Last week The New York Times ran an article by Tim Chartier of Davidson College — and math miming fame — and Sharon Jones of Central Piedmont Community College, all about using sports analytics in the classroom. For those interested in bringing sports or data analytics into the classroom, this article is a fantastic primer. It explains how to gather large data sets and then visualize, analyze, and interpret them.

This is of course a valuable skill in general for anybody looking to pursue a career in data science or data journalism. However, I would argue that sports analytics as a particular teaching tool for this skill can be a bit fraught. In my personal experience, the deeply gendered aspects of professional sport (the players, the spectators, the money handlers) can make it difficult for female students to, quite frankly, care at all. And I’m not suggesting that women don’t like (or play) sports, but one doesn’t need to look too far to find data sets that aren’t as *blatantly* gendered as NFL, MLB, and NBA sports stats.

But as an immediate counterpoint to what I just said, the ever-awesome Laura McLay of the blog *Punk Rock Operations Research* just posted about using game theoretic strategies or linear programming to decide if a football team should pass a ball.

So, I guess I should conclude by saying that nothing drives home a mathematical concept like an applied approach, but let’s just remember that WHIP, WAR, and VORP aren’t necessarily meaningful to everyone. Also, if you know the answer to my batting average questions, tweet it at me @extremefriday.

]]>Math3ma was originally created as a tool to help me transition from undergraduate to graduate level mathematics. Quite often, I find that the ideas of math are hidden behind a dense fog of formalities and technical jargon. Much of my transition process has been (and still is!) learning how to fight through this fog in order to clearly see the ideas, concepts, and notions which lie beneath. Throughout this process, I’ve found that writing helps immensely.

Math3ma would probably be a helpful study aid to a first-year graduate student or a nice preview for someone getting ready for graduate school. The posts include topics from first-year analysis, algebra, and topology, from Lebesgue and Borel Measurable sets to simple and non-simple groups to connectedness. I’ve been thinking about open sets recently, so I especially enjoyed her post about why open sets are everything. (They really are!) Her introduction to Galois theory is a nice overview and includes a suggestion for an abstract algebra TV commercial tagline. “Groups: it’s what they do.”

In addition to the main posts, keeps a few tidbits in her back pocket: an unspoken rule of algebra (Stuck? Try the first isomorphism theorem!), two ways to be small, and other helpful morsels.

All of Bradley’s posts can help students get from the “OK, I understand the words in the statement and proof” stage to the “how does this fit into the big picture” stage. For me at least, that is one of the biggest challenges when learning new math. In my first algebraic topology course, it took me a couple months to realize that the fundamental group wasn’t just something we defined in passing but a major focus of the subject. (In retrospect, the word “fundamental” should have tipped me off.)

Reading Math3ma, I wonder whether my first year of grad school and qualifying exams would have been easier if I had started blogging a few years earlier! A blog like this definitely would have made a good resource when I was studying for quals. I’m looking forward to reading more from Bradley as she advances in grad school and her career in math.

]]>Suppose you have some arbitrary sequence of 1 and -1, something like this

1, 1, -1, -1, -1, 1, 1, 1, 1, 1, -1, -1, ….

And suppose you start plucking entries from fixed intervals and adding them together. For example, if I just pick every third entry from the sequence, and add them all together, I would get

-1 + 1 + 1 + -1 + …

If I carry on doing that for some finite amount of time, will that sum get as big (positive or negative) as I want? Very simply, this is the idea behind the Erdős Discrepancy Problem, which claims that for any arbitrary sequence *{x _{1}, x_{2}, x_{3}, …}* where the

x

_{k}+ x_{2k}+ x_{3k}+ … + x_{nk}> C.

Sounds easy enough, right?

Not quite. And indeed this problem perplexed the great Paul Erdős to such an extent that he offered a $500 reward for a solution in the 1950s. And let’s hope that the late Erdős left a stack of cash lying around, because as of last week the problem has been solved by Terence Tao and the contributors to Polymath5.

In his blog, Tao gives a series of technical explanations of how he and the people of Polymath5 solved the famous problem. He explains how the key to cracking open the Discrepancy Problem was actually solving a totally different problem called the Elliot conjecture. This is such a common occurrence in math (and all fields, I suppose) that solving a really hard problem turns out the be *just* solving another really hard (but maybe a wee bit easier) problem.

The Polymath project — whose most well-known result to date is the tightening of the prime gap bounds of Maynard and Zhang — is a crowd-sourcing effort to drive mathematical breakthroughs. Mathematician Timothy Gowers, over on his webblog, gives a great commentary on the collaborative nature of this result. Although Tao was admittedly the one to put bring the whole thing to a culminating result, the terrain of the foundation would have looked much different without the crowd-sourced contributions. Gowers comments,

“My own experience of polymath projects is that they often provoke me to have thoughts I wouldn’t have had otherwise, even if the relationship between those thoughts and what other people have written is very hard to pin down — it can be a bit like those moments where someone says A, and then you think of B, which appears to have nothing to do with A, but then you manage to reconstruct your daydreamy thought processes to see that A made you think of C, which made you think of D, which made you think of B.”

This result is a marked victory for polymath, speaking to the impressive power of collaboration in mathematics, and an exciting success for Tao. And Tao will not be spending his well-deserved $500 on high-priced journal fees, since he will be publishing his paper in the new Arxiv overlay journal Discrete Analysis.

]]>Although I’m new to teaching using IBL, I’m not new to IBL. The class that made me want to be a mathematician was taught using an IBL model where students presented all theorems and proofs. The class structure was perfect for me, but it turns out my experience as a cocky college student in the class isn’t very helpful for me now that I’m a less cocky teacher trying to run the class. Luckily, between my facebook friends and math blogs, I do have a lot of places to go for guidance.

There are many different ways to implement IBL. I decided to emphasize student presentations of material. I give them notes with definitions, theorems, and exercises, and they supply the proofs in class. The notes I’m using are by Richard Spindler, and I found them in the Journal of Inquiry-Based Learning in Mathematics. I am of course making some modifications, but it’s been incredibly helpful to have a base to work from. I follow more IBL-ers than I can mention here, but for course structure, I am probably most indebted to Dana Ernst. I borrowed heavily from his past syllabuses when I was setting mine up. I have also benefitted from reading Dave Richeson’s posts, especially the one about his inquiry-based topology class. While I was preparing for the first few weeks of class, I found Carol Schumacher’s Instructor’s Resource Manual (pdf) invaluable. It is designed to be a guide to use with her abstract mathematics textbook *Chapter Zero*, but the first 30 pages are about course organization and strategies for various types of active learning in the classroom, and her advice in those pages applies to many different math subjects.

There’s a new IBL blog on the block that has also been interesting for me: the Novice IBL Blog. It’s a joint blog by David Failing of Quincy University, Liza Cope of Delta State University, and Nick Long of Stephen F. Austin State University that’s been running for about a month, the same length of time as my class. It’s nice to feel like there are a few other people with some of the same questions I have, and reading about their experiences. For example, I am currently not entirely satisfied with the quality of course presentations and of my assessment of them, so I am grateful that Cope has written about the way she evaluates presentations. I also keep an eye on Stan Yoshinobu’s IBL blog. I’m trying to incorporate more of his tips for positive coaching into the way I give feedback.

I’m not going to pretend things have gone entirely smoothly for me so far. I had more students drop the class at the beginning of the semester than I expected, and the drop rate for women was higher than it was for men. One of the reasons I decided to go for IBL was that many people believe that IBL and other active learning techniques are more fair to people who aren’t well-off white men. (See for example this recent New York Times article about whether lectures are unfair.) Of course I could not do exit interviews of students who dropped (they tend to ghost), so I don’t know why they left, but I am disappointed that my class has not retained women as well as men. If I set up a similar course in the future, I need to think carefully about how to get better student buy-in and assuage students’ fears that they are not prepared for the class.

So far, my semester has been a bit of a roller coaster, but I’m glad I’m trying something new. And I’m extra glad to be teaching complex analysis. For never was a math topic of more *whoa* than this of complex differentiation and its lovely Cau…chy-Riemann equations. (With apologies to Shakespeare and all people of good taste.)

To quickly recap, academic journals are ridiculously expensive to administer, subscribe to, and even publish in. This is all so crazy when you imagine that I, as a junior faculty looking towards tenure, am required by my institution to publish in the leading journals. So I submit a paper, the fruits of months of my own labor and hand it over to the Journal of Whatever and Such-and-Such. If it is accepted, I’m then given the option: publish it open-access for a cost of upwards of $1,500 (typically paid by the university, I guess, to be honest I’ve never picked this option) or publish it normally, in which case the only people who can read it are those who pay the — some huge amount of money — per year subscription fee to the journal. And in the age of the internet, what is it that you’re really paying for? Official journal typesetting, a meager 200 KB of server space to host the paper, and a whole bunch of pointless overhead.

But as long as you can use LaTex, the official typesetting really isn’t all that important. Reach some minimum baseline aesthetic and your paper is more than readable. And I can think of a place where you can upload all the KB of paper you want: the ArXiv. So really, all you’re paying for is the pointless overhead part. Enter Tim Gowers, host of the great *Gowers’s Weblog*, and the editorial team of the new journal *Discrete Analysis*.

What they are launching is called an ArXiv overlay journal. What this means is, from the editorial standpoint it looks exactly like a traditional journal. The editorial board deals with submissions, sending them out to be peer-reviewed by appropriate reviewers, and managing the quality of content of the journal. Accepted papers are edited and the typesetting is cleaned up, per the referee’s suggestions, and uploaded to the ArXiv. Then *Discrete Analysis* hosts a page of ArXiv links with abstracts. Your paper has gone through all of the rigors of the traditional peer-review process, and it will even get a special *Discrete Analysis* stylized bibtex entry, but you’ve saved everyone a lot of time and money.

I think it’s an interesting model. I mean, we all want to do math, we want to publish math, and we want to read math. It makes sense to keep the process as close to the mathematicians as possible. Right now the electronic end of the journal is being run on a relatively cheap platform that costs $10 per submission. That’s currently covered by a grant, but when that runs up, Gowers points out, “the absolute worst that could happen is that in a few years’ time, we will have to ask people to pay an amount roughly equal to the cost of a couple of beers to submit a paper, but it is unlikely that we will ever have to charge anything.”

Gowers, who won the Fields Medal in 1998, gives a great point-by-point explanation of every facet of the publishing process, and whether you agree with the feasibility of the model or not, I think it’s a worthwhile read.

]]>Several good math writers covered the story of the little pentagon that could. Alex Bellos wrote about it for the Guardian, NPR had a nice article by Eyder Peralta, Kevin Knudson wrote a post about it in his recently-launched Forbes column, Katie Steckles covered it for the Aperiodical, and Robbie Gonzalez wrote a short post for io9.

Lost in some of the news coverage was the fact that most of the pentagonal tilings are not individual pentagons that work but infinite families of pentagons, so there are infinitely many pentagon tilings, not just 15. It’s a subtle point but one that has caused a bit of confusion. Luckily, Wolfram has an interactive demonstration of the different types of tilings that helps clear things up. Reshan Richards also has a pentagonal tiling module on his Explain Everything app. (I do not have Explain Everything, so my mention of it here is not an endorsement or review.)

The question of pentagonal tilings is two questions in one: what pentagons can tile the plane, and how can they tile the plane? In other words, this is the distinction between tiles and tilings. The current discovery is of a new tile, a different shape that tiles the plane. Frank Morgan has a post about the other question: how can tiles be arranged into tilings? Specifically, he has mentored undergraduate students who are studying tilings involving two of the different pentagonal tiles. The post is full of pictures of these two tiles in all sorts of different tiling patterns with names like Christmas Tree, Toothy Smile, and Space Pills. His group’s AMS article about their work (pdf) has even more illustrations.

I first learned about the history of pentagonal tilings when Math Munch wrote about Marjorie Rice in 2013. Rice, one of the big names in pentagonal tiles, was not a mathematician but learned about the problem from a Martin Gardner article in *Scientific American*. It is rare that true amateurs make breakthroughs in research mathematics, but she discovered four new classes of pentagons that tile the plane, and her story is heartwarming and inspirational. Ivars Peterson, the Mathematical Tourist, has a post about her and the tiling of the MAA’s entryway.

While I was researching this post, I ran across another article from Peterson about pentagons and tilings. This one is about the seemingly paradoxical Biosphère dome in Montreal. It is a dome, hence spherical rather than flat, but it appears to be tiled with regular hexagons. “Where does the curvature come from?” Peterson asks. “I know the pentagons are there, and I have tried to find them, but I have had very little success in locating even one.”

I wonder if we’ve now found all the pentagonal tiles, or if someone else will have success locating another one.

]]>Unsurprisingly, as a professor I still experience many of those feelings. A new batch of classes, a fresh crop of students, I’m bristly with ideas about alternative assessment strategies and non-traditional classroom models. On the first days we feel each other out, as we jointly embark on this magical journey into the unknown.

Of course in our modern day the unknown quality of the journey is slightly compromised. The omnipresence of social media and anonymous online forums steal some of the mystery. And then there’s the greatest blight of them all: RateMyProfessor.com.

This is a site that collects brief narrative reviews of thousands of professors by their students. Some reviews are helpful “definitely buy the textbook, it helps a lot,” to lewd and ridiculous “her class was ok but mostly I just stared at her butt.” I’ve often wondered about some of the biases that appear in these reviews, for example, do butts come up more often in reviews of female faculty? Luckily for me Ben Schmidt swooped in this year with his blog *Gendered Language in Teaching Reviews.* Schmidt scraped the data from 14 million RateMyProfessor reviews to study the occurrences of particular words across genders and disciplines.

As reported on NPR earlier this year, men are far more likely to be “brilliant,” especially when they are philosophers, while women are more likely to be rated as “friendly.” In mathematics, we see some really egregious (although unsurprising) gender splits with the words “genius” and “funny.”

But don’t worry, not everyone thinks we’re too smart, math professors also have the highest incidence of the word “stupid,” with it showing up 160 times per million words of text regardless of gender.

And if you’re curious, the word butt doesn’t really seem to follow a distinct gender pattern, but mathematicians seem to rank quite low as compared to the other lab sciences.

]]>David Kung, math professor at St. Mary’s College of Maryland and director of Project NExT, gave an inspiring keynote address at the Legacy of RL Moore Conference in July. It’s a must-watch. I’ll wait.

In the talk, Kung calls on us to be honest about the current state of affairs, which is not good when it comes to representation of women and some minority groups in most STEM fields, and reminds us that math classes are sometimes gatekeepers for the rest of STEM. Our actions in the classroom can affect whether people become physicists, doctors, or engineers.

Kung’s talk is just one of the things I’ve seen recently about the intersection of math, teaching, and bias. Adriana Salerno of Bates College, in one of her last posts for PhD+Epsilon, writes about her desire for the classroom to be “structured in a way that empowers students or that makes them capable of resisting oppression and changing power structures.” Like her, I am tempted to think math is neutral and pure, not sullied by society’s prejudices, and I appreciate reading her ruminations on how to shake that idea off.

Both Kung and Salerno are critical of the way organizations that promote IBL use R. L. Moore in their branding. I was until recently unaware of his appalling treatment of black students. Raymond Johnson, the first African American to earn a degree from my grad school alma mater, Rice, writes about his undergraduate experience with Moore at UT:

Moore, his method and his work are highly thought of in the mathematical world. When he died, there was a laudatory article in the Math Monthly, a publication of the Mathematical Association of America. There is also a major MAA project on the legacy of R. L. Moore. The image of R. L. Moore in my eyes, however, is that of a mathematician who went to a topology lecture given by a student of R. H. Bing. Bing was a student of Moore. The speaker was what we refer to as Moore’s mathematical grandson. When Moore discovered that the student was black, he walked out of the lecture.

One person’s shortcomings in one area do not erase good things they do in another, but the pain and bitterness in Johnson’s writing made me think about the fact that attaching this person’s name to a pedagogy can be one of those little things that makes people feel unwelcome. Moore was not racist in some abstract way; he clearly and deliberately made it harder for some individuals to succeed than others, and those individuals have faces, names, and stories. They remember their treatment by this person others hold in such esteem.

Darryl Yong, mathematician at Harvey Mudd and the school’s associate dean for diversity, has re-launched his blog with a post about radical inclusivity. He writes, “My message to all educators: *not attending to diversity and inclusion concerns in the classroom is the same as allowing your classroom to continue propagating the discrimination and bias that exists in our society*. We have to actively combat discrimination and bias in our work as educators.”

None of us want to feel like we’re racist, sexist or anything else-ist, but we all have implicit biases. It hurts to be told or to admit to ourselves that we have these biases, but it hurts more to have your education and progress impeded as a result of these biases. These biases often don’t manifest themselves in overt racist or sexist actions, but small differences in how we treat different groups of students can accumulate into a force that pushes some students forward in math and some students out. And being a member of an underrepresented group does not mean you can’t be biased against that group. Last year, a story about sexism in hiring made a big splash. “John” got hired more often and with a higher salary than “Jennifer” did, even though they had the same resumés. One of the takeaways of the study was that both men and women on hiring committees were biased against Jennifer. That is a bit demoralizing, but it has a silver lining. No one needs to feel like they’re being singled out when we call on people to examine their biases. We can all fall prey to them, and we should all think about how we can do better.

If you’re reading this, I’m sure you would never tell a woman she shouldn’t be majoring in math because she’s a woman. (If you would, hi, nice to meet you, now stop saying that crap.) But you probably have some implicit biases that cause you to treat women a little differently from men and black people a little differently from white people. I don’t know how to stop doing that, but I honestly believe that being aware of our cognitive biases instead of soothing ourselves with the comfortable reassurance that we aren’t racist or sexist is a huge first step. Whether I am grading, having a class discussion, or writing a recommendation letter, I try to ask myself fairly frequently whether I would act or feel the same way if the student were a different gender or race. I hope that the small step of asking the question keeps me from unthinkingly slipping into biased behavior.

It’s all well and good for me to think about my actions in the classroom, but I also want my students to think about diversity and respect. Yong’s post about being welcoming on day 1 has gotten me thinking about how I should address the issue of diversity in my syllabus and on the first day of class. The fact is, in some ways this discussion is purely academic for me. I teach in Utah, and the demographics of my classroom reflect the (very white) demographics of the state. I can’t make my school more racially diverse, but in addition to striving to treat the few underrepresented minority students I do have equally, I can explicitly encourage my students to think about diversity and respect in my classroom and other aspects of their lives.

How are you promoting diversity and respect in your classroom?

]]>Math is full of laws: group laws in abstract algebra, the law of sines in trigonometry, and De Morgan’s law in set theory, to name a few. And occasionally, the law is full of math. That was the certainly the case in recent patent dispute at the London Court of Appeals, as covered by The Independent.

Here’s the TLDR: two drug companies were arguing over a patent. Company A has a patent for a solution containing between 1 and 25 percent of a certain compound. Now company B has manufactured a very similar solution, containing .95 percent of the compound in question. But everybody knows that .95<1 so company B is obviously in the clear, right?

Wrong. The judge eventually decided that any number larger than .5 is actually the same as 1, since we can round .5 up to 1, and apparently this judge has no love for non-integers.

My immediate reaction as a mathematician is that this could all have been avoided if Company A had just used interval and set builder notation. A quick recap in case it's been awhile since you've seen interval notation. There are two types of intervals, closed and open. The closed ones have square brackets, like [1,25], and the open ones have round brackets, like (1,25). The first contains all numbers between 1 and 25 *including* 1 and 25, and the second contains all numbers between 1 and 25 *excluding* 1 and 25.

The whole point of interval notation (in my mind) is that it takes away any and all possibility for ambiguity. If I say that my solution contains *m* percent of some compound, where m is in the interval [1,25], I truly mean that the smallest possible value for *m* is 1 and the largest value is 25. For example, the number 0.9999…9. (that’s just some long string of nines), which by any convention of rounding would round to 1, is still, itself, smaller than 1 and therefore not part of the interval [1,25]. Because of course, significant digits aside, you can round and truncate wherever you please. So to say that anything larger than .5 is really the same as 1 is a bit arbitrary, why not say anything larger than .49 or even .499, you get the idea.

So I guess the upshot is this: when making large business deals, use the most rigorous language possible to describe numbers, because you can’t count on some guy in a powdered wig to do it for you.

Correction: I initially said that .9999… repeating nines forever was less than 1, but as several apt commenters pointed out, if it really goes on forever forever, that’s just 1 — a true but unsettling controversial fact the internet loves to argue about! So let’s say it’s .999…9 for some really long but finite amount of nines, then we’re ok.

I recently ran across a similar idea from Nathan Yau at Flowing Data. “Data plus beer. Multivariate beer.” (By the way, if you don’t already follow Flowing Data, you probably want to rectify that immediately.) Fibonacci lemonade has two variables: lemon juice and sugar. Beer has a few more degrees of freedom in the types and amounts of grains, hops, and malt.

Many of Yau’s data visualizations involve maps and demographics, so it’s not a surprise that for his first foray into mathematical libations, he chose to make beer recipes based on statistics such as the ethnic makeup, population density, and education levels of different counties. In the end, he brewed batches that represented Aroostook, Maine; Arlington, Virginia; Bronx, New York; and Marin, California. He writes:

Here’s what I eventually settled on.

1. Population density translates to total amount of hops. The more people in a county, the hoppier the beer tastes.

2. Race percentages translate to the type of hops used. For example, a higher rate of white people means a higher percentage of the total hops (determined by population density) that are Cascade hops.

3. Percentage of people with at least a bachelor’s degree translates to amount of Carapils grain, which contributes to head retention.

4. Percentage of people with healthcare coverage translates to amount of rye, which adds a distinct spicy flavor.

5. Median household income translates to amount of Crystal malt, which adds body and some color.

Did it work? Yau didn’t run a randomized control trial, but he says the beers definitely tasted different, and he had some tasting notes notes relating to the population density, healthcare coverage, and median income of the counties the beers represent.

I am coming to this idea from the point of view of a mathematician rather than a data journalist, so something I love about the idea of multivariate beer, Hanna Kang-Brown’s census spices, and other data gastronomification, as Tom Levine calls it, is that it is a natural way to explore the idea of dimension without going the *Flatland* route. (*Flatland* is great, don’t get me wrong, but it’s good to have extra tools at our fingertips.) It seems that most practitioners are interested in the way such concoctions can help people understand real-world data, but I like the potential for use in the strictly mathematical realm. Who knows? Flavors that represent shapes or polytopes? Could you taste the prime factorization of a number?

Yau says he is out of the multivariate brewing game, but if anyone is interested in doing an experiment in mathematical flavor, I’m a willing and able taste tester.

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