So Long, and Thanks for All the Blogs

(You don’t know how long I spent trying to find a word related to math blogging that rhymes with fish.)

April 22, 2013, we launched the AMS Blog on Math Blogs with a calendrically appropriate post about the Mathematics of Planet Earth blog. I have written over 100 posts here about everything from World Tessellation Day to math poetry to specifications grading to Fibonacci lemonade. After five years here, it’s time for me to move on and focus on other projects.

Credit: denipet, via Flickr

I am deeply grateful to the AMS for the opportunity to work on my writing here, first as a postdoc and now as a full-time freelance writer. I got my start in math writing through a AAAS mass media fellowship sponsored by the AMS, and not too long after wrapping that up, they gave me the opportunity to continue blogging here. Being part of the Blog on Math Blogs has helped me get plugged into the online math world, become visible as a math writer, and hone my voice. It has been an honor to work with the AMS through this blog. I hope the AMS continues to prioritize math communication and new math writers through its platform and financial support.

My favorite number is six, so I’d like to share six of the posts I am most proud of from my time blogging here. In chronological order:

Mistakes Are Interesting
Math and the Genius Myth
There’s Something about Pentagons
Beyond Banneker: Resources for Learning about Black Mathematicians
Adding to the Faces of Mathematics on Wikipedia
What Are You Going to Do with That?

And because the Blog on Math Blogs is all about sharing the great things other people are doing in the math blogosphere, here are some of my favorite math blogs I’ve found through my work here. If you don’t have them in your feed yet, do yourself a favor and add them! (I couldn’t just stick to 6, so you get 12 instead. And there are so many other good ones! You should just go through our archives and add everything we’ve mentioned.) In no particular order:

Baking and Math
The Liberated Mathematician
Mike’s Math Page
Fawn Nguyen’s Finding Ways
The Aperiodical
Intersections—Poetry with Mathematics
The Renaissance Mathematicus
Michael Pershan’s many blogs
The Accidental Mathematician
Stats Chat
Mathematical Enchantments

Fear not! The Blog on Math Blogs will continue to bring you interesting writing from around the math blogosphere with Anna Haensch at the helm. If you’d like to keep up with my writing, I will still be writing the Roots of Unity blog for Scientific American. I’ve also started a monthly email newsletter collecting my writing and other links of interest. You can subscribe here. Thanks to my readers for making my time here fun and worthwhile.

Posted in Math Communication | 1 Comment

Radical Notation

There was one day in my life when I got a standing ovation in a calculus class. I’ll admit, it was an extra special group of students who were prone to spontaneous outbursts of enthusiasm. Business Calc, amiright? But it was a day that stands out in my memory. That was the day I went on a long notation based tangent and told them, among other things, the story of the radical symbol. One short version of the story, per Leonard Euler, has √ being modeled after the letter “r”, which is the first letter of the Latin word “radix” which means root. Conveniently, it is also the first letter of the english word root. Other versions of the story say that the shape is inherited from the Arabic letter “ج” and the Arab mathematician Al-Qalaṣādī. But the more interesting substory, is how often notation is arrived at in a totally roundabout or random way.

Folklore abounds, and notations evolve, and the origin of mathematical notation is an endless source of fascinating speculation.

As far as I’ve seen, the most frequently cited text on the subject is A History of Mathematical Notations by Florian Cajori. There’s a really entertaining Math Overflow thread dedicated to notation that makes people “uncomfortable.” It includes some favorites like why is




An inverse function, not a reciprocal, as you would expect if we were playing fair. I can’t blame students for feeling like we’re trying to Numberwang them.

On this blog Division by Zero, Dave Richeson gives a great account of the day the division symbol went viral. I remember that day fondly. Richeson reveals the real story behind that symbol that definitely looks like a fraction with dots in the place of the numerator and deniminator but is actually so much deeper and historically rich.

The notation for division in general is pretty fraught. I always notice my students struggle with the notation a | b for “a divides b” which means that b/a is an integer. It is a bit confounding. As was pointed out on Math Overflow, one should never use a symmetric symbol for an asymmetric relation.

Jeff Miller, a retired high school math teacher, maintains a nice page about first-uses and attributions of various mathematical notation, like matrices, relations and delimiters. For example, did you know the use of the Greek π for that number 3.14159… didn’t show up until 1706 when William Jones just offhandedly threw it into the mix? One guy, without preamble, forever altered baked good consumption in the month of march.

There’s a great post on the Wolfram blog all about the notebooks of Leibniz. It’s a long post, but it gives a great historical account of Leibniz and his relationship to notation and computation — specifically how Leibniz’ calculus ratiocinator is like a proto-wolfram Alpha — with great pictures of his notebooks. Nothing says living on the edge of human innovation like using alchemy symbols in your mathematical notation, while simultaneously laying out the schematic for a universal arithmetic machine!

In case you need to brush up on some of your fancy (non-alchemy) notation, and get that fraktur “g” just right, I am always happy to recommend Old Pappus’ Book of Mathematical Calligraphy.

And then there’s this, my favorite notation themed short story of all time.

Many thanks to everyone on Twitter who send me interesting notation links and anecdotes. Feel free to send along more @extremefriday.

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Math by the Book

Many mathematicians are familiar with Paul Erdős’s idea of a proof from The Book. The Book was God’s collection of the most beautiful, elegant, and deep proofs. (Never mind the fact that Erdős was an atheist.) In 1998, Martin Aigner and Günter Ziegler published Proofs from THE BOOK, a collection of these divine proofs, or at least an “earthly shados” of them. At Quanta, Erica Klarreich recently interviewed Ziegler about the book, which was awarded the 2018 Steele Prize for Mathematical Exposition by the AMS. She also posted about two of her favorite proofs from the book on Quanta’s Abstractions blog.

A collection of math books sitting on a shelf

Credit: the kirbster, via Flickr.

People tend to learn a lot of math from books. But in addition to The Book and the many other math textbooks we use, math also shows up in fiction. College of Charleston mathematician Alex Kasman maintains a website about fiction that incorporates mathematics. There are currently 1259 works on the list, so if you’re looking for a book recommendation, you have a lot to choose from. I recently wrote an entry for the site about one of the less successful (at least in my opinion) such books, Lost Empire by Clive Cussler and Grant Blackwood.

Fiction with mathematical themes and other non-textbooks can help people see math and mathematicians in a different light. KQED’s MindShift podcast recently posted about math teacher Joel Bezaire, who reads The Curious Incident of the Dog in the Night-Time with his seventh grade math classes, and Sam Shah, who has incorporated a math book club into his calculus classes. Math teacher and math education professor John Golden has also used a book club in his university math classes.

If you’d like to join a math book club yourself, the blogger behind Life though a Mathematician’s Eyes started a math book club group on Goodreads.

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Family Math With The Lawlers

Some duplos bring home a clever take on multiplication.

When I watch videos of Mike Lawler teaching math to his sons it makes me want to be a better teacher. Lawler, a mathematician by training and former academic, started Mikesmathpage to chronicle his lessons in homeschooling his kids, and his lessons are a master class in patient inquiry and the art of the slow reveal.

Lawler’s blog is a collection of videos, teaching ideas, tough math problems, and cool tools for bringing advanced mathematical concepts to beginner audiences. Last week I got Lawler on the phone and had a chance to talk to him about his work. Lawler started his career as a professor and quickly learned that the academic life was not the life for him. But his mathematical self found new life when he started homeschooling his two kids. Lawler says, “I lost interest in math, and the kids brought me back in!” He calls his family math, “the math world I would’ve dreamed about when I was in high school!”

And it’s true. While Lawler hits some of the high points of early math education — some of his most popular videos have been short lessons on dividing fractions and why a negative times a negative equals a positive — he and his kids typically are working well outside the realm of K-12 math standards. They are doing this kind of things that can’t help but spark some curiosity in even the most hardened mathphobe.

He typically finds his inspiration by checking what research mathematicians are up to, and seeing how that might be adapted to his kids. For example, he recently attended a lecture on developable surfaces by Heather Macbeth at MIT, and he adapted some of the ideas to do a lesson with his kids. I love when Lawler asks his younger son, “what are some shapes that you know how to make out of a piece of paper?” And his son bypasses the cylinder and goes straight for the Mobius strip.

Lawler has lots of posts and videos devoted to working through competition math problems. “I grew up in math competitions, I was on the MIT Putnam team, so I really enjoyed it,” says Lawler, “my kids are not big math contest kids. The reason I do a lot is because the problems themselves are really good.” One such problem that generated some great insights from Lawler and his kids was a problem from the European Girl’s Math Olympiad about snails in the plane.

Inspired by attending a talk by Conrad Wolfram at the Computer Based Math Education Summit, Lawler has also started doing some computer math with his kids. In one such post,“Computer Math and the Chaos Game,” he walks his kids through a cool coding exercise using Khan Academy’s coding interface (I didn’t know about this tool before today; it’s totally cool). The video, embedded below, of his kids playing with the chaos game and catching the surprising reveal (I won’t spoil it for you) actually made me laugh out loud with glee.

Visit Mikesmathpage and you will see that there is more of where that came from. If you’re ever having a day when you feel sad about pedagogy — sometimes I have those — a few minutes of family math will definitely get your head back in the game.

Posted in Math Education | Tagged , | 2 Comments

Genius Revisited

Three years ago, I wrote two posts (post 1, post 2) about math, the media, and the genius myth, the idea that in order to be successful in math, you have to be born with some particular talent. They’re good posts, if I do say so myself, and as math hasn’t rid itself of the genius narrative in the intervening years, they’re still relevant.

“The Inspiration of Genius” by Jules-Clément Chaplain. Credit: Public domain, via the Metropolitan Museum of Art

I have been thinking about the genius myth recently because of some posts I’ve read about genius and identity in the math blogosphere. Most recently, Jim Propp’s post “Genius Box” talks about the complicated relationship he has had with the concept of genius in mathematics. Another post I’ve been thinking about this this one from Piper Harron about her objections to being labeled as a genius.

Something I have been seeing more and more in writing about the idea of genius and in neighboring discussions such as #MeToo is an acknowledgment that it’s easy to focus on the art, math, or science created by those who were able to thrive in an environment and worry that changing practices would deprive us of those things, but it’s impossible to see the art, math, or science that would have been created by the people who were pushed out of the field. That is something that I wrestle with when I read about early women and members of other groups that are underrepresented in math and which I tried to flesh out in a post last year about Sophie Germain. And of course, our loss of the products people would have created is not the chief wrong in this situation, and thinking that way risks commodifying other people. People who wanted to be mathematicians but were pushed out were deprived of the opportunity to do activities they wanted to do and thrive in a way that they were interested in thriving.

Along with the genius myth, I have been thinking about the idea of identity in math and identities as mathematicians. Last fall, UK math(s) teacher Ed Southall, author of the blog Solve My Maths, wrote about his struggle labeling himself as a mathematician. The word has baggage related to genius, speed, and tricks that made him hesitant about whether he should call himself a mathematician. I have seen this same question come up on Twitter, recently from Kate Owens.

In departmental orientation in graduate school, the then chair of the department (who later became my advisor) told us all, “You are mathematicians.” We were paid to think about and tell people about math; therefore, we were mathematicians. Today I would probably not center the role of money; the facts that we were choosing to spend our time thinking about math and had been accepted into a program where we would be trained as mathematicians and teach math to others were the salient points. Regardless, my advisor’s framing of me, a naive first-year graduate student, as a mathematician helped me view myself that way. I won’t claim I never struggled to see myself in academic math research (and I eventually stopped doing academic math research), but I did not worry that I was misusing the word mathematician by calling myself one.

Another interesting post about mathematical identity from Piper Harron asks whether we can improve the way we tell undergraduates what it is their math professors really do. Too many students don’t consider a math major because they don’t want to be primarily calculus teachers. Can we tell stories about people’s different paths into math and mathematical careers that will broaden students’ conceptions of who does math and what mathematicians do?

How do you think about genius and identity in mathematics?

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Some Math About Guns

Trying to get a clearer picture of gun violence in the US. Image courtesy of Mike Maguire via Flickr CC.

Turns out it can be really difficult to understand our collective relationship to guns, gun violence, and gun control. What seems to be obvious to some, runs completely counter to others. This was illustrated nowhere better than in the recent report out of the RAND corporation on gun policy. It studies all sorts of relationships between our attitudes about guns and our impressions of the state of gun violence in the US. An article in Vox gives a really thorough summary of the RAND report, and leaves one with two major impressions: (1) we don’t have nearly enough research on gun policy, and (2) despite the fact that there should be plenty of data about this stuff, opinions about what makes us safe seem to be totally subjective.

When faced with something like the incredibly politically divisive debate around gun violence happening in the US now – and always – it’s helpful to quantify.

Mark Reid, who writes the matching learning and statistics blog Inductio Ex Machina, recently posted some data and plots relating gun ownership to gun violence. He sourced the data form Wikipedia and wrote the plots using the statistical computing software R. A quick glance at the plot below shows that the US owns a whole lot of guns and has a whole lot of gun violence.

Gun deaths per capita versus gun ownership in OECD countries compiled by Mark Reid for Inductio Ex Machina.

Reid’s post also includes several other plots, some that incorporate the non-OECD countries, and some that differentiate between gun deaths and gun homicides. The comments section of Reid’s post is also full of alternative questions prompted by the data – like what’s up with Switzerland? – and lots of useful links for similar analyses.

Based on the same data set, Kyle Kinsburg, who writes the blog Aphyr (pronounced “AY-fur”), recently published several plots relating gun death, gun ownership and economic inequality. In particular, he compares gun homicides to Gini index, which produces a linear looking relationship. This isn’t exactly news, we’ve known for a long time that income inequality is correlated to violent crime for all sorts of reasons. Kingburg does point out that there is something interesting to be observed here about gun homicides and prevalence of guns, namely, prevalence of guns doesn’t tell the complete story. For example, Brazil and Argentina have the same prevalence of guns, but Brazil has nearly 10 times more violent crime.

Graph of gun deaths versus the Gini index complied by Kyle Kingsbury for Aphyr.

The R code for Kingbury’s plots are available on his blog, and the data for gun ownership and gun deaths is available on Wikipedia or as .csv files on Reid’s blog. As Reid points out, and I feel obliged to reiterate, this isn’t a rigorous analysis, but it’s cool that we have the tools and technology to get a reasonably quick quantified sense of the problem.

If this sort of data interests you, last year the podcast Science Vs did an episode about guns that includes a good analysis of the data surrounding guns. It is definitely worth a listen, and it draws attention to the relationship between gun ownership and suicide.

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Blind Review Review

Theoretical computer scientists have been talking about double blind peer review, and it’s an interesting discussion. The current incarnation of this discussion started when Rasmus Pagh and Suresh Venkatasubramanian used a double blind refereeing process for submissions to the ALENEX18 conference they co-chaired. Venkatasubramanian posted about their motivations and how they pulled it off in two posts on his blog, The Geomblog (post 1, post 2, post 3).

Artist’s rendition of blinded peer review. (Justice, from the Cardinal Virtues by Nicolaes de Bruyn) Credit: Public domain, via the Metropolitan Museum of Art

Why double-blind? First, it’s the standard for computer science conferences outside of the theory subdiscipline. More importantly, many people worry that single-blind peer review, where the reviewer knows the identity of the author, leads to some objectionable outcomes based on implicit and explicit biases. More famous authors or authors from more prominent institutions may have their work reviewed more favorably, and more broadly, the bias in favor of these authors combined with other biases reviewers have can continue systemic bias against women and other groups that are underrepresented in the field.

Obviously, a major change in the paper submission system is not without controversy. The discussion has continued in posts by Boaz Barak, Michael Mitzenmacher, Omer Reingold, and Lance Fortnow. In general, the conversation I have seen has been civil and thoughtful. In one post, Venkatasubramanian writes,

First up, I think it’s gratifying to see that the the basic premise: ‘single blind review has the potential for bias, especially with respect to institutional status, gender and other signifiers of in/out groups’ is granted at this point. There was a time in the not-so-distant past that I wouldn’t be able to even establish this baseline in conversations that I’d have.

“The argument therefore has moved to one of tradeoffs: does the installation of DB review introduce other kinds of harm while mitigating harms due to bias?

A few math journals—mostly in math education and undergraduate research, as far as I can tell—do use double-blind peer review. But it is not standard. One of the biggest barriers to double blind reviewing in computer science, physics, or math is the fact that so many preprints are posted on arxiv or authors’ websites before they are submitted, making it that much more difficult for a reviewer to avoid knowing who wrote the paper. (Venkatasubramanian writes about how they dealt with that problem in his posts; one point he makes is that double-blinding the process won’t necessarily prevent reviewers from being able to determine authors eventually, but it could prevent some knee-jerk reactions. He also points to a post by Regina Barzilay that delves into the issue in more depth) In some fairly narrow subdisciplines, there are few enough researchers that even without seeing the paper online, others in the field will be able to tell who wrote it anyway.

While societies and individual humans in them have biases, there will be no way to completely eliminate these biases when people (or algorithms) make decisions about paper and conference submissions. It is important for academics to look at the advantages and disadvantages of different strategies to mitigate the effects of bias. I am looking forward to seeing how this conversation evolves.

Posted in Publishing in Math | Tagged | 4 Comments

Are Smart Cities Really That Smart?

The fun thing about a smart city is that when you watch it, it might be watching you back. Image courtesy of aotaro via FlickrCC.

Lately I’ve been reading a lot of science fiction, and also a lot of articles about smart cities. And the two seem to be converging to a single point. I’m not entirely sure what “smart city” means as a term of art, but it seems to have something to do with using technology to make a city more adaptive to its inhabitants, and thus to serve them up with a better, healthier, richer, and safer city. From what I’m reading, I’m not entirely sure this is happening.

On the one hand, increased technology has given us access to a better understanding of what humans need. One could easily argue that tracking the movement of people en masse, and studying traffic data available from smartphones should help develop better roadways and infrastructure to serve humans. And one would not be wrong. The MIT SenseableCity lab’s Global Mobility Index gives a gives cities a measure of movement that helps guide their funding towards the right resources: bike shares, car services, public transport. One could also see that knowing such granular details about where people are moving is not always in the best interests of humans. For example, look at the kind-of-hilarious-if-it-weren’t-so-scary Strava debacle of earlier this year.

Cyberpunk science fiction writer and Wired blogger Bruce Sterling says “stop saying ‘smart cities.’” Sterling argues that cities being touted as “smart”…well, they aren’t. They’re just a magnet for capital. He talks about the bygone notion that the internet boom was creating a “flat-world” where equal access to the internet would be the final and ultimate democratizer and unifier. Certainly we have seen access to wifi technology open up avenues for healthcare and create economic inroads in developing nations. But Sterling argues, that the so-called smartness is gutting cities by prioritizing the needs of big tech giants over the needs and wants of the citizens. Instead of using technology to tabulate citizen input and make decisions in accordance with their voiced wishes, they are using technology to track citizen movement and consumer habits and make decisions as their proxy (from which the big companies involved stand to profit in a big way). It seems I read a book about this once, it didn’t end well for those involved.

Sterling may be overstating the case a bit, but not by much. Already there are some mega-creepy surveillance programs being sanctioned in the smart cities of China. These programs follow people’s movements online and IRL to generate a “social credit.” Much like a traditional credit score — which BTW is already totally fraught on an Orwellian scale — will determine what sort of opportunities and freedoms a person is entitled to. And again, it seems I’ve seen a dystopian show about this somewhere. And again, it didn’t end well.

Having said that, technology can bring some good to cities. Ridesharing services, have proved to be a reasonable driver of infrastructure funding in cities. Chicago has already raised massive municipal funds by collecting a surcharge on all Uber rides, and New York is poised to do the same. Of course Uber is not without its share of gloom. A recent study out of Stanford giving some of the metrics on driving for Uber was the subject of a post on the blog TheRideShareGuy.

In my own smart-ish city of Pittsburgh — which has made it to the short list of possible new homes for Amazon’s second headquarters — I’ve seen technology bring a renewed vitality to the metropolis. The launch of the Steel City as a testing ground for self-driving cars was a mixed bag. You can consult with blogger Laura McLay on PunkRockOR on whether or not automation is really a smart choice.

For tons of spooky articles about our cyberpunk futures, The Atlantic is currently running a series on smart cities. And if you need me, I’ll be out in a field somewhere wrapping my entire body in aluminum foil.

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Gold Medal Math

For the past week and a half or so, my computer browser has been finding its way to NBC’s Olympics coverage while I’m supposed to be doing other things. I might have a different answer the next time I watch a Simone tumble through the air or shoot through the water, but right now after seeing Chloe Kim and Nathan Chen defy the laws of gravity, I’m inclined to agree with June Thomas at Slate that the winter Olympics are the best Olympics.

Credit: Jankenhoi, via Wikimedia Commons CC BY-SA 3.0

As a bonus, I can justify my short-term skate, ski, and snowboard obsession by reminding myself that winter Olympics events are exhilarating, gorgeous examples of applied math, physics, and engineering.

I wrote about the physics of figure skating jumps for Smithsonian, and I’ve enjoyed reading about the math and physics of other events as well. Jen Ouelette wrote about taking a curling expedition with a group of physicists for her blog Cocktail Party Physics. Dina Spector explained why speed skaters swing their arms back and forth for Business Insider. Larry Greenemeier at Scientific American wrote about how the U.S. skeleton team tested their equipment and body positions in a simulator at Rensselaer Polytechnic Institute. Big air snowboarding, a new Olympic sport this year, is a physics marvel, as Scientific American and Wired have explained. Teachers who want to use the Olympics in their classrooms have some suggestions from the New York Times Learning Network and the American Association of Physics Teachers.

The sports themselves are where most of the magic happens, but I have also enjoyed learning about some math and physics behind the scenes. For instance, did you know we don’t actually know why ice is slippery? F Yeah Fluid Dynamics explains some of the theories and controversies in the first post of her series about the winter Olympics. And the National Institute of Standards and Technology explores one of the most important behind-the-scenes parts of the Olympics: precision measurements. FiveThirtyEight has joined the fray with medal forecasting and data-driven stories about Olympic sports. I was particularly interested in Ella Koeze’s analysis of what might happen if men and women competed against each other in skiing events. Finally, the Olympic rings themselves have some math to them. I wrote about the topology of the connected sum of four Hopf links in the summer of 2016.

Enjoy the rest of the games!

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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.

Posted in Current Events, women in math | Tagged , , , | Leave a comment