Little did I know (real analysis edition)

I just finished teaching Math 521, undergraduate real analysis. I first took on this course as an emergency pandemic replacement, and boy did I not know how much I would like teaching it! You get a variety of students — our second and third year math majors, econ majors aiming for top Ph.D. programs, financial math people, CS people — students learning analysis for all kinds of reasons.

A fun thing about teaching outside my research area is encountering weird little facts I don’t know at all — facts which, were they of equal importance and obscurity and size and about algebra, I imagine I would just know. For instance, I was talking about the strategy of the Riemann integral, before launching into the formal definition, as “you are trying to find a sequence of step functions which are getting closer and closer to f, because step functions are the ones you a priori know how to integrate.” But do Riemann-integrable functions actually have sequences of step functions converging to them uniformly? No! It turns out the class of functions which are uniform limits of step functions is called the regulated functions and an equivalent characterization of regulated functions is that the right and left limits f(x+) and f(x-) exist for any x.

Compactness as groupwork

Compactness is one of the hardest things in the undergrad curriculum to teach, I find. It’s very hard for students to grasp that the issue is not “is there a finite collection of open sets that covers K” but rather “for every collection of open sets that covers K, some finite subcollection covers K.” This year I came up with a new metaphor. I asked the students to think about how, when a professor assigns a group project, there’s always a very small subgroup of the students who does all the work. That’s sort of how compactness works! Yes, the professor assigned infinitely many open sets to the project, but actually most of them are not really contributing. And it doesn’t matter if some small set of students could do the project; what matters is that some small group among the students assigned to the project is going to end up doing it! And this seems to happen — my students nodded in agreement — no matter how the professor forms the group.

емкость is great (or: what I learned at the Writing Scientists Workshop)

This semester I did something I’ve been meaning to do for a long time: I ran a writers’ workshop, modeled after the many many fiction workshops I attended in college and at the Writing Seminars. But this one wasn’t about crafting a short story that exquisitely limned the emotional landscape of people almost exactly like me and my friends; it was for early-career scientists, and it was aimed at writing the 1000-word general-audience science article, the kind of thing I’ve mostly been writing since I gave up prose fiction a couple of decades ago.

And it worked! Not thanks to me so much as to the committed, insightful, extremely-willing-to-think-hard-about-craft group of eight students I had working with me, on Zoom, from around the US and in a couple of cases elsewhere.

Why did I want to do this? Because over the years a lot of young scientists have asked me how they can get into science writing and how they can combine it with a career in research. And the answer is not so much “here’s an editor you can contact” or “here’s what goes in a pitch letter,” it’s “learn to write a very specific kind of 1,000 word chunk of prose.” And that’s what we worked on.

I will probably do this again. It was really fun. And my real hope is that, just as Math Circles went from being a thing a few devoted Russian expats did in Cambridge and Oakland to something that every self-respecting math department runs, there will be Writing Scientists Workshops that don’t involve me at all, where groups of grad students and postdocs get together and read each others’ work seriously and reflectively and train themselves to be outward-facing scientists.

With that in mind, I wrote a pretty thorough account of how I ran the workshop, what we did, what things might usefully be changed, and what we spent our time talking about, here:

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WSW Post-workshop summary 2022Download

I got a lot of useful feedback from the participants, but maybe my favorite was the student who sent back a bullet-point list of all the advice about writing I’d given, filtered through her paraphrase. She’s a Russophone, and one of the bullet points was “емкость is great.” What is емкость? I’ve been asking all my Russian friends. It seems to mean something like “putting a lot of meaning into a few words.” That is, indeed, what the WSW is going for, and it is, indeed, great.

Why won’t anyone teach her math?

Lots of discussion in my feeds about this Daily Princetonian piece, “Why won’t anyone teach me math?” by first-year student Abigail Rabieh. She just took Math 202, an intro to linear algebra, and the experience was so lousy she’s done with Princeton math for good. That’s bad!

So what was wrong with Rabieh’s class?

“Though I passed MAT 202 class just fine, my experience in it was miserable. The way the course was run did not at all set up students to succeed — or even learn math. For example, though we were provided with practice problems to prepare for our exams, we were never given solutions. My class consistently begged my professor for these, yet all he could say was that not providing them was departmental policy, and it was out of his control.

This begs the question: what interest does a department have in making it impossible to study? Study materials are given so that students can learn the course material and prepare adequately for the exam. Solution sets are part of this: to properly learn, one needs to be able to identify their mistakes and understand why they are wrong. This struggle was reflected in our exam averages, which were, respectively, in the 50s, the 60s, and the 30s.

I am far from the only person who felt this way about my class. MAT 202 has an abysmal rating of 2.71 on princetoncourses.com during the spring 2020-2021 semester. The evaluations on the Office of the Registrar’s website are no better. Students described the course as “disheartening” and said they “lost a lot of respect for the Math department after taking this course.” The advice that came up again and again in many reviews was: “Don’t take this class unless you have to.”

A lot of math teachers instinctively reacted to this with defensiveness, and I was one of them. After all, what’s so bad here? You hand out practice problems for an exam because you want students to do the problems, not because you want them to read the solutions; the mechanism is that the student works all the problems they can and then asks in office hours or review session about the problems they couldn’t do. I don’t think it’s bad to include solutions, but I would never say that not doing so makes it “impossible to study.” Student evals, well — the literature on their unreliability is so vast and I’m so lazy that I’ll only just vaguely gesture at it. And of course, most people taking Math 202 are not taking it for intellectual broadening, as Rabieh admirably was; they are taking it because somebody told them they had to. That makes the evaluations impossible to compare with those for a course people take on purpose. And as for those exam scores, well — a median in the 30s is too low, that would mean I’d made the exam much too hard. A median in the 60s, on the other hand, seems fine to me, an indication that I’d written a test with real challenges but which people could mostly do.

But you know what? Our students, especially our first year students, don’t know that unless we tell them! A student who got into Princeton, or for that matter a student who got into UW-Madison, has probably never gotten a 60/100 on a test in their entire life. No wonder it’s demoralizing!

What we have here, as they say, is a failure to communicate. Rabieh came away feeling like her teacher didn’t care whether she learned linear algebra. I’m sure that’s not the case. But I think we often don’t explicitly demonstrate that care in our classrooms. It makes a difference! We are asking the students to care about our stuff and if we want them to respond, we have to show them that we care about their stuff. What do I mean by that, explicitly? I mean that if we think the median score on an exam is going to be in the 60s, we should tell students in advance that we expect that and talk about our reasons for writing the exam that way! I mean that we should ask for student input on how the course is going before the semester is over — three weeks in, send out a survey asking for suggestions, and then talk to the class about the feedback you got, showing them you want their input while you can still make use of it. It means that if you teach a crappy lecture one day — it happens! — be open about that the next time and talk about how you intend to present a better one next time. And I feel like these are mostly “things I already do,” which could just be self-flattery on my part, so let me add this: it might be that showing students we care could mean making nice prepared slides like the professors in all their non-math classes do, instead of just showing up and writing on the blackboard. (Doing this would be a huge change for me and it exhausts me to think about it — but would it be the right thing to do?)

We don’t really talk about this stuff when we talk about teaching. We mostly talk about content and logistics; in what order should we present the material, how much should we cover, how many quizzes should we give, what should our grading policy be, should we hand out solution sets for practice problems? That’s how we think about what good teaching is, and that’s how our students think about what good teaching is, and that’s why that’s the language Rabieh reached for in her article when she wanted to explain why she had such a bad time. But I don’t think it’s actually the problem.

I’ll bet her teacher did care. Most of us do. But it often doesn’t show; let’s say it out loud! And strive for a classroom where we’re working as partners towards a goal, not just trying to get to the end without feeling like we’ve failed.

Pandemic blog 34: teaching on the screen

A small proportion of UW-Madison courses were being given in person, until last week, that is, but not mine. I’m teaching two graduate courses, introduction to algebra (which I’ve taught several times before) and introduction to algebraic number theory, which I’ve taught before but not for quite a few years. And I’m teaching them sitting in my chair at home. So I thought I’d write down a bit about what that’s like, since depending on who you ask, we’ll never do it again (in which case it’s good to record the memory) or this is the way we’ll all teach in the future (in which case it’s good to record my first impression.)

First of all, it’s tiring. Just as tiring as teaching in the classroom, even though I don’t have to leave my chair. This surprised me! But, introspecting, I think I actually draw energy from the state of being in a room with people, talking at the board, walking around, interacting. I usually leave class feeling less tired than when I walked in.

On the screen, no. I teach lectures at 10 and 11 and at noon when both are done I’m wiped out.

My rig, settled on after other setups kept glitching out: Notability open on iPad, I write notes as if on blackboard with the Apple Pencil, iPad connected by physical cable to laptop, screensharing to a window on the laptop which window I am sharing in Microsoft Teams to the class while the laptop camera and mic capture my face and voice.

What I have not done:

• Gotten a pro-quality microphone
• Set up a curated “lecture space” from which to broadcast
• Recorded lecture videos in advance so I can use the lecture hour for discussion
• Used breakout rooms in Teams to let the students discuss among themselves

All of these seem like good ideas.

So far (but I am still in the part of both courses where the material isn’t too hard) the students and I seem to find this… OK. My handwriting is somewhat worse on the tablet than it is on the blackboard and it’s not great on the blackboard. The only student who has told me they prefer online is one who reports being too shy to speak in class, sometimes too shy even to attend, and who feels more able to participate by typing in the chat window with the camera turned off. That makes sense!

I have it easy — these courses have only thirty students each, so the logistical work of handling student questions, homework, etc. isn’t overwhelming. Teaching big undergraduate courses presents its own problems. What happens with calculus quizzes? In the spring it was reported that cheating was universal (there are lots of websites that will compute integrals for you in another window!) So we now have a system called Honorlock which inhabits the student’s browser, watches IP traffic for visits to cheating sites, and commandeers the student’s webcam (!) to check whether their eye motions indicate cheating (!!) This sounds awful and frankly kind of creepy and not worth it. And the students, unsurprisingly, hate it. But then how does assessment work? The obvious answer is to give exams which are open book and which measure something more contentful about the material than can be tested by a usual quiz. I can think of two problems:

• Fluency with the basic manipulations (of both algebra and calculus) is actually one of the skills the class is meant to impart: yes, there are things a computer can do it’s good to be able to do mentally. (I don’t think I place a complicated trig substitution in this category, but knowing that the integral of x^n is on order x^{n+1}, yes.
• Tests that measured understanding would be different from and a lot harder than what students are used to! And this is a crappy time to be an undergraduate. I don’t think it’s a great idea for their calculus course to become, without warning, much more difficult than the one they signed up for.

Commutativity, with fractions

Talking to AB about multiplying rational numbers. She understands the commutativity of multiplication of integers perfectly well. But I had forgotten that commutativity in the rational setting is actually conceptually harder! That four sixes is six fours you can conceptualize by thinking of a rectangular array, or something equivalent to that. But the fact that seven halves is the same thing as seven divided by two doesn’t seem as “natural” to her. (Is that even an instance of commutativity? I think of the first as 7 x 1/2 and the second as 1/2 x 7.)

Teaching-track positions at Washington University: a testimonial

My Ph.D. student Silas Johnson is teaching at Washington University in St. Louis. This is a kind of job that’s getting more and more popular; teaching-focused, non-tenurable but also not on a limited term. I’m pretty interested in the nuts and bolts of how these jobs work, so I asked Silas to explain it to me. Take it away, Silas! The rest of this post comes from him.

Washington University in St. Louis is hiring two new Lecturers this year: one in math, and one in statistics. These are long-term teaching-track positions, meaning they’re intended to be permanent but do not come with the possibility of tenure. I’m currently a Lecturer here, and I enjoy it a lot. I get to teach a lot of interesting courses; so far, my 3-3 load has usually included two sections of calculus and one upper-division course, with the latter including everything from probability to number theory. I also support the department’s broader undergraduate teaching mission. For example, since arriving, I’ve worked on a project to streamline the course requirements for our math major, tried out new ideas for calculus recitations, and conducted teaching interviews for postdoctoral candidates. Most importantly, my colleagues have been wonderfully supportive as I adjust to the university, try out new teaching methods and techniques in my courses, and work on departmental projects.

These teaching-track faculty positions seem to be increasingly popular, though the nature and details of such positions vary from department to department. While non-tenured, my position is on a parallel promotion ladder from Lecturer to Senior Lecturer to Teaching Professor. This structure is fairly typical, though the titles vary (Assistant/Associate/Full Professor of Instruction is also common). In our case, the promotions also come with increased guarantees of job security.

Teaching-focused positions are sometimes stereotyped as a lesser option, perhaps even a backup plan for those who can’t find a research postdoc or tenure-track job. I disagree; I see this as a good path for mathematicians who, like me, have a genuine interest in teaching and want to make it the focus of their careers. Our department and college are clear about the value they place on teaching-track faculty, too; we vote in department meetings, serve on important committees, and are treated as equals in just about every way. (The only thing we can’t do is vote on tenure-track hiring and promotion.)

Overall, I really like it here. I’m happy with my decision to pursue a teaching career, and I’m glad there are other mathematicians out there who are interested in doing the same. I would encourage such people to apply for our position and others like it. If you’re a grad student, by the way, there are teaching-focused postdoc positions too!

Math!

I really like talking with AB about arithmetic and her strategies for doing problems.  All this Common Core stuff about breaking up into hundreds and tens and ones that people like to make fun of?  That’s how she does things.  She can describe her process a lot more articulately than most grownups can, because it’s less automatic for her.  I learn a lot about how to teach math by watching her learn math.

Dream

I’m at Disney World with CJ, on a Pirates of the Caribbean-style ride, a car careening through a tunnel.  On the wall of the tunnel there are math posters, the kind you’d see in a high school classroom, about Pascal’s triangle, conic sections, etc.  And I feel sort of annoyed and depressed, because I know that Disney is going to make a big deal about how educational this ride is, but actually, nobody except me is looking at the posters, nobody who didn’t already know the math could get anything out of the posters, the way the car speeds down the track.

Please interpret and derive relevant policy prescriptions for math pedagogy in comments.

Doctoral programs can have a strong influence on the weak-minded

Daniel Drezner:

First, I cannot stress enough the cult-like powers of a PhD program. Doctoral programs can have a strong influence on the weak-minded. Even if you’re pretty sure what you want going into a program, that can change as you’re surrounded by peers who want something different. You might think you’re strong-willed, but day after day of hearing how a top-tier research university position is the be-all, end-all of life can have strange effects on your psyche.

I really do feel this is something we handle well at Wisconsin.  Our Ph.D. graduates go on to a wide variety of positions, some in primarily teaching colleges, some in research institutions, some in industry, some in government.  We do not consider the North American research university the be-all and end-all of life.  We are not just trying to produce clones of ourselves.  We really do strive to help each of our students get the best job among the jobs they want to get.