What should Mathematicians do now?

​Mathematicians sometimes pretend we are above the everyday vicissitudes of life, preferring to inhabit a realm of abstraction and perfection, but that’s a lie. We live here too. We are voters, citizens, residents, and teachers. What happens in our country matters. I’m sure Anna and I will eventually get back to writing about other parts of the math blogosphere, but the election is still big news, and we as mathematicians need to ask ourselves what to do next.

I know our readers are not a monolith, but a large number of you are mathematicians at universities in the US. I’ve written this post with that in mind, though much of it will be relevant to people in other careers as well. I am also aware that though I did not support Trump, some of my readers probably did. I am not arguing with you about that. I trust that in spite of that difference, we have similar standards for how to treat others, and we are in favor of a strong, healthy culture of math and science research.

So what are mathematicians to do? Many of the actions we take are the same actions any citizens should take right now: talk to our representatives about issues that are important to us, donate to groups that need our help, reach out to friends and family who are feeling scared, and take care of ourselves so we can continue those other actions long-term. But I think there are a few ways to take action that relate specifically to mathematicians and the jobs they do.

1. Keep students safe

In the wake of Trump’s election, many people feel scared. Trump’s rhetoric energized some people who are racist, sexist, Islamaphobic, homophobic, and transphobic. Since the election, there have been numerous reports of hate crimes targeting people of color, religious minorities, and LGBTQIA+ people. Professors should be doing everything they can to make sure their classrooms and campuses are safe.

It’s tempting to think that math classrooms should be politics-free, but the right response to the election is probably not business as usual. Many educators have written about how they’ve talked with their classes since the election. I especially appreciate Jose Vilson’s post: Politics are always at play in our classrooms. We also need to continue promoting diversity in mathematics. One way of doing that is to cut back on the hero-worship of dead white men. Astrophysicist Chanda Hsu Prescod-Weinstein has a list of resources for decolonizing science that can help us do just that. I’ve also written posts with resources about black mathematiciansHispanic/Latinx mathematicians, and women in math.

One group likely to be at risk in the next administration is undocumented immigrants. If you are concerned about undocumented students, you might consider joining the hundreds of other professors who have signed this petition to extend the Deferred Action for Childhood Arrivals (DACA) program. DACA allows undocumented people who came to the US as children to obtain work permits and remain in the country.

2. Fight misinformation

As Anna mentioned in her last post, there is evidence that misinformation (“fake news”) may have affected the outcome of the election, thanks to the Facebook algorithm bubble. Since then, a lot has been written about how important the phenomenon was to this election and what we need to do to stop it. Cathy O’Neil’s book Weapons of Math Destruction feels especially prescient right now. (Read my review of it here.) Her blog mathbabe.org is one of my go-to resources, and she is part of a New York Times debate about how to best stop the fake news problem. Here are some other things I’ve read recently about fake news and the election:

This Analysis Shows How Fake Election News Stories Outperformed Read News On Facebook by Craig Silverman
Fake News Is Not the Only Problem by Gilad Lotan
The “They Had Their Minds Made Up Anyway” Excuse by Mike Caulfield
Factiness by Nathan Jurgenson
Post-Truth Antidote: Our Roles in Virtuous Spirals of Trust in Science by Hilda Bastian

Fighting misinformation is an area in which I think mathematicians are especially, though certainly not uniquely, equipped to take action. When we write proofs, we are trying to construct watertight arguments using pure logic. Ideally, we attempt to poke holes in our own work until we can ensure that it is impenetrable.

We need to use those skills when we read the news or the outrageous videos our friends share on Facebook, whether we agree or disagree with the conclusions of those stories or videos. Apply the same skepticism to the stories you want to believe are true as the ones you reject. Check Snopes, try to find the numbers instead of taking someone else’s word for it, listen to the full context of the quote, see how other sites are spinning it. Settle for an answer of “it’s complicated” if it is.

An example: in the past few days, a growing number of people have been calling for an audit of the vote in Wisconsin, Michigan, and Pennsylvania (update: as I’m posting this, the audit is looking more and more likely). Those of us who wanted a different outcome could latch on to the story that statistical anomalies make the election look “rigged.” There are a lot of numbers floating around in that article, and it sounds truthy. But J. Alex Halderman, one of the computer scientists urging Clinton to call for a recount, is more measured. “Were this year’s deviations from pre-election polls the results of a cyberattack? Probably not. I believe the most likely explanation is that the polls were systematically wrong, rather than that the election was hacked.” Zeynep Tufekci, a sociologist who studies our relationship to technology, wrote about voting machine vulnerability before the election. Her message is that it’s not likely that it affected this election, but we should be auditing the vote regularly and making sure we leave a paper trail. Halderman’s and Tufekci’s messages aren’t as sexy as “rigged election!” but we need to fight the urge to jump to the sexiest conclusions without sufficient evidence.

How else can we fight misinformation? By supporting real journalism. I recently subscribed to the Washington Post because I’ve found a lot of value in their coverage of Trump’s appointments and financial dealings, but there are many other media outlets that you might find equally or more valuable. The media certainly made mistakes in its coverage of the election, but we still need to support journalism. As subscribers, we should also hold media outlets accountable when they screw up.

We should probably also read more media we disagree with. Yen Duong of Baking and Math recommends the National Review. I recently read “You are still crying wolf” by Scott Alexander of Slate Star Codex. I don’t agree completely with his thesis in that post, but thinking about why instead of dismissing it outright has helped me think about where my preconceived notions come from and how to engage in this conversation.

3. Support climate change research

This is more specific than the above suggestions, but a Trump advisor recently suggested that we should defund NASA’s climate change research. Climate change is likely the most pressing issue of our time. We have to keep studying it and try to find ways to mitigate the damage it is causing.

4. Read history

I hope the people who are warning us that the US is falling into authoritarianism/fascism/kleptocracy are wrong. Or that their warnings help us avoid those dire predictions. But it has happened before, and it can happen again. I think mathematicians would do well to read up on the history of math in Göttingen in the 1930s, perhaps in this Notices article from 1995 by Saunders Mac Lane.

Finally, I’ll leave you with this post by Matilde of the blog Listening to Golem about the moral responsibilities of mathematics and science: “Pack all the tools you need in your bag: network theory, bayesian analysis, probability, differential equations, cryptography, computing, game theory, neural networks. We need them all and we need them now. Get down to work for the sake of our future.”


The Pseudocontext 2016 Deserves

2016 has been the year of the lolsob. I have my reasons for feeling that way, and I’m guessing you might too. In that light, I’ve especially started looking forward to Dan Meyer’s “pseudocontext Saturday” postsIn each one, he finds a picture from a math book and challenges readers to figure out what math concept is being illustrated or tested with each one. Is a rock-climbing kid illustrating a question about types of quadrilaterals or counting by tens? Does a picture of a dartboard accompany a question about probability, circle sector areas, sequences of numbers, binomials, or the quadratic formula? With connections this tenuous, even if you get the question right, you lose.

What is pseudocontext? Meyer writes, “We create a pseudocontext when at least one of two conditions are met. First, given a context, the assigned question isn’t a question most human beings would ask about it. Second, given that question, the assigned method isn’t a method most human beings would use to find it.” (For my money, the all-time prize for pseudocontext will always be this question from the New York Regents Exam shared by Patrick Honner, though as he states, the story is so flimsy it’s not even pseudocontext.)

Pseudocontext Saturdays don’t just give us an opportunity to lolsob about the bizarre and irrelevant “real-world” questions math textbooks often ask. Commenters can also suggest better questions to ask that go with the picture or that explore the concept the picture was trying to ask about. Felicitously, as I was working on this post, I read Dana Ernst’s post about students generating examples on the MAA blog Teaching Tidbits. That post isn’t about students asking real-world questions necessarily, but it makes me wonder if it’s possible (or desirable) to get students in on the pseudocontext joke: 10 points to Gryffindor for the best math question that would actually relate to the picture in question!

If you’re not already reading Meyer’s blog, there’s a lot more there to enjoy beyond pseudocontext. Meyer is a former high school math teacher who now works for the online graphing calculator Desmos. Though I haven’t spent much time talking math with high schoolers, I appreciate the thought and energy he’s put into figuring out what will reach students the most effectively and how to spur them to ask the questions we want them to be asking about math. As a bonus, his blog is also one of the few places where you can really read the comments. He encourages people to participate and have real conversations in the comments section, often highlighting selected comments in his posts. How refreshing!

The Lure of The Rubik’s Cube

The 3x3x3 Rubik's Cube, can you solve it?

The 3x3x3 Rubik’s Cube, can you solve it?

Who among us has not lost at least one afternoon of their life to that most seductive of toys: The Rubik’s Cube? Originally invented by the Hungarian architect Erno Rubik in 1974, this cube – although apparently not its patents – have stood the test of time.

The beauty of the Rubik’s Cube, much like the beauty of mathematics, is that it seems totally impossible at first. But as soon as you learn the solution, it becomes totally trivial. The problem is to take this jumbled up cube, and perform a series of permutations (by twisting across various axes) to get each face to display a single color. For a 3x3x3 cube there are 4.3252×1019 possible permutations to chose from. That’s quite a lot. But even so, computations taking 35-CPU years by a bank of computers at Google show that the worst possible jumbling of the cube can always be solved in 20 or fewer moves. This maximum number of moves to solve a Rubik’s cube is known as God’s Number.

So this means that for any jumbling, you’re always only 20 moves away from a solved cube. Now you see where things start to get tantalizing. Of course you may not solve the cube perfectly, that is, you might use an algorithm that ends up taking more than God’s Number. But just knowing the solution is so close at hand is already fun. The difficulty then is in coming up with an algorithm to solve the cube, and most methods do this by breaking down the algorithm in to several sets of moves, or “macros.” And these can be best thought of as operations in group theory. We can think of permutations of the cube as elements of a group, R, whose binary operation is concatenation of moves. Then building the macros to solve the cube can be thought of in terms of commutators and conjugates, see this great explainer for the full story.

So, if you are looking for a holiday gift to occupy please your mathematical loved ones: look no further! Math’s Gear will meet all of your Rubik’s related needs with competition grade speed cubes of all dimensions. They even have the really fun looking — but I’ll admit, slightly intimidating — Skewb puzzle cube. And what better to accompany this gift of procrastination than a paper on group theory to go with it, or for the les mathematically inclined, a plain english explanation of the macros and algorithm.


Best of 2016

There were several cool breakthroughs in math this year. My personal favorite involved the famous question of how to optimally stack higher dimensional spheres in space. This year Maryna Viazovska made a critical breakthrough, solving the 8-dimensional case, and several weeks later the 24-dimensional case tumbled too. This breakthrough is an important one because of its applications to coding theory and data transmission. When the result was announced Quanta published a very thorough history of the sphere packing problem that led to the breakthrough.

This year we also found some interesting (and huge!) new primes. The world record for longest known prime is now 22,338,618 digits. This bad-boy is a Mersenne Prime. In September there was also a new world record set for the largest twin primes. If we printed out all the new prime goodness we found this year it would take about 20 reams of printer paper.

My favorite math in pop culture this year was The Man Who Knew Infinity, the film about Ramanujan and Hardy. If you haven’t seen it yet, I urge you to. Several great books about math also came out this year, including Cathy O’Neil’s Weapons of Math Destruction about the dangers of data science, and Margot Lee Slatterly’s Hidden Figures about a group of African American woman mathematicians who contributed to the space race. I just received the latter as a gift for christmas, so you can expect a review of that in the next few weeks.

Worst of 2016

The real computational dunce cap of the year definitely goes to Facebook and their biased newsfeed algorithms that proliferated fake news during an historic and incredibly tense election. Cathy O’Neil did a nice job covering news of all things algorithmic this year before, during, and after the election. In general, this also reminds of the trouble we’ve had with bias in algorithms this year. For example, that algorithm that was supposed to help the legal system by predicting criminal behavior and instead has just contributed to our already incredibly racist justice system. I guess this was the year to remember that algorithms are run by computers, but written by humans.

On the theme of politics, it was a weird and bad year for polling too. I suppose we learned the value of 2 percentage points, and learning is a good thing, but I suspect we also had a false sense of reality going into the elections and that was a bad thing. While the speed with which we can consume infographics and data makes is quicker to digest numbers, it also leaves us with a pretty poor understanding of what’s going on in the margins. The lesson we learned here is that numbers need context.

And finally, the absolute worst of the worst this year (and perhaps a partial solution to the problem of the previous paragraph) was this craziness about the myth of algebra that just won’t seem to quit. I’m talking, of course, about Andrew Hacker and his infamous call to arms against mandatory high school algebra. This year he wrote a book on the subject, and I will concede that he makes a few good points about numeracy and problem solving. But he also makes dozens of horrible points about some made up algebra straw man that forces you to compute azimuths. So, I’m sorry Hacker, I just can’t. We need Algebra. So much Algebra.