Math Tea

My school has a longstanding tradition of a weekly math tea. And unlike other places I’ve been, where this time is a social hour before a seminar, at Hood it’s a time to play games, solve puzzles, or do some interesting math with students and faculty. When we divvied up departmental duties at the beginning of the year, I ended up as Math Tea co-czar (that’s the official title), and it’s ended up being one of the most fun parts of my week.

Each week, we choose some kind of activity. We’ve built up a pretty impressive of activities and games over the years, many of which are detailed on by our own Betty Mayfield on the MAA’s “Math Club in a Box” site.

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Our annual math pumpkin carving day. Carving done by student Justin, pumpkin gutting done by the author.

We have a shelf of games and puzzles, some of a mathematical bent like Set or Rush Hour, but some are just fun card games. We’ve also cribbed a few activities from the book Solve This: Math Activities for Students and Clubs, like one on cutting modified Möbius strips, and another on doing math on the surface of a donut (with the real thing for a snack, and a papercraft model for experimenting).

Bewildered by cutting weird Möbius-like constructions

Bewildered by cutting weird Möbius-like constructions

My favorite activities just explore whatever interesting topic I’ve found on the internet recently. Last week I’d seen Buffon’s Needle flying around, so we gave it a shot.

A small-scale test

A small-scale test

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Scaling up










The week before, we experimented with drawing 3.5-gons.

Our department chair's 3.5-gon.

Our department chair’s 3.5-gon.

And a while before that, we tried to figure out why coin flip probabilities are so frustratingly counterintuitive. I’m not sure we really resolved that one to everyone’s satisfaction, but it was fun to collect data.

The best part about our math tea is that the whole department comes, and we take care not to dominate the discussion. I think it’s incredibly helpful for students to see how professors approach difficult problems, particularly that we rarely know how to solve things immediately.

This takes very little time to put together, and it’s a valued part of our week. Since we do it in a public area of our building, I think it also gives our department visibility and makes us look fun, which probably helps attract new math majors. If you’d like to help develop a stronger sense of community with your department and your majors, I recommend starting a math tea.


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My Lecture-less Modern Algebra and Foundations Courses–Part I of an Ongoing Saga

This semester I decided to take a (maybe temporary) break from lecturing. Don’t get me wrong, I love lecturing, and I love going to good lectures. However, there have always been a few aspects of my own lecture classes that are not satisfying. If I take enough time to explain some topics very carefully, we don’t have enough time for other topics I want to cover. Many students in any lecture are focused entirely on copying down my blackboard work and are not actively engaged with the material. If I do group work in the classroom or try to incorporate interesting activities, we don’t get through enough material and I end up rushed later. If I assign reading to make up the difference, students will only rarely complete the reading and then often complain a lot about it. Also, articles like this one and this one also got me thinking that there might be good philosophical/ethical reasons to look beyond lecturing. So this semester I decided to go all the way and turn it around.

What instead? The best choice for me, and these classes, seemed to be something very structured, that made use of existing resources as much as possible. The plan:

  • Assign reading before each class.
  • Give students a reading assignment and check their notes for each section.
  • In class, discuss the reading assignment and take questions on the material.
  • Work in groups on homework assignments from the text.

I am currently teaching two classes this way. The first is Foundations, Villanova’s bridge to higher mathematics course. Second is Modern Algebra, required for all math majors, but not a prerequisite for many other courses in the catalog. In both cases, the students already have some mathematical maturity, so reading a book was not as distasteful to them as it might have been to a Calculus I class.

Another reason that this was a good semester to try something different is that my good friend and colleague Katie Haymaker is also teaching Foundations, and we both wanted to take a very hands-on approach to the course. As Katie says, “for me, one really important aspect of this format is that students are practicing proving things during class. I feel like intro to proof classes in particular suffer from the osmosis issue – students think that because they saw you prove something and understand every step, that they understand how to do it. But then when they are asked to prove something themselves, they freeze up and have no idea how to start. This is why I much prefer this format in the proof-heavy classes, but why I don’t feel compelled to do things this way in Calc 2 (in addition to the fact that no one wants to curl up and read Stewart after a long day of classes).” We have been planning our Foundations classes together and comparing notes on what worked and didn’t. This has been incredibly helpful. I can also get Katie to talk to me about my Algebra class, since she is teaching that class next year and is now considering how to organize that.

For Foundations, Katie and I selected Book of Proof by Richard Hammack, which we really like. All the students have reported that it is very readable, it has a lot of exercises in each section, and the odd problems have solutions, which provide many extra examples for the students. Also, it is available as a low-cost paperback or as a free pdf from the author’s website. Most of my students bought the paperback, but they always have access to the book online. For Modern Algebra, I chose A Book of Abstract Algebra, by Charles Pinter. (I apparently like these direct titles; titles that let you know “I’m a book! About this!”) Pinter’s book is also extremely readable and has many exercises. A few more solutions and examples would be great, but there are a few full solutions in the back of the book. My students have reported that they like the book. Both of these books are divided up into reasonably short sections, something I think is important for this model. The students can read an entire section or chapter, then answer questions relevant to just this material before we move on.

The reading assignments are generally short, and I aim for them to be relatively easy, given the reading. Sometimes I overshoot and ask more challenging questions, but it seems to work out since we go over the reading assignments in class. Each class I also check for 1-2 pages of notes on each section. The reading questions and notes are worth 10% of the students’ final grades. Weekly homework is worth 30%, tests are worth 30%, and the rest of the grades come from: presentations (10%), a proof portfolio (10%), and weekly-ish blog entries (10%). I am not sure now if I would distribute the points the same way (more on that another week).

The blog posts were inspired by a blog post (meta!). One of the main goals I had for the course was to help my students become great at both communicating mathematics and communicating about mathematics. Writing proofs is important, and we put a lot of energy into that, but I think it’s almost as important that the students can have a satisfying conversation about mathematics with either a mathematical peer or a non-mathematician. The blog posts ask them to practice explaining mathematical ideas in everyday terms, or reflect on especially difficult concepts in the course.

I got the idea for a proof portfolio from Chris Rasmussen at Wesleyan University. The assignment asks students to choose their ten best proofs from the whole semester, revise them to make them absolutely impeccable, and then submit the portfolio in the last week of class. I used proof portfolios in a previous number theory/intro to proofs course, and was really impressed with what the students submitted.

Teaching this way has been a roller coaster for me. I really like it, and think it is working great in many ways. However, it is stressful to ask students to invest in something they initially find foreign and annoying. If the students don’t buy in, it can’t work, and I often feel stressed that I am not really in control of this aspect of the class. It has required a huge investment from me, too, in developing new materials, lots of grading, and the straight effort of organization. That said, I think that teaching this way was a good choice. My students are taking responsibility for their own learning in a real way. They are submitting solid proofs and becoming clearer communicators. I just gave the students a mid-semester feedback survey, and their responses indicate that they find the general structure of the course really effective. Hooray!

As the semester winds down, I will report back with more in-depth discussion of the components of the course, what has worked and what hasn’t. In the mean time, please send me some good teaching vibes, and tell me what has and hasn’t worked for you in the comments!

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Writing Better Recommendations

So much of our daily to-do lists seem to be tasks for which we have little-to-no training, few direct guidelines, and practically no oversight (at least until mid-tenure review). I’ve just sent off several letters of recommendation for students hoping for REUs or transfers, and even though I’ve written a few of these over the years, I still feel like I have no idea what I’m doing. I’ve gotten pretty comfortable with that feeling in my day-to-day life, but usually it’s only my career I’m potentially damaging with a mistake. The thought that I could screw up the future of one of my bright young students with a careless letter is worrying, to say the least.

So I started researching. I still haven’t implemented some of the suggestions in the links below, but I will in the future. I hope this will be helpful for those of you who are in the same boat. For everyone else who already knows the drill, I hope you will chime in with your advice in the comments.

First, PhD+’s creator Adriana Salerno has some great tips. She says to be specific, and give examples of your interactions with the student. She also talks about the difficulties of addressing a student’s weaknesses when asked to do so, which mercifully I haven’t had to do yet. The comments are also extremely helpful, one giving a nice breakdown of how to structure a letter.

Another problem with letter-writing is how to get the right kind of information from the student in the most efficient way. Michael Orrison at Harvey Mudd wrote a nice piece for the MAA where he lays out his process. He has a separate page on his site directing students to send him not just the relevant practical details (due dates, envelopes/links, etc), but also more specialized information to help him write a great letter. The students answer questions like “For what classes have I had you, what final grades did I assign you, and how did you distinguish yourself in my classes?” and “What are your long-term goals and will this position/honor/award help? If so, how?” No matter how well you think you know a student, their answers to these questions can’t help but improve your letter. He also specifically requests that students send email reminders when deadlines approach. I think some students are nervous about appearing to nag, but I would welcome a quick heads-up to make sure I don’t forget due dates.

One more subtle concern is the effect of biases hidden in my letters. This handy reference from The University of Arizona Commission on the Status of Women is directed specifically at gender bias in recommendations, but it can also help us examine other lurking biases with a more critical eye. There are general suggestions (“Keep It Professional: Letters of reference for women are 7x more likely to mention personal life – something that is almost always irrelevant for the application”) as well as a list of words to use and others to consider avoiding. We all have biases – I’m growing more and more aware of mine – and the best we can do is to recognize them and confront them whenever we can.

The hardest thing for me to learn was how to diplomatically decline to write a letter. If the student seemed to lack both aptitude and drive in my courses, I won’t waste my time (or those of the eventual recipient) trying to find a positive spin, no matter how much I might like them. In one memorable case, my attempts to gently turn such a student away were too subtle for them to catch. Rather than deal with that again, I now say that I just didn’t get a good enough sense of their abilities in class, and because of that I probably wouldn’t be able to write an effective letter. So far, so good.

One last requirement, adopted from a colleague, is to mandate that the student update me when they hear the results of their application, good or bad. It’s harder for people to disappear at a school as small as Hood, but at previous institutions I would occasionally write a bang-up recommendation for a student, only to never see or hear from them again. I don’t think students intentionally neglect to tell us the results, but they seem to respond to making this expectation clear.

What are your recommendation tips? Especially those of you on the other side – what works? What doesn’t?


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