Teaching What You Really Don’t Know, Part II

This fall I’ll be teaching a new prep: our senior capstone class on the history of math, featuring an intense research project. The course also counts as a Global Perspectives credit for our students, meaning the class should broaden our students horizons beyond a classic western viewpoint.

Which is great. I’m excited for the challenge, and I think it’ll be interesting to explore this material with them and help train them to be more well-rounded. The only problem is…

I don’t know anything about math history.

I know the anecdotes and the just-so stories that get passed down as asides in lectures: Gauss summing  1 to 100, Euler and Hamilton crossing bridges, the cult of Pythagoras’ thoughts on flatulence. But until recently the closest thing I’d ever done to learning real math history was reading The Baroque Cycle. Which doesn’t count.

So now it’s almost August, my summer class is over, and I have a month to learn math history. I’m starting with a book I started awhile ago, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World by Amir Alexander. It’s a real gripping read, covering the state of mathematics in the 16th and 17th centuries (so far, I haven’t finished yet) and how the Jesuits’ abhorrence of the concept of the infinitesimal affected first the development of mathematics, and then the development of the Jesuits.

The mathematics in it aren’t completely dumbed down, but are still pretty accessible to a motivated layman, and the author goes through arguments for the utility of the concept of the infinitesimal, as well as the reasons why many thought such a thing is patently absurd. There are also some really relatable stories in the book: colleges throwing shade at their weird mathematics departments, academics struggling to get hired, and people claiming to read the standard treatises of the time, even though it was obvious nobody had the patience to slog through a particular author’s borderline-unreadable text.

I’m selecting some sections for students to read, and one thing I’m having to really pay attention to is the amount of cultural and historical context the author takes for granted. He does lay out a short history of the reformation, and the Jesuits arising in response, but if I pull other sections without that context students might get lost.

Two other books are on the docket once I’ve finished this one. One is Journey Through Genius, by William Dunham. I’ve read bits of this, and one of Dunham’s other books The Mathematical Universe, and I really enjoy his writing. The other is the textbook I’ll use for the course, adopted by the previous instructor, Math Through the Ages by William P. Berlinghoff.  The subtitle is A Gentle History for Teachers and Others, but I’ve been told not to let that fool me into thinking this is kid stuff. I’m sure I’ll let you all know how it goes.

Any other recommendations for math history? Books? Podcasts? Documentaries? Especially ones with a non-Western focus? Lay ’em on me, because I’ve got a whole semester to fill.

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Recreational Mathematics for Fun, Sanity, and an Sometimes Even Papers

The IBM card sorter! Why is this relevant? See exciting puzzle below.   Photo by waelder [GFDL (http://www.gnu.org/copyleft/fdl.html), CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/) or CC BY 2.5 (https://creativecommons.org/licenses/by/2.5)], from Wikimedia Commons

In life on the job market and pre-tenure academia, it can seem that no math is worth doing unless it results in a paper, preferably a very fashionable and serious one. This can be a real soul crusher when the inevitable setbacks occur. Pressure (internal and external) to produce serious mathematics, in order to build the CV, in order to get or keep a good job, can sometimes make the discipline feel like a monstrous machine instead of a wonderland. In that setting, it feels almost subversive to pursue problems that are just straight up entertaining, that may have been solved before but are truly fun to think about. So today I focus on this joyful act of rebellion against the machine: recreational mathematics.

Puzzles were my gateway drug to hard mathematics. They have also served to reinvigorate my interest in math when my research seemed too hard and when work that I thought was serious and useful suddenly looked trivial and unfashionable. Recreational math reminds me why I bothered doing this in the first place. Half an hour with a Martin Gardner book is better than a nap when I need a reset (and a nap is pretty good). And math packaged as a puzzle can certainly have application. Jennifer Beineke and Jason Rousenhouse remind us in the preface to the excellent 2015 collection The Mathematics of Various Entertaining Subjects: Research in Recreational Math, (volume 2 also out now)that probability, Latin squares, and graph theory were all born of recreation. Nothing useful there…

I feel like a jerk even bringing it up after bemoaning the pressures of “the machine”, but recreational math can even pay off in papers, even if it doesn’t become the foundation of a whole new branch of mathematics. Additionally, recreational mathematics problems make great gateways to research mathematics for undergraduates who might not have the mathematical tools to attack more formally phrased problems.  An example from my life: a few years ago, when I was in my second visiting position and felt extraordinarily stuck on two of my main research problems, I got an email from Judy Gilmore (my friend’s mom) about a problem she was having in her quilting circle. She and her four friends wanted to make quilts together in a sort of round robin.  Each person would start working on their own quilt, then pass it on to someone else in the group.  That person would add a border to the quilt, and pass it on again, until everyone had added a border on to every quilt. Judy wanted to set it up so that each person passed a quilt on to everybody else at some point during the process, but she couldn’t seem to make this work for the group of five.  There was a good reason Judy couldn’t do it: turns out there is no way to do this for five people. Oh, the fun I had proving that. Though I didn’t have the vocabulary or context to say it at the time, the fact that this is impossible is equivalent to the fact that there is no 5 by 5 row complete latin square.  My friend Katie Haymaker and I went on to look into the known and unknown aspects of row complete latin squares, essentially just for fun. We rediscovered a lot of known results and put together the pieces for a small extension of one, and eventually wrote a mostly expository paper about these objects and the quilt problem for Mathematics Magazine. This problem has given me a whole family of great research problems suited for undergraduate students who may not have had many upper-level math classes. Five years later, my students and I are still working on the questions that began with Judy’s email.

More examples of recreational math making papers: the whole family of puzzles that are based on shuffling cards. You could start with a pretty simple question (that a lot of people have, based on how quickly it comes up as an option when you start typing the query into Google): how many shuffles does it take to really randomize a deck of cards? As Diaconis and Bayer (here explained by Francis Su in Harvey Mudd’s Math Fun Facts) showed in 1992, seven shuffles is good.  But these are not “perfect shuffles”, i.e. perfect alternate interleaving for the cards in two halves of the deck.  Perfect shuffles insert no entropy, because they are fully deterministic.  In fact, 8 perfect shuffles of a deck will return it to its original order, as shown by Diaconis, Graham, and Kantor in 1983.

But again, you don’t have to write a paper about shuffling to get a lot of joy out of thinking about it.  Here’s a puzzle for you.  Like any good story, a quality puzzle gets passed around and changed, so can be hard to source. I got this from Joe Buhler, recreational mathematician extraordinaire and co-writer of the MSRI Emissary newsletter puzzle column, who got it from Stan Wagon, who got it from Colin McGregor. In any case, let’s say that a shuffle is any way of dividing the deck into two pieces and interleaving the cards so that all the cards in each piece stay in order relative to one another. Now say that you have a deck of n cards, and that you are a master shuffler—you can physically accomplish any shuffle with your deck of cards. What is the maximum number of shuffles required for the most efficient algorithm to put the deck in any requested order?  Every permutation will have some most efficient way to accomplish it, so we are basically looking for the worst case of this most efficient way over all permutations of n cards.

Okay, so that might be kind of hard to think about all at once.  So how about a warm up:

Q: How many shuffles are required to completely reverse the order of the deck?

A: I’ll first share my wordy answer to this simpler problem with you, then share someone else’s extremely elegant solution to the general problem. I spent a few very fun hours thinking about this, and here’s what I came up with: For n=2^k, you do the simplest thing possible–divide into two parts, and perfectly interleave the parts with the better card on top at each step. Repeat for a total of k shuffles to get the reversed deck. This is not so hard to prove (let’s call it an exercise!). If n is not a power of 2, you should be able to do this separately (ignoring the other cards) for the number of cards represented by each 1 in the binary expansion of n.  When each of these chunks are sorted, for every 1 in the binary expansion of n, we’ll have several chunks of cards such that the cards in each chunk are in order relative to one another. We can now interleave each smaller piece successively with the largest power of 2 part of the deck. So, this should take between log(n) and 2*log(n) shuffles, depending on the binary expansion of the number. I think the worst case scenario should be something like n=2^k-1.

After thinking about this for a while and talking with friends, I realized that you should really be able to do it in the ceiling of log(n) shuffles, because if 2^(k-1)<n<2^k, then you could just pretend like there were 2^k cards to sort, working with 2^k-n “phantom cards”.

Okay, so are you ready for the better answer?  Larry Carter explained it to me in about 30 seconds by explaining how an old IBM card sorter worked. These machines were physical implementations of radix sort.  In the machine, imagine that you start with a stack of n numbered cards in any order. The binary expansion of each number is at the top of the card, and this expansion is encoded using slots and punches.  A punch is a hole where you could put a rod through and pick up the card. A slot would be made by taking a punch and snipping up to the edge of the card, so you couldn’t pick up the card with a rod anymore.  For every 1 in the binary expansion there is a slot, and every 0 is a punch.  To sort the cards, line them all up.  Starting with the leftmost position (the ones place), slide a rod through the stack of cards and pick up all of the cards with a 0 in this position.  Bring these cards to the front of the stack.  Then repeat with the next position, and keep repeating until you have gone through all the log(n) positions.  Like magic, the cards will be perfectly sorted. All cards with 0 in the highest position will be in front of cards with a 1 in the highest position, and all cards with 0 in both highest positions will be in front of these, and so on. The genius idea is that because you could go from any permutation to sorted in log(n) steps, you could imagine just reversing these steps as shuffles!  So it can take at most log(n) steps to create any permutation of the n cards.

Alright, enough puzzle for one blog.  Hope everyone is enjoying summer, and finding plenty of great puzzles without my help.  What experiences have you had with recreational mathematics?  Please share in the comments!

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Summer Research with Undergraduates Plus Fun

Sam Kottler, Garrett Figueroa, Jerrell Cockerham, Hanqui Li, and Zhaopeng Li, standing on the shoulders of giants at Colorado State University.

I had my first-ever summer research students this year: sophomore Jerrell Cockerham and senior Zhaopeng Li worked together on a problem about row complete Latin squares, and senior Sam Kottler is working on a cool project in locally recoverable codes. They have all done very good work, and it has been fun for me, too.  I met with the students most days for five weeks, and they’ve written up their results, which I hope will find their way into published papers sooner than later.  I like my students and it was really worth it. It was also a lot more work than I had anticipated.  “What a deal! They’ll work on my research problems and learn a lot, and I’ll get the answers I want,” I thought. “I’ll meet with them each a couple times a week for an hour and I’ll get so much of my own research done in the rest of the time.” Indeed, they did work on my research problems, I think they learned a lot, and they did figure out some very nice things.  Sam is still working I have a good feeling that he will prove even more.  But the fact is that I worked really hard, didn’t get anything of my own done during those weeks, and I still felt like I wasn’t entirely keeping up as a research mentor.  Where did all that time go? I still don’t know.  I do know now that a couple times a week is probably not going to work when undergraduates are learning a whole new area and learning how to do research for the first time.  I also learned that it’s really hard to motivate myself to read drafts and make good comments when I just finished an intense bout of teaching. Next time I’m going in with my eyes open, and I’m taking a few weeks off between classes and student research responsibilities.  That said, I think it was a success, and I thought I’d share a couple of non-research activities that we did together that were really fun.

One of the best things I did with this group was to organize a lunch for all the research students in the department and their mentors, so everyone could talk about their progress and just hang out for a while. This was pretty fun. It was the first time that many of the students had tried to describe what they were working on to anyone else. As we all know, this can be really challenging, and it was cool to see them struggle with it and figure out how to tell the story. One of the mentors practiced a conference presentation on us, based on work that had been done with a student, and the students seemed to engage and be able to see themselves talking about their own work in the future.

Another fun activity: with some of the other research students from Colorado College, we took a field trip up to Fort Collins and met with undergraduate research groups at Colorado State University led by Rachel Pries and Patrick Shipman.  The students got to meet each other and again share what they were working on and hear about what it was like to study at a totally different kind of school.  Both CSU groups were doing really interesting things—studying curves related to coding theory for one group, and modeling a chemical/physical process using differential equations for another.  We had a lunch with everyone together, visited the dynamical systems group’s lab, and then met with the staff member who runs parts of the graduate program. The students had plenty of questions about the graduate school admissions process and some were interested to hear that yes, a teaching or research assistantship usually covers your tuition and a living stipend.  We met with the wonderful Research Scientist Elly Farnell, who told us about her work in the Pattern Analysis Lab and gave the students another chance to talk about their own research projects. Not only is her work fascinating, but somehow talking to Elly the students really hit their stride in their own exposition.  Having heard them talk about their work many times, I was silently cheering as they gave their best explanations yet.  Made me hope that my talks just get better and better as I give them, too.  Our final stop was visiting with my old friend, CSU Statistician Ben Prytherch (sharing his music page because why not), whose enthusiasm for his subject is infectious.  He asked the students what they were working on and explained a few of his favorite statistical ideas/techniques. At one point he said, “Oh, let me just make a quick applet for this,” and he whipped up some visualization with a sliding bar. Again, the students had lots of questions, and Ben was an amazing stats ambassador. Overall, it was a great experience and the students came out a little more connected to students and faculty at another regional school, and enthusiastic about research and maybe about graduate school. I am really grateful to everyone who took the time to hang out with us at Colorado State.

Back to work for me!  Let me know if you are doing any fun summer research activities in the comments.

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