Topology Teaching Blogs

Stereographic projection of a Clifford torus making a rotation through the xz plane in 4-space. Image: Jason Hise, via Wikimedia Commons.

I’m teaching topology for the first time this semester, so I’ve been poking around the blogosphere for ideas of different ways to explain some of the ideas in this class to my students.

Luckily, right before I started the semester, I ran across this post on College Math Teaching: Bad notation drove me nuts….(and still does). Quick, what’s the area of a circle? πr2. What’s the fundamental group of the circle? ℤ. Those aren’t the same circles. The circle with fundamental group ℤ has area 0. It’s the boundary of the circle with area πr2. Unlike the author of College Math Teaching, I wasn’t confused about the circle when I first saw it in topology, and it never occurred to me that that particular example of sloppy nomenclature could be a stumbling block for my students. I think that reading that post has helped me be more thoughtful and careful about notation and terminology when I’m teaching this class.

Another post that has shaped my thinking about teaching upper-level undergraduate classes is a guest post on the AMS math education blog: Mathematics professors and mathematics majors’ expectations of lectures in advanced mathematics. The short version: those expectations are different. For example, I (and apparently many other math teachers) think that leaving some details of proofs to students can be a good way for them to solidify their understanding of a proof in class. After reading that post, I’ve decided to include more of those details in classes if I think they’re important, and if I want students to fill in details on their own, I am having them do it in class or on homework.

I’m teaching algebraic topology, and the author of College Math Teaching is teaching point-set topology, so there aren’t a lot of posts directly about the material I’m teaching, but I’m enjoying a feeling of camaraderie when I read their posts about their class this semester. My students are fairly experienced proof writers at this point, but the author’s students are not. I can’t imagine trying to teach some of these concepts at the same time as teaching proofs writing! On a companion blog, there are course notes for the author’s class. I could probably use some brushing up on the separation axioms. I’ve been interested in counterexamples in topology recently because I wrote about the π-Base, an online version of the classic Steen and Seebach book. College Math Teaching has nice posts about two interesting counterexamples, the Alexandroff Square and the Topologists’ Sine Curve.

I’ve also found Math ∩ Programming, Jeremy Kun’s blog, to be valuable for my teaching. I read over his post about the fundamental group at the beginning of the semester when I was starting to teach it. I’m doing a lot of my teaching from Munkres, which can be a bit too formal at the expense of intuition, so seeing an explanation that is a little more idea-focused is helpful. Kun is a very good math writer, and I’m sure his primers would be useful for students and teachers of other subjects as well. There’s plenty more to enjoy in his blog, and I can’t help but point you to his guest post on Baking and Math. After all, torus knot baklava is much tastier than programming. (Sorry, programmers. I’m never going to be as interested in programming as I am in pastry.)

I recently ran across The Geometric Viewpoint, a blog about geometry and topology aimed at undergraduates and written by Colby College faculty and students. I thought about having some writing assignments in my topology class and ultimately decided against it, but maybe reading the student posts will make me think about it more for next time.

There are a few defunct or rarely updated blogs I’ve found useful. Sketches of Topology has some interesting topology constructions with great illustrations. Dan Ma’s Topology Blog has accessible posts about point-set topology. I haven’t seen much about undergraduate algebraic topology in the blogosphere, though.

Any other recommendations for topology blogs that focus on undergraduate topology?

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Math For Your Ears

Sometimes you just need to rock out to some sweet sweet mathematics. Photo: key-shine via Flickr Creative Commons.

Sometimes you just need to rock out to some sweet sweet mathematics. Photo: key-shine via Flickr

It is undeniable: podcasts are having a moment. The burgeoning podcast culture being shaped by the Radio Labs, 99% Invisibles, and Freakanomics Radios of the world, has gotten me thinking about some of the particular hardships of adapting pure mathematics to a strictly audio setting. As a subject so rich in notation and abstraction, it always struck me as ill-suited for ears-only consumption. But as it turns out, I was way wrong. So, even though this is a blog on math blogs, I want to take a quick detour into the world of mathematics for your ears.

This week I had occasion to speak to two mathematicians who have successfully navigated two very different realms of audio mathematics: the podcast and the local radio show. According to Samuel Hansen, the director and producer of several great math podcasts including Strongly Connected Components and the kickstarter success-story, Relatively Prime, the key to successful mathematical audio is to tell a story. “The way you do it is through metaphor,” he says, “focus on the things that radio and audio really are good at, which is sparking people’s imaginations.”

Instead of trying to use a podcast to teach heavy technical concepts, Hansen thinks we would perhaps be better served with a little bit of levity and mathematical entertainment, “we need people to think about mathematics not just as a school based or education based activity,” he says, “telling someone a story that is not educational but is interesting is a great way to do that.”

For Beth Malmskog, an assistant professor at Villanova University and former host of “Fort Collins Colorado’s premiere music and call-in math puzzle show,” the key was always to give people a relatable story — no matter the cheese factor. “Even if the story is dorky, you have to give them some kind of objects to hang the whole thing on so they can remember what’s happening,” says Malmskog.

As part of her radio show, Malmskog would offer up mathematical puzzles to her local listeners, who were not necessarily mathematicians. “If you’re going to actually get people to solve or understand some mathematical thing, you almost have to turn it into a puzzle,” she says, “if people haven’t spent their whole lives studying math and they haven’t built little personalities of their own for the objects, you’re just speaking gibberish.”

And, as someone who has on several occasions flung both hands dramatically towards the sky, tossed my head back and shouted “just tensor with Q_p and LIFT the whole thing,” I get it. Whether it makes sense or not, we fill our math with secret relationships and personalties all the time.

One point that Malmskog and Hansen were both quick to make: never assume that your audience is full of mathematicians. Turns out all kinds of people love to read, hear, and learn about math, which is something we can definitely get behind.

To learn about a whole slew of great math podcasts, check out Samuel Hansen’s guest post over at And now that you’ve started thinking about this, what sort of stories have you always wanted to tell about math? And what tricks do you use to explain your cool math ideas to the non-experts?

Posted in Math Communication, Recreational Mathematics | Tagged , , , , | 1 Comment

The Social Side of Mathematics

Samantha Oestreicher is a recent Ph.D. in applied mathematics. She’s been blogging about social mathematics since 2007, but I only discovered her blog in December when she started a series of posts about math and tap dancing. She describes a rhythm game she used to play in tap class in terms of modular arithmetic, and in a later post, she connects tap to her climate science research.

A tap dancing class, perhaps about to embark on a lesson about modular arithmetic.

A tap dancing class, perhaps about to embark on a lesson about modular arithmetic or climate cycles.

Oestreicher says she hated math in high school but eventually decided that she was just going along with the crowd. Her undergraduate degree was in theater, and she spent some time in grad school working as the coordinating director of Math Thespian Presentations at the University of Minnesota. I think the combination of a math-averse past and her experience in the arts give her a perspective not seen on many math blogs.

Oestreicher recommends the Elegance Series to new blog readers. Unsurprisingly, it’s about elegance in mathematical proofs. What is it? Why do we like it? Where does it come from? Many other mathematicians have written about this topic, of course, but I thought she had some interesting things to add. In one post, she compares mathematical elegance to elegance in fashion. “Elegance is a social construct, an adjective to describe a simple, well designed, full-package beauty.” She asks, “Is social elegance reserved for the rich? Is mathematical elegance reserved for the mathematically advanced?” As my answer to the question of whether math is only for “math people” (of course not!) has been one of my hobby horses lately, I was interested in the connection she made between that idea and the idea of elegance.

Browsing through the archives, I also came across some posts about proofs, belief, and communicating mathematics. One post in the proof series offered an unusual perspective about the opaque proofs we read in journal articles.

I used to think bitter thoughts about some author’s proofs. Then my real analysis professor said something fascinating.  I think he may have been quoting someone else (any one out there know who he was quoting?).  A masterpiece of architecture would never be opened to the public with it’s scaffolding still up.  You are to see the final product and wonder at it’s beauty as apposed to analyzing how it was built.  For the same reason mathematicians remove the scaffolding from their proofs once they are complete.  They do the final finish work and publish incomprehensible and beautiful proofs.

Because mathematical proof is about communication and context (the proof may change drastically based on the audience), I don’t think I agree fully. I think authors should leave more of the scaffolding up for readers. But sometimes it’s not clear what is scaffolding and what is the building. A related issue, the right way to talk about math with non-mathematicians, appears in a post about “Speaking Math.”

With over 7 years of archives, there are quite a few other series and freestanding posts about mathematicians (both social and antisocial) and the mathematics hidden in everyday life, including Sudoku, video games, and art. As Oestereich’s research is related to climate science, there is also an EcoMathematics category. I’m looking forward to reading more.

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Who Is The Anti-vax Movement? Data Science Explains.

It was all theoretical until Jenny McCarthy gave Sidney Crosby the mumps. Then it got real. Ok, I know that’s a sensationalist — not to mention flagrantly untrue — thing to say, but it’s how I suddenly felt a few weeks ago. The abstract notion of the anti-vaccination movement and the re-emergence of these dangerous all-but-forgotten diseases suddenly landed on my doorstep, and it got me thinking. Who exactly are these people who don’t vaccinate their children?

The great state of California, whose Department of Public Health recently made public all of their data surrounding childhood immunization levels, can help us begin to answer this question. Specifically, they’ve posted a child-by-child count of non-immunized California Kindergarteners who have opted out of vaccinations through the state’s Personal Belief Exemption (PBE) program. To get a lay of the land, we’ll start with a very straightforward plot directly from their 2014-2015 Academic Year Annual Immunization Report.

Percentage of California Kindergartners with Personal Beliefs Exemptions

This plot, from the CA Department of Public Health’s annual Immunization Report, gives the percentage of California kindergartners with Personal Beliefs Exemptions, by school type.

So it’s pretty clear that the rate of PBE’s has gone up since 2010. But as with any simple visual, this one raises more questions than answers. For example, we still can’t quite see where the high rates of PBE’s are coming from. From this information, it’s possible that there’s one gigantic mega-school housing all of the unvaccinated kiddos. And this is an important distinction to make, since what we know of herd immunity says that risks for illness increase when large numbers of unvaccinated kids interact regularly. In case you missed it, Evelyn Lamb gave a great run-down on how diseases spread even in populations that are partially immunized.

So, beyond this basic distribution question, it would also be interesting to see how PBE rates look across types of school. For example, are the rates at small schools and large schools comparable? Can we see consistencies across school types within a district? To help us tease out some of the finer details from the mountain of raw data, Kieran Healy at the blog Crooked Timber turned it into some helpful eye-candy.

The personal belief exemption rates for kindergarteners in California schools by school, based on data from the California Department of Public Health.  Courtesy of Kieran Healy.

The personal belief exemption rates for kindergarteners in California schools by school, based on data from the California Department of Public Health. Courtesy of Kieran Healy.

What we can see here is that larger schools actually tend to have lower PBE rates, that is, they have higher rates of immunization. So the unvaccinated-mega-kindergarten theory is out. This plot also gives a pretty clear answer in the public vs private school question. If you study the data points closely, it also shows some interesting inconsistencies within districts. Healy notes,

“The concentration of PBEs in smaller schools is evident, as is the concentration in private schools. Note that regions with high PBE schools can still show a lot of heterogeneity. For example, consider schools in Berkeley. On the one hand it is home to the school with the second-highest PBE rate in the state. On the other hand, six of its fifteen other schools have PBEs of zero, two more are at three percent or lower, and the remainder range from six to sixteen percent PBEs.”

Finally, we might want to get a more granular sense of what types of schools we’re dealing with. Whether parochial private schools and charter private schools have comparable rates, and what sort of public schools have large numbers of unvaccinated kids.

California Dept. of Public Health data gives the personal belief exemption rates for kindergarteners in California schools.  Courtesy of Kieran Healy.

California Dept. of Public Health data gives the personal belief exemption rates for kindergarteners in California schools. Courtesy of Kieran Healy.

I’ll let you draw your own conclusions from this. But I think this data paints an interesting picture of the vaccination culture — at least in California. Check your own state to see if they make their immunization data available to play with, and if you can’t find your own, Healy put the California data set up on Github, along with the R code for his plots. So with that, I wish you happy data mining. And remember, it’s still flu season, so probably go wash your hands right now.

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Math and the Genius Myth

Earlier this month, Science published a paper about the genius myth and gender. It found that when academics in a field think their discipline requires a special innate talent, that field tends to attract fewer women. “We’re not saying women [or African-Americans] aren’t brilliant or can’t succeed in a field that requires brilliance,” one of the authors told Science News. “It’s the culture of the field that undermines representation because of stereotypes.”

Where do you need to be to be a mathematician? Image: Mwtoews, via Wikimedia Commons.

Where do you need to be to be a mathematician? Image: Mwtoews, via Wikimedia Commons.

Bethany Brookshire, who wrote the Science News article, shared some more personal thoughts about it on her blog. She writes that we tend to attribute men’s success to their talent and women’s to their hard work. When combined with the perception that innate talent is necessary, and perhaps sufficient, for success in certain fields, this idea reinforces the stereotype that women are more suited for some fields than others.

It’s no surprise that mathematics is one of the fields where the genius myth is most pervasive. Mathematicians used to actively cultivate the idea of mathematics as a hallowed priesthood to which only a few are called. Paul Halmos wrote, “To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess.” As I read through that sentence, the idea that our innate mathematical talent is crucial feels sillier and sillier. Are any of us born with insight? Concentration? Taste? We might be born with some amount of aptitude for those things, but we cultivate them over time in response to what our family, friends, and society say is valuable. Why do we think mathematical ability is different?

Cathy O’Neil says that the Science study resonates with her personal experiences. She writes,

It’s just one study, and the response rate was small, so the word is not final. Even so, I think this proves that we should look into this more, gather more evidence, and see where it leads.

Personally, I have already spent quite a bit of time trying to deal with this very problem in mathematics. For example, I’ve explained before how I deliberately teach kids an introduction to proof that emphasizes practice over the silly and distracting concept of having an innate gift. It works, and it’s more fun too, for both men and women.

One of the common arguments about men’s and women’s aptitudes in mathematics is that while the mean and median of mathematical ability, and many other traits, may be the same for men and women, there are more men who are outliers in both directions. Therefore jobs that require outstanding work—such as tenured mathematics professor at a top university—go disproportionately to men. Izabella Laba is blunt in her assessment of that position.

From my professional point of view as a mathematician, here’s how I see this argument. Take a fluid, complex, multidimensional quality such as “math skills.” (Or such as “propensity for criminality,” for that matter.) Assume that this quality can be uniquely quantified, on some scale that covers all types and ranges of “math ability.” Assume further that the resulting distribution is described by a bell curve, because why not. Condition on events of probability practically zero, assume that the same generic, first-approximation model is still accurate on a scale and in a range where it was never meant to be applied, and draw your conclusions about women faculty at R1 universities.

Furthermore, the problem is even deeper than the bell curve outlier idea:

We’ve lived for centuries in a culture that has discouraged women from focused achievement–and by “discouraged” I mean “actively prevented”–directing them towards unassuming mediocrity instead. We’ve lived in a culture that has propagated the stereotype of a woman as an all-round dilettante, while encouraging men possessed of any discernible talent to pursue it to distinction.

The perception that math success is based on innate talent is hard to eradicate, but I am encouraged that today, people like O’Neil and Terry Tao emphasize the importance of hard work and enjoyment over native intelligence to making progress in mathematics. I hope that by changing our emphasis, we can encourage a more diverse group of future mathematicians to excel.

Update: I wrote about the media’s role in the genius myth at Roots of Unity, my Scientific American blog.

Posted in people in math, women in math | Tagged , , , , , , | 3 Comments

$${Mathematicians} \subset {Artists}$$

Simon Donaldson, Ampere's Law, 2014. Printed by Harlan and Weaver, Inc. New York. Courtesy of Parasol Press. Inc

Simon Donaldson, Ampere’s Law, 2014. Printed by Harlan and Weaver, Inc. New York. Courtesy of Parasol Press. Inc

Certain equations or concepts strike us as beautiful, stunning even. As she walked amongst the aquatints on the wall of Yale Art Gallery’s latest exhibit entitled “Concinnitas”, Jen Christiansen posed the title question of her blog post: “Math is Beautiful, But is it Art?”.  Concinnitas means “an elegant or skillful joining of several things”, and its Latin origin made me think about the Latin origins of the word “art”. In Latin, “artem” refers to a practical skill (think “art of blacksmithing”), and also “artus”, meaning “to join” (for instance, joining disparate pieces of information or matter to form a coherent whole, like a sculpture, or a completed puzzle). We now also think of art as stemming from a spark of inspiration that calls the artist to create. In each of these manners, I see the creation of mathematics as an art. Jen Christiansen, the art director of information graphics at Scientific American, also leaned in this direction as she considered the exhibit which consists of responses from mathematicians and scientists to a question: “What is your most beautiful mathematics expression?” The responses came from venerable mathematicians and physicists: Michael Atiyah, Enrico Bombieri, Simon K. Donaldson, Freeman Dyson, Murray Gell-Mann, Richard Karp, Peter Lax, David Mumford, Stephen Smale and Steven Weinberg. You can see their “answers” online at the website of the Greg Kucera Gallery, which is also on the list of galleries exhibiting this portfolio of prints. The idea for the portfolio came from Bob Feldman of Parasol Press, and it was curated by Daniel Rockmore, a math professor at Dartmouth. My favorite was Simon Donaldson’s. Most recently, Donaldson won one of the inaugural Breakthrough Prizes in Mathematics, and his expression is Ampere’s Law. Ampere’s law, which expresses some of the ideas with which a Physics undergraduate might be familiar, implies some connections between topology and physics with the knotting of the “wire” through which “current” is flowing, and with the physical incarnation of the mathematical equation.

As Ben Volta's Micro to Macro mural wraps around Morton McMichael school, wraps around the corner and moves north, the imagery becomes cosmic with solar systems and planetary orbits. (Emma Lee/WHYY)  Courtesy of

As Ben Volta’s Micro to Macro mural wraps around Morton McMichael school, wraps around the corner and moves north, the imagery becomes cosmic with solar systems and planetary orbits. (Emma Lee/WHYY)
Courtesy of

It should come as no surprise that certain individuals are working to make the connections between art and mathematics more apparent to the general population, as evidenced by the recently minted acronym S.T.E.A.M. (Science, Technology, Eduation, Art, and Mathematics). For example, in a recent interview (January 21st) of one the recent Fields medalists, Manjul Bhargava, at NDTV, Dr. Bhargava discusses connections between the Indian musical instrument Tabla and mathematics. He gives an example of the need of a musician to know all possible methods of partitioning eight beats into one- and two-beat sections. Dr. Bhargava uses this as one example of a way to teach mathematics in a more appealing manner that is less “robotic”. Another recent example of the intersection of mathematics and art in education is art teacher Ben Volta’s work with middle schoolers to create a giant mural inspired by the 1970’s video “Powers of Ten”.

However, this exhibit really turns the question away from finding intersections between math and art, away from how mathematics influences art or how art influences mathematics, and asks the more direct question of whether mathematicians are artists.  What do you think?

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A Different Perspective: Mathochism and the Calculus Diaries

I went skiing for the first time on Tuesday. As a native Texan, I’d never really seen the point of putting something slippery on your feet and then stepping on frozen water. But I live in Utah now, so ski I must. It was hard. It’s hard to do something you suck at, and it’s hard to fall down, pick yourself up, and do that thing you suck at again. Not everything in life is a metaphor for teaching or learning mathematics, but this one is. It’s easy for an experienced person in any field to forget how hard it is to suck at it, and math is no different.

Find x. Image: Aaron Rotenberg, via Wikimedia Commons.

Find x. Image: Aaron Rotenberg, via Wikimedia Commons.

Journalist A.K. Whitney is a “math phobe turned math phile,” according to her Twitter bio. Growing up, she enjoyed science, but bad experiences in middle and high school math classes kept her from pursuing a STEM major. A few years ago, she enrolled in a pre-algebra class at a local community college and started her math education basically from scratch. When a friend asked her why, she said, “I got sick of believing I suck at math.” In a series called Mathochism, she writes about her experiences working her way up through calculus. I first encountered the series on Medium, where she is posting an installment every Monday, Wednesday, and Friday, but it is also available on her blog. She also wrote an article for Cosmo about why and how she overcame her fear of math.

Another perspective on learning math as an adult comes from Jennifer Ouellette’s series that eventually spawned The Calculus Diaries. Ouellette is a well-known science writer who focuses on physics, but she had never taken calculus. While she finds conceptual explanations helpful, she writes,

I must confess that on the rare occasions when I’ve bothered to put in the effort to understand a basic euqation or two–and they must be basic, given my functional innumeracy–it has deepened my grasp of the essential concepts in ways I don’t entirely undrstand, yet can’t deny. It’s like some final piece clicked into place that I never even knew was missing.

It’s really interesting to see the points of view of students who start without a strong math background or confidence but genuinely want to learn—and learn to like—the subject. We sometimes assume that students in introductory math courses just want to fulfill a university requirement and don’t really care about learning the material. Those students do exist, and we probably won’t really be able to reach them, but it’s valuable for us to see what makes it easier or harder for a motivated student to make progress. For Whitney, teachers and textbooks are important, but she also writes about broader issues: male dominance in classrooms, our society’s attitude towards women STEM majors, and the myth that you have to be a genius to be good at math.

Her comments about her teachers, from the “dapper professor” of pre-algebra to the “calculus dementor,” are interesting because we get only one side of the story. Sometimes I try to fill in the gaps and figure out what might have been going on from the instructor’s perspective. In a post called Show Your Work, Whitney expresses her frustration at worked examples that skip steps. She writes,

It’s particularly galling when an instructor skips steps on examples in lectures (and so does the book in its ridiculously expensive solutions manual that promises to solve all the odd problems yet leaves out a third of them), then expects me to “show my work,” all my work, and even takes points off for paraphrasing a little on a definition or for not writing f(x) on every step.

I wish I knew specific examples of what steps were omitted and what definitions were paraphrased. I’m sure the instructor thought he or she was doing the examples fully and that his or her rules about how work had to be shown were clear. And what seemed like “paraphrasing a little on a definition” to Whitney may have changed the meaning of the definition. It’s probably impossible to avoid those problems in any class, but it’s good to remember that students who make mistakes like that aren’t being sloppy or lazy. It’s just hard for students to evaluate their own work and understanding.

Unfortunately, Whitney’s calculus class ended on a sour note. But this time, she left with a different attitude than she had in middle school. She writes,

But unlike previous bad experiences, this didn’t sour me on math. Nor did it take away from my interest in learning more calculus, which was actually one of the most satisfying chapters in my short math career.

Yep, I really love calculus.

In 2013, she took a calculus MOOC to solidify and expand on her previous calculus experiences, and for her, it was a good fit. It gave her more motivation than just sitting at home with a calculus book had, but she had the flexibility to choose when to watch lectures or take quizzes. She ended the MOOC on a high note.

At the end, I was tired but very happy. The course confirmed that I really did learn calculus, and that I wasn’t wrong to love it.

And my grade? It doesn’t matter, but I wound up with an 89.3 percent. My B streak continues, and illustrates what I have learned about my math ability: It will never come easily, but I can understand it. I can appreciate it. I can learn it.

I don’t suck at math.

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Return of the Statistics Blogs

When I shared a few of my favorite statistics blogs over a year ago, Thomas Lumley self-promoted his blogs in the comments, and I’m so glad he did! He is the ringleader and a contributor to the University of Auckland statistics department blog StatsChat, and his personal blog, Biased and Inefficient, has some not-quite-Stats-Chattable posts that can be a lot of fun. Last time I wrote about statistics blogs, I found a great one to follow. Will lightning strike twice?

Image: xkcd.

Image: xkcd.

Lumley is particularly interested in the way the media reports medical statistics. Recently, he wrote about the cancer study that has been going around with headlines like “Most Cancers May Simply Be Due to Bad Luck.” He also expands on some of the data in a supplemental post on Biased and Inefficient. Overall, he is critical of the hype but says that the study itself was important.

In contrast, some of the articles he discusses are quite silly: using lipstick during pregnancy, charging your cellphone in your bedroom (if you are a rat and the cell phone is the absence of melatonin), and eating chocolate to help your memory. It’s a bit depressing to see the same errors over and over, but the critiques can be enlightening and funny. I also appreciate his interesting comments about data visualization. Caution: last link includes objects that look like 3-d pie charts (but aren’t) adorning a tree for no apparent reason. Click at your own risk.

Another statistics blog I’ve been reading lately is A Little Stats, written by statistics teacher Amy Hogan. I particularly enjoyed her recent post about a few of the words she thinks can be stumbling blocks to people who are trying to “translate” statistics back into their normal vocabulary. She highlighted percent, which was timely for me, having just been annoyed by someone using “percent” for numbers smaller than 100, a practice I find unhelpful and somewhat deceptive. Hogan doesn’t mention that issue specifically, but I think her comments about some of the other potential pitfalls of percentages are helpful. In the wrong hands, percentages can be very misleading.

It never would have occurred to me to include age on a list of tricky statistics concepts. Age is pretty straightforward, right? Hogan writes, “If someone is 19 years old, for example, it can be confusing as to whether that means the person has finished their 19th year of life or is starting it. Sometimes people round ages, often if asked about the age of a relative. This is further complicated because in different languages how one says his/her age varies. While I don’t think that the true definition of age varies too greatly, good surveys avoid this issue by asking people for their birth date.” It’s such an easy fix, but it’s one I wouldn’t have thought to do.

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In Memoriam

Last week, I was shocked to learn of the unexpected death of Tim Cochran, a topologist from my grad school alma mater, Rice University. In addition to being a well-respected mathematician, he was an advocate for women and other underrepresented groups in mathematics and a beloved advisor to his many students. Since hearing the news, many of my friends have posted their remembrances of Tim, and it is clear that he cared about helping his students and colleagues find their way not only mathematically but also personally. Daniel Moskovich wrote a post about knot concordance in memory of Tim on the Low Dimensional Topology blog. It’s hard for me to imagine the Rice math department without him.

In the rest of this article, I am collecting posts about some of the other mathematicians who passed away this year.

Alexander Grothendieck, who passed away on November 13 at the age of 86, was the most famous mathematician who died this year. The combination of his profound mathematical brilliance and his political activism and eventual withdrawal from society made him a legendary figure in mathematics. David Bruce and Peter Woit were two of the first English-language bloggers to write about Grothendieck’s passing, with short posts containing links to other information about him. Steve Landsburg also wrote two posts about Grothendieck and his mathematics shortly after his death, and Ken Regan’s tribute is a blend of biography, philosophy, and mathematics. Grothendieck was known for making the mathematics he worked on as general and abstract as possible, and his work is quite difficult for even other mathematicians to understand. Recently, David Mumford posted an obituary he and John Tate wrote for Nature that attempted to describe not only the man but also some of his math; however, the article was deemed too technical for Nature, and it was rejected.

Lee Lorch passed away this February at 98. Like Grothendieck a politically active mathematician, he remained in society and worked for desegregation in several different communities where he lived. He taught many of the first African Americans to earn PhDs in math. Unfortunately, some of the institutions where he worked—and the House Committee on Un-American Activities—did not appreciate his activism, and he was pushed out of several jobs in the US before getting a post at the University of Alberta and eventually settling at York University in Toronto. York science librarian John Dupuis has a remembrance of Lorcho that includes links to other information about the remarkable man. JoAnne Growney also wrote a post on her math poetry blog in memory of Lorch. It concludes with “The Locus of a Point,” a lovely poem by Lillian Morrison.

Italian math educator Emma Castelnuovo passed away in April at age 100. I have only been able to find posts about her in Italian, but the IMU recently named an award for “outstanding achievements in the practice of mathematics education” after her.

Ken Regan and Dick Lipton wrote touching remembrances of mathematician Ann Yasuhara and computer scientist Susan Horowitz, both of whom died on June 11th. Like all their posts, these articles do an excellent job of telling us about both the people and their work.

I learned about Dame Kathleen Ollerenshaw, mathematician and politician, in 2012, the year she turned 100. Sadly, she passed away in August at 101. I found her story very moving. She dealt with some very difficult circumstances but seemed to be resilient and optimistic through it all. Her autobiography, written at age 93, describes someone who never stopped being curious.

UCLA mathematician Geoff Mess also passed away in August. Danny Calegari remembered him with a post about groups quasi-isometric to planes, the subject of one of Mess’s most important papers.

If you have written or know of a blog post about a mathematician I have neglected to include, please share it. If you would like to learn more about mathematicians who have passed away recently, the AMS maintains an “in memory of…” page.

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Can Specifications Grading Cure What Ails My Syllabus?

An alternative grading system. Image: Sage Ross, via Flickr.

An alternative grading system. Image: Sage Ross, via Flickr, h/t Robert Talbert.

I love teaching, and I hate grading. I know I’m not the only one. This semester, my math history course posed new grading challenges to me. Grading writing assignments is much more subjective than grading traditional math homework and tests, and the wide range of prior experience (some students’ most advanced math class was calculus 1, and some have taken abstract algebra or topology) proved a challenge for the math-heavy assignments. I have never been completely satisfied with my grading systems, but this semester convinced me that I really need to rethink my approach.

As Robert Talbert wrote in a recent post about grading, “Traditional grading systems work against my goals as a teacher.” Because this was a writing course, editing and revision were important parts of the process. I felt good about the way my feedback helped students improve their work, but I felt like assigning points was petty and antithetical to the collaborative atmosphere I wanted to create.

Enter specifications grading. Last month, specifications grading started popping up in my blog feed. Specifications grading is based on a recent book by Linda Nilson, founding director of the Office of Teaching Effectiveness and Innovation at Clemson University. Talbert interviewed Nilson on his blog, Casting Out Nines. The basic idea of specifications grading is that the syllabus for a class will outline exactly what students need to do to get a desired grade, be it a D or an A, and all assignments are graded pass/fail. Students who want to get a higher grade will have to do more and possibly better work, but all students will have to do acceptable work on some assignments in order to pass the class. Nilson also advocates giving students “tokens” at the beginning of the semester that can be exchanged for an extension or a second chance on an assignment.

Talbert and T.J. Hitchman had a Google hangout on the subject of specifications grading that is now available on YouTube. One thing Talbert said that stood out to me was, “You have students basically opting in to the grade and the work load that they want to take on.” That opting in is what appeals most to me about specifications grading. Some students just want to pass a class, and some want to get an A. In practice, at least for classes I’ve taught using traditional grading schemes, this means that the students who just want to pass do a mediocre job on all the assignments. Wouldn’t it be better if students had to do acceptable or even good work on the assignments they chose to do but could choose which assignments to do? Then I wouldn’t waste my time pushing and prodding students who aren’t interested in putting forth the effort necessary to get a high grade.

I have been thinking about using standards-based grading for a while, but the endless cycles of reassessment that Hitchman mentions in the hangout have been deterring me. I’m sure this is about my struggles to think creatively about standards-based grading, but specifications grading just feels more straightforward to implement. Bret Benesh wrote an interesting post comparing specifications grading as he understands it to the system he currently uses, which he calls accumulation grading, and I think his experience will help guide me as I start to think about reassessing my assessment method. I’ve checked Nilson’s book out of the library, and I hope to incorporate specifications grading into my courses next semester. I know that it will not be a magic bullet, but I think the ideas will help me create a syllabus that better serves both me and my students.

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