On Michael Atiyah and the Riemann Hypothesis

Posted on October 1, 2018 by racheljcrowell
Riemann

Riemann has been back in the news lately, thanks to an announcement that his nearly 160 year old hypothesis might be solved.
Public domain image courtesy of Wikimedia CC.

At the 2018 Heidelberg Laureate Forum (HLF), Sir Michael Atiyah gave a lecture in which he claimed to have found a proof for the Riemann hypothesis. If Atiyah’s proof holds up, then the nearly 160 year problem concerning the distribution of primes will finally have a solution.  It’s on the Clay Mathematics Institute’s list of seven Millennium Prize Problems and just one of those — the Poincaré Conjecture — is listed as solved on the institute’s website. However, the $1 million prize for a proof of the Riemann hypothesis is yet to be awarded, and some folks doubt that the long-open problem is finally solved.

It’s unsurprising that the world is watching Atiyah — and waiting to hear the verdict about the proof he proposed. Atiyah received a Fields Medal in 1966. In 2004, he and Isadore Singer were jointly awarded the Abel Prize for their discovery and proof of the Atiyah-Singer index theorem. Yet many other supposed proofs for the Riemann hypothesis have been proposed, only to fall apart under further scrutiny.

The Clay Mathematics Institute’s “Official Problem Description” for the Riemann hypothesis is 11 pages long (with nearly two full pages of references), while Atiyah’s current write-up of the proof is five pages long. “Atiyah attributes much of the theoretical work that underpins the proof to a paper of his own that has been submitted to the Proceedings of the Royal Society A,” Frankie Schembri wrote for Science. “That paper has yet to be published,” she added. In fact, Atiyah’s list of references includes just three works: the aforementioned unpublished paper, his 2018 Abel lecture at the ICM in Rio de Janeiro and Friedrich Hirzebruch’s 1966 Topological Methods in Geometry.

News of Atiyah’s claim has reached far and wide since his announcement. Articles about the development appeared in Science News, Gizmodo, Popular Science, NBCNews.com, the Irish Times and more. Bloggers also covered the announcement, including Katie Steckles and Christian Lawson-Perfect for the Aperiodical, John D. Cook for his consulting blog, as well as Dick Lipton and Ken Regan for Gödel’s Lost Letter and P=NP.

John Baez also wrote a lengthy thread about Atiyah’s claimed proof on Twitter, starting on September 23 (the day before Atiyah’s lecture at the HLF). He wrote “I bet that Atiyah’s claimed proof…will not convince experts. In 2017 he claimed to have a 12-page proof of the Feit-Thompson theorem, which usually takes 255 pages: https://www.maths.ed.ac.uk/~v1ranick/atiyahtimes2017.pdf. He showed it to experts, and… silence.” What’s more, “In 2016 Atiyah put a paper on the arXiv claiming to have solved a famous problem in differential geometry. The argument was full of big holes: So, I’m not holding my breath this time. But of course I’d be happy to be wrong,” Baez added.

Steven Strogatz tweeted “Uh oh. I have a bad feeling about this. Famed mathematician Michael Atiyah claims proof of Riemann hypothesis,” along with a link to the September 21 New Scientist article about Atiyah’s claimed proof. When asked by Twitter user Jonathan Horrocks “Why bad? I take anything Atiyah says very seriously,” Strogatz responded “I take him seriously too. It’s the same bad feeling I have when a diver attempts an extremely difficult dive: afraid, yet hoping for success.”

The skepticism surrounding the claimed proof doesn’t appear to faze Atiyah. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he said to Gilead Amit​, author of the New Scientist article. “People say ‘we know mathematicians do all their best work before they’re 40,'” Atiyah said, adding “I’m trying to show them that they’re wrong. That I can do something when I’m 90.”

Regardless of whether Atiyah’s proof holds, he has already done something phenomenal: reignited a worldwide conversation about the Riemann hypothesis. That alone is something to celebrate.

What do you think? Share your thoughts in the comments below or reach out to me on Twitter @writesRCrowell! As always, I’m also happy to hear ideas for future blog posts.

Posted in Current Events, people in math, Uncategorized | Tagged , , , , , , , | 16 Comments

The Thing Last Week With That Sexist Paper

Posted on September 24, 2018 by annahaensch

Once again the mathematical world is rocked with scandal. Let me get you quickly up to speed. It started when a controversial paper on the variability hypothesis was accepted to the Mathematical Intelligencer. Shortly thereafter, University of Chicago mathematics professor Amie Wilkinson sent an email to the editor in which she “criticized the scientific merits of the paper and the decision to accept it for publication.” The paper never appeared in the Mathematical Intelligencer, but did eventually turn up (one author less) in the New York Journal of Mathematics.

Without going to deeply into the details, the paper proposes mathematical models to support the variability hypothesis, which essentially says that if one gender is more selective in mating, then the opposite gender will have greater genetic variability. For us humans, it’s thought that females are more choosy with their partner since they spend more time (for reasons biological, social, historical, etc) nurturing the offspring, and consequently the males end up with more variability. This is why — allegedly — there are more men with very high and very low intelligence, and women just tend to clump around the middle. You probably remember when Larry Summers suggested something along these lines a few years ago. That landed like a ton of bricks.

And so too has this recent paper. Beyond the weird publishing bait and switch, the disappearing coauthor, and the sort of dubious job of editing and refereeing, the paper seems to be just fraught with baggage of every possible sort. First, there’s the sexist baggage that comes promoting a controversial theory in what seems like a pretty unsubstantiated way. Terry Tao wrote a short post for What’s New about the whole affair, and if you’re not convinced of the existence full-blown gloves off misogyny, go spend five minutes in his comments section. Timothy Gowers also takes a deep dive into the whole mess on his blog. Again, check out the comments over there but take a deep cleansing breath before you start.

In the meantime, Hill has published a post on Quillette giving his own blow-by-blow of the events. Not only that, but Hill has set up a public dropbox full of documentation and correspondence relevant to the episode.

Oh, the intrigue!

And even later in the week an inspired rejoinder came from Lior Pachter, a computational biologist who writes the blog Bits of DNA. He starts by really taking the math community to task saying, “imagine the hubris of mathematicians spewing incoherent theories about sexual selection when they literally don’t know anything about human genetics or evolutionary biology, and haven’t read any of the relevant scientific literature about the subject they are rambling about.” Ouch!

But Pachter has a good point. Mathematicians and biologists don’t communicate enough, despite the fact that there are so many fundamental and interesting conversations we could be having. He also goes on to make many very thoughtful points about the dangers inherent in “hijacking” the publishing process to push a personal agenda. Specifically a sexist one. The post is great, and the comments section will have you either grimacing, fist pumping or high-pitched squealing, depending on your temperament.

So the conclusion I draw based on the posts I’ve seen is that it’s a confluence bad math, bad biology, an epic fail on the part of the publishing process, and some people just need to delete their account. You disagree? You can let me know @extremefriday.

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On The ‘Math Section’ Blog

Posted on September 17, 2018 by racheljcrowell
Elias Wirth

Photo courtesy of Elias Wirth

Swiss mathematician Elias Wirth created the “Math Section” blog earlier this month. Even though the blog is new, he’s already written several interesting posts, like this one about using the mean value theorem to catch speeding motorists. In an interview conducted over email, Wirth shared more about his blog and recommended resources for other math bloggers. (Note: The following interview has been edited for length and clarity.)

Rachel Crowell: What would you like to share about your own math background, including your interests and research?
Elias Wirth: I just finished my bachelor’s degree in mathematics at the University of Berne and am now looking forward to beginning my master’s degree in applied mathematics at the ETH Zurich. During my years in Berne, I developed a fascination with analysis, complex analysis especially, and applied mathematics. As a preparation for my master’s degree at the ETH, the applied mathematics professor and I then decided that I was going to write a thesis titled Linear Multistep Methods to improve my knowledge of numerical analysis.

At the moment, I am mostly interested in finding wonderful applications of mathematics in everyday life. Partial differential equations are also something that I want to spend more time on because the topic seems to be extremely diverse.

RC: I read that you launched your blog in 2018. How long has it been since you first launched?
EW: The blog was released on September 2. Everything is new to me, but the worst part is over and I can now focus on writing articles.

RC: What inspired you to start a blog?
EW: I already mentioned my bachelor’s thesis. I wrote the first draft in a week and a half, not because I had to,  but because I wanted to. During the writing process I was able to enter a state of concentration that was really satisfying. I woke up in the morning and started writing and would not stop until I went to bed. (During my last semester, I didn’t attend many lectures. Thus, I was able to basically take two weeks off and focus on writing.) I had never before learnt so efficiently and with so much joy. The idea of starting my own blog probably originated during that time. All that I needed to really go for it was one last push. This final push came from my co-worker telling me about his fashion blog and how much fun it was to write for others. It was at that moment that I knew I just had to go for it.

RC: What are some of the things you have learned about communicating about math since you first started your blog?
EW: There are mainly three things that I learned. First, I learned that it is really important to write a good title. People get flooded with information on a daily basis and the title needs to capture their attention. Second, I am surprised by the number of people that are really interested in reading about applications of mathematics. A lot of people from all over the world are eager to broaden their knowledge, which is just wonderful. Third, there are a lot of people that have already established their own blog, website or Facebook page that are very friendly. They are willing to support a new blog like mine if they feel like one is serious about informing people about mathematics.

Continue reading →

Posted in Applied Math, people in math, Uncategorized | Tagged , , , , , , | 1 Comment

Seeing The Future From The Past

Posted on September 10, 2018 by annahaensch
Hop in my Delorian and we’ll travel back in time and appropriately tweak our predictive models.  Imagine courtesy of William Warby via FlickCC.

I just finished reading The Signal and the Noise, a book about predictions by the American statistician and blogger turned big time data journalist Nate Silver.  I highly recommend it.  The book came out in 2012 and there was some sort of meta-instructive quality to reading a book whose main theme is using the past to predict the future, written in a past that was (mostly) unaware its own future.  Still with me? 

For example, reading anything written about correctly interpreting political polling in the pre-2016 world induces the reader to struggle under the weight of her own misconceptions.  This in turn causes her to stare off into the distance for one dim moment and think about all of the futures that could have been but at the same time were never destined to be.   Then she shakes it off and keeps reading.

A theme that Silver returns to repeatedly is the impact of our own biases on our ability to interpret and deliver statistical predictions.  Our biases almost inevitably seep into our mathematical models, like which variables we choose to include, how heavily we choose to weight them, and how willing we are to adjust models as we move through time.  Bias has been a hot topic for Google lately, who has been in some hot water for its biased human-programmed (and not actually a magical oracle at all) search results that autocomplete and prioritize some really racist search results.  This could be compounded as Google and others become more reliant on AI that’s trained on fallible human data and the output gets simultaneously farther from an actual human brain but somehow more deeply steeped in human thought. 

The blog Overcoming Bias, written (mostly) by Robin Hanson explores a lot of the places where bias interferes with our understanding of predictive modeling.   In his most recent post Hanson explores the idea of introducing what he calls news accuracy bonds to combat the spread of fake news.  The basic idea is that each article comes paired with a token, or bond, and a reader can get the bond by provably demonstrating that the article is false.  The article goes into more of the details on how this might work, but the basic idea is that an article with a high value bond is more likely to be true.  A high bond values indicates a high degree of certainty on the part of the publisher.  

A related idea has been put forward by Facebook to introduce reputation scores, where effectively users’ scores get decreased every time they post content that the company considers suspicious.  It sounds a little bit spooky — and maybe a bit too reminiscent of a certain episode of Black Mirror — but the idea is similar to Hanson’s in that number attached to an article or user profile will act like a confidence interval by the platform or publisher.  

What do you think?  Would you trust Facebook to responsibly sort dubious information?  Do you ever think about how your biases from the past make it into your assessment of the present or predictions of the future?  If you need me I’ll be in the garage tinkering with my time machine, or as usual, you can find me on Twitter @extremefriday. 

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On Tricurves

Posted on September 4, 2018 by racheljcrowell

rainbow triangles

If the straight edges of these colorful triangles were made curved, they would become tricurves, a shape which Tim Lexin described in a recent blog post. Image courtesy of Solkoll via Wikimedia Commons.

Tim Lexen, a mechanical engineer in Cumberland, Wisconsin, wrote a post about tricurves for the Aperiodical. As their name implies, tricurves are sort-of triangle cousins which have three sides, but instead of having three straight edges, each of their sides are curved.

“Of the two ancient tools, I preferred the compass over the straightedge. I was fascinated with the classical geometric constructions, the intersecting circles and arcs. As a simple personality test, preferring a compass over a straightedge might mean something: maybe roundabout-holistic-intuitive more than straightforward-linear-realistic. At any rate, the pursuit of curves eventually led me to this topic,” Lexen wrote.

He invites us to imagine a plane which is tiled with repeating identical triangles. The picture starts to get interesting when he begins adding curves.

“But something unexpected happens when we add curves. For tiling we need equal amounts of concave and convex arc. The only way to do that is with two shorter concave sides joining a longer convex side,” he wrote. His post shows a variety of tricurve constructions, from the 30-60-90 tricurve to the 45-135-180 one.  He shows beautiful examples of periodic, non-periodic and radial tiling of tricurves.

Last year, Paul Bourke wrote a post inspired by Tim Lexen on his website. “There are a number of ways one can define a tricurve, the one used here is to start with an arc of some angle, replicate two identical curves ard [sic] rotate each about some angle about the ends of the arc. The Tricurve is the enclosed area,” Bourke wrote. Besides showing more examples of tricurves, he also links to a 2017 paper by Lexen about the shape, it’s tiling and variations, a method for finding its area, and more. (“This is intended as an informal paper. I am freely presenting the idea, for what it is worth; and I am soliciting feedback from any interested readers,” Lexen noted, adding that he can be reached via email at novustcl@charter.net.)

In a different post for the Math ∞ Blog, Lexen wrote about these and other curved shapes from the perspective of designing a flat puzzle that’s more interesting than one “with dozens or hundreds of identical pieces [which] may sound a little dull and predictable.” In fact, Cherry Arbor Designs now offers a wooden tricurve puzzle.

(While a few of the above mentioned posts provided a link to the National Curve Bank entry on tricurves, I was disappointed to find the link broken. This left me wondering if another curve bank has taken its place or if we are now living in a curve-bank-less world…Anyone know of similar current resources? If so, please reach out in the comments below, on Twitter @writesRCrowell or via email RachelJCrowell@gmail.com.)

John Golden used GeoGebra to create a related applet for tiling lenses. Check it out here.

Circling back to the Lexin’s original statement about preferring the compass over the straight edge, my question to you is “Are you Team Compass or Team Straight Edge?” Let me know in the comments below or on Twitter.

As for myself, I’m Team Straight Edge. This position dates back to my preschool years when I told my sister “I may not be a good draw-er, but at least I can draw a straight line.” (My drawing skills have since improved, at least marginally.)

 

 

Posted in Mathematics and the Arts, Recreational Mathematics | Tagged , , , , | 1 Comment

The Fat Tech Cat Diet

Posted on August 28, 2018 by annahaensch
This got me thinking about the things that live inside my phone.

Like much of the world, I seem to live in a permanent state of vexation about technology, privacy, and how to survive in a world where so many access points are guarded by hungry algorithm crunching data trolls. This is not a new anxiety for me, I’ve blogged here before about some of the privacy concerns of smart cities, the privacy choices made by Apple in the past, and even the benevolent hackers protecting us.  But technology moves on, and the fat cats of the tech sector seem to be getting fatter on a steady diet of user data. 

Well this certainly doesn’t inspire confidence.

This topic has been especially hot on my mind this summer, because in an unexpected move (precipitated by this) I switched from an iPhone to a phone with an Android operating system and have been questioning the implications to my privacy ever since.

I mean, wow, I’ve had to click “I Agree” to a whole lot of things the last month.

But as it (shockingly) turns out, it doesn’t quite matter whether I agree or not.  A report from the Associated Press revealed that Google apps store a time-stamped register your locations, even when you specifically turn off location services.  So, for example, even if your “location history” is turned off, every time your gmail app pings a tower, the time and location is saved in your history.  That’s frustrating.

So maybe you think it’s smarter to use an iPhone and stay away from Google’s proprietary apps.  Well, I have bad news for you.  Apple, who has always differentiated itself by loudly proclaiming its commitment to locking up all of your private data inside your phone that it can’t be touched even by the engineers at Apple, has a new (as of iOS10) privacy scheme.  And the math around it doesn’t look very good.  The scheme is called differential privacy, and in a nutshell, Apple now sends all of your local data back to the mothership, but mixes it with enough noisy data that (in theory) your data could never be tied back to you personally.  This is an opt-in scheme, and when you agree to it, you are agreeing to a data sharing budget of epsilon per day.  That is, there is an upper bound, epsilon, to the amount of data they will harvest each day.

However, mathematicians have shown that even for a fixed epsilon, the amount of privacy being lost is not really something to be proud of, in fact they show that the amount of privacy you can lose each day is unbounded.  A post from Andy Greenberg at Wired gives a good rundown on some of the research that has been done on the shortcomings of the algorithm.

What also really bugs me about this, is how apparently nonchalant Apple is about dealing with the criticism.  I mean, I’m torn.  On the one hand, Google is standing there out in the open with its grabby robot hands taking all of my data and I can’t stop them.  And in some sense, Apple is doing the same thing but just pretending it isn’t.  

Cathy O’Neil, the longtime blogger and now frequent contributor to Bloomberg has done a lot of writing about big tech companies and their questionable algorithm practices.  Recently, O’Neil wrote about a set of proposals from Mark Warner regarding data privacy.  Specifically, what the government might do to limit who gets to access your data and what they get to do with it.   For the algorithms that have big control over your life, Warner recommends a system of algorithm auditing by human (because in case you missed it, algorithm bias is a very real thing). Most recently, she wrote about what Zuckerberg and his fellow cats can do to reel in the powerful and dangerous beast they’ve created.

What motivates your personal decisions on privacy?  Do you think about what operating system you use and does it vex you every day?  Let me know in the comments, or as usual I’ll be tweeting from my underground bunker @extremefriday.

Posted in Data Science, Mathematics and Computing | Tagged , , , , , | Comments Off on The Fat Tech Cat Diet

Musings on a Mathematician’s Duties

Posted on August 20, 2018 by racheljcrowell
Diver figurines on fruit

Come with me to take a dive into the abc conjecture and the ethical duties of mathematicians! Image courtesy of HaPe_Gera via Flickr CC.

As I mentioned in my last post, I wish a genie would grant me thorough understanding of the proof Shinichi Mochizuki proposed for the abc conjecture. Much of this wish is motivated by a desire for the divisive debate to be settled concerning the more than 500 page text. Last year, it was announced that Go Yamashita had written a summary of Mochizuki’s proposed proof, but that summary was still 294 pages and didn’t settle debates about Mochizuki’s potential proof.

In July, Ivan Fesenko, who has organized conferences on the inter-universal Teichmüller(IUT) theory that underlies Mochizuki’s proposed proof, released a document titled “Remarks on Aspects of Modern Pioneering Mathematical Research,” which heavily focuses on Mochizuki’s IUT theory and the abc conjecture.

Continue reading →

Posted in Math Communication, Number Theory, Publishing in Math, Uncategorized | Tagged , , , , | 1 Comment

On Math Anxiety

Posted on August 14, 2018 by annahaensch
Just below the placid surface is an infinite wellspring of anxiety. Image courtesy of Christopher Michel via Flickr CC.

Math anxiety is so real.  We’ve always kind of known it, but a study confirmed it in 2017. By reading the brain functioning of math anxious and math non-anxious people while performing simple arithmetic problems, the research shows that people are better at automatic problem solving when the parts of their brains associated with math anxiety aren’t activated.  So that feeling you get when someone asks you to quickly multiply two numbers and you just stare at them, tearing up, like a sad deer in headlights…that’s real. 

So we know it’s real.  And we know that a lot lot lot of people feel it (disclaimer, sometimes I have really had bad math anxiety…sometimes math still makes me cry) but that doesn’t change the fact that everyone has to get through some amount of math education.  This means we need to teach math in a way that minimizes the stimulation of that anxious brain and maximizes the release of those glorious math fueled dopamines. 

Stan Yoshinobu, who writes The IBL Blog, published a collection of student voices on math anxiety. The student voices will likely sound familiar to you, and what doesn’t surprise me at all is the the fact that the quotes Yoshinobu collected come for college students but I’ve heard the same words coming from the mouths of people 50 years removed from formal education.  People carry that anxiety with them forever and it’s so intense. 

This is bad situation, since studies have also shows that math anxiety is contagious; math anxious parents pass on their attitudes and mindsets to their children and perpetuate the problem even with the best of intentions.  Also, a recent study of elementary school teachers showed the impact of mindset on student outcomes. Teachers with a growth mindset produce more successful students.

As a way to mitigate the anxiety, Yoshinobu blogged about the iceberg diagram for recognizing and addressing student anxiety. In the post, he points out some of the nonverbal and non-obvious ways that student anxiety manifests and how they can be addressed.  One point he brings up is the interplay between math anxiety and other attitudes, beliefs, and mindsets that can inhibit learning.  Looming large among them is stereotype threat, the idea that people “like you” aren’t good at math so you won’t be either.

An interview with one of the authors of the 2017 study, Sian Bailock, also discusses stereotype threat and how to avoid introducing it to children.  Bailock’s research focuses on the various types of situations —  whether in math, sports, or life — that can “rob us of the cognitive horsepower that we need to succeed.”

Whether you’re teaching university level abstract algebra, or helping your kid add fractions at the kitchen table, the anxiety can be just below the surface — yours or theirs, depending on your own level of mathematical trauma.  This semester I’m going to try and pay more attention to student buy-in and intentional mindset conditioning throughout the semester.  So I’m curious, what sort of exercises do you do at the beginning, middle, end of the semester to help improve attitudes and beliefs in your classes?  Let me know here in the comments, or on Twitter @extremefriday.  

Posted in Uncategorized | 2 Comments

What Wish Would You Ask a Math Genie to Grant?

Posted on August 6, 2018 by racheljcrowell

Genie figurines with a magic lamp

Credit: Image via Flickr CC Brian Neudorff

“If a genie offered to give you a thorough understanding of one theorem, what theorem would you choose?” blogger John D. Cook recently asked on his @AnalysisFact Twitter account. Responses ranged from the names of theorems to questions about the genie’s potential trickery to creative ways of gaining insights beyond an understanding of one theorem.

Cook summarized the results on his blog. In an interview conducted over email, he answered my questions about his genie idea. (The following interview has been edited for length and clarity.) Continue reading →

Posted in Math Communication, people in math, Recreational Mathematics, Theoretical Mathematics | Tagged , , , , , | 6 Comments

Welcome To The Blog

Posted on July 23, 2018 by annahaensch

Our new blogger takes a walk with her service dog Delilah.

If you’re new to the blog, welcome. We are the Blog on Math Blogs, your premier destination for blog reviews, tours of the mathematical blogosphere and all the cool math stuff that lives on the internet (omg there’s so much of it this internet thing is never-ending). And beginning next month we’ll be joined by a new co-editor, Rachel Crowell!

Just like Evelyn and me, Rachel is part of the proud lineage of AAAS-AMS Mass Media Fellows. Every summer the AMS sponsors one mathematician to work at at a mass media news desk. Something like NPR, Wired, Scientific American, or in Rachel’s case, the Oregonian. It’s good for science and it’s good for the world, since I’m sure we can agree there should always be more mathematicians everywhere all the time. But it’s also a really great experience for anyone with a technical background interested in writing, as Rachel was when she applied. Over the years the AMS has collected essays on the experiences of past fellows.

After studying math at the University of Missouri, Rachel was the 2015 fellow working at the Oregonian in Portland, Oregon, where she was part of the storytelling team. She wrote a bunch of articles while she was there, including a lot of coverage for the local science going on at the universities around Portland. Since then Rachel has been writing for the AMS Math in the Media blog, as well as freelancing at other venerable science news purveyors.

Rachel writes for Science News for Students, a super resource for plain english science articles about all sorts of current research, scientific and mathematical. Rachel wrote a fun piece about different jobs that use geometry to study motion, from crash test dummies to video game design.

She’s also been writing for the millennial focused PBS affiliate site Rewire.org. Over there she was working on some really fun Buzzfeed style quizzes, like “Which Top New Species of 2018 Are You?” I’m the Swire’s Snailfish. Depressing since this makes me the top predator among bottom feeders. Not exactly how I prefer to self-identify.

Rachel has a soft spot for applied math, and likes thinking how different types of math affect our day-to-day life. She’s also really into the craft of science and math writing, and wants to share more resources for people who want to get in on that.

I asked Rachel about her ideas for the blog moving forward, and she said she really wants to talk to you, the bloggers and readers and citizens of the internet, to cover your blogs and your interests. If you have ideas for Rachel or a blog you’d like to share, she encourages you to reach out by email (racheljcrowell at gmail dot com) or you can find her on Twitter @writesRCrowell.

And of course you can always find me on Twitter @extremefriday.

Posted in Math Communication | Tagged , , , | 1 Comment