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.
Monday, August 30, 2010
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.)
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.
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.
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.
Monday, November 30, 2009
Retrospective
Well that's it. This marks the end of National Blog Posting Month (which I mentioned earlier), or at least my participation. So what have the results been?
- It turns out that over the course of November I wrote about 10,000 words for my blog. That's fully 1/5th of the goal for NaNoWriMo (National Novel Writing Month), so who knows, maybe someday I'll try that too. I can finally write my novel about the smart, hip, jet-setting, handsome mathematician who has men fawning all over him. (Hey, this is my Mary Sue, right?)
- I started this blog August 2005, which is about 4 years and 4 months ago, or 52 months. At the start of November, I had 162 public posts. That means I was averaging around 3 posts a month, until this month when I produced ten times as many. Or, over the course of this month, I've increased by almost 1/5th my total postings.
- Sometimes I faced some writer's block of course, but usually once I picked a topic, I found I had things to say.
- I've written about an enormous variety of things. I've talked about students and teaching (including the good, the bad, and the ugly), holidays, dreams, visiting a conference, and random musings of my own mind. Some days I wish I had a more interesting life, but often I like it fine.
- Not everything I've written has been a gem. But I've definitely written things I've liked, and some of them seem to have inspired other people to chime in. It does definitely convince me that I should write more often. Having an explicit (and public) goal definitely spurs some creative work. I'm not sure if I'll participate again in the project or not in any sort of semi-official way. (As I noted, I wasn't doing anything official this time; I just thought I'd try it.) For the moment, 'though, I may take a pause before I write again.
Sunday, November 29, 2009
Hibernation
I don't care for the cold. (I don't know what I think I'm doing up here. Who am I, Nanook?) But I do like burying myself under a heavy pile of blankets to go to sleep at night when it's cold. Flannel sheets, a big artificial down comforter, maybe a quilt, 'til I'm nice and toasty, and trying not to let any part of me poke outside my nest. It's wonderful. The only problem is that it's still cold in the morning when I wake up and have to get out of bed. I think bears are onto something. If I could just sleep through the winter and wake up again when it's spring, I'd be all set.
Or I just need to live someplace warmer.
Or I just need to live someplace warmer.
Saturday, November 28, 2009
What students hear
I sometimes think about what students hear when we (their teachers) talk. I don't think undergraduate students (even good ones) manage to grasp the essentials of mathematics. I don't think they see what we are doing as logical reasoning based around some fundamental principles. They hear the specific words and explanations for a particular problem, but don't grasp that what we say comes from some coherent system. This is not necessarily because we don't say that this is so. It just seems that this concept doesn't completely register in the time we have them. It might later, or it might not.
When, for example, we want to find all solutions to x2 = x, I might suggest dividing both sides by x, getting x = 1, which is one solution. I then note that we can only divide by x if we assume x is not zero, and in fact x = 0 is the other solution. We then have all solutions to the equation. To me, at this point in my life, I see division of both sides of an equation as a legal operation with very specific restrictions (namely, that we cannot divide by zero). I also recognize that since we can view both sides originally as multiplied by x, that x = 0 is certainly a solution. But I remember when things were not quite so coherent, and then the step back to note that we cannot divide by zero seemed like a trick just to justify a zero solution. It seemed that a lot of algebra (and some other mathematics) was made up of a bunch of special rules and exceptions, and it seemed like teachers had a never ending supply of these to pull out to justify whatever they said the answer was. It seemed a bit like playing pretend with a small child, where there is an exception to everything, to be made up on the spot: "Oh yeah? Well I had my invisible anti-force field magic belt on, so I could escape from your force field!"
How do people learn to see mathematics (or any field) as a unified whole? Perhaps part of it comes from time and experience. It may just take a certain amount of time working with the concepts before they become solidified and can be deftly manipulated. It may be similar to the feeling I can still remember in college, when I became sufficiently comfortable with algebraic manipulations that I could use them to do faster mental arithmetic, by disassembling and reassembling the numbers in convenient ways, making the numbers dance as needed. It's not that no one had ever suggested the idea before; it's popular to present these ideas to students, but until those concepts are internalized, it doesn't make mental arithmetic any easier. It just seems like a trick--and a somewhat painfully difficult one, at times.
Or maybe we need to give students more opportunity to see mathematics as a unified whole from the beginning. That would be one reason I've for years pushed for depth over breadth. Whenever a curriculum issue comes up, my first thought is usually, "What can we cut out?" Skimming quickly over dozens of topics and techniques encourages that sense of mathematics as a big collection of tricks, rather than something that has meaning and can be reasoned about. We need time to think about ideas, process them, wrestle with them, and make them our own. If students can understand a few concepts deeply, they'll have a better chance of figuring out something new on their own.
For the same reasons, I also feel drawn to inquiry based learning or Moore method teaching, where students are asked to figure things out with minimal guidance. We start with a few ideas--say, some definitions and a few axioms--and students are asked to build on that framework. Each new step has to be justified, and students need to figure out what works and what doesn't, without being told specifically what to do or what's right and wrong. They should rather be led by careful questioning to notice for themselves what works and what doesn't. If instead of just saying, "No, that's wrong," the teacher can simply present another problem and let the students figure out that their previous approach is flawed, they may internalize the structures much better. After all, structures you yourself have built are already essentially internalized. And it's hard to view a subject as filled with arbitrary tricks when you yourself have built up those "tricks" because they were needed.
In any case, what I would most like is for the students to come away with a sense of my subject (and others) as a unified whole, as something they can investigate and reason about, rather than just a collection of tricks and techniques. The techniques are useful, and the tricks are powerful, but without a strong foundation, it all falls apart.
When, for example, we want to find all solutions to x2 = x, I might suggest dividing both sides by x, getting x = 1, which is one solution. I then note that we can only divide by x if we assume x is not zero, and in fact x = 0 is the other solution. We then have all solutions to the equation. To me, at this point in my life, I see division of both sides of an equation as a legal operation with very specific restrictions (namely, that we cannot divide by zero). I also recognize that since we can view both sides originally as multiplied by x, that x = 0 is certainly a solution. But I remember when things were not quite so coherent, and then the step back to note that we cannot divide by zero seemed like a trick just to justify a zero solution. It seemed that a lot of algebra (and some other mathematics) was made up of a bunch of special rules and exceptions, and it seemed like teachers had a never ending supply of these to pull out to justify whatever they said the answer was. It seemed a bit like playing pretend with a small child, where there is an exception to everything, to be made up on the spot: "Oh yeah? Well I had my invisible anti-force field magic belt on, so I could escape from your force field!"
How do people learn to see mathematics (or any field) as a unified whole? Perhaps part of it comes from time and experience. It may just take a certain amount of time working with the concepts before they become solidified and can be deftly manipulated. It may be similar to the feeling I can still remember in college, when I became sufficiently comfortable with algebraic manipulations that I could use them to do faster mental arithmetic, by disassembling and reassembling the numbers in convenient ways, making the numbers dance as needed. It's not that no one had ever suggested the idea before; it's popular to present these ideas to students, but until those concepts are internalized, it doesn't make mental arithmetic any easier. It just seems like a trick--and a somewhat painfully difficult one, at times.
Or maybe we need to give students more opportunity to see mathematics as a unified whole from the beginning. That would be one reason I've for years pushed for depth over breadth. Whenever a curriculum issue comes up, my first thought is usually, "What can we cut out?" Skimming quickly over dozens of topics and techniques encourages that sense of mathematics as a big collection of tricks, rather than something that has meaning and can be reasoned about. We need time to think about ideas, process them, wrestle with them, and make them our own. If students can understand a few concepts deeply, they'll have a better chance of figuring out something new on their own.
For the same reasons, I also feel drawn to inquiry based learning or Moore method teaching, where students are asked to figure things out with minimal guidance. We start with a few ideas--say, some definitions and a few axioms--and students are asked to build on that framework. Each new step has to be justified, and students need to figure out what works and what doesn't, without being told specifically what to do or what's right and wrong. They should rather be led by careful questioning to notice for themselves what works and what doesn't. If instead of just saying, "No, that's wrong," the teacher can simply present another problem and let the students figure out that their previous approach is flawed, they may internalize the structures much better. After all, structures you yourself have built are already essentially internalized. And it's hard to view a subject as filled with arbitrary tricks when you yourself have built up those "tricks" because they were needed.
In any case, what I would most like is for the students to come away with a sense of my subject (and others) as a unified whole, as something they can investigate and reason about, rather than just a collection of tricks and techniques. The techniques are useful, and the tricks are powerful, but without a strong foundation, it all falls apart.
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