The One-to-Many Problem

I've been thinking about a class of related phenomena recently: those where a one-to-many relationship leads to inappropriate feelings or behaviors. I'll give a few examples.

Often, some shared cultural item (The Chronicles of Narnia, Amsterdam, Ernest Hemingway, chopsticks, etc.) comes up, and I feel as though I have a special relationship to it. I want to impress upon others my special relationship, to tell them the stories about myself and the item. And then I realize how extremely common it is for someone to have specific personal memories of reading a popular book series or using or watching others use chopsticks or whatever. Just because I visited Holland in the mid-90s doesn't really mean I have a special relationship with the country.

Professors commonly get a lot of annoying emails from their students. If you think about it, a student generally has at most five or six professors at a time, while a professor may have anywhere from 20 to several hundred students. Three or four emails per semester to each professor is easily manageable to the student, and overwhelming for professors. Students also feel that their own experiences are more unique than professors find them to be.

There are a few well-known blog authors whose blogs I have read for years. (John Scalzi and Andrew Sullivan, for instance.) I feel like I know these bloggers, have a sense of their personalities and experiences, etc. If I saw one of them, it would be easy for me to assume some kind of mutual familiarity that does not exist at all; they don't know me from Adam. I think people often have this feeling about celebrities.

Of course, it is always tempting to regard one's own experiences and situations as unique and special anyway, given that one's world revolves around oneself. But I do think these one-to-many situations, where something is more unique for you than you are for it, are especially prone to provoking such feelings. Perhaps this is why people like to have known indie bands before they became popular; it makes their special relationship to the band more plausible because it existed when the many wasn't so multitudinous.

Grad Apps In

As of today, I have done every single thing for my graduate school applications. One of my recommenders has submitted all of his letters, and the other two have shown signs of working on theirs, so everything seems to be well in hand.

I'm glad to be done with the process. I have applied to the following schools (in no particular order):

• Colorado State University
• New Mexico State University
• Texas A&M
• University of North Texas
• University of Florida
• University of Tennessee
• University of Kentucky
• University of Pittsburgh

I think I have a decent chance at some of these schools, but we'll see. If I don't get in this year, I will take some more classes (perhaps a graduate-level one in the fall, though I could also use a couple more undergrad classes to improve my background), take the GRE mathematics subject exam, and apply again next year.

Fake English

I really enjoy listening to people do "fake English" - I mean things like improv comedians in foreign countries doing gobbledegook that is supposed to represent someone speaking English. This guy did a whole song, and to me, the "English" is really so convincing that it's almost frustrating that I can't understand it. It's actually a pretty good song too, of a particular, easily recognizable American genre. I've seen this a few times, anyway, but ran across it again today, so here it is:

See more funny videos and Music Videos at Today's Big Thing.

Netflix to Blockbuster

I've had a Netflix subscription for several years now, but I've never gotten as much out of it as I could. It's very hard for me to mail movies away, since I can't get anything right away. And then when I do want a movie, I can't have it right now. I probably watch 4 or 6 movies a year from Netflix, which is ridiculous considering that I could buy a couple of DVDs every month for what I pay them.

Lately we rent a lot at Blockbuster. (By "a lot" I mean once or twice a month.) That's very convenient and nice, except that I never take the movies back, so I end up owning them. They are only $10 or $12 each, so it's not that expensive, but...still.

So, I have cancelled Netflix (and I give them major props for letting you do this on the website; so many services make you call them) and opened an account with Blockbuster online. What I got, for $12/month, is 1 DVD at a time by mail, and up to 5 in-store exchanges per month. This means that for less than the cost of one rental (when you figure in the high probability that I will never return the movie), we can just go to Blockbuster any time and switch out our movie. Or we can get one by mail if we want one that isn't available locally.

I'm pretty happy with this decision.

Attitudes

For weeks, two of my coworkers (let's call 'em James and Morgan) and I have been speculating over the situation with bonuses, which are typically given out in December at my company. James has been saying all along that he didn't think we'd get a bonus this year. We had been assured (since our president heard about these rumors) that a bonus with forthcoming, but James pointed out that any token amount could qualify as a "bonus" so the assurances were meaningless.

Last year, my first year here, I got quite a magnificent bonus. I was told that 1/3 of it came from the group that had recently purchased our company, and the rest was from our company. It was awesome. My offer letter says that the company typically pays bonuses of around 10%, depending on company profits and individual performance. I didn't have much hope of getting the same bonus as last year, but I was hoping for around 10%. That is still a really large bonus, in my book, despite what we may have seen last year.

Santa Claus finally came today, in the form of our president telling us our amounts. Mine was slightly less than last year, and well above any line at which I would have felt disappointed. I am pretty thrilled with it.

I made the mistake of going to lunch with James and Morgan. James is angry - he saved the company some large amount of money this year, was told this would be remembered at bonus time, and yet he didn't get a proportionately large bonus, he didn't feel. Morgan was not dissatisfied with her amount (she also reported getting slightly less than last year, but none of us shared our amounts or percentages), but was fretful over various things she's heard. She fears that because she is sort of mousy and doesn't often work directly with the people who decided the bonus amounts, it might be easier to "screw her" by not giving her as much as she should get. She had a lot of process worries over how these amounts were decided, and was also upset because of some conflicts over other people's bonuses. (Apparently one employee was upset that another employee got a larger bonus - something she was in a position to know about - and left for the day.)

It was basically non-stop negativity from those two. The president had apparently made a comment to James that, given that she didn't have as much total bonus money as she would like, she trimmed down the bonuses to some people who are highly paid in order to boost bonuses for people who make less. He was very angry and upset over this prospect, calling it "unprofessional" and commenting, "A company that knew what they were doing would pay people their market value," and then going on for at least 5 minutes explaining how this idea that some people make a "premium" could be applied to anyone and bitching how there are no standards and so on and so forth. Morgan mostly cavilled about this, that, and the other.

Morgan also pressed me about how I felt, and though I didn't want to be self-righteous about it, I did have to make it clear that I thought the amount was very satisfying and generous, and that I have chosen not to concern myself with the sausage-making aspects of how the company is run, both in general and with regard to bonuses. (I have found that worrying or becoming annoyed at the ways of corporations only leads to madness, especially since I'm never privy to the real details of what is going on. Best to just let them run the company, assume they know what they're doing, and look out for my own interests in more productive ways.)

I did not say any of this, but I find it really unproductive and, frankly, immoral to have the attitude (in life in general) that, while what you got might be OK, someone else might be unfairly getting more. Our whole economic system invariably leads to a lot of "unfair" things that people at my company, at least, are benefitting from. I have friends who are harder working, probably smarter, and definitely more educated than I am, and who make less money than I do even though their jobs are not in any sense easier. That is not "fair." It is not supposed to be fair. Fair did not come into it at any point.

Morgan was concerned that some people might get higher bonuses because they are more liked rather than because they are better employees. On what basis is an employee liked? How do we judge "better"? Why worry about the parts you can't control? Why choose to be dissatisfied when things are, actually, really good?

I don't think either of these people has had a serious job working for another company, or they would know that all companies are fucked up, to a lesser or greater extent, and that this one is particularly great to work for, and very generous in every way. And, you know, if you don't think you're being paid your "market value," there is an easy remedy for that. (That would be the, um, job market.)

If I do go to grad school, and if my cohort is large enough that there are subgroups to it, I really need to be part of a subgroup of people who work hard and have good attitudes. Because this stuff does influence me and I don't want to be around a bunch of whiners who think life is unfair all the time.

(NB: I don't think people are wrong for complaining about legitimate grievances. If you work at Walmart and can't afford to take your child to the doctor, rail all you want - I'm with you. But if all you have are upper middle class problems, then STFU with your BS about "unfair.")

Taking the Putnam

Yesterday morning, I went to school to participate in the Putnam Competition. This is a six-hour math exam (if you want to call it that), split into two 3-hour sections, with a lunch break in between. Each section has six questions that require college mathematics and a lot of creative thinking to solve. Each question is worth a maximum of 10 points, with partial credit given, and the median score is typically 1 or 2 points (out of 120).

I didn't really have time to go, but I didn't see how I could miss it. The competition is only for undergraduates, so this was my last chance, and I'd never done it before (nor had an opportunity to, that I was aware of, though you can take it four times overall and needn't be a senior nor a math major). I wanted to support our math department as well; I knew some people wouldn't show up, and I wanted to be able to say that, yes, we can at least field a Putnam team. (I wasn't on the actual team, which is three people from each school, in the end. But that's fine. You don't work together anyway - it's a purely individual endeavor.)

The morning session was fun. Of the six problems, one looked tractable, but I didn't get anywhere with it. I turned to another question that looked less tractable and ended up writing out an answer. (You have to write a full proof for the answer, not just solve the problem.) I am pretty sure a central assertion in my proof was wrong, though, but I haven't had a chance to check it yet. (It involved an 18x16 matrix, which I'm sure was not how the problem was meant to be solved.)

What was fun was that I was only trying to score any points at all. On a normal exam you're trying to get all of the points, or fall short as little as possible, but I was aiming to just get above 0, so it wasn't really stressful. I had three hours to work on as little as one problem.

The professor running the show bought us lunch - we all walked over to Old Chicago. It was actually pretty blissful. There were six of us students, of whom two are in my advanced calculus class, one is in my "senior seminar" next semester (but I hadn't met him yet), and the other two were unknown to me. (One I'm not so sure about - he argued on the way to lunch that irrational numbers can be accurately represented as fractions, using 22/7 - a classic approximation for pi - as an example.) We talked about the test, and other math topics, all through lunch. The professor kept quiet and just let us talk, which for all I know might have been out of peevishness at our annoying qualities, but felt gracious.

I found myself hoping that this is what grad school is like - that there are other people around and you can talk about math with them sometimes. I realize undergrad is like that for some people, but it hasn't been for me. I really enjoyed it. I also realized that I hope my graduate program does not have a competitive feel to it, because I wouldn't have enjoyed lunch nearly as much had we all been trying to one-up each other.

After lunch, we had six new problems. Several of them seemed tractable but I couldn't gain any traction for a long time. I was really tired from my week (I've been exhausted pretty much all week, and had to get up extra early for this thing), and that started to kick in, and I had had too much iced tea at lunch, so I had that nervous/sick kind of feeling, plus I kept having to pee. I finally did get an answer to one question. I now think that part of my answer was wrong, but I was really happy with the way that I approached it and the style of proof that I wrote for it. Still, I did not enjoy the afternoon session very much.

In the end, I am really glad that I went, and I'm hopeful that I might have scored 2 or 3 points on the exam. I'll post when scores come out.

Mathematics Genealogy

I was looking at The Mathematics Genealogy Project today. It's a project that kind of sounds like what it is - for (many) math professors, it lists their advisor and students, so you can trace up and down. And it's kind of astonishing.

I did a search on one of my professors, and this is what I found as I went up the chain of advisors (leaving out some initial steps for anonymity):

1. my Professor - no known students (makes sense; my school doesn't have graduate programs)
2. his advisor
3. the advisor's advisor, PhD from Indiana University, 1960.
4. Tracy Yerkes Thomas, Princeton, 1923. I knew I was getting into the past because the dissertation title was "The Geometry of Paths," which is just way too basic and short to be modern.
5. Oswald Veblen, U of Chicago, 1903.
6. E.H. Moore, Yale, 1885.
7. H.A. Newton, Yale, 1850.
8. Michel Chasles, École Polytechnique, 1814.
9. Simeon Denis Poisson, École Polytechnique, 1800.
10. Poisson had two advisors - Joseph Lagrange, and Pierre-Simon Laplace.
11. Lagrange's advisor was Euler, and Laplace's was d'Alembert.
12. Euler's advisor was Bernoulli.

Anyway...it's kind of amazing how few steps it takes to get from anyone to someone famous (even famous to me).

Another of my previous professors led me upward to Darboux (of Darboux sums fame, presumably) within a few clicks.

I got from one of Sally's professors to Isaac Newton! (It did take a few clicks, though.)

Pandora on xkcd

This is so true:

Grad School Applications

As of today, I have applied to four graduate schools. I've finished the online applications for all four, submitted my Statement of Purpose (except to one school, pending an email inquiry), sent my GRE scores, and emailed instructions to my professors on how they can send/submit their letters of recommendation for me.

I can't send my transcripts from Metro until the semester ends, because I need those grades on there, and that will be mid/late December, which is in time for all of the deadlines, but still makes me feel a little bit nervous. I don't want to send my other transcripts until then either, because I don't want them to be looking at all my crappy older grades without the better, newer grades. (That's probably just irrational, but whatever.)

I still have five more schools to finish my applications to, but I'm glad that I've made progress and have some apps in. I still have no idea how good or bad a candidate I am; I can't tell if I'll be lucky to even sneak into my lowest-rated programs or if I can expect several programs to invite me. But all I can do at this point is apply and wait.

Math Gaps

I read something recently where a person was talking about her approach to teaching math - that she reassures her class that they are not "bad" at math, but that somewhere along the way, they had a bad teacher, or they missed a day, and the lacking knowledge accumulated so that now they are confused by various things. And I was thinking of how many gaps I have in my own knowledge of math, and how crazy these gaps drive me when they show up.

It's hard to even recognize that something is caused by a gap. A certain topic will show up (infinite series, say, or rules of limits) and my mind will just go "that's too hard" or "I can't do those" without asking why. Are these topics somehow such that I alone cannot learn them? Do I have some tiny genetic flaw that has knocked out the part of my mind that would let me understand what a Taylor series is?

No. I just don't understand something because I have some gaps, and the thing to do, then, if I'm serious about math, is to figure out what the gaps are and fill them in. It's unlikely I'm missing anything that I can't learn, so I need to just get on that, as it comes up. And now that I'm finally more of a badass about reading and understanding math, I'm in a perfect position to do so.

Gendered Language

Sometimes my grandmother used to tell me, "Act like a little lady," and it always rankled me. Even when I was four years old I didn't like the word "lady" and didn't want to be one. But it strikes me that I have an unusual aversion to unnecessarily gendered language.

I noticed right away that my new therapist tended to say things like, "You're clearly a very [adjective] woman," and that it rubbed me completely the wrong way, even though the adjectives were positive things like "intelligent" or "passionate." (I mentioned it and he seems to have stopped.)

I never say things like "when I was a little girl" -- it's always "kid" or "child." I don't think I ever refer to myself as a woman unless there are situations that clearly call for it ("I'm not sure how other women manage their facial hair").

I found myself saying the other day that Ed would probably not be a good partner for the type of person who prefers not to know certain things. I don't think Ed will ever date a man, so I'm not sure why the language is gender-neutral except that "the type of woman who prefers not to know certain things" seems slightly offensive to me, like it invokes a stereotype, while "type of person" does not.

I don't think it's only female gender that I tend not to specify. I would never say, "What were you like as a little boy?" instead of "kid." I wouldn't call someone a sensitive, smart, fair, or caring "man" instead of "person." (I would only use "man" if I intended a constrast, e.g., "You're a really maternal man," and even then I'd probably say "person" most of the time.)

I don't do this gender-neutrality on purpose, I don't think. I use common words like "waitress" and "actress" and "handyman." I certainly use pronounce like "he" and "she" in the normal way (though I'm also a fan of the unfairly-maligned singular "they").

I wonder if this is a function of my feminism (using the word in a very basic sense to incorporate the feminism I already had when I was 4 years old), something that is more accepted in my social environment, or a result of some personality trait.

Idiot David Lane

So, Richard Heene pled guilty today to a felony. He's the guy who, along with his family, engineered the hoax that caused a big rescue operation to save his son who was supposed to have accidentally taken off in a helium balloon, but who was actually hiding in the garage as instructed by his parents. A lot of Americans watched this rescue effort anxiously, including some people in my office.

From the Denver Post article, we get this:

After the hearing, Richard Heene's attorney David Lane said that the seriousness of the charges reflects the anger Americans felt after learning they had been duped by the parents into fearing for Falcon's safety.

"Don't mess with America's emotions," Lane said. "America has the emotional instability of a hormonal teenager."

I can't believe the man's lawyer would say something so offensive and unprofessional. It is not "emotional instability" that causes people to become angry when they've been tricked into fearing for a young child's life, or to want restitution for a foolish crime that wasted public money on an unnecessary rescue operation.

Thomae's Function and Integrability

Thomae's Function is something we've discussed in advanced calc a few times - it's basically a sort of pathological function that is discontinuous at every rational number. I've drawn a picture here, along with the definition:

For irrational numbers, the value of f(x) is zero, and for rational numbers (other than 0), it's 1 over the denominator in reduced terms, so for instance f(1/5) = f(2/5) = f(3/5) = 1/5.

Between any two real numbers, there is a rational number, and between any two real numbers, there is an irrational number, so there are two things to notice here, to wit:

1. the x-axis is extra black because there are infinite irrational points on this interval [0, 1] that are irrational so that f(x) = 0
2. points below 0.1 on the y axis aren't shown, but would continue this sort of ever-thickening pyramid structure all the way down

Now let's talk about integration.

When Newton and Leibniz invented caculus, they conceived of integration solely as using antiderivatives. That is, if you have two functions, F and f, such that the slope of F at a point x is the value of f(x) -- i.e., F'(x) = f(x) -- then f is the derivative of F, and F is the antiderivative of f. By the Fundamental Theorem of Calculus, you can use an antiderivative to calculate a definite integral of f - that is, to find the area under the f(x) curve between two points (the area of S in this picture):

Now, obviously, you can't conceive of integration in quite the same way for something like Thomae's function. It can't possibly have an antiderivative - even if you can make a function with that kind of infinitely discontinuous slope, which I can't imagine you could, it wouldn't have a derivative.

So in the 19th century, a different way of conceiving of integrals was conceived. In calculus, you learn about Riemann sums. Basically, if you divide the area under the curve into vertical boxes, and then calculate the area of each box, that's an approximation of the area under the curve. The more boxes you have, the closer your approximation gets to the actual value, and the limit of those approximations is the area under the curve, as in this picture:
It turns out that using a method similar to this, Thomae's function can be integrated on the interval [0, 1]. The method is called "Darboux Sums."

One question when you use Riemann sums is how to calculate the height of each box. An easy way to do it is to use the leftmost or rightmost edge of the box as the height. As long as you use some method, it actually doesn't really matter what it is -- the approximations will still get better and better -- but Darboux does it a particular way.

Specifically, for Darboux sums you get an upper sum and a lower sum. For the upper sum, you use, as the height of each box, the highest value inside of that box, and for the lower sum, you use the lowest value inside of each box. You then add up the areas of all of the boxes, and that gives you an upper sum and a lower sum for the integral.

What next? Darboux's Criterion for Integrability is a theorem that says that if, given an error amount (epsilon), there is a way to divide up the function (i.e., a placement of the boxes - they don't all have to be the same width) such that the difference between the upper and lower sums is less than epsilon, the function is integrable.

If we apply this method to Thomae's function, no matter how we partition the function, the lower Darboux sum will always be 0, because every interval will contain irrational numbers. Since, if it's integrable, the upper sum has be able to be made arbitrarily close to the lower sum, it's clear that, if we can integrate Thomae's function on [0, 1], the result (the area under the "curve," if you will) must be 0.

And it turns out that we can indeed get the upper sum to be as close as we want to 0 (though never actually 0 - but that is never required in calculus, which deals in limits). I won't go through the proof of that, but it relies on the fact that, basically, if you draw a horizontal line somewhere above 0 in this function, there will only be a finite number of points above that line. Then you can just make the boxes that contain those points skinny enough that their overall area is arbitrarily low, and there you go.

Google Images as Visual Dictionary

One of my favorite ways to use Google lately is to look up things I'm unfamiliar with in Google Images. It doesn't work for straight-up words (like "mendacious"), but it works great for things like "hipster" or "umbros" or "engineer boots." You almost immediately find out what the thing is without the annoying mediation of words.

It's like the best visual dictionary of all time.

And Then What?

In a recent Since You Asked column, a woman asks, "How can I detach from my mother without feeling like a horrible person?" Cary counsels:

Cognitive therapy lets us ask, OK, so, What if, indeed, I were to feel like a horrible person? What does that really mean? What would be the end result of that? Would I die? Would I feel intense pain? Would others be harmed? Maybe we have had this voice in our heads, this little voice, saying, You can't do that or you'll feel like a horrible person! If we write these thoughts down, and see them, we see that they are not so accurate. We can ask ourselves, OK, how long would I feel like a horrible person? Would it be momentary? Would it last an hour, or days? And just how horrible a person would I feel like?

I have discovered lately that this general technique - going down the "what if" path rather than treating it as intolerable - works for a lot of fears. Sometimes it is the key to resolving insecurities.

For instance, if I am worried that Ed is mad at me, it can make me very upset. I might then expend mental energy trying to figure out if he is, indeed, angry. Maybe he says he's not, but I think he's lying, or he's angry and doesn't even know it. (A lot of this is hypothetical, but such concerns do come up around some of my other insecurities. For another person, the question might be, "What if he/she is cheating on me?") There's no way to be sure.

I have to stop and say, OK, so he might be angry. What then? Everyone gets angry sometimes. It's not a big deal to have someone angry at you. It's not like he's going to physically attack me. If he yells I'll just wait until he stops, or I can always leave. He probably won't yell anyway. And who is he to be angry? Why do I care? Fuck him if he's mad, I didn't do anything. Whatever.

Sometimes consequences really are pretty catastrophic. What if I have an incurable cancer? The best I can do there is perhaps to consider that I knew I would die of something someday in any case. It's not much comfort. But most fears don't have consequences that are actually that bad. What if I have a panic attack on the airplane? Well, then, I'll feel absolutely horrible for a while, but at some point it will end, and I'll still land on the other side and go on with my life. What if this woman cutting my hair accidentally nicks my ear? Well, it'll hurt a little and then heal.

So I think the key to a lot of those "Oh my god, what if...?" moments is to go ahead and answer the question.

Fenceless Confusion

Every Monday and Wednesday when I go to my class in the science building, I park in a corner lot, walk up to and then through the North Classroom building, and when I emerge, walk around the gigantic fence that has surrounded the Science building for probably about two years while they've been working on the new building.

Monday night I came out of North Classroom and...what? Was I in the wrong place? Everything looked different.

Oh. The fence was gone. Amazing! Now you can walk right up to the new science building:


Later, when I came out of class, I headed down the sidewalk, and automatically turned left where the fence has a corner, except...there was still no fence, and I was automatically dodging nothing.

So I walked over to the new building instead and looked through the windows. They're not quite done finishing the inside, and it's not open yet (to judge by all the signs direly warning you not to go in), but it looks pretty great. One thing I was happy to see was that at least the classrooms I could look in had tables with simple chairs rather than one-piece desks. I find that type of arrangement far more comfortable.

Hopefully at least one of my three classes next semester will be in there so I get to experience it before I graduate. It's the least I deserve for having to walk around that fence for years.

Severely Awesome Response

On this Ask MeFi thread, a woman asks

Sometimes my partner says silly things. I forgive him because, after all, we do all have our faults--and mine is leniency. However, sometimes he makes statements like,

"I'm glad I didn't go; if I had gone, and seen hipsters running amok, I would have cried severely."

He uses 'severely' in that fashion ALL THE TIME. I finally took issue with his grammar, & he said that it can mean "to a great degree, or requiring great effort," in which case I put up with him (severely), but that's hardly a good explanation.

I understand that 'severely' is an adverb modifying 'crying.' I still think he's ENTIRELY INCORRECT. MeFi, help a girl out?

When asked later whether she's objecting to the grammar (e.g., the word order or something) of the statement or the semantics, she clarifies that it's the semantics.

I loved this response by ROU_Xenophobe:

I feel that, semantically, one cannot cry severely.

If it's just a light slang usage of a perfectly cromulent word, sure you can.

I mean, say that he was using "thermonuclear" instead. As in, "I would have cried, thermonuclear" or "You hurt me thermonuclear bad." Clearly, he cannot actually cry thermonuclear because his body could not survive converting matter to energy inside his tear ducts.

Or say that he used "level 10" instead. I would have cried level 10. This also can't be, because crying does not involve discrete levels.

Or say he used "filthy." As in "I would have cried filthy" or "I kill you filthy, Vorga." Obviously you can't cry filthy, unless you consider tears to be filth, which is verging on the pathological.

But in all cases you know what he means. He sounds a bit like... dude... but in most circumstances that's not a big deal. If he can speak proper when he needs to, and if he'll stop if this is causing you actual no-shit consternation, all is well.

I love those examples so hard.

Wavelets

When I looked up my professor for next semester's "Senior Math Seminar" (the 1-hour class I have on Fridays), I saw that he had written a book about wavelets. And apparently that is going to be the topic for our class, and we'll be using his book.

I don't think I've had a textbook written by my actual professor since...well, Rice. And I only know I had one at Rice because of Jason's famous comment (on a course evaluation he happened to be present for) that the lectures were much better than the textbook (which had been written by the professor). And come to think of it, that may have been Sally's class, so I'm not sure I've ever had this particular experience.

Next Semester

I was only going to have two classes (one 3-hour and one 1-hour) next semester, but when I checked the schedule, I saw an opportunity to take Abstract Algebra, which, along with Differential Equations, is one of the most noticeable courses missing from my schooling so far. As of this moment, I am registered for the following classes next semester:

Advanced Calc II (Mon/Wed 7:00-8:15 PM)
This is the continuation of my current class, but unlike my current one, it is 3 hours instead of 4, which will be easier on the schedule.

Abstract Algebra I (Tue/Thu 5:30-6:45 PM)
Yes, once again I will be going to school every day of the week. But wait! There's more!

Senior Mathematics Seminar (Fri 10:00-10:50 AM)
This is a one-hour course required of all math majors. I've looked up past syllabi, and it looks like the professor just picks a topic of some kind and you do projects or something like that. It's convenient that this only meets once, for an hour, and I'm glad it's not on the same day as any of my other classes, but of course it's a bit inconvenient to go to campus in the middle of my morning at work.

Anyway, it should be a fun semester. It's technically 1 less credit hour than I have this semester, despite that I'm taking one more class. We'll see. And, of course, I am supposed to graduate at the end.

"Bayesian" Approach to Lie Detection

You may have heard about this guy, Richard Heene, who played a hoax in which he and his wife claimed that his young son Falcon had stowed aboard a balloon. It was quite the national news item, with TV stations playing footage of this saucer-shaped craft as it flew around, while various rescue crews tried to figure out what to do. It turned out that Falcon had been hiding in the garage, and the next day, Falcon spilled on TV that his dad had said they were "doing it for a show," and now it is generally known that it was a hoax.

Slate has an article today about why we couldn't tell that Heene was lying. It is an interesting look into how we try (and often fail) to detect emotional falsity, for instance judging the Heene family's fear for their son's life, versus how we detect lies. People are pretty bad at both, generally speaking, though of course some people are better liars than others.

My best way of guessing about lies is to take what I think of as a Bayesian approach to it.

What I think of as the default, non-Bayesian approach to lie detection is to watch and listen to the person making the statement and try to evaluate directly whether they are lying. Often we're not even listening for lies, so a ton of lies can pass completely unnoticed, but if the truth is important and you're not sure, you might be paying close attention to the teller.

I find it more useful to reason about the entire situation. For instance, say you are selling a car, and a man who lives in Kenya contacts you, wanting to buy it. He will send you a money order for the amount plus $600 and he needs you to pay the $600 in cash to his man stateside so that the car may be shipped. He sounds perfectly professional and nice and you have not heard of this particular scam before.

Still, you can ask yourself, "Is it likely that a person in Kenya wants to buy my specific car? Why would that be? Don't they have cars in Kenya?" and then proceed to this question: "Is it more likely that a perfectly nice gentleman in Kenya wants my car, and needs to handle the finances is this way, or is it more likely that someone is trying to scam me in some way?"

I was dating a guy once, and things weren't really going anywhere, but I was having a good time. He disappeared for a bit and then told me that he wasn't going to continue seeing me because he didn't want a commitment. I was kind of boggled because I hadn't said anything about a commitment, had shown no signs of wanting one, and was just having fun. Yet he seemed sincere about it, so it was kind of confusing.

Then I realized it was far, far more likely that he just didn't want to see me, for whatever reason ("just not that into me"), and he made up the commitment thing as a plausible and not hurtful thing to say about it.

I guess one way to go about this is to consider the alternative scenario and what it would look like. One time two friends of mine, a couple, were arguing. The woman had forgotten to get a (psychiatric, I think) prescription refilled and the man was insisting that she do so, and saying he would go do it for her, and she was being fairly belligerent in return. At some point, it crossed my mind that, had she refilled the prescription yet not wanted to take the pills, it would explain her behavior quite well. And indeed it turned out that she was lying and the pills were in her jacket pocket.

Since we understand the world, we can often imagine what scenarios might be in place around us, and starting there and moving down to people's behavior is, I think, a more accurate way to detect lies than starting from the behavior and trying to reason "up" to the scenario.

School Motivation

I've been thinking about all of my years of schooling and the difference between times that I excelled and times that I failed or just got by. There are some of each.

In grade school, there were structural changes that made a difference for me. In elementary school, I generally had good grades just because I was academically ahead of my peers and homework wasn't such a big deal. I also did well in the gifted program at my main elementary school, where we had a certain number of tasks to complete independently each week, the completion of each of which led to a box being hilited in your folder. If you finished early you could play (educational) games on Friday, and I loved my teacher very much.

Middle school was more of a struggle, because suddenly things like keeping your folder in a certain order were important, and homework began to be more of the grade, which was calculated in a more fixed fashion than before. Some teachers were more flexible than others if you got high test grades but didn't turn in a lot of homework.

And yet, in 7th grade, the middle year of middle school, I got all A's the entire year. I remember doing the same things I had always done - leaving assignments until the last minute if I did them at all, being smart but disorganized, etc. - but I think what made the difference was that I really loved some of my teachers. I had a giant crush on my earth science teacher (Mr. Garrett) and my English teacher (Mrs. Agrons) was the bomb as well. (She taught us to write essays. I remember a whole board filled with statements about Rikki Tikki Tavi that were and were not thesis statements, e.g., "Rikki Tikki Tavi is a weasel" - not a thesis statement - vs. "Rikki Tikki Tavi succeeds through cleverness.")

So, clearly I do better when grading is flexible, and I do better when I love (and thus want to please) my teachers. But those things are pretty much beyond my control, and in particular, the flexibility of grading becomes much less of an issue once courses become hard enough that I actually need to do the homework in order to succeed. (At that point, graded homework becomes a help to me, since it's slightly harder for me to do otherwise.)

If I go to grad school, there probably will not be classes where I can ace the tests while blowing off the homework.

But, more interesting than this stuff is something that I have noticed only recently, though I think it is a pattern of long standing. When I specifically desire to excel at a class, then I do; when I view the class as something to get through, or something I don't want to fail, then I tend to do poorly. I think most of the classes I've taken in my life I've viewed in the latter way - as obligations, basically, or something I just needed to survive - and so I haven't done very well at them.

When I find a class very tricky or puzzling (like Logic at Rice), or I love the teacher (like Mrs. Agrons), or I want to defeat my classmates, or somehow or other I really want to do well, then I generally perform at or near my best. By contrast, when I don't really see an upside to the class, I have a hard time even meeting the minimum standards.

I am not sure how much this generalizes to life, but it strikes me that one of my problems at work is that I don't really see much point to excelling at my job. When I apply a very moderate amount of effort, people are very impressed (probably because most people who have my position are not very talented, or they'd be engineers or something instead). There are not spot bonuses or anything like that. Mostly they pay me a fixed amount of money as long as I do whatever is basically required of me.

I am led to understand that some people have an internal drive to excel (or to be professional or work hard), but I seem to be much more likely to respond to external incentives like grades or professorial approval. (I do respond to some internal things, like feeling brilliant - which is part of why I can excel at math - but doing a good job at work doesn't usually make me feel brilliant since most of my job is pretty easy.)

This is probably why I like school so much, just in general. You get constant feedback (I love getting graded things back) and there are always new people to please and impress.

I wonder if I can find a way to internalize the desire for excellence such that it applies to more situations and is less reliant on external motivators.

Tokyo Joe's

Recently, Tokyo Joe's has changed the way they handle vegetables. It used to be that you could either get broccoli or a vegetable mix with your bowl. Now you can get up to five vegetables from a long list, or you can get a mix of popular vegetables. It's so, so nice.

I had lunch there today - brown rice with dark meat chicken (they also have steak, white meat chicken, salmon, tofu, etc.), my five vegetables (broccoli, red peppers, bok choy, snow peas, and red onions), and curry sauce. The curry sauce is good there.

The real downside to Tokyo Joe's is realizing that you could make their food at home very cheaply, if you wanted several portions of the same kind. There isn't really that value added of "I could never make this" or "This is too much trouble to cook" (e.g., how I am never going to make a chile relleno). But for convenience, cost, and health, it's hard to beat paying $7 for the bowl I just described plus a tall cup of very good iced tea.

Self-Motivation

In therapy lately, we've been talking (of course) about my unusual degree of inability to make myself do things. Of course, everyone struggles with self-discipline to some extent - working is hard - but I seem to struggle less successfully than most.

I got frustrated with my therapist because he was doing more talking than listening and saying things about setting goals for yourself and finding out what is holding me back from performing and blah de blah, and I felt like a person missing a leg being asked why they don't want to walk and whether they've tried building up to it with just a few steps at a time. Like, no, you don't fucking get it, that is not what is going on here.

I specifically do not believe (though I would love to find out that I'm wrong) that something is blocking or preventing me from applying myself. I think I have an actual deficit of whatever it is that people use to get things done.

Way to take responsibility there, Tam.

One of the ways people motivate themselves is by setting small (at first) goals, and then building on their success in meeting those goals. But I have set goals so many times, and not met them so unbelievably many times, that I don't even believe myself when I say I'm going to do something. I mean, I can tell myself I am going to do something that takes 5 minutes later that very same day, and I know all along that I'm probably not going to actually do it.

How many times does a person have to let you down before you stop thinking they might step up? When the person is yourself, it seems the answer is "quite a few times," but those quite a few times have long since passed by. (I'm tired tonight, so this is a slightly more negative view than I usually have, but not far off.)

However, obviously I do in fact do some things. I go to work every day, get some work done on most of those days, and I keep up with my classes well enough to ace most of them, which does require some work. I attend classes more than half the time. I sometimes clean the kitchen or wash the towels. I have clean laundry to wear every day. I am able to present as a functional person.

There are strategies I use to get myself to do things. First, I'll tell you what I don't use:

• Ongoing To-Do Lists: I often make one for the next few hours, but never one for days from now, because then I will just avoid even looking at or thinking about the list or anything on it.
• Small Goals: Discussed above.
• Rewards: I know I won't honor this type of promise to myself, so there's no point. I'll get the reward later whether I did the thing to earn it or not, if I want it. This includes very short-term rewards like "I'll work for 30 minutes, then relax for 10 minutes," because I won't follow through on those either.
• Punishments/Consequences: I definitely won't honor these.

So, let's look at some specific things I do and see if there is a pattern. For work (at my job) or homework, I turn on music when I am working, and only very rarely at other times. (I also listen to music in my car, but that's about it.) Sometimes I will start working because the idea of putting on some music is appealing. The appeal of the music kind of bleeds into the appeal of the work.

At home, for homework, I almost always sit down with a glass of iced tea, which I brew first. I keep regular and decaf tea bags for this purpose. The brewing time lets me goof off while knowing I'll soon start working. I can let the tea go for a while but eventually I need to go pour it over the ice. I really enjoy the tea and, although I sometimes drink it when I'm not doing homework, I mostly have it with homework (to the point that I feel cheated when I try to do homework with just a glass of water). Tea time = homework time.

Some household tasks are naturally appealing to me, like washing the towels. I don't have too much trouble at least getting that started, though sometimes I fail to ever fold them afterwards. Other household tasks, like doing the dishes, I just push myself to do in whatever way I can. "This will only take 5 minutes," I tell myself. "All you have to do is put them in the dishwasher. Look, it's 9:17. By 9:25 you'll be done. And Ed will be really happy." Sometimes that works.

There are two things that need to happen in order for me to do some work. First, I have to actually decide to do it. That may seem really basic, but sometimes I can feel myself, in my mind, simply refusing to do something, even a task at my actual job, where they pay me to do things I don't necessarily want to do. Dishes is a hard task to decide to do. Sending a letter or calling someone on the phone is hard. Homework, by contrast, is a very, very easy task to decide to do - there is almost no barrier there at all. I am always open to doing homework.

The second thing that needs to happen is for me to get around to actually starting to do the task. That is also often a challenge. Sometimes I start and then drift off to doing something else, if I'm not careful. Sometimes hours go by while I just don't quite get started. (Brewing tea helps this problem with homework, since it puts a soft, flexible time limit on goofing off. Of course, sometimes it takes me a while to get up and actually start the tea brewing.)

Then, of course, you have to stick with the task. Some tasks, like washing dishes, are so short and different from the rest of life that they're easy to stick with. Working at my job is the hardest task to stick with, because I'm almost always using the computer, which makes it very easy to slide over to looking at things on the Internet instead. Homework is intermediate, because I take the keyboard off my desk and usually only surf the net intermittently when I need a little break; I can't get too absorbed with all the papers between me and the monitor. Also there is my glass of tea looking at me, saying, "Don't finish me too soon - you still have a lot of homework left."

So I guess I kind of try to condition myself to work by associating work with other pleasant things - not future rewards, but things going on at the same time, like iced tea and music. (I sometimes use hot herbal tea at work in a similar way, but it's not as rewarding as iced tea, which is too much trouble to make at work.) I am like a bad little mule, led on with a lot of kind pats and the simple conditioning of, "OK, here's your lead, must be time to get on with things."

Refraining from doing things is another story entirely, and I have almost no success whatsoever with that.* Occasionally I can substitute one pleasure for another ("instead of going out to eat, I could read this book, that sounds great") but otherwise it's hard to associate refraining with an immediate pleasure.

Despite my gloomy outlook here, it should be pointed out that I am far, far better at this than I used to be, so it's possible I will continue to improve over time.

(* "No success" relative to most people, that is. Obviously I refrain from doing quite a lot of things all the time.)

Obnoxious

I was talking to Ed tonight about that famous psychology experiment where one person (the subject) is put in a room with a bunch of confederates and a researcher asks the group a question with an obvious answer and has them vote, at which point all of the confederates vote the wrong way, and the subject usually follows along despite the group being obviously wrong. (Sorry I don't have the details, but it's a widely known thing.)

My natural tendency is to scoff at the subject, but in truth, that's probably the right thing to do. It seems to me that that probability of a bunch of other people being wrong about something completely obvious is lower than the probability that you have misunderstood the question somehow. (And by "obvious" I don't mean like "It is obvious that rent controls only serve to make housing more scarce, thus exacerbating the underlying problem," but more like, "It is obvious that this line is longer than the other.")

Nevertheless, Ed and I are both the kind of obnoxious jerks who think we are right despite this kind of external evidence. When my mother taught me to write the numbers, right before Kindergarten, I argued with her that she was writing the 5 and the 6 backwards. I argued with my 5th grade teacher that "colonel" - one of our spelling words of the week - was clearly incorrect. I do not hesitate to argue with my professors now about factual issues within their areas of expertise, although I am not nearly as obnoxious now as I was as a kid. (To be clear, I don't argue with professors about matters of opinion unless it seems wanted.)

I am almost always willing to admit that I may be wrong, but I always actually believe that I'm right despite this theoretical possibility, and even in situations where the odds are against it. And I don't really mind being shown how I am wrong. (It occurs to me that this is similar to how I am with games - tending to be a bad winner, but nearly always a good loser.)

At any rate, this is probably why, when our linear algebra professor realized there was a lot of confusion over the question of whether the column space of an mxn matrix A was a subspace of Rⁿ or R^m, and had us vote, I was willing to raise my hand for m even though 17 or 18 of my 19 classmates had voted for n. (Reminder, for those who can be reminded: the column space of a matrix A is all of the vectors that can be formed of linear combinations of the columns of A, or in other words, all of the vectors b satisfying the equation Ax = b for some x.)

I thought that I must be wrong, because even though my classmates are often confused, that was an overwhelming majority against me, and yet, what I had on my paper sure made it look like m, so that's what I went with. I turned out to be correct.

She next had us vote on the same question pertaining to the nullspace of A. (The nullspace is the vectors x satisfying Ax = 0.) This vote didn't go against me quite as strongly - it was more like 17-3. But I was again correct.

(I should note that I was perfectly capable of getting either question wrong - this is not a question where the answer was obvious to me at the time, and this is the type of thing I am often wrong about, usually because I have made some simple mental error. This post is about my psychology, not about my besting everyone in feats of dimensionality.)

I suspect that I am different from most people in this non-majority-joining respect because I am not very bothered by being mistaken, and I see the possibility of being right (i.e., winning) as having a pretty big payoff. And although holding the minority position makes it more likely that I'm mistaken, it also raises the payoff for being right. (After all, being the only one who is right is much more awesome than being right in a crowd.) Also I am just unwilling to let things go until I see why I am wrong.

When I Grow Up

I've been considering a lot of career changes and such lately, and one idea that has arisen, that I'd previously dismissed but am now considering more closely, is going to graduate school in math full time, probably hoping to get a PhD, but at least for a Master's. There are several reasons that this appeals to me, among which are

• I want to study math full time, and
• I'm tired of having a job where people are surprised that I'm smart.

There are a lot of questions about the grad school idea. Can I get accepted somewhere with funding? (Based on my research, I believe the answer is yes.) Can I be a successful full-time student? (I've never managed it before, but I haven't tried since ~ 1996, and I have demonstrably changed as a student since then.) Can I live on a greatly reduced income, or will I end up in horrible debt because I can't rein in my spending?

If I do go with the full-time grad school idea, there are three general paths I could take.

Pure Math

This is the most academically appealing path. What I seem to like best in math is stuff that involves a small number of axioms and seeing what comes of them. (This could change; I don't have a lot of maturity in math yet.) I think I would enjoy (as much as one ever does) writing a master's thesis or doctoral dissertation in pure math, once I got far enough along for such a thing to become possible. (My paper on Laguerre planes last semester was as close as I've gotten, and that was thrilling and I found I could easily work on it for hours and hours.)

Having this degree would broaden my career prospects in the regular world to a degree, and enable me to try to get a job teaching at a college or, if I want to continue doing research, at a university. Academic jobs (other than horrible, low-paid adjunct positions) are often difficult to come by, however, even if you want "only" a community college job.

Applied Math

Something in applied math - perhaps Operations Research - would be the most career-applicable path. Applied math interests me somewhat less than math with no conceivable application, but is still very interesting, and, though the thought doesn't thrill me, it seems very possible to imagine writing a paper in an applied math area. As far as academic careers go, a higher applied math degree would be about the same (as best I can tell) as a higher pure math degree. They have professors in both, of course, at universities, and for lower-level teaching-oriented positions I doubt it makes any difference.

Math Education

Sally joked the other day that I could go to her interim school and get my PhD there, where they have a math education doctoral program that seems very research-oriented. I find this a very intriguing option (in general, not just at that school) because it has such strong highs and lows to it, in my view.

A PhD in math education is not at all helpful for a regular job in industry, as best I can guess, even though you do learn a lot of advanced math in the course of getting one. It pretty much limits you to the education field, if you want your degree to really count for something. It also does not feel as prestigious to me, which I admit is a consideration. ("I have a PhD in math" just sounds so much better to me than "I have a PhD in math education.")

Yet the topic of math education interests me greatly. I feel I would happily read any number of books or papers about it.

Yet I'm not sure the idea of doing original research in math education sounds that great. It doesn't really sound much better to me than the idea of doing original research in say, psychology, and I wouldn't go do that, and not just because my background for it is wrong. I like the research and want it done, but I'm not sure I should be the one doing it.

If I were to get a PhD in math education, the kinds of jobs that would be available to me would be the usual academic jobs (where I could teach either math - using my math education focus to do so more effectively, perhaps, especially if I specialized in post-secondary ed - or math education itself, like to future teachers), or I could probably get some kind of job in the public school system doing something like program or curriculum development, etc. (I'm not sure exactly what jobs exist, but there are surely jobs along those lines.)

A PhD in math education, or really any PhD, would probably make it a bit harder for me to get a high school teaching job, at least in a public school, since I would have the deadly combination of no teaching experience at that level + being required to be paid more than someone with a Bachelor's. But maybe a fancy private school would like to hire me, and I'd certainly have teaching experience in general, since teaching college courses generally happens as part of a program like that. (It's explicitly required at Sally's interim school, and would be part of TAing in any program.)

One question central to all of these considerations is really what kind of career I want to have, or at least, what kind of career I want to try next. I worry that I am not really suited, psychologically, to the general type of work I do now - i.e., working in an office. I seem to be an inveterate slacker in those types of jobs. And while having a more interesting and challenging office job - as I might after getting a higher degree - might help, it might also involve elements that I find even more impossible than what I face now, like project management.

I have never tried teaching, but I think it might be a kind of job that I would be good in. I enjoy teaching people things quite a lot, certainly. It seems like it would be very engaging and, while you can slack off as a teacher, it wouldn't have the same kind of slippery-slacker-slope thing going on, I think. I also fundamentally really like school - almost everything about school - and it would be nice to be a part of that kind of system.

Ultimately, then, I really don't know what I want to do and how I want to go about it, at all. My current plan is to apply to various schools for Fall 2010 and try to decide what to do between applying and getting the results of those applications.

Stupid

If I were asked to describe myself, and if I were being honest at all, the first word out of my mouth would be "smart." It is reliably the first word to come to mind about myself, and carries more weight than any other I can think of. Being smart is central to my identity.

I am also weirdly, inexplicably defensive about being smart. When I take a class, I don't feel comfortable until I am sure that the professor has figured out that I am smart. It is the first thing I want new coworkers (people I work for, at least) to know about me. The idea that anyone - friends, bosses, or teachers - might think I am stupid is very concerning to me. I think this is odd given that it's not that likely that someone will conclude that I'm stupid. But I am very afraid of appearing stupid, probably because being smart is such an important part of my identity.

When I was a little kid, and it started to become apparent that I was "gifted," this delighted my mother. In addition to the usual delight people take in their children's positive attributes, I think there are two things about my mother that made this so. First, she's an intellectual snob, valuing intelligence and learning over most other things, and second, she herself never felt like one of the smart kids. (One of the things that attracted her to my father was that he seemed so smart.)

In therapy the other day, I was talking about what my therapist characterizes as my mother's negative attitudes towards a lot of things - for instance, the way she was so clear to me that my 2nd grade teacher's insistence on my copying my spelling words 3 times each, despite that they were very basic words I knew how to spell a hundred times over, was stupid.

Later, I was thinking about my mom's attitudes towards a lot of things, and the word "stupid" came up in my mind over and over. She especially had a lot of contempt for my dad's family, and nearly everything I told her about them was dismissed as stupid. (To give an example, I once told her that they had those rough-textured flower-shaped stickers in their bathtubs - the kind that are supposed to keep you from slipping - and she told me those were stupid.)

My grandmother once chided me, saying, "It's more important to be nice than to be smart." When I told my mom about this (I had thought it stupid), she was absolutely indignant about my having been told such a stupid and insulting thing. (Let's try to be fair, and note that my mother was probably not objecting to the idea itself, but more to the fact of its being used to chide me, given that it suggests that my intelligence was not as important as I thought, and that I wasn't as nice as I ought to be.)

Stupid. Stupid stupid stupid. Everything bad is bad because it is stupid.

I was also talking in therapy about my early years of school. Was I popular? I was not. Most other kids didn't like me, as best I can recall. Why might that be?

I can't remember the details of my interactions with other kids in elementary school - the ones who weren't my (few) friends, at least. But I do remember that the other kids were mostly stupid. They couldn't read out loud without pausing at the ends of the lines. They couldn't spell words. (One time in 3rd grade, one of my classmates - a friend, actually - asked me how to spell "I'll" and I told him "a-i-s-l-e" because it hadn't even occurred to me that someone wouldn't know how to spell "I'll.")

It must be hard to like a weird kid who thinks you're stupid.

I felt bad, in therapy, reporting that I thought my classmates were stupid. I felt like an adult picking on little kids. I wouldn't describe a 3rd grader as "stupid" now. I tried to make that clear to my therapist.

Intelligence and ability are intrinsically good things. Most people would choose to be more capable in any way they could - smarter, faster, fitter, stretchier, more charming, more dextrous, you name it. But the value system that equates stupid to bad is wrong. (In my head, I say it is stupid. Out loud, I'm saying it's wrong, unethical.)

Lileks once wrote that his young daughter said of Spongebob Squarepants's friend Patrick something like, "He's kind of dumb...but he has a good heart." Lileks was happy that she put it that way, and not the other way around. And I agree, but as a kid I would never have said something like that.

I think there are really two problems here. One is being taught that stupidity is the ultimate form of bad. The other, perhaps worse, is being taught to hold others in contempt. (Contempt comes naturally enough in adolescence; it doesn't need to be taught to toddlers.)

I wish I'd been raised with different values, because I find that adopting them as an adult is possible but difficult.

Christmas in October

This morning when I woke up, everything was covered in a light layer of snow. It looks oddly a bit like a spring snow, since everything is lush and green, but the trees are actually too lush and green for spring, so...well, an early fall snow after a quite wet summer, I suppose.

Ed and I spent essentially all day together. We had lunch out, and on the way home stopped to rent a movie. I mentioned my desire to buy a Wii (which I shopped for but didn't buy the other night, unsure whether I really wanted a Wii more than I wanted the $300 I would spend, including games), and he said, "Want to get one on the way home?" So we went to Best Buy, where I got the console and Mario Galaxy, and he got an extra set of controllers and the Metroid Prime Trilogy (for us to play together).

At home, he checked out some things for work and I hooked up the Wii. The smell of the plastic, computer parts, styrofoam wrapping paper, etc., reminded me of Christmas. And it was a Christmasy day outside, with snow still visible on the trees. I said this to Ed, and he said the day really felt like Christmas all the way around, and somehow his saying it made it so. The rest of the evening really felt like we were living outside of normal time, the way that holidays sometimes do. We didn't give any thought to shopping or chores or homework or any weekend tasks, just played with our new toy. And watched The Triplets of Belleville.

Buying the Wii was also a kind of budgetary triumph. I currently put some money aside every month that I'm officially allowed to spend however I want, but that I have to conserve one month of to cover budgetary excesses. After a few months of doing this, I had more than enough to buy the Wii and the game without hitting this month's savings at all. It's the first thing I've bought with that money. I feel really good that the plan worked out, then.

Ed kicked my ass (unsurprisingly) when we played Metroid Prime - he had played the game before and I had not, and I had a lot of trouble figuring out how to do anything properly while he blew me up a lot of times. But later I got to play the regular game myself and start picking up some skills. (But damn, kids, that game is kind of hard.) After Ed went to bed, I started playing Mario Galaxy.

It was really fun and great to take a day off from life and spend it playing with another kid. (Plus we're having a sleepover!)

Facebook Poke Arrest

I found this story really funny to read, even though it seems perfectly reasonable that you could be arrested for this, actually:

Shannon D. Jackson, 36, was arrested Friday, Sept. 25 for allegedly violating an order of protection.

According to the affidavit filed in Sumner County General Sessions Court, Jackson is accused of using the “poke” option on Facebook to contact a Hendersonville woman, thus violating the terms of the order of protection, which stipulates “no telephoning, contacting or otherwise communicating with the petitioner.”

Poking is a feature unique to Facebook that conveys no other message but informing a user they have been “poked” by another user.

Jackson declined to comment Thursday afternoon.

Hendersonville police have made copies of the page in which the alleged victim is shown to be “poked,” according to the affidavit.

I think it's the repetition of the scare-quoted "poked" that gets funny.

Bait

This month's Mathematics Magazine (Vol 82, No 4) has an article featuring

• a familiar university
• game theory
• probability distribution functions

"To Buy or Not to Buy: The Screamin' Demon Ticket Game" by Frederick Chen and Alexander Kays

(This journal assumes undergraduate math knowledge.)

Unfortunate Political Positions

I have finally been convinced, by much argumentation here and there (mostly by Megan McArdle, if I'm honest), that health care reform as currently envisioned is a bad idea. Unfortunately, I also believe that the current system is untenable, and the only kinds of solutions I can imagine are not currently under consideration, and don't seem likely to become so.

I remain reluctantly in favor of the passage of a health care form bill along current lines.

Weird Coincidence in the House of Nerds

Sunday night...

Me: I have a math question for you. I was just thinking about this.

Ed: Sure, OK.

Me: What's so special about linearity? Why is that so salient, or different, or important, or whatever?

Ed: You know, I've been working on that particular question all day.

Me: ?!

Ed: Yeah.

[Discussion ensues.]

Motivation

In therapy yesterday, Gary (my therapist) and I talked about some of the issues I have with work and motivation. He observed that I am very dependent on the opinions of others, have an external locus of control (duh), and also, in an insight that was a bit new to me, that I depend on a lot of external pressure to get things done. I would take that last bit a step further and say that I am sort of fear-dependent.

Most of that is all sort of "well-known facts about Tam."

But he asked me to think this week about what my motivation to change is. If I want to change the patterns of behavior that I exhibit, what is my motivation for that?

And for me this tied into something I've thought about a lot in the past year or so in reference to my childhood, which also relates to what I wrote about, and that is that I was never (as far as I can recall) encouraged, as a child, and (hence?) usually fail to, as an adult, consider positive motivations for anything.

That's not entirely true. I did hear, "You're so smart, you can do anything you want, if you just apply yourself," which of course is a message with its own downsides. But I never heard anything like, "It's good to be an honest person," or, "You should take pride in your work" (or appropriate formulations of those ideas).

What I heard was, "You shouldn't lie. Don't be a liar. People won't ever believe you." Or, "If you don't do your homework you won't get into college." Or, "Nobody is going to hire you looking like that" (with reference to my weight).

This doesn't sound like a big deal, but what I find is that, as an adult, I rarely ever consider positive motivations, and when I do it can be kind of a revelation. I rarely think something like, "I want to get a lot of good work done today" in reference to my job; I generally think something more like, "If I don't get caught up on a lot of this work today I am going to be doomed."

The problem with thinking only in terms of bad consequences is that it then becomes an issue of whether those bad consequences are fearful enough to motivate you. You're kind of continually trying to scare yourself into doing things. Yet, I guess bad consequences haven't actually happened to me so very much (I did get into college; I have worked for over eleven straight years without ever being laid off or fired; I flunked out of Rice but this did not doom me to work at McDonald's), and I'm not naturally all that fearful about things like this, perhaps because of all the empty threats over the course of my life.

I am curious to know how typical or atypical my fear-based motivations are. Like, when Sally thinks about sitting down to study, is she thinking mostly something like, "I want to learn this material and do well in the course," or something more like, "If I don't do this I will flunk my exam and never get into a PhD program"? (Of course, a lot of actions become habitual and don't have thoughts with them. I don't have either fear-based or positive motivations for showering every day; I just do it. Maybe studying is like that for Sally, but surely not everything is.)

I do seem, these days, to have positive motivations towards school. I want to absolutely excel at my courses, not just avoid failing or doing poorly. But the question of whether I can excel in hard classes definitely helps motivate me, and I'm unattracted to studying topics I don't feel that fear about.

So, I don't really know. What is my motivation for wanting to be better?

Sally's Christmas Present

The Styroclock.

More About Everyday Math

The other night, I went and read pages and pages of a forum in which parents talked about their bad experiences with the Everyday Math curriculum. Remarks like this I had expected:

We got my 9 year old's progress report on Friday and it wasn't good. He scored in the 99th percentile in math all the years in Catholic school, but is slipping in public. He's great with multiplication, division, measurement, graphs, etc., but neither he nor I are any good at "strategies." He can sit there and perfectly well multiply three or four digit numbers, but he doesn't have to do that. He has to create a LATTICE with the numbers. On a diagonal. With lines. And then estimate. Why can't he multiply? Why isn't the RIGHT answer good enough? It's been fine up until this unit, they've dont the usual stuff and even more geometry than his sisters did, but now he has to strategize rather than calculate, and he has to write sentences and explanations. 2+2=4 because it does, that's why!!!!!
Arrrrghhhhhh!!!!!!!!!!!!!!!!!!!!!!

A lot of parents expressed frustration on behalf of their kids for having to estimate things they could easily calculate, and were perplexed by their young children being assigned problems that they, the parents, couldn't solve.

I'd be surprised if my 2nd grader came home with a math problem I couldn't solve, definitely, but it also occurs to me that there might be problems that seem quite advanced but that a 2nd grader could solve by methods that aren't the ones I would use. For instance, there might be a problem I would algebra for, but that would also succumb to other methods that I wouldn't readily think of (since I have mad algebra skillz) (relative to a 7-year-old).

I don't want to dismiss these complaints out of hand, at all. Also, when I say I would like this program, I say that from my current perspective. Perhaps as a child I valued much more highly than now the ability to get a definitive right answer to a problem using a method that worked every time. But I did find one class of anti-everyday-math posts interesting. Here's an example:

We had a bad experience with Everyday Math also in 1st/2nd grade (private school). The teacher was excellent, specially trained and very enthusiastic - can't blame her. Fortunately, she was also very cooperative. When it became apparent that D simply was not going to be able to manage even rudimentary skills such as two column addition, I intervened. I was able to convince the teacher that D could master addition and get correct answers (!) with me teaching her my old fashioned method. Teacher had no difficulty with me "homeschooling" that skill (and others). She did indicate that she would continue working with D the EDM way - I think she eventually gave up! D moved to a different school after 3rd grade. They didn't use EDM and she was a solid B - B+ math student through high school. Math concepts don't seem to come intuitively, but explanation, demonstration and practice in a more traditional format served her well.

And another, perhaps clearer:

I spent 3 years campaigning against the "new" math in our district. When my D had Mathematics in Context in 6th grade, I met with the teacher, principal, district math consultant, assistant superintendent, superintendent, and school board ... in that order ... giving them specific examples of its weaknesses and citing info from experts to support my concerns. I was blown off every single time. I finally put my kids in private school, even though it was a major financial strain. I just didn't see any other way to do it. It was just too important.

My D had multiplied & divided fractions in elementary school. In 6th grade, she was being asked to compare fractions by putting water in a tuna can & pouring it into a soup can. It was so 1st grade!

D was stuck with it for 3 years, and when I put her in private high school, she scored 99th percentile on the math portion of the entrance exam...

or

Its funny, I was just thinking about Everyday Math a couple of weeks ago. S1 started it in 2nd grade, when it was first brought to our district (he's a college soph now)... I thought it was ridiculous then. He was never a spectacular math student, and I couldn't understand how this was better than memorization. S2 began it the following year, when he was in kindergarten. I had to admit, that seeing the program from day 1 -- it made more sense than starting it in 2nd grade. But S2 (who turned out to be a very good math student) had no troubles with it. He zipped right through the worksheets. A few weeks ago, I was doing hw with my 1st grade nephew, when he pulled out his math assignmnet, and there was an Everyday Math worksheet.... I swear! I began to get heart palpitations at the sight of the In/Out charts! I just don't agree that any of those manipulatives help them with higher level math. You just can't beat memorizing the multiplication tables and basic rules of algebra. And I am what they'd call mathematically-challenged.

or

My D, now a college freshman, went to a small private K-8 school which initially had a particularly week math program. When she was in 3rd grade, they "upgraded" to Everyday Math. At the time, I was going to graduate school to be certified to teach HS math, and I didn't see much improvement by going to Everyday Math. Fortunately, she had a great math teacher in middle school, where they used something called Transition Math for pre-algebra and algebra. The biggest frustration of the math teacher was that the students didn't come to her with "math sense"...they hadn't developed a good "feel" for the right answer, weren't comfortable with simple mental math, and the like. Fortunately my daughter found she liked math and was good in it in high school, but it was the only subject where she ever got less than an A.

Now there are a lot of different kinds of posts in that thread, but there are intermittent ones like this that seem to argue something like the following: my kid struggled with Everyday Math, then when he/she got into a normal math curriculum, he/she turned out to be mathematically gifted! The Everyday Math was holding him/her back!

All of which admit a different interpretation: Everyday Math was hard but made my kid good at math.

There are plenty of counterexamples in the thread - people who said their kids had Everyday Math (and the programs that come after that and are similar) and then turned out to be behind when they joined up to regular math programs. But I do wonder if some of these parents are misinterpreting what happened with their kids.

In my best math classes I spend a lot of time confused and struggling. There is really a very direct relationship between being confused and struggling and learning a lot of actual math. (Where "a lot" is defined in psychic weight, I guess - learning a lot of what is important to me to learn about math. YMMV.) This is why I've basically decided to boycott classes that teach me how to solve certain types of problems without rigor (i.e., without insisting that I understand why the solutions work).

To be any good at math, you definitely need both things - a firm, rote knowledge of algorithms and rules, on the one hand; and on the other hand, an understanding of why things work and how to explore math. I doubt any one curriculum is best for all kids. Of course, one advantage of weirdo exploration-based math is that the parents can supplement with plain old-fashioned "this is how you multiply numbers" at home. It's much harder to supplement with fancy strange problems.

The Bolzano-Weierstrass Theorem

Don't worry - I'm not going to try to explain the Bolzano-Weierstrass Theorem. I just like the name. Don't you?

Bolzano-Weierstrass.

We use it a lot. There is something really fun about using a theorem with a name like that, and it only improves with repetition.

Bolzano-Weierstrass.

Advanced Calculus was really hard last night. What's happening these days is that we suddenly have a whole bunch of theorems, each of which relies on the previous theorem, going back to the aforementioned Bolzano-Weierstrass. (Of course, Bolzano-Weierstrass has its own antecedents, but I understand them already.) The last in this series is the Intermediate Value Theorem, which I think is the first time we've had a theorem I recognize the name of from calculus. So, we're finally up to calculus. Epsilons and deltas roam freely across the land.

Yes, I am a bit loopy with exhaustion today. Why do you ask?

Anyway, what's challenging about doing a bunch of theorems in a row is that, when you barely understand one theorem, using it to prove another theorem leads to confusion. And when you do five or six theorems in a row that way it starts to suck, and you sit like a zombie in class, mechanically copying from the board.

Hopefully tonight or this weekend I can sit down and go through all of the theorems, starting with Bolzano-Weierstrass, and their proofs. By "go through" I mean "write down from scratch, making sure that I understand what the theorem is saying and every step in its proof." I think this has to take priority over the homework due next Wednesday for the sake of my understanding of the material yet to come.

Everyday Mathematics

Denver Public Schools uses the Everyday Mathematics curriculum from the University of Chicago for elementary math instruction, which apparently causes a lot of conflict and consternation. When I was describing it to Ed the other day, he pointed out that his mother, who is one of those people who rants against New Math, would consider this to be everything that is wrong with education today.

Personally, I think it sounds pretty cool.

From what I've read, the basic idea is that, when a new math operation (for instance, adding 3-digit numbers) is to be introduced, what happens is this process:

1. The kids work to develop an algorithm to accomplish the task (e.g., they are told to add two 3-digit numbers without being taught how to do that).
2. Once the kids have had a chance to develop their own methods, several different algorithms are taught.
3. One specific algorithm, out of those taught, is considered the basic one, and kids are required to learn and demonstrate that they can carry it out.

(I am probably misstating details. Please check out their website if you want more information.)

I think that, had I been taught with this type of approach in elementary school, I would have enjoyed math more and developed much greater math skills and understanding.

Of course, I turned out just fine in math anyway, and presumably math curricula to turn me into, say, Ed, are not nearly as much in demand as those to raise the performance of average or below-average kids. There is also the question of whether this system is too difficult for elementary school teachers to carry out properly, since many seem math-phobic and have poor math skills. (We can wish that weren't the case, but there's no magic way to change it.)

Something that annoys me about both Ed's mom's (hypothetical) rejection of this system and my own delight in it is that the effectiveness of the curriculum is an empirical question. The U of C website lists some empirical studies that (of course) support their program. In my alternate life I'd like to study these issues in more depth.

An Event of Unfortunate Rarity

This morning, the president of our company beeped me on the phone (not sure what you call that, when it doesn't ring but the person is just there).

"Hey Tam," she said. "Do you know the status of the Grablagher project?" This was something she left me to do while she was gone. (Project names have been changed to protect, well, me.)

"Yes," I said. "I made your changes, re-ran the economics, and I emailed you the new summaries and oneline yesterday afternoon."

"Oh. I haven't gotten that far in my email yet. I'm sorry. Thanks!"

It's nice when it goes that way, versus the much more common way where I tell the person I haven't finished their thing yet.

"Getting" Math

Tonight in linear algebra, we started talking about vector spaces. A vector space is a set of objects that you call "vectors" (whether they are traditional vectors or not does not matter), with "vector addition" and scalar "multiplication" defined however you want, and satisfying some axioms (like commutativity of the addition and having an additive identity and scalar multiplication distributing over the addition and so on).

According to our prof, this is the topic that differentiates* this course from the lower-level matrix algebra class that my school also offers.

Now, this is exactly the kind of thing I think is the most fun in math. "Here is this tiny set of rules - go see what it does." Those of you who read my Laguerre Planes paper of last semester may recall that I spent a lot of that paper discussing how various systems were, in fact, Laguerre Planes. I'm into that.

But, man, were people confused. Some of them had never had abstract algebra, or apparently even the rudiments thereof (I haven't taken the class myself), and so the idea of taking some arbitrary operation and calling it "addition" really screwed with their heads. Calling something that wasn't a vector a vector, and calling (-1, -1) "the zero vector" (as happened in one example) - all confusing.

A few semester ago - perhaps as long as two years ago - Ed told me that when you get to a certain point in math, it all starts to kind of become One Thing, and I feel like I'm reaching that point. I think what I feel is actually an illusion, but it seems like the same concepts and ideas come up over and over again in my classes. I can draw on multiple sources of knowledge for the same concepts.

When people in my classes don't "get" something that seems readily comprehensible to me, it is very hard for me not to think it's because they're stupid. Yet I know that comprehending a math concept is not completely, and probably not even mostly, about some generic intelligence. One way I can prove this to myself is to consider that I can remember being frustrated and confused by math at various points in my life, for instance many times in middle school, that I easily and completely understand now. And I have not actually gotten intrinsically smarter over time.

It feels to me like I've simply been exposed to a lot more math than some of my classmates. The kind of generic abstract algebra stuff that allows me to easily understand vector spaces comes to me from

1. Proofs (a prereq for linear algebra), where we did some abstract algebra
2. Discrete Math, where I think the idea of groups/fields came up somewhere
3. The first time I took linear algebra, where we had this exact topic, and
4. Higher Geometry II, where we did a lot of field/group stuff in the beginning

That's really quite a lot of times that I've been exposed to material that was similar enough to make this a cinch. It's no wonder I don't have any trouble with it.

(* oh no! calculus!)

Surrogates

Tonight, Ed and I saw the movie Surrogates, which is a near-future sci-fi movie in which most people stay at home in comfortable chairs while sending their more attractive robot alter egos out to live life for them. It's kind of like if Second Life were the real world. It has Bruce Willis as a cop.

It is not very like Minority Report in any particular, but the audience of people who would like one is probably near-identical to the audience of people who would like the other. It's a kind of sci-fi action movie that also has personal relationships in it.

I have found that I really like Bruce Willis in sci-fi movies. Maybe I like him in general and just don't realize it because I don't tend to watch the other kinds of movies he's in. But he's really very funny, attractive, and entertaining to watch. (They did well to choose him for this role, since he looks better when he's not prettied up - i.e., the man is more hunky than his robot double.)

I thought it was reasonably well done overall, so if you like this sort of thing, go see it or put it on your Netflix queue.