Cook’s Take on Benford

The leading digits of the heights of the world’s tallest buildings satisfy Benford’s Law.

Lately, I’ve been having fun reading John D Cook’s Blog. Cook is an applied mathematics consultant who blogs and tweets up a storm about all sorts of topics mathematical, statistical, computational, and scientific. He maintains 18 daily tip Twitter feeds giving daily facts about…well, everything…and one personal feed.

A leading digit 1 is expected to appear a whopping 30% of the time.

But what I like most are his mathematical blog posts. He writes short easy to digest posts about reasonably accessible topics in math, often with a computational bent, and I always walk away feeling like I learned something. This past week he revisited the topic of Benford’s law, which is this totally weird and strange thing that I’ve always wanted to understand more about. Benford’s law says that in many naturally occurring data sets the leading digit is more likely to be small. If the leading digit d from the set {1,…,9} were distributed uniformly you would expect each digit to show up about 11.1% of the time. But in reality, the leading digit is more often distributed according to the chart on the right. Cook can fill you in on some of the more precise formulations of Benford’s Law.

In his blog, Cook describes how the leading digits of factorials satisfy Bedford’s Law, and even gives some tips on how you can use Python to compute leading digits up to 500!! (One of those exclamations is a factorial, the other one is for my excitement.) He also show that the collection of SciPy constants follow Benford’s Law, which Cook explains and computes using Python. Cook blogged about how samples from the Weibull distribution satisfy Benford’s Law, and most recently he even showed that the iterates of the Collatz conjecture seem to follow Benford’s Law.

And you know a party is getting good when the Collatz Conjecture shows up.

These posts just give a small flavor of Cook’s writing. I also really enjoyed his recent posts on harmonic numbers and golden angles (largely because it prompted me to check out the work of the visual artist John Edmark), the lesser known cousin of the golden ratio.

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Growing Up Gifted

It seems that Hollywood can’t get enough of mathematicians. Most recently, Gifted hit theaters. It’s the story of the mathematically gifted seven-year-old Mary who is living with her uncle in Florida. We follow Mary’s struggle adjusting to a typical public school classroom, while the conflicting desires of the adults in the film — her uncle, teacher, grandmother and neighbor — to allow her a normal childhood while making sure to nourish her talents, play out around her. As the film progresses we learn that Mary’s mother was a genius herself, who died while on the cusp of proving the existence and smoothness of the Navier-Stokes equations, one of the as-yet outstanding Millennium Problems.

The movie didn’t actually involve all that much math, save for occasional references to differential equations and some teary-eyed discussion of the problem that got away. But it did capture something charming and lovely about the sometimes non-trivial dynamics of teaching exceptionally gifted children and the captivating allure of mathematics.

I stumbled upon a blog written by several educators and researchers at Duke University’s Talent Identification Program, who are not necessarily experts in mathematics, who write about whether its depiction of giftedness in the classroom was accurate and well-handled. They bring up several good points, including how different the landscape can be for a student depending on whether their parents and educators are completely aware of all of the resources available to them. They also bring up an important point that I think the movie very conspicuously missed: being mathematically gifted and being social are not necessarily in opposition to one another.

The movie concluded with a cameo from mathematician, math blogger, and recent Erdős-Bacon number-haver Jordan Ellenberg, who consulted on mathematics in the film.

In 2014, Ellenberg wrote an essay for the Wall Street Journal, The Wrong Way to Treat Child Geniuses, (sorry about the paywall) about the disproportionate and sometimes wrongheaded way that society thinks about genius in children. Ellenberg cites a Vanderbilt University study that tracked the achievements of a cohort of children identified as gifted at an early age. He was part of this cohort, a fact he discusses in a recent interview with math blogger Anthony Bonato. Ellenberg and his cohort do have a disproportionate amount of success, especially as success is defined in the academic realm, but he points out, “most child prodigies are highly successful—but most highly successful people weren’t child prodigies.” The cult of genius, he claims, might do more to scare otherwise top-notch people away from math and science than it does to foster the geniuses.

The idea that math is an area strictly reserved for super geniuses is generally speaking, a very bad one. Evelyn Lamb wrote about some of the specific problems in the genius myth as it corresponds to the retention of women in STEM fields. Lamb also wrote about how the media contributes to this stereotype.

Fields Medalist and math blogger Terry Tao, who also consulted on the film, has written about the short-sightedness of over-hyping giftedness when it comes to mathematics. Tao writes, “I find the reality of mathematical research today – in which progress is obtained naturally and cumulatively as a consequence of hard work, directed by intuition, literature, and a bit of luck – to be far more satisfying than the romantic image that I had as a student of mathematics being advanced primarily by the mystic inspirations of some rare breed of ‘geniuses’.” Tao has also written about strategies for gifted education, and points readers to several articles about his experience growing up gifted.

These are all interesting points, and as a mathematician and educator I would strongly recommend watching this movie, if only as a well-scored and reasonably entertaining springboard to launch into all of the rich ideas surrounding giftedness, the cult of genius, and the strange otherness of mathematics.

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Recommended Reading: Euler, Erdős

Have you ever used an analogy in a conversation only to have the conversation derailed as the person with whom you’re speaking points out that the analogy is not quite perfect in some way? Of course it’s not perfect! If it were perfect, it wouldn’t be an analogy. It would just be the thing itself. Or maybe you’ve been the one nitpicking an imperfect analogy. I was that nitpicker in a recent Facebook conversation, and it reminded me of a blog post aptly titled Analogies are the Worst! from Jean Pierre Mutanguha’s blog Euler, Erdős. In it he uses analogies to explain why he doesn’t like analogies, or at least the way many people use analogies in arguments. He also writes about how his mathematical thinking influences the way he converses and thinks through arguments.

Mutanguha is a graduate student in mathematics at the University of Arkansas. I took a peek at his blog when he followed me on Twitter (you can follow him here), and I added it to my feed because I enjoy the way he writes about math with enthusiasm and humor. Math is clearly a joyful subject to him, and he wants to share his insights about his favorite topics rather than trying to impress you with how much he knows.

One of my favorite posts is the fanciful Annotated history of the reals. Math textbooks can give one the impression that math came to humans perfect and immutable, created by the hands of a divine being. Mutanguha takes that idea and runs with it: “In the beginning, there was nothing, 0=∅. Then we realized it was something, 1={∅}. Then we wondered, why not have another thing? 2. And another thing, 3. And another, 4, and another, 5, and another, 6, etc. Just like that, we had the natural numbers!”

“In the beginning…” Image: The Creation of the World and the Expulsion from Paradise, Giovanni di Paolo. Public domain, via the Metropolitan Museum of Art.

Mutanguha started his blog in 2013 because he felt like there weren’t enough math blogs for undergraduate-level math. Some of his early posts are explainers about topics from pentagonal numbers to ordinal numbers to the inclusion/exclusion principle. He’s tended to incorporate more of his own insight and voice over time. Recent offerings include Fermat and his missing proofs, mathematical crafts, and randomness. I’m always excited when I see a new post from him in my feed, and I think people who read this blog will enjoy adding it their internet mathematics diet. So why not surf over to Euler, Erdős and start reading?

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Divorce And Margarine

A spurious correlation involving a spurious butter substitute. Via tylervigen.com.

The correlation between the divorce rate in Maine and the per capita consumption of margarine, though compelling, is totally spurious. This is just one of the many such correlations that Tyler Vigen explores on Spurious Correlations, and in his book of the same name.

I’ve been thinking a lot about fallacies in statistics this week, every since I read Stephen Woodcock’s article Paradoxes of Probability and Other statistical Strangeness in TheConversation. This article gives great examples and graphics to explain some of the weirdness of statistics like Simpson’s paradox and my personal favorite, the base rate fallacy.

Perhaps there’s more than one reason to refrain from eating cheese and crackers in bed. Via tylervigen.com

The extent to which two variables, X and Y are related is most often measured using the Pearson correlation coefficient. Formally, this is just the covariance of X and Y divided by the product of their standard deviations. Practically, it is some number between -1 and 1, where a correlation coefficient of 1 means total correlation, 0 means no correlation, and -1 means a negative correlation.

For example, the number of people who tripped over their own two feet and died has a correlation coefficient of 0.9 with the number of lawyers in North Carolina. Which means that they are very closely correlated, which in reality means absolutely nothing. Compare this to 0.8, the correlation coefficient for the number of people who tripped over their own two feet and died compared with Apple iPhone sales.

Although that correlation seems a little bit less spurious.

Recently fivethirtyeight also explored the prevalence of spurious correlations in nutritional studies in You Can’t Trust What You Read About Nutrition. Nutritional data, which is largely gathered through food diaries and eating questionnaires, leads to all sorts of crazy correlations like cabbage and innies and nuts and immortality.

If you’re teaching a course in statistics, Vigen’s website would be a really fun place to pick up data sets and cautionary examples for your students. Vigen includes links to all of the data he uses in his charts.

Vigen’s website also allows you to play with variables and find your own totally spurious correlations. Have a go at it and let me know if you spot any good ones @extremefriday.

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Maps and Math

Gauss’s Theorema Egregium was in the news recently! The news articles didn’t quite put it that way, though. Their headlines were more like, “Boston public schools map switch aims to amend 500 years of distortion.” That’s right, they’re switching from using the Mercator map projection to the Gall-Peters projection in their classrooms.

The Gall-Peters map projection. Credit: Strebe, via Wikimedia Commons. CC BY-SA 3.0

The Theorema Egregium is the theorem that states that Gaussian curvature is an intrinsic property of a surface, not a result of how it is embedded in space. An application of the theorem is that accurate maps are impossible. The surface of the earth is positively curved, and a plane has zero curvature. So any function that maps the surface of the earth to the plane must distort something, whether it’s area, shape, or distance.

The familiar Mercator projection projects the globe onto a cylinder. It preserves shapes fairly well (circles near the poles and circles near the equator are all circular), and it is useful if you’re navigating across an ocean using a compass. But the area distortions are severe. In real life, Africa is 14 times as large as Greenland, a fact that is not clear on the Mercator map. (Check out The True Size to compare the actual sizes of different countries by dragging them around a Mercator map.) The Gall-Peters projection replacing it preserves area but distorts shape dramatically.

As a representative of the fictional Organization of Cartographers for Social Equality explains on an episode of The West Wing, the Mercator projection can make places close to the Equator, such as Africa and South America, look smaller and therefore less important than northern North America and Europe. An equal-area map makes Africa in particular seem proportionally much larger than it seems with the Mercator.

I personally find the Gall-Peters projection bracing. It looks strange and a little wrong, which helps me think about the assumptions we’ve internalized about what the world looks like. Even though I know in my head that the Mercator is inaccurate, I’m exposed to it so much that it seems like the real thing to me. But I wonder why Boston schools and writers for The West Wing chose Gall-Peters instead of a different projection that distorts shapes less. There are so many other options!

Ernie Davis, a computer science professor at NYU, wrote a guest post on Cathy O’Neil’s blog mathbabe.org about the recent map projection discussion. He’s a member of the pro-globe camp. But it’s not always practical to have a 3-dimensional globe handy. Luckily there are several posts in the science blogosphere that give nice overviews of some of the common projections. Dave Goldberg votes for the Winkel-Tripel, which also happens to be the projection used by National Geographic, on his blog A User’s Guide to the Universe. Max Galka at Geoawesomeness votes for the Authagraph, which I had never seen before.

Mike Bostock has a great animation of dozens of the most popular map projections so you can see how they compare to one another. After you’ve found your favorite, you can see what it says about you in this xkcd comic.

Several map-related blogs have found their way into my (mostly mathematical) blog reading list. They don’t often mention math specifically, but they all combine geometry, data analysis, and data visualization with culture, history, and sociology. A few of my favorites are Musings on Maps by Daniel Brownstein, All Over the Map by Betsy Mason and Greg Miller, and Adventures in Mapping by John Nelson. I also enjoy William Rankin’s website radicalcartography.net. A particular favorite is the map called Actual European Discoveries. Today these territories have a combined population larger than all of Connecticut! For dessert, I’m partial to the Twitter account @TerribleMaps, which shares helpful charts such as this map of the population per capita of countries in Europe.

Do you have a favorite map blog?

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Does This Make Sense?

What would it look like if I plotted the temperature of my morning coffee with respect to time? Image courtesy of Flick CC via Jen.

Some of my favorite questions to ask in class involve drawing up some sort of a mathematical model for my students and asking: does this make sense? Whether matching curves to the heating and cooling laws of my morning coffee, or studying how our shadows grow relative to our speed, it’s always a nice way to play with students’ quantitative intuition.

Recently, Dan Meyer of the blog dy/dan, wrote about this idea in The Difference Between Math and Modeling with Math in Five Seconds. He shows a video of a dog setting a world record for balloon popping (yeah, I know, I’m also simultaneously thrilled that’s a thing and deeply concerned about the fact that dogs love to eat deflated balloon bits) and then asks students to model the situation. He teases out the differences between linear and non-linear models, and the difference between just doing math and interpreting the realities of a situation.

I think that people underestimate the element of intuition and human touch in mathematics. Students ask questions like, how do I know which model to use? How do I know which integration technique to use? How do I know which convergence test to use? And I always feel like a bit of a jerk (slash, I also feel a bit like a wise old sage) when I say the only possible correct answer, namely “You use the one that works.”

But you have to develop some intuition for these things.

Models are interesting because people are typically using them to describe phenomenon that they already see happening. Trying to quantify the meaning of several different variables in the context of a larger system, and to forecast where that whole system might go if the variables change. Recently, I podcasted about another great example of mathematical modeling in several variables, sharing the recent work of Steven Strogatz and some folks at the MIT Sensible City Lab about the effectiveness of ridesharing.

Teaching calculus, the closest that we get to mathematical modeling is in studying the laws of natural growth, but this is already rich territory. If you really want to scare your class, have them watch this video from NPR abut population growth, tell them about the human carrying capacity of the earth. Then have them calculate how long we’ve got left.

For the less macabre among us, Meyer also points us towards the reindeer of St. Matthew’s Island.

Someone once said to me that computers will never put mathematicians out of business, because a computer can never match our human intuition. A computer can never look at a model with our human eyes and human sense of the world and say, does this make sense? I’m not sure if I believe that, what do you think? Let me know on Twitter @extremefriday.

Posted in Applied Math, Biomath | Tagged , , | 1 Comment

Adding to the Faces of Mathematics on Wikipedia

For better or for worse, Wikipedia is the first place most people look when they want to learn about someone or something online. I don’t use Wikipedia as my sole source for important facts, but it’s a great first stop when I’m researching a topic, and it often helps me find the more reliable resources I end up citing in articles I write.

Mathematician Vivienne Malone-Mayes, a professor at Baylor University from 1966-1994. Her picture was just added to her Wikipedia page. Credit: The Texas Collection, Baylor University. CC BY-SA 2.0

Last month on her PLOS blog Absolutely Maybe, Hilda Bastian wrote about her efforts to improve the Wikipedia pages of black women in science by both expanding on the articles and by adding pictures to ones that don’t have them. She writes:

As I’ve been combing through what seems like a bottomless pit of digitized old black and white photos of white scientists, the Black History Month stories and tweets about African-American women scientists were mostly about the same small group – although this year, plus the fabulous supersonic boost by Margot Lee Shetterly and the women of Hidden Figures.

That’s not because the supply of African-American women scientists from the past with gripping stories is tapped out. It’s not. Rather, when it comes to the stories of black women scientists, Diann Jordan writes, “The harvest is plentiful but the laborers are few”.

There is some randomness about whose stories have been told, who had compelling, high quality photos taken, and which images have surfaced online. Mostly it’s not random, though. The odds are stacked against visibility in the historical record, as it was in life – and for many of the same reasons.

Vivienne Malone-Mayes is one of the women Bastian mentioned in her post. Malone-Mayes got her Ph.D. in math from the University of Texas in 1966 after navigating challenges such as being ostracized by her fellow students and being barred from classes taught by R.L. Moore, venerated pedagogue and notorious racist. She was the first African American professor at Baylor University. Her Wikipedia page was bereft of a photo because there wasn’t one available with the correct license. I am a Baylor alumna, so I wrote to my alma mater about making a picture they have in their archives available for Wikipedia. Hours later someone from the photo collection had changed the license uploaded it to Malone-Mayes’ Wikipedia page!

It only took me a few minutes to compose the email that liberated the photo, and it gave me a nice feeling of accomplishment for the day. Bastian has another post on her blog about how people can help get missing scientists’ faces added to their Wikipedia pages with tips you can use whether you have minutes or days you can devote to it. She also started the @MissingSciFaces Twitter account to encourage people to help write and share stories of scientists from underrepresented groups. *Update: Bastian also started a blog, Missing Scientists’ Faces, to document progress and ask for pictures of specific scientists.*

I just signed up for a Wikipedia account, and I’m hoping to start contributing to Wikipedia pages of mathematicians from underrepresented groups and other math articles. I’m new to Wikipedia (as a contributor; not as a consumer), so I’ll be using WikiProject Women Scientists and Women in Red for inspiration as I get started. If you are so inclined, I hope you’ll join me.

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Diversify Your Blogfolio

No matter what we write with, we all approach math from a different angle. Image courtesy of lisamikulski via Flickr CC.

It’s March. As the sun sets on black history month and rises on women’s history month, I feel inclined, as I do every March, to draw attention to some of the great women who blog about math as well as several blogs that address diversity in mathematics.

The most recent exciting news on that front is the launch of inclusion/exclusion, a new blog in the AMS family. The blog is edited by Edray Goins, Piper Harron, Brian Katz, Luis Leyva, and Adriana Salerno (former editor of another great AMS blog PhD + epsilon). The mission of the blog is brought into clear focus in their first post, Inclusion/Exclusion Principle. The editors aim to change the common notion of what it is to be “professorial,” to include a diverse array of views through conversations about the profession, and to provide us with strategies for addressing the diversity in the classroom and in the field.

One of my favorite posts so far came from Piper Harron, who formerly (and perhaps still?) blogged as The Liberated Mathematician. In her post, Hands Off My Confidence, Harron attacks the common “women-lack-confidence narrative,” by describing her own relationship with confidence through the arc of her life. She makes important points about the difference between lacking confidence and making calculated decisions within a system.

It’s often a challenge to understand how a person’s life experiences have shaped their outlook and relationship to math when we haven’t necessarily experienced the same life. Recently my co-blogger Evelyn Lamb, who also blogs for Scientific American at Roots of Unity published Being a Trans Mathematician: A Q&A with Autumn Kent. In it Kent admits that her story of transition is not necessarily the same one shared by all trans women, but it’s very worthwhile to read about her story of coming out occurring in parallel with her trajectory as an academic mathematician.

Finally, this year at the JMM the outgoing MAA president Francis Su gave an address, which is now posted to his blog as Mathematics for Human Flourishing. His message of inclusions is a poignant one, and he speaks to the experience of minorities in math. In his post he addresses how important mathematics and a sense of belonging in the mathematical community are to our ability to flourish as humans. Whether we explore math for play, beauty, justice, or truth, math has something to teach us about the larger world. In it, Su says, “justice is required for human flourishing. We flourish—we experience shalom—when we treat others justly and when we are treated justly.”

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A Thrice-a-Day Complex Analysis Infusion

Sometimes I like to sit back and take in math via pictures only. The newest addition to my math picture blog stable is kettenreihen.wordpress.com where three times a day, a beautiful picture appears.

Credit: Thomas Baruchel

These are graphs of complex-valued functions based on continued fractions. White points are where the function takes on real values, black are imaginary, and the other points are colored according to angle. On the blog’s “about” page, Baruchel explains the notation, which is not the most common continued fraction notation. 

It’s fun to look at a graph and try to figure out why it looks the way it does: why are there purely real or imaginary values in particular places, why a graph has horizontal or vertical symmetry, and so on. These are not straightforward functions, so I must admit I don’t always get anywhere, but it’s fun to think about anyway. If I were teaching complex analysis, I would probably try to work this blog into my class somehow, if only to gawk at pretty things with my students.

I’ve been trying to decide which kinds of graphs are prettiest. “Flowers” are nice.

Credit: Thomas Baruchel

But so are hearts.

Credit: Thomas Baruchel

Graphs that evoke mathematical symbols are always on trend.

Credit: Thomas Baruchel

Simple ones are nice.

Credit: Thomas Baruchel

And so are the wild, busy ones.

Credit: Thomas Baruchel

Luckily you don’t have to choose. Just add the blog to your feed, put your feet up, and relax with some gorgeous mathematical illustrations.

Posted in Mathematics and Computing, Mathematics and the Arts | Tagged , , , | 2 Comments

A Circular Approach To Linear Algebra

Sadly, this is not actually the way linear transformations work, from xkcd.

This semester I’m teaching Linear Algebra for the first time, so naturally, I am constantly on the prowl for all of the linear algebra resources the internet has to offer. To begin with, I’m using a free online textbook called Linear Algebra Done Wrong by Sergei Treil. I’ve found that it’s a bit…intense. As a person who understands linear algebra the book is very nicely written and has a logical presentation and abundant clever examples. But for a person who has never seen linear algebra, well, let’s just say it’s a bit like diving into the deep end with no floaties on while someone shoots you with a paintball gun.

Consequently, this semester has left me foraging the world wide web for supplementary resources to help my poor flailing floatieless students as they try to navigate the waters of vectors and matrices.

A great place for students to begin if they are totally lost is a series of wonderful YouTube videos called the Essence of Linear Algebra, from 3Blue1Brown. The animations really help to bring out some of the geometric intuition behind vector spaces, which can seem abstract (and sometimes totally impenetrable!) to students seeing them for the first time. Of course Khan Academy also hosts a linear algebra series, but my students haven’t found them as helpful.

Don’t like the dog? You can also toggle between a cat and a mouse. Screenshot from Wolfram Demonstrations Project.

On the theme of helpful animations, which teaching linear transformations, I found some really great demonstrations on Wolfram that let you transform a dog, more specifically, a Scottish Terrier, by a personalized 2×2 matrix. You can stretch, flip, and shear the Scottish Terrier by changing the values in the accompanying matrix. Somehow this is way more convincing than just drawing pictures and waving your hands around. The Wolfram Demonstrations Project is packed with great demos for transforming vector spaces, and you can share your own.

Finally, when I ask former math majors what most mystified them about Linear Algebra I almost always hear something about eigenvectors. It’s shocking how many students get in and out of Linear Algebra and have no intuitive idea what an eigenvector or eigenvalue are. And I’m not passing judgement here. When I took Linear Algebra as an undergraduate I was in the same boat! I knew how to compute them, but I had no idea what I was really looking at.

Not just a beautiful pirouette, but also a great example of a linear transformation with eigenvector e2 and eigenvalue 1.

Luckily, we have Steven Strogatz to the rescue with a most concise and intuitive explanation of eigenvectors and eigenvalues. He compares a linear transformation of 3-dimensional space to snapshot of a dancer, arms outstretched spinning in a pirouette-like motion. Her arms (a vector in the x direction) are moving, her gaze (a vector in the z direction) is moving, but the leg she’s spinning on (a vector in the y direction) stays fixed. This fixed vector is an eigenvector. And if she comes down of her pointed toes, then there is some element of scaling. This is an eigenvalue.

Mathew Simonson, who wrote about his own “eigenightmares” for the AMS Grad Student blog, proposes a spiral approach to pedagogy to combat those eigenfears. Students learn early on to express linear transformations as matrices, at this point they already can get some sense of eigen-type behavior just by acting on a simply figure in a vector space. Say, maybe, a nice little Scottish Terrier. In this way, students can see that eigenvectors are happening, before formally knowing what they are. This puts the intuitive before the formal, which I like.

Do you have any favorite online resources for teaching or understanding Linear Algebra? Let me know on Twitter @extremefriday.

Posted in Math Education | Tagged , , , | 4 Comments