Beyond Euro-American Mathematics

A New York Times op-ed by  earlier this month calls out university philosophy departments for their lack of diversity. “We therefore suggest that any department that regularly offers courses only on Western philosophy should rename itself “Department of European and American Philosophy.”

The Monte Albán observatory in Oaxaca, Mexico. Image: Rob Young, via Wikimedia Commons.

The Monte Albán observatory in Oaxaca, Mexico. Image: Rob Young, via Wikimedia Commons.

Garfield and Van Norden take the position that philosophy deserves to be singled out in a way mathematics and science do not. They write,

Others might argue against renaming on the grounds that it is unfair to single out philosophy: We do not have departments of Euro-American Mathematics or Physics. This is nothing but shabby sophistry. Non-European philosophical traditions offer distinctive solutions to problems discussed within European and American philosophy, raise or frame problems not addressed in the American and European tradition, or emphasize and discuss more deeply philosophical problems that are marginalized in Anglo-European philosophy. There are no comparable differences in how mathematics or physics are practiced in other contemporary cultures.

Putting aside the fact that it is strange to ignore literature, art, history, or religion, departments that are frequently Eurocentric in ways that mathematics and physics are not, it may not be as clear as Garfield and Van Norden think that mathematics departments should not be criticized for being Eurocentric. The apparent universality of mathematics is one of the things that draws people to mathematics, but nothing takes place in a vacuum. Michael Harris, author of the book Mathematics without Apologies and a blog of the same name, writes,

What is or is not ‘comparable’ is in the eyes of the comparer, of course, and it’s no doubt true that cultural differences are no barrier to communication between contemporary mathematical practitioners in Asia and the rest of the world.  Historically, however, mathematics developed around the world in conjunction with a variety of metaphysical traditions, and this has inevitably affected the approaches to foundational matters.

In another post, he suggests that “the most interesting problem currently facing philosophy of mathematics is to clarify how or whether Chinese and European mathematics differ and how or whether these differences reflect differences in the respective metaphysical traditions.”

I taught math history for two semesters, so I’m hardly an expert on how the subject is taught in general, but I did struggle with how Eurocentric my own math history background and the vast majority of math history resources I came across were. Sometimes it seems like the dominant math history narrative is “Greeks (nevermind that many of the ‘Greeks’ were from North Africa and the Middle East, we call them Greeks so you’ll think of them as European) invented mathematics, it died out around 500 CE, and then Italians started doing it again in the 15th century.” If we’re lucky, the narrative might mention Al-Khwarizmi, whose name gave us the word algorithm and whose book Hisab al-jabr w’al-muqabala gave us the world algebra.

Unfortunately, my math history class fell into the Eurocentric model more than I wish it had. I felt I did not have the knowledge base necessary to teach a class specifically on non-European math well, but I did require my students to do projects on mathematics from “non-western” sources. (It’s difficult to figure out the right label here. I wanted my students to research mathematics from someone whose culture is not well represented in math history books. Non-European is not quite right, because many so-called Greeks were from Africa and Asia. Non-western is not quite right because mathematics from the Americas before European conquest very much counts. In the end, I went with “non-western” in scare quotes and a long explanation of what I meant.)

One difficulty we encountered in researching non-western math sources was that my students and I are all products of the same metaphysical tradition, as Harris would call it, in mathematics, and it was difficult for us to understand mathematics from other traditions on their own terms rather than viewing them through our own cultural lens. Another, as I’ll come back to later, was the dearth of documents available for them, especially if they were interested in math from pre-Columbian America, Africa, or Oceania.

Eurocentricism in mathematics is on my mind right now not only because of Garfield’s and Van Norden’s New York Times article and Harris’s response to it but because I’m on vacation in Oaxaca, Mexico, home to several impressive ruins from pre-Hispanic civilizations, including Zapotec and Mixtec. These civilizations are not as well known as the Aztec or Maya, but they, like those more famous Mesoamericans, were accomplished astronomers. (In the ancient world, astronomy and mathematics went hand in hand in a way they don’t today.) On a tour of Monte Albán, the remains of a Zapotec city, we saw buildings oriented exactly to the cardinal directions and an observatory that occasionally aligns with the sun perfectly. (Perhaps we should call it Albánhenge.)

Heartbreakingly, the destruction of indigenous populations and documents from indigenous cultures means we have very few resources for learning about the astronomy and mathematics of ancient Mesoamerican people. I learned this when I saw how limited the choices were for my math history students wanted to find Mesoamerican math sources for their projects, despite the sophisticated astronomical calculations they did. (Go ahead, try to understand the Maya calendar system!)

I would love to share some good online resources on non-Euroamerican mathematics, but sadly, I don’t have many. Offline, The Crest of the Peacock seems to be one of the best books about non-European mathematics out there, and the North American Study Group on Ethnomathematics publishes a Journal of Mathematics and Culture.

Online, the award-winning MacTutor math history archive has some articles about the mathematics traditions of different cultures. (If you’re wondering why they have an article specifically about the mathematics of Scotland, note that the site is hosted by the University of St. Andrews.) The Story of Mathematics, an online math history site, also has some articles about Maya, Chinese, and Indian mathematics. On blogs, the pickings are a bit slim. I do want to toot my own horn a bit and point you to my students’ math history blog, 3010tangents. There, my students wrote about a lot of topics, including the amazing navigation devices of the Marshallese, the number zero in Babylonian, Indian, and Maya mathematics, and The Nine Chapters on the Mathematical Art, a Chinese math text.

Do you have any more suggestions on where to learn about mathematics from cultures who are often left out of the history mathematics? Please share them below.

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Functions Too Cool For Facebook. But Don’t Worry, We’ve Got You Covered

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

This map captures the web of contributions from over 80 participants in the LMFDB project over several years.

Today is the official launch of the L-functions and modular forms database. The LMFDB is a database containing all the relevant information about millions of mathematical objects. Set up like a Facebook for mathematical objects — by objects I mean curves, functions, special equations and structures — the LMFDB lets us see which objects are related to each other, which ones share a common ancestor, and which ones can at least play nice.

But maybe you, like nearly all people who aren’t seeped in a daily brew of number theory, wouldn’t recognize an L-function if it walked into the room right now. Even so, I promise this database has some exciting implications for you. Yeah, you. Understanding how the social network of all these millions of objects looks can give a huge kick in the pants to the famous Riemann Hypothesis. But even for those of us who don’t run around muttering about zeroes on the critical strip, we still profit, perhaps unwittingly, from this and other really hard number theory problems every day when we use the internet. Knowing more about he universe of the LMFDB can help find vulnerabilities in encryption, keeping our private data and transactions safe.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

The connection between elliptic curves and modular forms is just a small part of the L anglands Program, a vast web of conjectures proposed by Robert Langlands, at the Institute for Advanced Study, in the late 1960s. Image courtesy of David Dumas, Timothy Boothby, and Andrew Sutherland.

But much more broadly — and perhaps more importantly — one of the motivating goals of so much mathematics of the last century has been to find a so-called grand unifying theory of mathematics which we call the Langlands Program. In the mathematical universe we deal with all kinds of seemingly unrelated objects, like those curves and functions and other things I mentioned earlier. “The connections between these classes of objects lie at the heart of the Langlands program,” explained the Fields Medalist Terry Tao in a blog post about the LMFDB today. The LMFDB teases out a lot of surprising relationships between theoretical objects, ones that aren’t so easy to see when you look at these things one at a time.

And even if you aren’t chasing the grand unified theory, if you work in certain areas of math, these objects come up all the time, and having an atlas to this mathematical universe can be incredibly helpful. As Emmanuel Kowalski wrote on his blog today, the LMFDB can help us understand their “random and possibly spooky” behavior.

Another huge boon of the LMFDB is that it stores billions of time intensive calculations for immediate retrieval — literally thousands of years worth of computations — saving our future selves huge time and effort. Tim Gowers, Fields Medalist and proponent of effort-saving tools, wrote about the LMFDB on his blog today, saying “I rejoice that a major new database was launched today.” This frees us up to do other things, like prove deep results.

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Straws Thingys and Other Mathematical Sculptures

I love an abstract math pondering session as much as the next mathematician (or at least within epsilon), but there’s something immensely satisfying about coming back down to earth and using your hands to make something. At some point last November, I stumbled onto Zachary Abel’s blog series about Straws Thingys, and I knew a Straws Thingy had to be mine.

A Straws Thingy. Image: Zachary Abel.

A Straws Thingy. Image: Zachary Abel.

A Straws Thingy is indeed a thingy made of straws. In this case, it happens to be a compound of five intersecting tetrahedra, popularized by mathematical origami guru Thomas Hull. After several months of forgetting to buy the recommended brand of straws from Target, I finally managed to make my very own Straws Thingy last week.

My Straws Thingy, before and after savagely breaking free from its scaffold. Image: Evelyn Lamb.

My Straws Thingy, before and after savagely breaking free from its scaffold. Sadly, the straws I bought only came in four colors, so my fifth tetrahedron is multi-colored. Image: Evelyn Lamb.

Abel is a graduate student in mathematics at MIT as well as a mathematical sculptor, and I just loved his Straws Thingy posts. HHe had challenged himself to write a blog post a day for the month of November (#NaBloPoMo, a more modest undertaking than #NaNoWriMo), so the instructions are conveniently served in bite-sized pieces.

I tend to be something of a mathematical social butterfly. I like to learn a little about something and then flit away to the next thing. Abel, on the other hand, is much more thorough. His month-long Straws Thingy series explores the subtle asymmetries of various Straws Thingys, eventually building to a five-dimensional hypercube of thingys (conveniently immersed in three-dimensions). 

If you have a mirror, you only need 16 Straws Thingys to make a 5-dimensional hypercube of them. Image: Zachary Abel.

If you have a mirror, you only need 16 Straws Thingys to make a 5-dimensional hypercube of them. Image: Zachary Abel.

He also gives us a peek behind the scenes to show us the origin story of the scaffold he designed to make the Straws Thingy easier to assemble.

A Straws Thingy scaffold, ready for action. Image: Evelyn Lamb.

A Straws Thingy scaffold, ready for action. Image: Evelyn Lamb.

If you want to make a Straws Thingy (or 32) of your own, his scaffold is available as a pay what you want download, and the instructions are easy to follow on his blog. I poked around the blog a little bit after building my Straws Thingy. He has a lot more fun posts about geometry and mathematical sculptures, including the Penny Pincher (made with $20.00 in pennies!) and instructions for a “potentially lethal” Impenetraball. Maybe I’ll get there eventually, but right now I still have a lot of leftover straws.

A Straws Thingy in Waiting. Image: Evelyn Lamb.

A Straws Thingy in waiting. Image: Evelyn Lamb.

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Math and Verbal Gymnastics

We are coming to the end of Math Awareness Month, whose theme this year was The Future of Predictions. A clever theme name, indeed. I do love when mathematics and verbal gymnastics come together. And on that theme of math and words, you should know that April is not only math awareness month, but it’s also National Poetry Month! In honor of this double-whammy, I thought we could take a moment to explore the intersection of math and poetry.

As it turns out, there are actually quite a few blogs dedicated to poetry and mathematics. If you, like me, are a mathematician who is new to mathematical poetry, a good place to start is with “Five Types of Mathematical Poetry” on the blog Mathematical Poetry.

If the strictly lexical type of mathematical poetry is what you prefer — that is, mathematical poems constructed from the written word and influenced by ideas in math — then I suggest JoAnne Growney’s blog, Intersections — Poetry with Mathematics. In the theme of math awareness month, Growney posted a beautiful poem by Joyce Nower, inspired by prediction, fate, and the tragic story of the mathematician Hypatia.

Another type of mathematical poetry melds the written word with mathematical symbols, not necessarily following any mathematical rules. The late Bob Grumman, a pioneer in visual mathematical poetry, described it as “poetry that does mathematics, rather than merely discusses mathematics.” In a post on the Scientific American Guest Blog, Grumman discusses the state of the art form and the work of fellow visual poet Karl Kempton.

Magic Square By BabelStone - Own work, CC BY-SA 3.0,

Magic Square By BabelStone – Own work, CC BY-SA 3.0,

Pure mathematics can also be seen as poetry. The patterns and repetition in numerical and symbolic mathematics do echo those in traditional lexical poetry. To the right you can see a magic square, an ancient example of pure mathematical poetry.

In the spirit of poetry and math awareness, let me close with a terrible haiku that I just wrote in honor of the close of the semester.

reaching the limit
harmonic series diverge
and so too do we

Share your #mathematicalhaikus with me on Twitter @extremefriday.

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Fold Your Way to Glory

Yesterday, I led a meeting of a Teachers’ Math Circle about the fold and cut theorem. This theorem says any region with a polygonal boundary can be folded and cut from a sheet of paper using only one cut. I learned about the theorem last year when Numberphile posted this excellent video featuring a virtuoso performance from Katie Steckles, who folds and cuts every letter of the alphabet from memory.

I was a little nervous about leading the program because I had prepared almost nothing to say. Everything I thought about saying was boring, so I decided the best way to approach the activity was to just get people started on it. Luckily for me, the group was ready to jump right in. I dumped a bunch of paper into the middle of the table, and people started folding.

I encouraged people to try the most symmetric shapes first, but other than that, I didn’t have to give them many suggestions. I was prepared for some frustration when they started trying the scalene triangle because it’s a big step up in difficulty, but several of them got the scalene pretty quickly, and no one seemed to give up. In general, the strange shapes people got when you mess up were amusing rather than frustrating.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

My favorite fold-and-cut mistake. I was trying to make a rectangle with a smaller rectangle inside it. Image: Evelyn Lamb.

Participants almost immediately started asking mathematical questions and trying to extend the activity: do we have an existence theorem? Must we always fold along every angle bisector? Is there a general theory of folding? I liked Anna Weltman’s suggestion of trying to make things without drawing on the paper, and I spent some time trying to fold stars without drawing them, but the teachers didn’t really bite on that. Instead, some of them started thinking about minimal folding numbers for different shapes, and some of them worked on developing a folding algorithm.

Erik Demaine is one of the pioneers of fold-and-cut theory and the mathematics of paper folding in general. His page about folding and cutting has links to all the gory mathematical details as well as some templates. I ended up bringing copies of his swan to the teachers’ circle. They are beautiful, but I had mixed feelings about bringing them because they have the fold lines marked on them already. I didn’t hand them out until one group had started talking about how to use angle bisectors and perpendiculars in their folding algorithm, and I thought the swan template might give them some ideas. Because I gave them only the template, not any explanation of how it was made, I think it didn’t take away too much of their fun.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb.

My fold-and-cut swan now enjoys pride of place on my new hexagonal shelf. Image: Evelyn Lamb. Swan template: Erik Demaine.

In addition to Demaine’s swan, I brought templates for lots of different shapes from Patrick Honner and Joel David Hamkins, who uses hole punching symmetry activities as a warm-up for cutting. I also got ideas from Mike Lawler, who has done fold and cut activities with kids, and Kate Owens, who ran a fold-and-cut workshop for teachers.

I’ve done a little bit of origami, but I’ve never gotten good enough to feel like I had geometric intuition for doing it. I’m still at the level where I follow directions and get what the book says I should. Making these fold-and-cut shapes, though, is an easy way to start thinking about paper folding mathematically and creatively. Thanks to the resources I mentioned above, you too can easily introduce people to the joys of mathematical paper folding.

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Packing Heat: The World Of Sphere Packings Is On Fire

Suppose somebody hands you a bunch of oranges and asks you to stack them on a shelf, I’ll bet I can guess how you would do it. You’d build a pyramid by laying down a base layer and then fill in the upper levels by placing oranges in each of the divots provided by the layer below. If you’ve done this before, you might have noticed that the oranges in the base layer create a repeating hexagon pattern. In case you don’t have a crate of oranges next to you right now to try this out, check out the photo below.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing.  Image via Jeremy Jenum Flickr Creative Commons.

A hexagonal packing of oranges creates the base layer on the left, and then the pyramid continues to rise using the hexagonal close packing. Image via Jeremy Jenum Flickr Creative Commons.

This is called a hexagonal circle packing, and it’s the densest way to pack a bunch of circles together. By densest, I mean that any other way you pack together circles is going to have much more empty space left over. When you place the subsequent layers on top by filling in the divots, what you’re doing is creating a well-studied arrangement called the hexagonal close packing of spheres. Just like the hexagonal packing in 2-dimensions, the hexagonal close packing is the densest way you can pack 3-dimensional spheres together. This was a result proved by Thomas Hales in 1998.

These both belong to the broader family of n-dimensional sphere packings, and it’s been a long standing open problem to find the densest sphere packings in each dimension. While we have the nice orange stacking analogy to help us visualize dimensions 2 and 3, in higher dimensions we can’t visualize things in the same way. But here is the essence of the problem. In any dimensions, a sphere is just a set of points that are equidistant from some center point, and a dense sphere packing is just an arrangement of non-overlapping spheres that fills up as much ambient space as possible.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

Maryna Viazovska unlocked the solution in 8 dimensions, paving the way for the 24-dimensional solution. Photo Courtesy of Oberwolfach photo archives.

A few weeks ago Maryna Viazovska, currently a post-doc at the Berlin Mathematical School and the Humboldt University of Berlin, solved the sphere packing problem in 8-dimensions. Erica Klarreich, a math journalist for Quanta Magazine gives details on how Viazovska arrived at her solution, and some stories about the people she met along the way.

And then not a week went by before she and her coauthors, Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Danylo Radchenko generalized her methods to solve the sphere packing problem in a 24-dimensions. On his blog, mathematician Gil Kalai gives some historical background for the 8- and 24-dimensional sphere packing problems.

In a video posted by the Institute for Advanced Study, Stephen Miller gets into the details of the proof, he says “there’s something very special about 8 and 24, we can’t expect every week to keep proceeding like this.” Although the dimensions 8 and 24 might seem totally random, the reason these solutions came so close on each other’s heels is that these sphere packings — unlike those in other dimensions, as discussed on the n-category cafe — are related to two special lattices, E8 and the Leech lattice. Having this connection to lattices, which, full disclosure, I’m obsessed with, means that there is a world of machinery in the realm of modular forms for dealing with the packings. In a very broad sense, solving the packing problem came down to finding some suitable modular function that satisfied an appropriate list of properties that are derived from methods in harmonic analysis.

Sphere packing problems, of course, have many interesting applications, but the one that has always fascinated me is the link between dense sphere packings and error correcting codes. Trying to pack n-dimensional spheres as close to each other as possible is like trying to find points (namely, the center point of the sphere) that are as close to each other as possible, while maintaining some prescribed amount of distance between them (namely, the buffer created by the sphere around each center point). This acts just like an error correcting code, in the sense that we want to find code words that are similar enough that we can build a language out of them, but far enough apart that they can be transmitted over noisy channels and not be totally degraded by interference.

Like all good problems, sphere packings touch on many branches of mathematics: number theory, geometry, analysis. The fact that this problem has so many approaches and that its solutions are simultaneously so diverse in flavor, John Baez points out so perfectly in his blog post, “hints at the unity of mathematics.”

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How to Celebrate Square Root Day

Apparently today, 4/4/16, is Square Root Day. (I supposed we could also have celebrated 4/2 to have a long Square Root Weekend.) How should a math enthusiast celebrate this holiday, which won’t come again until May 2025?

8 squares arranged in a curve of pursuit. Image: Evelyn Lamb.

8 squares arranged in a curve of pursuit. Image: Evelyn Lamb.

Of course, one option is to be a curmudgeon. As a curmudgeon myself, I heartily support you in this endeavor, and Michael Lemonick of Scientific American does as well. If, however, you are not as cold and dead inside as he and I are, I’ve got some lovely square facts and activities for you.

I’ve got to lead off with one of the coolest things I learned last month, courtesy of Matt Baker.

Let n be a positive integer.  It is easy to see that a square can be dissected into n triangles of equal area if n is even (Exercise).  What if n is odd?  If you play with the question for a bit, you probably won’t be surprised to learn that in this case it’s impossible.  But you may be surprised to learn that this result was not proved until 1970, that the proof involved p-adic numbers, and that no proof is known which does not make use of p-adic numbers!

I find it shocking and delightful that a fairly simple question about plane geometry requires p-adics to solve. If, like me, you’re a bit uncomfortable with p-adics, cut-the-knot math has a p-adic page where you can learn about these strange completions of the rationals.

A more traditional way of celebrating might be to ponder the wonderful Pythagorean theorem. Cut-the-knot has more proofs of it than you can shake a stick at, and I’m still enchanted by Albert Einstein’s elegant proof, which Steven Strogatz wrote about last November.

Arts and crafts have many opportunities for square-making. One of my favorite square-centric designs is a curve of pursuit: out of tilted squares, curves seem to appear. The excellent mathematical knitting site Woolly Thoughts has some information about how to knit curves of pursuit. A few years ago, my grandparents celebrated their 82-th anniversary, so I made them a tablecloth with 8 squares arranged in a curve of pursuit. You can see it in the picture at the top of this post or read about it here.

Last year, Math Munch shared a project called SquareRoots by John Sims. Inspired by the quilts of Gee’s Bend, he made mathematical quilts based on the base 3 digits of pi. And for more mathematical quilts, check out the gorgeous ones on Elaine Ellison’s website.

If you want to take your square explorations up a dimension or two (because who has time to wait until 4/3/64 or 4/4/(2)256?), Numberphile has a lovely video about higher-dimensional Platonic solids, including the hypercube, and Mike Lawler has been making hypercubes with his kids.

It’s baseball opening day today as well (√ √ √ for the home team!), and Patrick Vennebush of Math Jokes for Mathy Folks has combined the two topics for a guess-the-graph game.

How will you be square today?

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A conversation with The Accidental Mathematician

Izabella Laba is a fantastic blogger. She is also a professor at the University of British Columbia. She is widely published in harmonic analysis, geoemtric measure theory and additive combinatorics. And most recently, she is one of the founding editors of the new ArXiv overlay journal, Discrete Analysis. But her blog, The Accidental Mathematician, was how I first became familiar with Laba, and it was the reason that I called her for a conversation last week. I was curious how she found her niche in the math blogosphere, and to hear her thoughts on the state of the art.

photo courtesy of Izabella Laba.

photo courtesy of Izabella Laba.

One of the pillars of Laba’s blog is the issue of gender imbalance in mathematics. I spoke with Laba about tackling this topic, “It’s never just gender inequality by itself. It’s not separated from everything else that happens to us. Gender inequality does not exist in a vacuum. It manifests itself in specific ways. And both depends on the rest of who we are and what we do.” It’s important to put things in a rich context, “if I were to just write a post about ‘men think that women are worse at math but they really aren’t,’ then there’s like one sentence I can write about that. And that’s where I have to stop,” Laba says, “it’s not a terribly interesting thing to either read or write.”

The context tells the story, and accordingly, the post of Laba’s that has stuck with me the most is “Gender, conferences, confrontations, and conversations.” For anyone considering organizing a conference, attending a conference, existing as a woman in math, existing as a man in math, seeking equality, recongnizing inequality in all its shadowy forms, or just generally getting it, Laba takes you there.

As a full professor at a top research university, Laba is able to scope out the gender terrain from a unique vantage point. Although it’s 2016, she says, “people get the impression that this progress is happening really quickly and this problem is going to be all fixed in a few years, and I don’t think that’s going to happen.” Laba cites the high numbers of women who fall through the so-called leaky pipeline, an idea that is confirmed by the AMS Annual Survey of the Mathematical Sciences. While 32% (which has held relatively stead over the past 10 years) of the mathematics PhD recipients in 2014 were female, when you compare the number who eventually get tenure track jobs at large research universities, that number is much smaller.

Sometimes it can be hard to see the big picture when we are so focused on our home institutions, Laba says, “blogging is important that it allows people to make those connections, by reading a lot of blogs and communicating with people you get a bigger picture than you would have on your own.”

Blogging, Laba says, is an important tool to shed light on all aspects of the profession, particularly for those who exist outside of the narrow confines of academia. “I don’t really think that I write to humanize mathematicians, maybe it has that effect to some people, but that’s not something that I aim for,” rather, Laba says, “if I decide to write a post it’s about a specific issue that I’ve been thinking about or discussing with people.” Laba has written about the duties of an academic mathematician, the way we choose speak as mathematicians, and the pervasive kookification of mathematicians in the media.

It took Laba some time to find her voice as a blogger, always an outspoken person, she says “I’m actually really embarrassed to look at some of my earlier posts. I’m sometimes tempted to delete all of that. Leaving that there for other people, especially other women who think that they might want to start blogging but they don’t write so well. Ok. Look at what I did!”

But the medium of blogging, Laba claims, is a good and important one. “It was a long time ago I came across personal blogs, political blogs, academic blogs, and it was just amazing how much people could do with that form,” she says, “you could speak your own voice, you could speak for yourself.” Even better, she says, “you could develop your voice gradually, you did not have to write a book, or do something big right away. You could start with small posts and work towards something bigger.”

Posted in people in math, women in math | Tagged | 1 Comment

All the P-values Fit to Print

A bell curve. Image: Cris, via Flickr.

A bell curve. Image: Cris, via Flickr.

I feel like I’ve seen news stories or blog posts about p-values every day this month. First, Andrew Gelman reported that the editor of the journal Psychological Science, famous to some for publishing dubious findings on the strength of p<0.05, will be getting serious about the replicability crisis. (The editorial he referenced came out last November, but Gelman tends to write posts a few months in advance.) Then the American Statistical Association released a statement about p-values, and a few days later, the reproducibility crisis in psychology led to some back-and-forthing between groups of researchers with different perspectives on the issue.

At the heart of much of the controversy is that much-maligned, often misunderstood p-value. The fact that the ASA’s statement exists at all shows how big an issue understanding and using the p-value is. The statement reads, “this was not a lightly taken step. The ASA has not previously taken positions on specific matters of statistical practice.” Retraction Watch has an interview with Ron Wasserstein, one of the people behind the ASA’s statement.

At 538, Christie Aschwanden tries to find an easy definition of p-value. Unfortunately, no such definition seems to exist. “You can get it right, or you can make it intuitive, but it’s all but impossible to do both,” she writes. Deborah Mayo, “frequentist in exile,” has two interesting posts about how exactly p-values should be interpreted and whether the “p-value police” always get it right. Mayo and Gelman were also two of the twenty people who contributed supplementary material for the ASA statement on statistics.

Misuse and misinterpretation of p-values are part and parcel of the ongoing reproducibility crisis in psychology. (Though some say it isn’t a crisis at all.) Once again, Retraction Watch is on it with a response to a rebuttal of a response (once removed?) about replication studies. The post goes into some depth about a study that failed to replicate, and I found it fascinating to see how the replicating authors decided to try to adjust the original study, which was done in Israel, to make it relevant for the Virginians who were their test subjects. Gelman also has three posts about the replication crisis that I found helpful.

One of the underlying issues with replication is something a bit unfamiliar to me as a mathematician: inaccessible data. Not all research is published on the arXiv before showing up in a journal somewhere, so there are still paywalls around some articles. More troubling, though, is the fact that a lot of data never makes it out of the lab where it was gathered. This makes it hard for other researchers to verify computations, and it means a lot of negative results never see the light of day, leading to publication bias. The Neuroskeptic blog reports on a lab that has committed to sharing all its data, good bad and ugly.

So what’s the bottom line? It’s easy to be pessimistic, but in the end, I agree with another post by Aschwanden: science isn’t broken. We can’t expect one experiment or one number to give us a complete picture of scientific truth. She writes,

The uncertainty inherent in science doesn’t mean that we can’t use it to make important policies or decisions. It just means that we should remain cautious and adopt a mindset that’s open to changing courses if new data arises. We should make the best decisions we can with the current evidence and take care not to lose sight of its strength and degree of certainty. It’s no accident that every good paper includes the phrase ‘more study is needed’ — there is always more to learn.

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Algebra: It’s More Than Just Parabolas

Our old pal Andrew Hacker is back at it again. With the publication of his new book and a spate of recent media appearances, he is a man on a mission. A professor emeritus in the Department of Political Science at Queens College, Hacker first rose to fame notoriety when he called for an end to high school algebra requirements in his 2012 op-ed, “Is Algebra Necessary?” This year, he’s doubled down on that demand in his book “The Math Myth and other STEM delusions.”

Naturally, mathematicians and educators have had a lot to say about this. Hacker’s main thesis is that we need to get away from this idea of teaching the arcane rules and formulas of algebra, and instead, replace it with something more intuitive and relatable which he calls “numeracy.” In an excerpt from his book, Hacker says, “Calculus and higher math have a place, of course, but it’s not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads.” Numeracy, he explains, entails a broader sense of quantitative literacy and ability to interpret numerical information without all that rote memorization and jargon.

Hacker is right, cramming rules and pointless seeming equations into the brains of young people is painful for everyone and it totally misses the point of math, which contains so much beauty, utility, and historical context — or ideally at least two of those three. Instead, says Simon Jenkins in The Guardian, “the prominence of maths in the curriculum is education’s version of Orwell’s imaginary boot, ‘stamping on your face … forever’.” Sounds pretty grim. In a post on Math With Bad Drawings, Ben Orlin claims, “Andrew Hacker has a coherent and lovely vision for how to teach mathematics. But to treat his work as a blueprint for all of mathematics education is to make a category error.” Orlin points out that there’s a looming backstory to why we teach the math that we do, competing interests from teachers, future employers, governing bodies, administrators, you name it. To malign the subject of algebra is to ignore the fact that the way we teach those rules and equations is actually more relevant than the rules and equations themselves.

And this is why I will now point out that Hacker is wrong. And the basis of his wrongness, as Keith Devlin breaks down in a post on Devlin’s Angle, seems to stem from the fact that he doesn’t really know what algebra is. Algebra, according to the Khan Academy, is simply “the language through which we describe patterns…Once you achieve an understanding of algebra, the higher-level math subjects become accessible to you. Without it, it’s impossible to move forward.” Algebra is not, as Hacker tries to claim, the inane study of parametric equations, polynomial functions, and vectorial angles. Algebra is the way that we learn to wrangle not just numbers, but concepts and unknowns. It is an ancient art form that allows us to frame questions about numerical things we don’t totally understand and march slowly towards an answer using a systematized approach. For example, suppose that a movie studio earned $15 million with 2 million total transactions. Part of that coming from $6 video rentals, and the other part from $15 video sales. How could you find out how many videos were rented versus sold? Algebra! It is incredibly powerful and as far as gaining a sense of numeracy and quantitative literacy, it can’t be beat.

Personally, I think the part that really sticks in my craw is that Algebra, when taught properly, is just not that hard. I am totally on board with toppling calculus from its place of prominence in the high school math echelon. Because it’s true, not all people need calculus. In fact, most people don’t. But Algebra? I think it’s ok to suffer through a year of Algebra I just to be aware of the fact that we can talk about the general behavior of math using equations and unknowns. To me, that’s the mathematical equivalent of learning how to read a chapter book. If we were willing to accept an equivalently low bar for literacy, then most of our nation’s high schoolers wouldn’t even be able to read Hacker’s book. Or any grown-up book for that matter. And what a scary world that would be.

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