Michael Pershan’s Problem Problems

I have enjoyed math teacher Michael Pershan’s work for a long time. I follow him on Twitter, and I wrote about his website Math Mistakes a few years ago because, darn it, mistakes are interesting! A couple years ago, he started another blog, Teaching with Problems, at a URL I love, problemproblems.wordpress.com.

I never taught the elementary and middle grades Pershan teaches, and I’m out of the classroom altogether now, but I am always excited to see one of his posts in my blog feed. I find his writing extraordinarily thoughtful, and he is humble, passionate, and thorough in his posts. In a recent, slightly meta, post, he wrote about one reason that might be. Some math bloggers write quick posts that deal with one smaller idea at a time, and that’s great. But he prefers to “slowly, painstakingly, dutifully carve out posts.” He loves taking the writing seriously, and it means great, but not always frequent, posts from him. It also means you want to read them slowly and thoughtfully rather than skimming.

I had somehow missed the launch of Teaching with Problems, but I started reading it after finding — and being blown away by — this post about a student he calls Rachel. She is a smart kid who has a strong command of math concepts and a lot of trouble with basic arithmetic. I don’t want to try to summarize the post. Just go read it. Another 0f my favorites is this post about whether third graders think fractions are numbers. Point: “NO a fraction is not a number a fraction is only part of a number.” Counterpoint: “Fractions are a certain category of numbers because without numbers fractions would just be lines.” 

Last month, I spent a lot of time thinking about ancient Mesopotamian mathematics because researchers published a new paper about Plimpton 322, a tablet I was familiar with from my math history teaching days. (I wrote about why I don’t agree with their interpretation here.) So when I was looking through the problemproblems archive, I was happy to see that Pershan had coincidentally written about Plimpton 322 and Mesopotamian mathematics as a teaching tool in July. In his post, he writes about how some of the earlier mathematicians who studied Plimpton 322 and other tablets imposed their more algebraic view of mathematics onto the tablets in anachronistic ways and homes in on a dilemma of looking at ancient mathematics from the point of view of a modern math teacher: “The historical question is whether this mathematics would have been meaningful to the ancients. The pedagogical question is whether it could be meaningful to our students.” He ends the post pessimistic about whether the geometric Mesopotamian methods can help students with the algebraic concepts and notes that perhaps, “It’s only when you understand both that you can look back and see the connections between them.”

A week or so ago, the New Yorker shared one of their old articles, the one about how political science professor Andrew Hacker thinks math is about nothing, in a tweet. It caused a bit of a dustup in the math Twitter world, as it tends to. As high school math teacher Patrick Honner pointed out after attending a debate last year between Hacker and mathematician James Tanton, it’s frustrating that we’re listening to Hacker and not math teachers here. I was thinking about that as I read Pershan’s blog and thought about writing this post. When he writes about whether ancient Mesopotamian tablets can help teachers communicate the difference of squares method to students, he has a much more realistic understanding of what students can make of that than I do. If you’re looking for math teachers to listen to, he’s a great one to add to your list.

Posted in K-12 Mathematics, Math Education | 1 Comment

That Neural Net That Predicts Sexual Orientation

Image via Flickr CC courtesy of Stefano Mortellaro.

What does a computer see when it looks at a face? Image via Flickr CC courtesy of Stefano Mortellaro.

A neural network is one way to achieve machine learning. Modeled after the human brain, a neural net teaches a computer how to do some task by processing a huge set of training data. The data passes through the network training thousands of nodes how to react to future data of that type. Some machine learning can lead to interesting if hilarious results, some of which Evelyn blogged about earlier this year.

This week a more questionable use of neural nets hit the newsstands with the announcement of a Journal of Personality and Social Psychology publication, Deep neural networks are more accurate than humans at detecting sexual orientation from facial images, and I hope you don’t mind me saying that the mere existence of this technology is frightening in the utmost.

As their training data the authors used several thousand photos of men and women who self-identified as homosexual or heterosexual from an internet dating site. After the training, they found that their computer could correctly detect sexual orientation 81% of the time for men and 71% of the time for women. This was compared to a 61% accuracy for men and 54% accuracy for women when detected by humans employed by Amazon Mechanical Turk.

There are many reasons why these numbers could have emerged, several are summarized on the blog ScatterPlot in a guest post by the sociologist Greggor Mattson. My first thought was that it may well have something to do with the provenance of the training set. Jesse Bering blogged about a similar but low-tech version of this type of study for Scientific American several years ago. However, the authors seem to dispatch with this idea in Study 5 when they feed Facebook photos into the trained computer.

According to the authors, the success of the neural net may have something to do with the (their words) gender-atypical features of gay men and women. And this, they claim, has something to do with prenatal hormone exposure.

A good analysis of this (erroneous) claim and publication overall was given in a blog maintained by the professors of a (really incredible looking) University of Washington course Calling Bullshit: Data Reasoning for the Digital Age. In it, the authors claim that even if we assume the neural net was set up in a totally reasonable way, and that all algorithms are mathematically sound, it’s still easy to see that the conclusions the authors draw are (in their words) not parsimonious. I appreciate their willingness to treat the technical business as a black box and nevertheless analyze the good-sense of the findings. “black boxes should not be deterrents,” they argue, “one doesn’t need extensive technical training in order to think critically about even highly technical analyses.”

But the mathematical pith shouldn’t always be ignored, since it often it takes a bit of pulling apart of the apparatus to see where things go wrong. At the same time I am also sensitive to the fact that the mere presence of math can sometimes bully people into believing.

I just finished reading Weapons of Math Destruction, the book about dangerous algorithms by the blogger Cathy O’Neil of mathbabe.org. First of all, I can’t endorse this book strongly enough. But also, this book really hammers home the idea that we mustn’t just accept things on face value because they are rooted in math. The assumptions that go into programming an algorithm are just as biased and fallible as humans, and the way we interpret the outputs of algorithms (or neural nets in this case) also require some critical thought.

With all of this, it’s just sobering to recall that whether the conclusions are specious or not, the tool now exists. And in this year 2017 we should know enough to believe that even the most critically flawed tools of math can be used against us.

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Public Domain Math

Many pieces of mathematics — for example, simple geometric shapes and some mathematical formulas — are uncopyrightable or unpatentable. You can’t copyright a square or patent the area formula for a circle. Anyone can use them. But this post is not about the intricacies of patent or copyright law as they apply to mathematics, as fascinating as that can be. This is about different public domain math.

An image from Max Brückner’s 1900 book Vielecke und Vielflache: Theorie und Geschichte (Polygons and Polyhedra: Theory and History). Credit: Public domain, via Internet Archive

The Public Domain Review is a website and nonprofit project that highlights weird and wonderful work that is in the public domain. (The definition of public domain varies by country; Public Domain Review labels their posts with further information if necessary.) I love seeing their posts in my blog feed because they are so varied and interesting. Some are beautiful, some are strange, some are funny. And some are math.

C. H. Hinton used multicolored cubes to illustrate the tesseract. Credit: Public domain, via Internet Archive

For instance, this article by Jon Crabb looks into some interesting late nineteenth and early twentieth century writings on dimension. Edwin A. Abbott’s Flatland: A Romance of Many Dimensions is fairly familiar to mathematicians. It’s some of the most effective math communication I’ve ever read, and as a bonus, you get vicious satire of Victorian social structures. I was unaware of C. H. Hinton’s 1904 treatise The Fourth Dimension, which runs away with the idea of a fourth spatial dimension. But it turns out I had already interacted with him, or at least a piece of his legacy: he coined the term tesseract for the four-dimensional analogue of the cube (also called a four-dimensional hypercube). Hinton believed the fourth dimension had psychic as well as physical implications, and his ideas about the fourth dimension influenced artists and writers including Marcel Duchamp and Gertrude Stein. Hinton’s book uses colored cubes to visualize the many cubes in a tesseract. Hinton was married to Mary Ellen Boole, one of the remarkable daughters of George and Mary Everest Boole.

Then there’s one of my favorite examples of design meeting mathematics: the 1847 Oliver Byrne edition of the first six books of Euclid’s Elements. And Ernst Chladni’s figures illustrating the nodes of vibrating plates. And Étienne Léopold Trouvelot’s astronomical illustrations, including a beautiful depiction of a total solar eclipse.

Credit: Public domain, via New York Public Library

Browsing through the mathematics tag, it’s fun to see work from people with enduring legacies as well as some things that are a little out thereMathematics shows up in some unexpected places. You can try to find frieze and wallpaper groups in an 1863 book of French textile samples or identify the curves in a 1919 book of Japanese wave and ripple designs. You can learn the correct proportions for Buddha and Bodhisattva depictions from an eighteenth-century book from Nepal. You can take a peek at early twentieth century data visualization in the infographics W. E. B. DuBois and his students created depicting various facets of African American life.

The Public Domain Review publishes a lot more than just math and science. It’s a worthy addition to the blogroll for all the interesting artifacts it brings to light, from math and science to art and religion.

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Some Stories of Journals Behaving Badly

Turns out money can buy a lot of things, even an expression for π as an algebraic number. Image via flicker CC: reynermedia.

Hoax papers have a long and time-honored history. Ten years ago a group of students from MIT wrote a program that randomly generated totally nonsensical computer science papers. One of their bogus papers was accepted by a conference and it caused enough of an uproar to prompt IEEE to pull its sponsorship of the conference. But this wasn’t before Springer had already accepted 120 papers generated by the program.

Needless to say, this caused academics and the bloated body of academic publishing to take a long look at themselves and wonder what the heck was going on with their standards. It’s no secret that predatory publishers exist. These are publishing outfits that usually charge a high fee for publication, may (or may not) promise some sort of referee process, and often have names like “Journal for Advances of Algebra, Number Theory, Biology, Chemistry and Electrical Technology.” Sometimes they’re easy to spot, sometimes they aren’t. Sometimes they ask you to submit a pdf to a gmail address, but sometimes they don’t.

But as we saw above with the 120 papers retracted by the venerable publishing house, Springer, detrimental and predacious practices aren’t just restricted to the obviously pay-to-play scam journals.

The blog Math Scholar recently wrote an interesting essay about a slew of predatory journals that published papers (eight of them!) claiming that π=(14-√2))/4. The author rightly argues that this is a terrible thing. The author speaks of the “collapse of peer-review,” and I just need to express my own mildly divergent view at this point. The peer-review process has not collapsed; peer-review is still a very good and necessary thing. The problem is just that journals are promising peer-review and not doing it. It’s a bit like using nacho cheese Doritos as evidence against the declining quality of cheese. We can’t besmirch all cheese based on the fact that those promethean deities of food science over at Frito-Lay use the word “cheese” on their devil corn chips.

Fields Medalist turned blogger Tim Gowers is at the forefront of a movement to call out the predatory practices of the so-called legitimate journals. In a recent post Gowers makes the case for uprooting existing journals and replanting their entirety — editorial boards, content, reputation –in an open-source context leaving behind empty hulls called “zombie journals.” This move was recently carried out by the journal formerly known as the Journal of Algebraic Combinatorics.

Unsurprisingly, math is not the only field feeling leery of its publishers. The field of gender studies also saw an interesting dust-up this summer with the publication of a hoax article (ostensibly) about toxic masculinity followed up by a take-down of the field of gender studies and its publishers by the authors of the hoax. The whole saga is interesting in our context because it points to the blurry line between legitimate and illegitimate publishing. The blogger Ketan Joshi does a nice job teasing out some of the universal-yet-nuanced complications of this particular hoax. In particular, Joshi gets at the question: how damning should a hoax paper be to the publishes versus the field itself? It’s an interesting read.

And just for fun, in case you have any secret messages that you need to send, the same group from MIT that wrote the hoax paper generator also have a program that can encode a secret messages as a bogus spam conference announcement. And that, my dear most esteemed sir/madam, is certainly one way to guarantee that nobody will be interested in reading your email.

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

Back-to-School Blogs, 2017 Edition

Today, I’m taking my chances with traffic and driving up to Idaho to try to get in the path of eclipse totality. (Fun fact: according to my back-of-the-envelope calculations, if everyone in the country went to the path of totality, its population density would be a bit higher than that of Salt Lake City. Get more eclipse math from Bedtime Math and NASA. Don’t have eclipse glasses? Ask the past how to behold it safely.) While I’m on my sojourn, my spouse and millions of other teachers and students will be getting back to the classroom. Last year around this time, I posted a roundup of some of my favorite math education blogs. It’s a good list, and you should check it out.

Welcome back to school! Image: US Department of Education.

This year, I want to add a few more recommendations for good math resources for parents, teachers, and students.

Not awful and boring ideas for teaching statistics I’m a sucker for this blog name. Are you teaching statistics? Do you want some ideas that aren’t awful and boring? Here you go! The author, Jessica Hartnett, is a professor in the psychology department at Gannon University. She finds interesting data in the news and other places and gives her advice about how to use it in a statistics class. She’s recently posted about a Harry Potter sorting quiz, unpopular wedding songs, and the statistics that indicate that ride-sharing apps might curb drunk driving. (I’ve idly wondered about this myself while waiting for a ride after pub trivia night.)

Math Hombre John Golden is a math teacher and math education professor at Grand Valley State University. Follow his blog for posts about teaching teachers and making cool math art, games, and designs.

On This Day in Math Pat Ballew rounds up math-adjacent births, deaths, and events every day. I also enjoy the quotes he includes at the top of each post.

MatthewMaddux Education University of Saskatchewan math education professor Egan Chernoff compiles this feed of articles and blog posts related to math and math education. It helps me keep up with the latest news, especially in mainstream media outlets I don’t normally check for math stories.

inclusion/exclusion My co-blogger Anna Haensch wrote about this new AMS blog here a few months ago. It’s not specifically about teaching, but as math professors make up a lot of the intended audience, it discusses aspects of teaching that intersect with its mission of addressing diversity and inclusion in mathematics. In light of the heartbreaking, infuriating events in Charlottesville earlier this month, Brian Katz wrote about how to discuss justice on the first day of class.

Have other suggestions? Feel free to add them in the comments. If you’re a teacher, parent, or student, have a great beginning of the school year!

Posted in Issues in Higher Education, K-12 Mathematics, Math Education | Leave a comment

With Profound Sadness

It was an incredible day in 2014 when Maryam Mirzakhani became the first woman to win the Fields Medal. I remember feeling absolutely overwhelmed with emotion and thinking to myself, alright, beginning today winning the Fields Medal is officially something that women do. It went from something impossible to something possible, just like that. It felt like the breaking of such a monumental glass ceiling, and like the opening up of this entire alternate universe of possibility. It was huge. At the time, Erica Klarreich wrote about Mirzakhani’s early life and work for Quanta Magazine.

It was with almost unbearable sadness that news of her passing broke last month. Mirzakhani died at only 40 years old, and the world became dimmer one shining star.

The weeks since have seen an outpouring of writing celebrating the life and work of Mirzakhani and mourning her death. Fellow Fields Medalist Terrence Tao shared a post the day after Mirzakhani’s passing, highlighting her contributions to the field and his own experiences in meeting her. Blogger and mathematician John Baez also wrote a thoughtful piece about Mirzakhani’s life and mathematics. The blog Mathsbyagirl featured a tribute post to Mirzakhani with links to expository articles on the key areas of Mirzakhani’s research.

Our own inclusion/exclusion blog posted about how Mirzakhani shone in all of her various roles, as mathematician, mother, trailblazer, and role model, with reflections on her life by various notable women in math. From Tai-Danae Bradley, “While reading through the many beautifully written tributes to Maryam, I am especially touched by one theme that pervades them all: her character. Words like persistent, determined, and resolute appear time and time again. And her humility and modesty seem to have garnered as much attention as her mathematical accomplishments.”

Mirzakhani working through ideas on a large sheet of butcher paper. This image is a screen shot from the Simon’s Foundation short film on Mirzakhani.

RAGE of the Blackboard, a blog exploring the bridge between scientists and artists, featured an illustration of Mirzakhani and spoke about the importance of drawing in her work. Mirzakhani often said in interviews that she enjoyed doing math by writing, drawing, and doodling on large pieces of butcher paper.

The AMS has collected a full list of tributes and obituaries for Mirzakhani, including words from AMS president Ken Ribet and a short video from the Simon’s Foundation.

This month has also seen the passing of several other notable women in math. Cathleen Morawetz, who did pioneering work related to airflow at supersonic speeds, died last week at the ago of 94. Earlier in July, Marina Ratner, who found acclaim later in life, died at the age of 78. Mathematician Amie Wilkinson wrote for the NYTimes about the shared aspects of Ratner’s and Mirzakhani’s works, despite very divergent lives. This month also saw the passing of Marjorie Rice at the age of 94. Rice was an amateur mathematician who made a big discoveries in the study of pentagonal tiling.

It is with tremendous sadness that we bid farewell to these mathematicians who inspired us and gave us so much. The impact of their work will surely continue to resound for many decades to come.

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Conversations with Women of Color in STEM

I online-met Williams College mathematician Pamela Harris last year through Lathisms, a Hispanic Heritage Month project that highlights Latinx and Hispanic mathematicians. She was one of the organizers of the effort, and I spoke with her and another organizer, Gabriel Sosa, for a post about it. It wasn’t until a few weeks ago that I learned more about her background via an interview for the Vanguard STEM blog. Unusually for a professional mathematician, she started college in intermediate algebra and worked up to a Ph.D. from there. It got me thinking about how mathematicians could help more students excel in math after coming to it relatively late in their academic careers. I also enjoyed reading about her childhood encounters with infinity and one of her hobbies, jiu jitsu.

Just a week later, Vanguard STEM published an interview with Alicia Prieto Langarica, a mathematical biologist at Youngstown State University in Ohio, who was one of my colleagues in the Project NExT class of 2013. (Go Brown 13s!)

Seeing two math friends in a row featured on the site convinced me that I really needed to add Vanguard STEM to my blogroll. Vanguard STEM was launched a few years ago by Jedidah Isler, an astrophysicist at Vanderbilt and TED Fellow. She created it as a place to facilitate conversations between established and up-and-coming women of color in STEM (science, technology, engineering, and math) fields, particularly African American and Hispanic/Latina women. The site has monthly Google Hangouts in which women of color discuss different aspects of their lives and careers in STEM in addition to articles with advice for people considering or working in STEM careers. But my favorite is the weekly #WCWinSTEM series where I read the interviews with Harris and Prieto Langarica. WCW stands for “women crush Wednesday,” and #WCWinSTEM is a great series of interviews and articles about STEM professionals in many different jobs and stages of their careers.

Vanguard STEM has featured mathematics and mathematicians several times on the site. Last winter when Hidden Figures fever was gripping the world, the site had several articles about the black women who worked for NASA as “human computers.” It was great to see Katherine Johnson, Dorothy Vaughn, and Mary Jackson portrayed by movie stars on the big screen, but there’s something very special about seeing their real faces, too, even if they weren’t quite as glamorous as Taraji P. Henson, Octavia Spencer, and Janelle Monáe. I especially enjoyed the Vanguard STEM article about Christine Darden, one of the heroes of the Hidden Figures book whose story was not included in the movie.

In the past, the Blog on Math Blogs has featured several other websites that provide information about and/or resources for women and people of color in math and other STEM careers. Check out our recommendations. (Warning: there’s a lot of celebrating going on.)

Celebrating the Grandmothers of STEM
Celebrating our Sisters in STEM
Beyond Banneker: Resources for Learning about Black Mathematicians
Celebrating Latin@s and Hispanics in Mathematics
Celebrating Black Mathematicians
They Answered the Call of Numbers
Diversify Your Blogfolio
Adding to the Faces of Mathematics on Wikipedia

You can also subscribe to the AMS inclusion/exclusion blog, which covers diversity and inclusion in mathematics.

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Searching For Einstein

No, not Einstein. We’re searching for einstein. Literally “ein Stein,” which translated from German means “one stone.” The one stone we’re looking for is a very special type of tile which, when repeated, can cover an infinite floor without leaving any gaps and without admitting any sort of pattern.

We call an arrangement of tiles that covers the plane without any gaps or overlaps a tiling, and a tiling is called non-periodic if it has no translational symmetry. That means, if I pick the tiling up and move it in any direction, I won’t be able to fit it back down on itself. A nice example of a non-periodic tiling where we allow two types of tiles is the Penrose tiling. And if we loosened our restrictions slightly to allow tiles which are not connected, Socolar and Taylor found such a tiling in 2010. So more formally, the search for einstein is the search for a single connected tile that tiles only non-periodically. Recently, in the quest for einstein, some interesting progress has been made.

First some basics. Let’s think about tilings that only use a single convex polygon, that is, a polygon whose angles all bulge out instead of in. If we allow patterns and periodicity, then it’s easy to imagine how you could achieve a non-overlapping gap-free tiling that with a square, triangular, or even hexagonal tile. Even though sometimes they can be in disguise.

We now know that these are the only three possible regular hexagonal tilings. They were first discovered by Karl Reinhardt in 1918. Image via Wikimedia Commons.

Things get a bit more interesting when we consider pentagons. In the early 1900’s Karl Reinhardt found five examples of families of pentagonal tilings. Several more were found by various people over the years, including 4 families which were found by housewife and mathematical enthusiast Marjorie Rice, who recently passed away. And just last year, as reported on this blog by Evelyn Lamb, another pentagonal tiling was found, bringing the total number of known families of pentagonal tilings to 15. For some fun teachable moments involving pentagonal tilings, check out mikesmathpage.

These are the 15 families of tilings with convex pentagons. Here colors indicate the number of edges touching each pentagon. Image courtesy of Wikimedia Commons.

In breaking news, mathematician Michaël Rao of France’s CNRS proved that these are precisely all of the convex tilings of the plane. There are just the 15 known families of pentagonal tilings, 3 hexagonal tilings, and all triangular and quadrilateral tilings. Of note, is that Rao’s work involves a computer assisted proof, which allowed him first to establish some bounds via theoretical methods and then do an exhaustive search. Rao’s conclusion: there are no convex polygons that admit only non-periodic tilings, that means, the einstein tile must not be a convex polygon.

This means, if we want to find einstein, we need to start looking at concave tiles.

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Resources for People Who Wanna Present Stuff Good and Do Other Stuff Good Too

Presentations are hard. You’ve been thinking about something for a long time, and you can get tunnel vision. What do you mean, everyone looking at your poster or going to your talk doesn’t already know why you care about the components of the representation space of π1(M) into PSL(2,R) with extremal Euler characteristic??? Luckily, if you want to up your presentation game, you’re not entirely on your own.

One good way to improve your posters and presentations, of course, is to go to lots of poster sessions and talks and keep an eye on what’s working and isn’t for the presenter and their audience. But you can also get advice from around the math and science blogsophere.

For presentations, Dan Meyer has a good post about how to prepare for a talk. It boils down to “testify and practice,” but he gives a lot of specific advice on the nitty gritty details of how exactly he does that. He’s writing specifically for math teachers, but his presentation tips will be applicable to math research talks and other professional topics. The comments also have some feedback, and he wrote a follow-up post of some advice from 14 of his favorite math education speakers.

Zen Faulkes, an invertebrate neuroethologist (I had to look it up too) at the University of Texas Rio Grande Valley has a free e-book (pdf) of presentation tips from his blog, NeuroDojo. He also has a blog devoted to helping people make better posters. I especially appreciate the constructive critiques of real posters, like this one about two posters by mathematical biology graduate student Chris Miles.

When I asked for suggestions for this post on Twitter, astronomers stepped up. Thanks, astronomers! Meredith Rawls, an astronomy postdoc at the University of Washington, wrote a blog post about how she made an award-winning poster for a conference. She and other astronomers also pointed to tips from Kimberly Cartier and Jason Wright and this list from a blog called Astrobetter whose goal is “to provide information and tips about streamlining all the things we need to do Astronomy well.” People also suggested Edward Tufte, particularly for presentations that have a lot of data visualization.

I got great suggestions from friends on Facebook as well:

  • Georgia Tech mathematician Dan Margalit has a page of talk tips, which includes articles and blog posts from Paul Halmos, Jordan Ellenberg, Bryna Kra, and other mathematicians.
  • The LaTeX package tikzposter was designed specifically for conference posters.
  • Technically Speaking, a page by Lewis D. Ludwig of Denison University, has videos illustrating common presentation pitfalls and how to avoid them.
  • University of Waterloo mathematician Chris Godsil has a webpage of math presentation tips. I particularly appreciated his pointing out that giving a research talk or presentation is not the same as lecturing or teaching. Of course there are skill overlaps, but the goal of a research talk is usually not for an attendee to reproduce your proof later. 
  • Evolutionary biologist Colin Purrington’s website has “geeky tips for scientists,” including a page on designing conference posters.
  • Stephanie Evergreen is a data reporting and visualization expert with a blog of presentation design and visualization suggestions.
  • The NYU library has a good page of poster design tips.

If you have other presentation or poster design tips, please share in the comments!

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Take The Math Less Traveled

Mathlesstraveled is a blog “dedicated to exploring beautiful mathematics.” The blog is written by Brent Yorgey, an assistant professor in the department of math and computer science at Hendrix College, who lives closer to the computer science end of mathematics. As such his posts are often somewhat computational in nature. He has a whole zoo of good looking graphics and everything is easily digestible by anyone interested in learning a bit of math.

Factorization diagrams for the numbers 1 through 100, courtesy of Brent Yorgey.

One ongoing series featured in the blog are Posts Without Words. These are just graphics depicting some mathematical idea, sometimes it’s easy to see what the pictures are describing and sometimes it’s more difficult. I really like Post Without Words #5. The explanation is a doozy as it involves Hilbert space filling curves and the Thue-Morse sequence. Although I’ll bet you can come up with a simpler explanation!

As an aside, if you like mathematical ideas in graphic form, you should check out Mathematics in the Eye of the Beholder.

Yorgey also has several other series of posts, including those in which he discusses the irrationality of pi, and more recently, the curious powers of 1+sqrt 2. The later series aims to answer the following question posted on Mathstadon: What is the 99th digit to the right of the decimal point in the decimal expansion of (1 + sqrt 2)500? After stating the problem, Yorgey comes up with a reasonable conjecture (motivated by some computational examples), states a clever solution, and then goes on to explain an alternative approach.

Another great offering on the blog are Yorgey’s factorization diagrams. These are pretty pictures generated using diagrams in Haskell, that give a visual representation of prime factorization. These reminded me of a clever little book my brother gave me recently, You Can Count on Monsters. Yorgey sells the factorization diagrams as a deck of cards along with some fun game ideas for teaching factorization. I’m into it.

Yorgey maintains a second blog, aimed (as he says) at his peers. These posts tend to be slightly less accessible but would probably be of interest to anyone studying type systems, category theory, or combinatorics.

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