## Sunday, October 24, 2010

### New Vegetarian Restaurant

We have a new restaurant in town, a vegetarian place which (apparently) moved down from Erie. I went to the grand opening with a few folks from the department this past week.

I was excited about, but the actual experience left me underwhelmed. The food was good, but it was basically all just standard sandwiches of some sort with the meat replaced with fake meat. So you can get a variety of veggie burgers, or a tofurkey club with tempeh bacon, and so forth. I can see why this would appeal to a vegetarian, but as a non-vegetarian, if I want a burger or a turkey club, I'll just go get a burger or a turkey club.

It's a college town, so there might be enough vegetarians to keep it running, but I don't think this is getting added to my rotation. I'm generally interested in more creative vegetarian options.

## Friday, October 15, 2010

### The Antiphany

I'd like to coin the term "antiphany" or anti-epiphany. It represents a sudden insight or understanding which is not in any way transcendent, but in fact horrible and gut wrenching. A moment in which you put the pieces together, and feel not enlightened, but as though you understand how much you've missed, and that things are much worse than you ever thought. Not just when you feel your stomach drop, but when you know it just entered the express elevator to hell.

The lesser was a simple matter of realizing how little success my precalculus students ever seemed to have in understanding transformations of graphs. I thought about the fact that none but the best ever understood this topic very well, and that perhaps a tiny percentage of those going on to calculus would understand or remember virtually any of this. And I thought about episodes in my calculus classes when I would casually, as an aside, mention how some particular result could be viewed in terms of transformations of graphs. Calculus is filled with lovely insights which blend the tools of geometry and the tools of algebra. In many ways, calculus is the blending of geometry and algebra. Analytic geometry makes calculus possible. So I try to share these insights with my students whenever possible. I try to point out the wonderful connections which seemed so magical and delightful to me when I first studied calculus (and which I still find magical and delightful, truth be told). And I thought about how little my precalculus students understood about transformations, just one semester before they might enter my calculus course.

And then... Antiphany. My calculus students don't have the foggiest idea what I'm talking about when I share those insights, do they? They're missing a basic facility with graphs and functions to be able to hear anything other than an impenetrable wall of words, and without that basic facility, there is no way they can hear it. Unless they are able to grasp the basic facts of analytic geometry intuitively, they can't hear an insight about those facts.

The second antiphany was an odder moment, and much worse. Without going into detail, it was the sudden understanding, in a brief revelation, that one project I've been working on for a few years was doomed from the beginning. The effort had generated stress and frustration for me (and for others), and I hadn't been able to succeed. I'd had to deal with the effects of the failure, and keep struggling to try to make it succeed, and then in a single moment, I saw why success was impossible. I had in fact been sabotaged before I ever started, without knowing it. And I found out how I'd been sabotaged, all at once. I could see the past few years flushed down the toilet, wasted effort worrying about trying to push the damned rock up the hill when it kept being shoved back down from the other side.

Antiphany. It's a good word, even if I did make it up.

(By the way, a brief Google search turns up some previous coinages of antiphany, based on the same idea of an anti-epiphany. But I intend a somewhat different meaning.)

## Tuesday, September 14, 2010

### Ponderings on Geometry

At a conference this summer I saw an interesting invited talk about teaching geometry. The speaker discussed some of the historical challenges to Euclidean geometry, and discussed how these were eventually overcome. The problem for schools being that while mathematicians like Hilbert managed to fill in all the gaps left by Euclid, the resulting geometry was very complex. It took a great deal of work to get from the axioms up to the interesting Euclidean results. In fact it took too long for a single year high school course to even get to the beginning of Euclidean geometry. As a result, the speaker claimed, most high schools eliminated axiomatic and proof-based geometry from the curriculum, replacing it with practical, computational geometry only. (He went on to detail a new approach, which allows a more rigorous approach to Euclid's geometry in a high school without starting from scratch, and using a book which is designed to be used in an inquiry based class. It looks fascinating, and I'm hoping to teach a class based on this book soon, but that's a digression.)

What caught me off guard was his statement that rigorous axiomatic approaches to geometry using proofs had been all but eliminated from most public high schools due to difficulty dealing with the challenges to Euclid. I had to wonder, was this actually true? I certainly had a proof based and axiomatic geometry in high school, and that was... ok, a little over twenty years ago. (Geez I feel old now.) And while my high school was mostly fine, it wasn't really an elite school. (When I taught an analysis class, I actually used several problems phrased in the form: "My high school calculus teacher said ___. Prove rigorously that he was wrong.") So had schools actually eliminated proofs from geometry since I had taken it? Or had my school been an odd stand out for not eliminating it?

Tonight I was about to start a geometry unit with a class. I asked the students for a show of hands of how many people remembered doing ruler and compass constructions. Less than a third--and maybe less than a quarter--raised their hands. Now I know from experience that some of the students may very well have seen constructions and just have forgotten everything five minutes later. But still, it seems that a number actually didn't do constructions. And I'm inclined to think that if they didn't do any constructions, they probably didn't do any proofs, although I could be wrong on this. (And of course some may have done constructions but not proofs.) So maybe the summer speaker was right, and axiomatic geometry is mostly gone.

But I question his reasoning about why the proofs are gone. He believed it to be because it proved too difficult to fill the gaps found in Euclid. I can't imagine boards of education, state legislatures and such largely care--or even know--about that. My first suspicion was that someone, somewhere decided it was "too hard", or that students just couldn't do it, and replaced it with other things.

But why too hard? Abstract reasoning and logic is hard, but given time and effort, most people can make progress. Ah, but there's the problem: Time. To do a good job with a class about proofs, it takes time spent working on problems, trying ideas, failing, and trying again. Learning to reason is a long and difficult process. And curriculum tends to fill up with all sorts of nonsense as everyone and their dog proposes new lists of things that "everybody" ought to know. Most of those things are lists of facts and formulas. I can almost hear the litany start for a geometry class: "Students must be able to give the formula for the area of a circle, a semicircle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus; Students must be able to find the perimeter of a circle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus..." (And it's worth noting that the only really interesting things in the list I just gave are probably the area of the circle and the rectangle, and perimeter of a circle. The rest should be easily derivable from some good geometric thinking. But the students will instead be given a list of formulas to memorize.)

We add and add to curriculum, and nothing is ever taken out. We add numerical approximations and work with computers. We add calculators, then graphing calculators, then geometry software. We add three dimensional shapes, and trigonometric functions, and whatever else anyone can think of because ... well, because someone else happened to remember it and thought it would show how rigorous we were being if the list of things for students to know was really long. (It should be noted that I am not opposed to any of these topics in high school math classes. But we must recognize that we cannot do every possible topic all at once.)

But a long list of topics is not rigor. That's just memorizing a bunch of stuff. Take any of those topics and spend some time with the students doing a long and careful analysis of some challenging problems, and you'd have a recipe for a great math class. The central questions will always be the following: What do you think is true? How do you know that? Why? Is this like anything we have done before? Can we generalize this result?

All of this of course misses the biggest elephant in the room as to why high school math has dropped most reasoning and replaced it with lists of tasks and formulas: It's very easy to write a statewide multiple-choice test which to see if students can choose the formula for the area of a circle. It's very difficult (or perhaps impossible) to write a state-wide multiple choice test which determines how skilled students are at reasoning and solving complex problems. But I'd much prefer a student who can work out how to find the area of a trapezoid based on what she already knows than one who has only memorized the formula. I regularly have students who have memorized a formula corresponding to a figure, but can't figure out if it's the formula for the area or the perimeter. (When I ask how to find the area of a circle, I can guarantee about half the class will respond "2 pi r.")

And I'd really love to have students who learned enough reasoning to write proofs in their geometry class, but apparently that's rare now. No wonder we have trouble teaching proofs later in college.

## Monday, August 30, 2010

### Morning scene

The garbage truck is outside rumbling and beeping, and I squint ferociously enough to make out the time on my alarm clock.

I realize I have one hour of summer vacation left. The sheets are still comfortably cool, and soft from yesterday's laundry, so I doze a bit. The window is open to make the most of a cool night, and light is sneaking past the mostly closed blind.

Roll over. *squint* I have half an hour of summer vacation left. I could reset the alarm a bit later, but I have things to do. Stuff. Stuff starts today. What did I get finished last night and what's still waiting for me? I should have enough time to get the last syllabus copied this morning and check over everything. What the heck is the deal with the new online course system, anyway? I have no idea what the students are going to actually see on that thing....

I have fifteen minutes of summer vacation left.

Sleep is a long gone specter, but I can at least enjoy the idea that I don't have to get up yet. Not really. Not yet. Summer isn't over until I actually get up. What happened to the summer, anyway? They just slip past faster and faster, and I've finished not a tenth of what I'd hoped to.

I have ten minutes of---\$*@# it, I'm getting up.

## Friday, July 02, 2010

### The Last Airbender (Movie review)

I went to see The Last Airbender last week. As a long fan of the Avatar series, I thought I'd write down my thoughts on the movie and how it stacks up to the original. Overall I thought the movie was OK, although it felt awfully hurried to try to fit into one movie what originally took place over a season of a television show. As I feared, character development suffered a bit from this; I thought both the characters of Aang and Zuko were pretty well fleshed-out, but most of the remaining characters were pretty flat, which is unfortunate. In some ways, the movie does a better job of pulling in some major themes (such as the Avatar's relation to the spirit world) than the early series did. On the other hand, it should be able to draw in some of these later-developed themes better, since it has the completed series to draw off of. But ultimately I'm not sure I can see the movie as hugely successful. It felt rushed, most of the characters felt under-developed, and somehow even the bending scenes failed to be compelling. So that said, let me say a bit about what I liked and what I found lacking. And I guess from this point out, there be spoilers--both for the movie, and for the rest of the series.

I liked the fact that the relationship of the Avatar to the spirit world was spelled out early on and woven deeply into the plot. As I recall, this was something which was introduced a bit later in the series. The dragon as a sort of spirit-guide works. (I'm assuming it's still Roku's dragon, although this is never mentioned in the movie.)

The movie tells us that water bending is strongly tied to emotions, which I think helps us understand Aang's overwhelming grief at the loss of all of his friends, guardians, and mentors, and his guilt over abandoning them. It's a nice device to get the idea across quickly. (But I have to wonder: What are the other elements linked with?) The betrayal of Aang by the Earth villager, who tells him he spent a life in poverty because of the Avatar's absence, deftly underscores the idea that some resent the Avatar for disappearing. An episode in the first season of the series dealt with the theme of the Avatar abandoning the world to the Fire nations ravages, and I remember that episode impressed me--and caught me off guard. It was a hint that the series was going to be deep and interesting. It's difficult to pull off the same trick in a single movie, since we have to identify with Aang before we can see this more complex (and uncomfortable) shading of his character. The movie tries (I think) to develop sympathy with Aang by having him start helping Earth nation villages right away. This works, but on the other hand is a little weird, since from Aang's perspective, he basically ran away from his responsibilities as the Avatar yesterday. So why exactly does he suddenly decide to accept those responsibilities and start acting like the Avatar?

The same series episode which started the development of Aang also started the development of prince Zuko, who wants to regain honor in his father's eyes by capturing the Avatar. The movie does a fairly good job of outlining Zuko's history and motivations. I think they are a little too quick to dive into the character of Iroh, Zuko's uncle, 'though. I recall him being a rather enigmatic figure for some time into the series. His reactions to and willingness to turn on other Fire nation troops so rapidly in the movie I think lead to a lot of loss of subtlety for the character. The movie watcher would have no real reason to question the Fire Lord's assessment when he calls Iroh a traitor, but this is a mischaracterization of Iroh. Iroh is proudly and fundamentally of the Fire nation; he's just not insane or evil, and tends towards a more harmonious blending of the all the nations rather than a desire to have the Fire nation completely dominate. There are certainly strains of "with us or against us" totalitarianism in the Fire nation, but I don't think there is enough in the movie to undermine that idea with a viewer unfamiliar with the series.

Unfortunately, despite some careful work on Aang and Zuko, and some attention to Iroh, most of the rest of the characters in the movie are pretty flat. The romance between Yue and Sokka is pretty much handled by saying, "Um, and they're in love, OK?" When Yue sacrifices herself to save a spirit, it's hard to pull much emotion out of the scene since we pretty much met Yue fifteen minutes ago, and much of that fifteen minutes was spent watching Aang try to learn to water bend. Katara isn't developed much either. After the movie harps on the idea that Aang abandoned his responsibilities because he was told he could never have a family, the scene at the end where Katara bows down before Aang as the Avatar--just like everyone else--could have resonated strongly, reminding us of what Aang was giving up, had the movie (like the series) suggested that Aang might have feelings for Katara.

The only other characters of any significance in the movie are Zhao (portrayed as a simple villain) and the Fire Lord (portrayed as, well... not much of anything, I think). Most of the other characters don't even get names. (I will say that I think the decision to have the monks in Aang's memories never speak is interesting; it keeps that past distant from us, and leaves us with just impressions about Aang's feelings about them, which are pretty clear. It helps emphasize the idea that these are memories of long gone people. Not that I think there would be a problem if they are to speak in a later movie either; it just worked well here.)

Partly the movie suffered from trying to cram a season worth of episodes into one movie. The first half of the movie left me trying to catch my breath; it seemed like a constant series of rapid scene changes. Now go here! Now there! Now Aang's captured! Now he's escaped! Now he's visiting a temple! Now he's in the spirit world! Now he's leading a rebellion! Now he's captured again!... To have been a successful translation to the big screen, I think the movie would have to have been twice as long (at least). Unfortunately, no one would have gone to see it.

Then there are distracting and weird changes made for no real reason I can see. Why in the world would you change the pronunciation of some the characters' names from the series? And fire benders seem to need an external source of fire to bend now, which I suppose makes them more like the other benders, except it also seems to put them at a huge disadvantage: While it's hard to go anywhere that has no earth, air, or water, it's not that unusual to be places without fire. (Of course, this also leaves me wanting to scream in the battle scenes with the water benders: "Just focus on putting out their fire source in the first place!") I suppose it gives an easy hook to emphasize the idea that Iroh is a truly gifted fire bender, or a way to explain what the comet will accomplish, but it mostly just seems unnecessary.

And one last minor complaint: I'm not impressed with the bending scenes. The animated series did a better job on two counts. First, the bending in animation seemed more controlled. The bending in "live action" (in reality computer animation) ended up looking more like a vague suggestion to the element, probably in an attempt to make it look more "real". Secondly, the bending motions in the movie rarely seemed to actually do anything. Most of the time the characters seemed like they flailed around for a few minutes, then finally something would happen. Sometimes. In the series, the element seemed to respond immediately to the work of a bender, and it seemed linked and connected with their motions. The movie bending looked more like a bunch of magic passes, followed by a brief magic trick. Perhaps this is partly a problem in coordinating the computer animation with the choreography, but there never seemed to be much connection between a bender's motions and the response of the element.

I'm not sure what the overall reaction to the movie will be, but I'm not feeling it's been a very successful retelling of the tale, and I'm not sure it will have the same appeal. (Of course, I'm sure more people will see the movie than the animated series; it's being sold as a summer blockbuster and there are plenty of people who just won't watch animation anyway.)

## Tuesday, January 26, 2010

### When do they turn off Disneyland?

As it turns out, an hour and a half after closing.

I went to the Joint Mathematics Meetings in San Francisco about two weeks ago. Since I was flying all the way to California anyway, I arranged to spend a few days at Disneyland first (of course). I stayed close to the park so I could walk in and out. (I discovered two years ago when the meetings were in San Diego and I visited Disneyland that this is a great way to visit. The whole resort is pretty walkable, even if I do end up with blisters on my blisters after a few days there.)

As it turns out, I stayed even closer to the park than I thought. I could see Space Mountain from my hotel room:

After the first night in the park, I left at closing and noticed when I got back to my room that I could still see Space Mountain. But at a later point, I looked out and it was gone. I eventually pinpointed the time on a later night: At an hour and a half after closing time, the mountain suddenly winks out of sight.

So I figure that must be when they turn off Disneyland.

## Monday, January 25, 2010

### The semester so far

Today began the second week of a new semester.

I have two particularly small sections of one class, in part, I think, because I'm the only person teaching the second semester of the course who was not teaching the first semester in the fall. I love having small classes, so I'm not complaining. One of my students who didn't show up on the first day claimed she had switched to Professor D's section, but was still showing up in my roster. I double checked with Prof D, since today was the last day of add/drop. It turns out that Prof D had signed an override to allow her to enter his already overfull section. He now has 33 students. I went from 16 to 15. I'm laughing. I'm not sure Professor D was when he found out what had happened.

On the other hand, my Gen Ed class (aka, "So you think you can math?") is not small at all, and has stayed firmly at the enrollment limit of 40. Students have some incentive to pass the class this semester, since in the fall both the difficulty of the course and the prerequisites will increase. I'm being very straightforward (and maybe just a little easier than usual) to give them the best shot of getting through before the new course requirements kick in. We have just finished the second class (for a total of 2.5 hours of classroom time) doing nothing but unit conversions. Some students are completely lost.

I had a student show up in my office today wanting to do an independent study this semester. We discussed some possibilities, but I said we should talk to the chair to find out if it was possible such a study could be approved this late. The chair responded by putting her head in her hands and making a sound like a expiring mongoose. Since the student still needs to be in the chair's good graces, he wisely rescinded his request for an independent study.

But my joy this semester will be teaching the "Intro to Proofs" class for majors. I'm teaching the course using a technique commonly known as a modified "Moore method", which I've been interested in for some time. I went to a workshop for new practitioners two summers ago, and last semester I applied for and received a mentor so I could start trying it. It's the ultimate in student centered instruction, where most (or sometimes all) class time is spent with students presenting proofs to the class and the class dissecting the proofs as needed. I was prepared for all sorts of disasters to occur, but to my astonishment, my first two classes have gone extraordinarily well. My current major concern is to make sure all of the students stay involved in the class, rather than just a subset. But so far the discussions and presentations in class have been wonderful, and in fact beyond what I had hoped for. I still expect plenty of challenges ahead, but at least I feel like I've been well prepared for them, so I'm cautiously hopeful that we may have a really good semester.

Oh, and the unseasonably warm weather we have been enjoying is drawing to a close. Looks like we'll be down below freezing for a while now. Drat.