcoins that they use to buy expensive sneakers. Truly the stuff of dystopian science fiction!

It’s crazy. I’m totally into it. And I just know there’s got to be a better way.

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Unsolved Problems in Math Class

Credit: Randall Munroe. CC BY-NC 2.5

A few years ago, I directed a high school summer math program. Half the day was devoted to exploring the delights of modular arithmetic—we ended the summer with a cake decorated with Fermat’s Little theorem!—and half to learning to program in Python, with number theory questions as motivation. One Friday afternoon, we included these questions in the programming part of the day.

Goldbach conjecture: Any even number larger than 2 is the sum of two prime numbers.

  • Is there a counterexample to this conjecture for an even number less than 10,000
  • Prove this conjecture.

Collatz conjecture: Choose some number a0.
Define an by a
n=3an-1+1 if an-1 is odd or an-1/2 if an-1 is even.
Then an will be 1 for some n.

  • Is there a counterexample to this conjecture for a0<10,000?
  • Prove this conjecture.

Perhaps it was a tiny bit evil to give these longstanding open problems to high school students without warning them, but it was a lot of fun to watch them come up with programs to search for counterexamples and brainstorm about ways of approaching the proofs. (And yes, we did eventually tell them the questions were still open. We didn’t want to ruin their weekends completely!)

Math teachers Annie Perkins and Sam Shah have written about the benefits of exposing kids to advanced math concepts early rather than waiting until they’ve mastered all the easy stuff. If you too would like to torture your students kindle your students’ curiosity and challenge their intuition, with unsolved math problems, there are lots of places to go for inspiration.

The MathPickle site (tagline: “Put your students in a pickle!”) has puzzles organized by grade level, board game suggestions, and a blog. I’ve seen this site mentioned in a few places, including a discussion on Dan Meyer’s blog.

Lior Patcher has a list of suggestions for how to use unsolved problems in K-12 classrooms at his computational biology blog Bits of DNA. I was especially excited to see a question involving Namibia’s mysterious “fairy circles,” circular patches of bare ground surrounded by vegetation. It’s nice to see some modeling and applied math get some love there. Why should number theory have all the fun?

Mike Lawler often discusses advanced and unsolved problems with his kids, and the Collatz conjecture has made several appearances on his blog. In his most recent post on the topic, his kids make music with John Conway’s “amusical” variation of the problem. (As a violist, I’m delighted that one of them does so in alto clef!)

Ben Braun writes about using unsolved problems in his college math classes at the AMS math education blog On Teaching and Learning Mathematics. He highlights some of the benefits for his students, including mindset shifts away from answer-getting and toward seeing failure as part of mathematical productivity.

It goes without saying that your students probably won’t solve the Goldbach conjecture or get a definitive answer about fairy circles in one or two class periods, but you never know. Some open problems might end up being easy to solve, or at least easier than we might think. The Gödel’s Lost Letter and P=NP blog has a fun post about open problems with short solutions.

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