Some Revelations In My Tech Free Adventure

On the ferry to Dar Es Salaam, contemplating pedagogy, accessibility, transportation, supply chain management and sunsets. Photo Courtesy of Frank Neikes.

I’m still in Tanzania, still with limited access to technology resources, so I wanted to take this post to share with you a few technology-free mathematical revelations I’ve had during my time here.

First, the pedagogical revelation. I’m teaching a Complex Analysis course to a group of 150 non-native english speakers in an acoustically challenging space. Imagine a large hall made of concrete, with lots of metal crossbeams running widthwise, and open air windows that allow all the noise in from outside and allow all of my careful words to escape into the schoolyard.

It has never been put in such stark relief how I take for granted my ability (and the ability in others!) to hear well and understand me when I speak. I have a great great deal of difficulty both hearing and understanding students when they ask questions, due to barriers of a linguistic and acoustic nature. And I’m constantly worried that they can’t hear or understand me. It’s completely changed the speed and cadence with which I deliver a lecture and engage with student questions. And it has really gotten me thinking, how many non-native english speaking or hard of hearing students have I had in my career who could have benefited from this?

It reminded me of this important microphone related PSA I read on Twitter a few weeks ago from writer @SarahPinsker. She wrote, “Abandoning the mic at a panel/ reading causes an accessibility issue for audience members. Even if you ask “y’all can hear me?” you don’t know if there’s someone with hearing loss who came out for you & has now been put in the uncomfortable position of speaking out or losing out.” I will never again eschew a microphone and dismissively say “It’s ok, I have a loud voice.” Although, to be honest, I never would have done that in the first place.

The AMS Inclusion/Exclusion blog has done some great coverage on accessibility concerns in mathematics, including ableism and inclusive pedagogy. If you’re interested in getting in on some of these conversation, I’ve found that Piper Harron’s Twitter feed is often full of thoughtful and nuanced takes on the subject.

The other revelation I’ve had is more mathematical in nature, and it involves what makes math interesting to humans, and it’s occurred to me that there are two main camps (feel free to fight me on this one, I’m not committed to this idea). There are the math-is-beauty-and-beauty-is-math-and-we-should-learn-number-theory-because-it’s-written-into-the-very-bones-of-our-being types, call then Type A. And there are the math-is-money-and-we-should-learn-complex-analysis-because-it-will-help-us-become-engineers-so-we-can-build-big-things-and-get-money, call them Type E.

I’ve been a Type A for most of my life, and I think that most of my students have been Type A as well. People who are just drawn to the wonder and beauty of it all. But suddenly I’m seeing a world full of nails, and math is the hammer, and I’m wondering if maybe my students were really Type E all along and maybe I wasn’t paying close enough attention. And I’m wondering if maybe I’m not also secretly a Type E…

It’s not that I think these two types necessarily need to exist in conflict with one another, nor are they necessarily mutually exclusive. And it’s not as simple as just a pure/applied split, because of course we all know that all math is eventually applied math. I’ve just noticed that math means different things to different people, and that’s pretty amazing. It’s the ultimate multitool. Whether you coming at math from hardcore operations research like PunkRockOR or mathematical essentials like From Fish to Infinity, there is an entry point for everyone.

My job as a lecturer is just to find out people’s preferred point of entry and guide them there. And my job as a mathematician, I guess it to make really good math regardless of which Type I belong to.

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Teaching Offline

The campus at Marian University College (Maruco) in Bagamoyo.

I’m in Bagamoyo, Tanzania at the moment teaching two summer courses to a group of undergraduate students at Marian University College. This experience is different from my typical teaching experience along several dimensions. I am teaching Complex Analysis to a group of 150 students. This is a course I’ve never taught, and it’s a group of students 5 times the size of my typical class. In order to deliver a message to the entire class at once, I need to write the message on a piece of paper and pin it to the main bulletin board. Moreover, this is an entire room full of non-native english speakers learning a subject full of technically dense jargon. And to take my challenge to the next level, I have one whiteboard that’s roughly…well, it’s small (it’s about the size of what I would have in my office).

As a blogger the internet is obviously critically important to my productivity, and so much of my teaching and blogging inspiration comes from the #MTBoS. And as a professor I am naturally pre-occupied with using technology in the classroom. This is such a hot topic, anyone who has ever applied for a job, or tenure has given it at least passing thought. The AMS blog PhD Plus Epsilon has featured several posts on technology for teaching and here we’ve done a couple of posts on teaching with technology. But this visit has got me thinking about teaching without technology. What does a technology-less classroom look like, and what are the advantages?

Getting ready to learn some Complex Analysis.

While tech resources some almost impossible to get away from in US schools, a few years ago the New York Times profiled a screen-free school in Silicon Valley of all places So it must be possible. Several years ago the Chronicle of Higher Ed published an article about efforts to “teach naked.” That is, to teach without have the power of machines to lean on and hide behind and how this stands to benefit students.

One important incentive for pumping our classrooms full of technology is that technology is like a language and it’s important that students be fluent its the language of technology by the time they graduate. To this important point, the Remind Blog has a post about teaching digital literacy without actually using technology. Students can learn about blogging, commenting, and online etiquette through well moderated discussions, and hashtags on the blackboard can actually work just like #hashtags.

The Flip’d Blog gives several good suggestions for using a technology-less environment to capitalize on student engagement, including several links to studies supporting the merits of good old fashioned pen-and-paper note taking.

I gave my first quiz today in Complex Analysis and without any prior discussion, each student showed up with a pencil, a ruler, and a scientific calculator. I have to say, this warmed my heart, since straight lines make graphing so much lovelier and I often find myself smh at the primacy of the TI-83.

I hope they learn so much! Wish me luck!

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

Summer Time is Puzzle Time

Courtesy of

It’s Mid-May, that means it time to put away your serious things and time to start thinking about (what else?) math puzzles!

Alexander Bogomolny, of CutTheKnotMath, has curated an amazing collection of math puzzles, problems, and interactive lessons. I always love to do geometry problems to get by brain working in the morning, and Bogomolny has plenty to spare. For example, this morning I picked up Two Equilateral Triangle on Sides of a Square. I solved it quickly, and then checked the (6 different!) solutions he gives on the page, all of which were different from my (sort of lame) approach.

Here I am in economy class, meanwhile he’s using quadrances and the triple spread formula. Bogomolny’s clear and varied solutions make me recall how much geometry has passed through my brain at one point or another, and also make the problems an interesting tool for exploring different math concepts in the classroom. He posts a lot of his problems to Twitter; you can follow him @CutTheKnotMath to get your daily dose of puzzles.

For the more tactile puzzlers among us, several people have sent me links to the infinity puzzle, which are “a new type of puzzle inspired by topological spaces that continuously tile.” There is a nice write up from some of the the makers at The Nervous System Blog, a blog about generative design. The puzzles are designed that they can be done right-side-up and upside-down and have no edges, but rather edge identifications.

Recently our friend of the blog, Mike Lawler from MikesMathPage, did a lesson in topology with his kids with the infinity puzzle as a jumping off point. Lawler and his crew determine the actual edge identification for their puzzle. Is it a Möbius strip? Is it a Klein bottle? Find out.

At the blog, Oliver Roeder curates a column called The Riddler, where really tricky but fun math problems are waiting to nerd snipe you. I particularly enjoyed the minimal urinal problem: what’s the minimum number urinals so that N people can optimally urinate? It’s much harder than it looked at first. This optimal urinal problem also showed up on the companion blag to xkcd a few years ago.

I appreciate that Roeder splits his riddles into Riddler Classic (to do while you’re on a long flight) and Riddler Express (to do while you eat breakfast). You might try the classic problem from a few weeks ago, which asked, how many decimal numbers are equal to the average of their digits?

Anyhow, if you’re headed to the beach, meeting friends for a beer, or just going for a walk in the woods, these puzzles are sure to delight everyone you meet, without exception. So definitely keep a few in your back pocket at all times, starting now. And please pass on worthy puzzles to me @extremefriday.

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Arts And Crafts Night

One fish had some extra parts, one fish was missing some parts, together they make a reasonable fish.

This week I rounded up several of my colleagues at the Max Planck Institute for Mathematics for a night of mathematically inspired paper crafts from the website The website site features an impressive collection of “interesting things to make out of paper,” and lots of them are mathematical. And while we won’t be setting up an Etsy shop anytime soon — I think perhaps we’re better at math than paper crafts — we did have fun experimenting.

The first one we tried was Dudeney’s classical construction of a square from an equilateral triangle. This one wasn’t too difficult (only one star out of five), and the result was a really fun hinged square/triangle. You can see a video of our completed construction here.

Emily Norton displays her hyperbolic paraboloid.

We also tried (to mildly mixed results, as you can see the the header photo above) to cut out an angelfish with one straight cut. Ideally, an angelfish would be obtained by folding a single sheet of paper in just the right way, and administering precisely one straight cut. Ours have a a bit of extra, erm, character. But actually, the anglefish itself isn’t particularly special. Any figure drawn from straight lines can be achieved in this way. Erik Demaine gives a good history of the mathematical properties of this problem on his website.

The hyperbolic paraboloid, which Emily Norton constructed out of a Wheatabix box and some colored yarn, turned out very beautifully. By building up some tension on the cardboard she made a perfect saddle point. It is really cool when you realize this shape with such a particular curvature is constructed entirely out of straight lines.

The Rhombic Polyhedra.

By far the most difficult, and most impressive was the rhombic spirallohedron. This is a polyhedra composed entirely of rhombic faces. You can see it in motion here, it would make a really nice piece of hanging art. Actually, I feel like polyhedra are having a bit of a moment right now. I’ve been seeing polyhedra all over the design scene, like here and here.

Now I really want to build the geodesic dome (large enough to sit inside!) out of newspaper. And then I want to sit inside my geodesic dome and construct this torus. Who’s with me?

Check out these math blogs too: Scientific Kirigami, “Dr. Alan Russell leads instruction in the art of kirigami during #MakeElon workshop, and Quarto Knows.

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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 | 3 Comments

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.

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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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