Sunday, July 22, 2012

Finished on the 50 (Harvard Pilgrim 10k at Patriot Place)

(Results) I had heard of this race for a few years (especially since two of my Williams teammates went 1-2 in it a few years ago), and I wanted to do it, but I didn't want to pay $50 for the privilege. Happily, I did a photo shoot for DMSE, and in return they gave me a free entry to any of their races, so I was able to do this one. (I would rather have done Mount Washington, but I was in Utah on race day.)

 The race was in the evening, so it was hot. Alan and I warmed up by running around the parking lots for the football stadium. There were about 7,000 people in the 10k and 5k races combined. I saw Bill Rogers on the starting line!

I also saw a BAA runner (Lisa) who I knew had run under 3:00 at the Providence Marathon, so I knew she was very fast. I figured she would be my main competition, and I was nervous. There was also a very skinny high school girl listening to music, but I was not concerned about her.

Rightly so. When the gun went off, the skinny high schooler sprinted to the lead -- of the entire race, ahead of everyone in both the 5k and 10k, including all the men. That didn't last very long. I passed her, and then waited for another woman to catch up to me. Where were they? I wondered. I didn't feel like I was going that fast.

A BAA guy passed me and I asked him where Lisa was. He told me she was planning to run 6:20 for the first mile. Just then, we hit the mile mark. 5:45! Hoo boy, that's a little quick. That's my cool evening track 10,000m pace, not my hot summer road 10k pace.

To make a long story short, that was it for the racing. I had maybe 20 seconds on the second female at that point, and I would extend it to about a minute by the end of the race, despite slowing down to a more reasonable pace. In the second mile, I passed a guy in red (Breno), and he immediately surged to pass me again, so I decided not to play that game, and instead just stuck behind him for the rest of the race. I was afraid that Lisa's race plan was some sort of cat-and-mouse deal where she would start slow and then chase me down (which is why I ended up with more than a minute lead; I didn't want to slow down too much just in case), but it didn't happen.

As we neared the stadium, I got a biker! This guy biked next to me and radioed back to the finish line about my number and what I was wearing. Also, I passed a guy, still on his way out of the stadium, who was walking with bare feet. Imagine paying an entry fee to walk a 10k barefoot on hot pavement! Not my idea of fun.

My idea of fun was to run into a dark, humid tunnel, and then emerge out of a giant helmet onto the football field.

I ran out of the big helmet!

I had this grand idea that when I emerged onto the field, there would be the 50-yard line and I would cross it -- you know, "Finish On The Fifty." But it was like a beach at high tide, people everywhere. The announcer pointed out that the first female had entered the stadium, and instinctively I put up my arm, to identify myself -- with the idea that anyone who paying attention to the announcer might need help trying to pick me out of the mob.

Only after I finished the race did I remember that no one really listens to the announcer, except to hear their own name. So the waving probably came off as self-promoting. But seriously, my cognition is severely impaired when I'm running (remember, I can't even subtract). I was just trying to help people.

Anyway, I finished! On the 50! And I ran 38:29. The last time I ran 10k on a hot day, I also ran 38:29, and it almost killed me. This time, I was just bopping along. I think that's a great sign!

Miss Massachusetts is holding the finish tape! 
Maybe a Patriots football player is holding the other end, but we'll never know for sure.

The post-race refreshments were good; I had a nice time talking with Lisa and Breno; I briefly met Dave McGillivray himself; and I was happily surprised to win a nice gift bag with socks, a hat, a shirt, an American flag, algae bits, $25 to the adjacent mall, and an overnight stay at the adjacent hotel!

Thursday, July 05, 2012

Teaching the teachers

Last week, I taught at the Exeter math conference for high school teachers. The teachers come for a week and take two courses, which each meet for two hours per day, on subjects such as using technology in the classroom, creative ways to teach math, and learning new mathematics.

My courses fell into the latter category. I taught one section of the Exeter math curriculum course, which someone else designed years ago, and then I created my own course, "Greatest Hits of Higher Mathematics." It was the first time I'd ever created my own course, and it was quite successful. In this post, I'll reflect on things that went well and things I'd like to change for next year.

The goal of the course was to introduce the teachers (hereafter, "students") to the college math major or graduate school math curriculum: specifically, analysis, algebra, topology and number theory. I chose these topics because the first three are standard core curriculum, and number theory seemed like the most compelling other topic (rather than algebraic geometry, partial differential equations, etc.).

When I chose the course materials and the exercises to assign, I had no idea what the participants' level would be. I was afraid that they would not be able to understand the textbooks that I used as an undergraduate, because they would only have one night to read it, and concepts like an "open ball" took me weeks to fully understand. Thus, I paired each textbook reading with an essay by Stephen Strogatz, which presented similar material in a way aimed at the general public.

I structured each class so that we spent the first half going over the previous night's homework, then had a short break, and then had an introduction to the next topic. This way, they were introduced to the ideas before doing the reading on their own.

As it turned out, the participants were really sharp. (This is probably because of the selection bias for people who sign up for a course on higher mathematics.) They understood the textbook readings and delved into the homework problems. Because I didn't want them to have no idea how to do the homework, I had assigned mostly easy problems, even some that were just repetition of examples from the text, such as proving that the fourth root of 2 is irrational after seeing the proof for the square root. I wish I had assigned some that were more difficult, because they could totally have handled it.

Here are my reflections on the individual topics.

Real analysis: This went well. On the first day of class, I asked them to prove (as a group) that the square root of 2 is irrational. Most of them had seen the proof before, but all had forgotten the details, so this was a good exercise in constructing a proof, and also in having a group discussion. Then we talked about open and closed sets (as intervals on the real line). Is the union of two open sets always open? How about the intersection? How about for closed sets? Can the infinite intersection of a collection of open sets be closed? Can the infinite union of a collection of closed sets be open? They loved thinking about these problems and were very engaged.

The homework also consisted of compelling problems, including some that were too hard for them to do for homework. We spent most of the class period going over solutions, and by the end we were able to solve even the hardest one (my favorite concept, "dense"). That was a good class.

Topology: I had trouble finding a good book to use. Everything I found started by talking about sets. I wanted something that talked about shapes and fundamental groups. I ended up using just the first chapter of a book on intuitive topology, which had a series of exercises that were pictures of knotted-up shapes that asked students to figure out how to unknot them.

My goal for the topology section was for the students to learn to imagine the shapes in their head, and to not use any numbers, equations or writing. This was successful, and we again spent most of the class period working on the hardest exercise, this time in small groups. However, it took the groups a very long time to untangle the object, and I was afraid that it wasn't the best use of time. But I decided that as long as they were thinking hard about math, it was a fine use of time.

Next time, I think I would actually do point-set topology, defining a topology and talking about the topology of various spaces with their open sets defined in various ways. I actually think the students might prefer that.

Algebra: My preview for algebra was to define a group, and then give a proof that the identity is unique. I didn't even have to give it, as someone told me what to write as I was writing it. The book I chose ended up being perfect for this course, as it was full of interesting exercises that were easy to state. I only photocopied small sections of the book, but I should have copied more, as some of the exercises I didn't assign referred to sections I hadn't included, so the students couldn't try them (because, for instance, they didn't know the definition of U(12).)

We went over the homework exercises quickly, and then spent the rest of the class period working on more problems in small groups. Algebra really is fun, especially at the beginning. This topic went well.

Number Theory: This was the weakest topic. I was using a book that explained the major theorems well, but lacked compelling exercises. I recently heard someone say that the exercises you choose make the course, and that was definitely true here. The homework exercises were not proofs, just verifications like showing that a whole bunch of even numbers could indeed be written as the sum of two primes, so we went over the solutions quickly in class and then had 90 minutes left without much to do. I will definitely choose next year's book based on the exercises.

Choice topic: Since the four topics I chose just scratched the surface of higher mathematics, on the last day I asked each student to pick a topic in higher mathematics and then give a 10-minute presentation telling the class what it is about, and maybe doing an example of the sort of problem the field looks at. I brought lots of books, and the students borrowed them throughout the week. They were also welcome to use other resources like the Internet, of course.

My hope was that we would see presentations on things like knot theory, graph theory, the real analysis behind calculus, and point-set topology. In fact, the presentations were about the game Nim, RSA cryptography, the math behind the win/loss predictor in baseball, the Euclidean algorithm, and subgroups. They were getting tired and overworked by the end of the week, and in any case, the presentations were good.

I'm looking forward to offering this course again next year, and I hope people sign up!