Remembering what you didn’t know

First of all, thank you to everyone who reached out after my last post. I’ve started replying, but even if I don’t get to yours, please know I read every one and took them to heart. This is a weird and occasionally toxic career that we’ve chosen for ourselves. It doesn’t have to be that way, but change won’t be fast or easy. Thanks to everybody out there who’s trying anyway.

Another fall is well under way here at Hood, and as the semester rolls on I’m realizing I’m fighting a new battle. I recently heard about a veteran teacher who had a sign at the back of her classroom where only she could see it that read “Remember: this is the first time they’re hearing it.”

Spring will be the start of my 10th year of teaching at the collegiate level; next fall will be my 15th year teaching. To my horror, it’s getting harder to remember what I didn’t know when I was in my students’ place. I sometimes toss out big words, or use notation for something that I forgot to introduce. I’ve probably even misused the word “trivial” without realizing it.

I’m becoming that which I once feared.

It’s getting even harder to remember the day-to-day college stuff that students often don’t know, especially first-generation college students who haven’t had good mentoring. A twitter thread last month, from an academic sending her son off to college, reminded me of a few:

I’m not comfortable with students calling me by my first name. And to me, Mrs. Malec will always be my (wonderful) mother-in-law. So I’ve had to dance around titles a lot. When I was in grad school, I would correct students who called me Doctor. But I never knew what to do about students who called me Professor before I was one. They don’t know there’s a distinction between a person who teaches college and a professor, and I’m not sure it’s worth teaching them about all our weird academic castes. Once I got to my postdoc, if I asked them to refer to me as Doctor instead, they’d see it as lording my title over them and not as a gesture of humility. I just ended up going by my genderless, titleless last name a lot, which suits me fine.

3) What office hours are, why profs have them, when and how to contact profs. His high school *texted him* reminders of homework.

This one hit home. Every year some student tells me they never came to my office hours because they’ve always got a class or another obligation, even though I say over and over that I’ll meet with them outside my posted office hours. Even if this is just an attempt at deflecting responsibility, I should make it more clear that this is a pretty lousy excuse. Also, I give students my cell number and tell them to text me if they need to get me quickly. I’ve done this since I first started teaching, and nobody’s ever abused the privilege. I can see how you might be horrified at the thought, but I’m a big fan.

8) How to take notes. THIS IS VERY IMPORTANT. No one teaches kids how to take notes. The tool is not the issue, whether keyboard or pen.

Does anybody have resources for how to teach this? Notetaking in a math course is a completely different animal than most other classes. I’m still not sure I ever found a good technique when I was a student. I try to do a lot of inquiry-oriented group work in class, and I’m never sure students are documenting their thought process well enough. I’m getting them to think the right way in class, but how can I get them to leave a better trail of breadcrumbs to do it again at home?

None of this stuff is in our job descriptions. None of it counts towards our tenure dossiers. We shouldn’t have to teach students how to be students. But if we don’t, we won’t just be driven crazy by unprofessional, unprepared students. We’ll also be systematically depriving capable people of their desired education, just because they were never taught these skills. There’s a reason children of academics are dramatically over-represented in academia: they learn all these things from a young age. It’s long past time to bring everybody else into the fold.

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Another First Day

The view from the Mathematics and Computer Science Department at Colorado College.

Monday was the first day of classes for me, and also the first day of a new job.  This fall I started as an assistant professor at Colorado College, a liberal arts college of about 2000 students in Colorado Springs, Colorado.  Colorado Springs is about 60 miles south of Denver, at the base of the truly awesome 14,000-foot Pike’s Peak.  Colorado College is great: beautiful setting and campus, my colleagues are interested in their work, but also laid back and simpatico, and the students are creative, curious, and motivated.  It is like many other schools in those ways, though of course CC has its own distinctive personality that I particularly like.  However, one truly unusual thing about Colorado College: the school year here is broken not into semesters or quarters, but into 8 blocks of 3.5 weeks each.  Students take one class per block, with all of their attention on that one course (at least in theory).  Professors teach only one class at a time, too.  But the teaching is intense—most classes meet for around 3 hours every weekday morning, with office hours, problem sessions, and/or labs in the afternoon.  It’s easy to spend 5-6 hours a day in intensive student contact, plus there’s always preparation for the next day and grading.

The intensity is intense.  Did I mention it was intense?  That word comes up constantly when describing the “block plan”.  No question, this schedule can be tiring.  Luckily, each block is followed by a block break, two and a half days plus a normal weekend, when students are completely free to go camping, read novels, play video games, binge watch TV, or whatever.  Of course, if you’re a professor, you might spend a couple of those days grading and a couple getting ready for your next course, which doesn’t leave much of a break when you’re teaching successive blocks.  Again luckily, professors don’t teach during every block—different departments on campus have different teaching loads, but professors generally teach from 4 to 6 out of 8 blocks.  In non-teaching blocks, professors keep up with service, but have a remarkable amount of freedom to travel for research, go to conferences, or do field work.  So professors also have some real breaks built in.

When I tell people about this system, they often tell me that they think it is crazy.  I assure you, I have thought it was crazy at moments as well. However, I knew what I was getting into when I came, since I spent two years as a visiting assistant professor at Colorado College.  That was a really positive experience, and I am very happy to be back here this fall.  Yes, there are aspects of the block plan that are difficult, for both students and teachers.  I think exhaustion is the biggest issue, followed by time management. Many people ask me whether knowledge retention is a problem—how do students remember anything when they completely switch focus every 4 weeks?  My experience has been that this is no more of a problem than on the semester system; Calc 2 students everywhere will claim that they have never heard of the chain rule.

The fact is that there are also many aspects of the block plan that are deeply rewarding and advantageous to teaching and learning. For example, the fact that students have only one class means that very few students totally blow off the class they are in. It happens, but very rarely.  Taking students on field trips and getting out of the classroom is much easier.  Something about the bigger chunks of time spent together can make it easier to build strong relationships with students. If a student is really passionate about math, they can take several courses in succession, moving through the entire calculus sequence, linear algebra, and number theory in one year if they like.  Though students have other appointments, it’s generally ok if class goes over by a few minutes sometimes.  Seeing the students every day means I get far fewer emails.  Non-teaching blocks make extended research travel possible during the semester, without having to cover or cancel classes.  For me, all these positives more than make up for the difficulties.


Getting off campus with classes is a great perk of this job. Colorado College has a satellite campus near Crestone, Colorado. I took this picture there a couple years ago when I took my students to “Number Theory Camp”.

So, to return to the moment: I have been teaching for 5 days and I am more than a quarter of the way through a Linear Algebra course.  I have 23 students, I know all of their names, and class has been exceptionally fun so far. We have covered solving linear systems, row reduction, geometric interpretations of solution sets, spans, linear independence, and linear transformations, and we have already done one Mathematica lab.  The first test is Thursday of next week, and will also cover matrix operations, invertibility, and determinants.  I have done almost nothing for the last week and half but prepare, teach, meet with students, and grade. I am soooo excited that it is Saturday.  But, of course, I woke up at 7 AM thinking about how to explain linear transformations differently.  I always obsess a bit when I am teaching, and it is one of my great life projects to learn to let go of whatever just happened in the classroom and live in the present.  Maybe that’s one reason the block plan works for me, though—it leaves me less time to obsess, since I’m spending so much of my time during the block actually teaching.


Really missing my friend, collaborator, and Villanova office neighbor Katie Haymaker right now.

One week in, things are mostly excellent here.  I love being back in Colorado. I have to say that I also miss my friends, colleagues, and students from Villanova.  I miss being part of the Graterford program, my students at Community Learning Center, the energy and food of Philadelphia, and the dense mathematical network of the east coast.  I LOVED living in Philly and working at Villanova. Starting over is exciting but moving is horrible.  It takes a long time to make connections in a community, to have real friends and feel at home, and, having finally made those connections, it is really sad and hard to leave those people and start all over again in a new place.  So why did I leave, even for a great job like this?  A big reason was to be close to my partner.  I had been in a long-distance relationship for the last three years, which was simply not sustainable for me.  I am also much closer to my family here.  The importance of these factors overrode the sadness of leaving for me.

This constant starting over is one of the worst parts about early career life in an academic job.  It is poignant to move again, when many of my friends from grad school and all my past jobs are buying houses or having babies or forming bands.  And here I am again, giving away my plants, shoving half my belongings into storage, moving into a new office, and going to new faculty orientation to meet the rare other people my age that are looking for new friends.  However, there are so many great people in exactly this position, and at least this time I have already started over once in this place—I mean, I have a dentist at least!  Friends, community, and a sense of belonging can’t be too far behind.

Starting a new job? Interested in the block plan or teaching in some other unusual way?  Please share in the comments.

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Category Theory and Context: An Interview with Emily Riehl

Emily Riehl

Emily Riehl is an incredibly accomplished early-career mathematician, working at the interface of category theory and homotopy theory. She is also a stunning number of other things, including a creative interdisciplinary scholar, working musician, and high-level athlete. A brief career outline: she did her undergraduate work at Harvard University, graduate work at Cambridge and the University of Chicago, was an NSF and Benjamin Peirce Postdoctoral Fellow at Harvard from 2011-2015, and is currently an Assistant Professor at Johns Hopkins University. Emily has been awarded an NSF standard grant and a CAREER award to support her work. She is the author of 21 published research articles, two books (Categorical Homotopy Theory and Category Theory in Context), and many other expository works. All this, and she also performs as a rock/alternative bass player and plays on the US women’s national Australian Rules football team. I recently learned about Emily’s work and profile while looking for women mathematicians to interview for the Association for Women in Mathematics newsletter.  I thought maybe PhD+epsilon readers would also be interested to hear about an early-career mathematician doing some really cool things.  The following interview is a compilation of email and Skype conversations from August 2017, while Emily was in Australia to compete in the AFL International Cup.

A longer version of this interview will appear in an upcoming issue of the Association for Women in Mathematics newsletter.

Question: How and why did you get into category theory? Is there a basic result that you can share that gives the flavor of what you love about it?

Emily Riehl: For graduate school, I deferred from the University of Chicago for a year to go to Cambridge and do what they call a Part III. One of the courses they offered at Cambridge was in category theory, and I liked it instantly; I fell in love. I feel like it chose me as much as I chose it. And it was for the reason that I think that everyone chooses their field, ultimately: the proofs felt like the right way of thinking about mathematics. I felt right away that this is the sort of argument that I wanted to delve into.

Category theory can sound intimidating because it’s highly abstract, but it’s actually not that hard. Several of the most important definitions are quite elementary, and you can start stating and proving the theorems pretty quickly. Indeed, there’s a common belief in category theory that once you understand the statement of the theorem, you can probably supply the proof yourself. Identifying the correct definitions is really the harder thing. The only reason that you typically don’t learn category theory until graduate school is that it requires a rather high degree of mathematical sophistication to appreciate what it’s for.

One of my favorite theorems in category theory is that right adjoints preserve limits—or, since you always get a dual theorem in category theory by simply “turning all the arrows around”—that left adjoints preserve colimits. This result specializes to explain why tensor products distribute over direct sums, why inverse images preserve intersections and unions while direct images only preserve unions, why quotients of topological spaces are formed by first identifying the appropriate points and then topologizing this quotient set. It’s not so much that I appreciate having one proof instead of having to repeat the argument in each context but I feel that the category theoretic proof—which uses the fact that limits are characterized by a “mapping in” universal property, while colimits are characterized by a “mapping out” universal property—is the right one.

Q: You are early in your career, but you have written many, many papers, two books, and a lot of shorter expository work (like posts on the n-category cafe).  How do you do so much stuff?  Do you have any insights into how/why you are so productive?

ER: I read Hardy’s A Mathematician’s Apology in high school and my main takeaway was from the forward written by C P Snow, who described Hardy’s typical day: he devoted four hours in the morning, from 8-12, doing math, and then spent the afternoon watching cricket. It struck me as a particularly aspirational life style and so I’ve always focused more on working well than on working long hours. My main time management strategy is to start work on the thing that is due the soonest last, when I’ll be the most focused. So, for example, if I have a referee report due in three months, I wait until almost three months have passed, and then start to read the paper. I also do the preparation for my teaching in the hour or hour and a half before class, in what often feels like a race to figure out how to prove all the theorems before I rush across campus. Occasionally this gets me in to trouble, for instance when I was trying set up a transfinite induction over the reals and couldn’t understand why the intermediate stages were all “countable” (aside: I’m now firmly in the camp that believes that the axiom of choice is clearly true, while the well-ordering principle is clearly false). But this approach is very effective at reserving time for research and other long-term projects.

Q: What do you think are the best/worst parts of a life in math overall?

ER: The worst thing is how intellectually isolated we all are, how few people there are with whom we can share the insights that we find the most exciting, even among other mathematicians. For me personally I feel very frustrated that there is this huge part of my emotional life that most of the people whom I care about have no access to.

My favorite part of my job has always been giving talks. Research talks are my favorite, for the reasons alluded to above, but I also get some of that same thrill from giving colloquia or even from teaching. Even in high school, I enjoyed the performative aspects of lecturing. When I ran for student body president, my only real interest in the job was to give the campaign speech in front of the entire school.

Q: You begin your book Categorical Homotopy Theory with a quote from ‘On proof and progress in Mathematics,’ by William Thurston: “…what we are doing is finding ways for people to understand and think about mathematics.” How has Thurston’s perspective on mathematics as a community endeavor, with human understanding at its core, influenced your mathematical life?

ER: I’ve wondered at various points whether I should be concerned about the amount of time I end up devoting to expository projects, such as the books, because it does certainly eat into research time. This is one of many instances where I’ve found Thurston’s essay, which I’ve re-read a few times now, to be helpful for keeping these kinds of projects in perspective. The passage you quote above is his definition of mathematical progress, which he sees as much broader than simply proving theorems. I happen to particularly enjoy mathematical exposition, so I think it makes sense—or as the economists would say, is a comparative advantage—for me to play that role in the broader community.

I read from a different section of this essay—on the difficulties of mathematical communication—at the introductory meeting for an AMS sponsored Mathematics Research Community workshop in Homotopy Type Theory that I co-organized this past June as a way of framing our goals for the week, which were largely to provide an opportunity for people who are not currently a part of that community (e.g., because they’re doing their PhD at a place that doesn’t have a faculty member working in that area) to find their way in.

Q: What is next for you, in math and life?

ER: One of my favorite things about academia is that the job changes all the time, or at least it can, if you want it to. Right now I’m focused on growing the category theory group at Johns Hopkins and a few long-term research projects that I’d love to get through before an MSRI semester on Higher Categories and Categorification that will take place in 2020. In a decade’s time, I hope I’m working on projects that I can’t even imagine now and have found a way to be a part of larger mathematical and public conversations.

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