How many people are in the room during these interviews? Just you and the applicant?

Maybe the boys would be more verbal if they had a male interviewer.

There's four people total - myself and two students who are prior scholarship recipients. Admittedly, the past two times I've done this, the two students have both been female as well. Maybe that's it.

But we're so smiley and determined to set the interviewee at ease! Perhaps that is actually undermining the boys?

So the applicant is being interviewed individually, by three women at once? No surprise that female applicants are more at ease in that situation, more able to presume that you understand their situation and can identify with them, and so forth.

I mean, um, yeah, well yeah, that right.

In my discussion classes, I find that women as a group tend to be more verbal - they talk more altogether - but that individual boys (generally the high-achieving ones), tend to talk more than any one girl.*

*This impression comes after filtering out science majors who clearly don't want to be in class and adjusting slightly for ethnicity/immigrant status.

:: blushes, averts eyes ::

3: But they're terrible. Like, "What does a typical weekend look like for you?"
"Uh......I sleep in....not much....homework."
"Are you involved in any extracurricular activities?"
"Uh......church youth group.....I work.....uh......soccer...."

Church youth group wouldn't figure in to your typical weekend in any way?

I am just surprised by the post's description of the situation. In middle/high school I think there was a shared presumption that girls were smarter and better organized than boys...but this didn't go along with a greater desire to talk about oneself, quite the opposite.

8: The girls just come off as normal; able to chat about what they are looking for in a college environment, able to describe an event that shaped their academic interests, that sort of thing.

3: dead right. I mean, it's admirable that heebie is so gender-blind and all that, but c'mon.

But, you know, my school wasn't very religious and conservative.

7: It's weird to think that this is coming from specifically Christian young men, because from my experiences in Christian youth groups (which is much more than you'd think given my heathen state) you're encouraged to speak A LOT, and deeply, about your emotions and your relationship with Jesus, and it's a place where men are distinctly encouraged to talk.

I wonder if it has something to do as well with the perceived educational/class/political/religious differences between them and university professors.

Huh, maybe it is the reverse minority-effect situation, akin to how women underperform on math tests when there are few other women in the room and the stakes are high, and the material is hard. And how that effect disappears when you tell them that the test has been calibrated to be gender-neutral. Perhaps I should tell the boys that the following interview has been calibrated to be gender-neutral.

I'm guessing it's partly regional. In my classes I tend to see that very high-achieving boys talk the most fluently, then the very-high-to-low-achieving girls, and then the low-achieving boys talk the least. That is, girls seem to talk some no matter how they perceive themselves, but boys only talk if they're very verbal and confident.

As far as grades go, it also tends to fall out that way. Boys either get high As or low Cs, while the girls are all over the map, but with the most getting A-/B+s.

The hypothesis that it's about discomfort being in a room with three women sounds like a strong possibility.

To the extent that you think this is related to these kids having been brought up in very traditional gender roles, it might have something to do with differing levels of comfort with a situation where you're expected to be deferential; the girls have been socialized to be comfortable deferring (including to a female stranger), but the boys find it distractingly uncomfortable.

OTOH, girls are almost always better at one-on-one interactions, and may be shyer in class, while boys who engage vigorously in class discussions would never, like, talk to me in my office.

Uhmmm... no. Stuff, I guess.

16: Hmm, I find that men and women talk equally well in office hours. And that the men are more likely to come to my office just to talk (focused on the course material of course) while the women only come when they have a distinct question. But only the highest-achieving ever come at all, so.

16: Yeah, I'd tend to read that as a deference thing as well; a boy being talkative in class is expressing dominance over the quieter class members, and is putting himself forward as your equal; in a one-on-one, you're the teacher and he's the student, no question about it. Someone concerned with their interpersonal dominance is going to be more comfortable in the former than the latter situation.

I don't think this is innate -- I communicate kind of like the boys I'm describing here (very aggressive in group settings, kind of tongue-tied in one-on-ones where I need to telegraph a proper respect for my interlocutor's authority) -- but I think there are socialized gendered patterns of interaction along those lines.

Could it be that the girls are just better coached or more likely to listen to the coaching? I mean, you have to train a teenager to get them answer questions plaubisly and I can see, in retrospect, how my parents did this to me and my siblings. I took more effort to train than my sisters.

However: I don't think the girls would underperform as markedly as the boys, were they being interviewed by an all-male team. But this is pure conjecture.

Are you the only one doing the interviews, or is there a man you could compare notes with?

I can compare notes; it's a pretty large-scale operation with lots of interview teams. I'll bring it up in conversation with male colleagues and report back.

I notice this daily with boys and men even older than college students. I have also wondered about how their parents and schools effected their verbal abilities. On the flip, boys with experience in therapy are better. Also, your general high school verbal activities-- debate, theatre, model UN, etc.-- seem to improve them somewhat. And this isn't just an employer thing. Imagine trying to date one of these stoics.

I'm not sure I understand the "discomfort being in a room with three women" hypothesis. Is this supposed to be "gosh I hope they don't notice my erection" discomfort, or "how dare mere females sit in judgment and authority over me" discomfort?

Obviously, with interviews like this, girls may do better because we're taught from a very young age that being "likable" is a terribly important part of how we get ahead in the world. A girl who is ruthlessly intelligent but not sociable gets a lot of advice about how brute smarts isn't enough. (This happened to me a lot, anyway, especially when I was little.) But a boy might be able to get to college thinking that his academic achievements speak for themselves; why should he also have to talk about film or playing guitar or how he spends his social time?

I did a LOT of interviews for undergraduate college apps, and I'd say about 1/10 of that time was spent talking about my work in biochemistry or my schoolwork. I spent hours, though, talking about my obsession with French cinema, ideas I had about avant-garde theater, and music. But I was told ahead of time that everyone who went to those interviews had 1500+ SAT scores, was in the top five in their graduating class, and had professional internships, so the only way to distinguish myself to them was to seem like a potentially interesting and interested young person.

Do these boys really think the interview is going to be about how they did so darn well on the ACTs, their study skills and so forth? Are they afraid that revealing they have hobbies and extra-curricular interests will make them look "soft" or not committed to their schoolwork?

3 to 25.

27: 3 doesn't make much sense to me, honestly.

Feeling uncomfortable because you're the "one of these things is not like the other ones" in the room doesn't sound plausible to you?

Maybe they do not want to talk because they do not know the right answer.

I think some of it must have to do with the particular genre of talking about yourself in a situation where you're being formally judged. It seems to me that being strong and authentic and tangibly awesome is coded male, and that persuading someone to look kindly on your less tangible assets is coded female. If that makes any sense at all.

It does sound plausible, but it's not hard to imagine a girl working an interview with three men better than a boy would, if only because she has spent her whole life being told that being pleasant is at least as important as being smart. Lifelong sexism hurts us in some ways, but in situations like this, it really really helps.

29: Unless there are some sexist attitudes re: women and power, or there is some sexual tension in the situation, no, not really. I mean, 3 or 29 could be applied to the only blue-eyed person in the room, as well.

No it couldn't.

31: Right -- this sounds like a more coherent version of what I was groping toward with my 'deference' comments above. These boys are in the position of being a supplicant, and that's something that boys are socialized to manage less well than girls.

I'd guess AWB's got it right.

And again we see that "Patriarchy hurts men too"!

32 seems very likely to me, too. The "how to chat pleasantly and put others at ease" skill. Which would be over-stressed in girls from these parts.

I e-mailed a colleague. I'm curious to what his experience has been.

Also IME, boys do better in situations like "Talk about whatever you want" but girls do better in response to specific questions. In class, I try to mix up vague questions like "What do you think about this poem?" which the boys seem more eager to answer with their own theories, and "How does the trochaic substitution in line five affect the mood of the poem?" will be a more attractive question to the girls. I think this might be adjunct to JM's 31 theory.

I wonder what the effect would be in an interview situation if the interviewer mixed up questions like this. "Tell me something you think we should know about you" and "What do you do during a typical weekend?" might appeal differently to interviewees, possibly aligning with the gender split.

Also, because I think women are using to having deal with groups of men in authority positions, especially if you've worked in a job, ever. But perhaps the "woman asking you questions about your life" think triggers the "mom" response and you start behaving like a sulky teenager. Whereas young women might be used to fielding questions from women who are not their moms.

Dadgum!

25
Is this supposed to be "gosh I hope they don't notice my erection" discomfort, or "how dare mere females sit in judgment and authority over me" discomfort?

The first one seems totally plausible to me, allowing for hyperbole in your phrasing. He makes a conscious effort to avoid looking at cleavage and overcompensates by not talking to that interviewer at all, he wonders just a little bit about whether the smileyness Heebie refers to in 2 is flirtation, he's more worried about potential disapproval from the female students because they're potential future dates than he would be from their male peers... it could add up.

The conventional wisdom is that male children and teens are less verbal (a vague term, by the way) than females, and I'm not surprised that being interviewed like this makes it worse.

My experience matches Parenthetical and AWB: the most verbal people in class are the high achieving males, the least verbal are the low achieving males, and the women form an ordinary bell curve, with most in the middle.

Interestingly, IQ tests tend to give males a much wider bell curve--with long, thick tales--than females. Of course, they might not be measuring anything important.

44 is indeed what I meant.

Another point: the highly verbal men are much less likely to back up their blabber with preparation for class. The high achieving women who might be the second and third most frequent talkers are far more likely to have done the reading and know the answers to factual questions. They are also more reluctant to state opinions, while the talky men are really good at having opinions about any topic that comes up.

males a much wider bell curve--with long, thick tales--than females.

Much? I thought any difference was supposed to be pretty tiny (and the argument was that it became important several SD out from the mean).

Interestingly, IQ tests tend to give males a much wider bell curve--with long, thick tales--than females.

I thought this was just a meme for sexists to parrot when conversations like "why aren't there more women in physics / math / etc?" come up.

I don't think men and women are "naturally" different in terms of IQ and academic performance---teaching at a women's college gives me a lot of evidence against this as, in the absence of men, women take over exactly the same high-achieving talkative roles men have at my other school. But I do think that socialization leads boys to perceive themselves either as exceptionally brilliant or dumb, while girls are taught to act "normal," even when they're not. Girls are "too smart" or "too dumb," while boys might just see themselves as smart or dumb without feeling the need to regulate in either direction.

I've always liked long thick tales, myself

(Or actually, boys seem to self-regulate toward the extremes--dumb boys acting like total morons, and smart boys acting like geniuses.)

I hate it when I google for answers, and can't find anything -- has anyone done any better at finding research on whether there's any kind of consensus on this male/female IQ variance difference, and how big it is? I'm finding a whole lot of non-quantitative references to the difference, but they all dead-end in for-pay academic papers.

In 49 I am not, of course, intending to imply sexism from r h-c. I just doubt that this meme is credible since I see it deployed so often by people who lack credibility. Even if it is credible, I doubt that IQ is correlated with anything very meaningful (especially that far out on the tails). And even if it is, long tails of the IQ distribution are going to be almost completely irrelevant in the typical classroom, which is going to sample mostly from the middle of the distribution.

"I doubt that IQ is correlated with anything very meaningful" really requires a lot more clarification, which I'm far too lazy to think hard enough about to do.

Males of a certain age are especially awkward, and eye contact averse, among the visibly impregnated subspecies of the female. ("can't look at those huge boobs . . . can't look at that belly . . . So she's done the deed . . . hope I didn't make one of those Saturday night . . . uh, my extracurricular activities?").

The other thing is, this pattern really only seems to hold for co-ed classes of 18-22-year-olds. Returning students, even slightly older, do not seem to follow these gendered patterns at all. The men are more flexible about answering specific questions, and the women are often unabashedly talkative and brilliant.

Do these boys really think the interview is going to be about how they did so darn well on the ACTs, their study skills and so forth?

I feel like I was kind of an odd duck growing up, so I am hesitant to universalize my experience, yet this is about where I might have been at that age. Or rather, while I was with it enough to know that we wouldn't be able to fill that time slot with a discussion of my SAT scores, I had given little thought to what we would be discussing. One instance that stands out was in my junior year of HS. I had applied for a short homestay in Spain, and as a part of this application process, a couple from the program came to our house to interview me. In retrospect, the main reason they were there is obvious (to vet me as a normal-ish human being who could function in some strangers' house), but during the interview I had no idea what answers they were looking for.

They would ask what I was hoping to get out of this experience, and I would respond that I wanted to improve my Spanish language skills. They asked in several different formulations whether there was anything else I was hoping to get out of the experience, and I would respond, "Um, uh, I don't know." Eventually they tried to spoon feed me the answer—"Do you want to meet new friends in a foreign country?"—and even then I sort of flubbed it. "Um, yeah, I guess," I think I said.

My parents were present for the interview, and afterward they kind of half-smiled and said, "In these situations, you just have to tell them what they want to hear." I had the sense that they thought I was being recalcitrant, but really, I had no idea what was going on!

50 accords with my limited experience of single-sex education.

I wouldn't have thought that was the answer they wanted to hear. "Meet new friends"? Sounds more like what you would want to hear from a 10-year-old.

I didn't try to meet new friends when I was in a foreign country for two months. What would be the point of that?

58 is very common, of course. The question, I think, is whether it's more common among boys than girls. And I think the answer is yes, for some of the reasons given in this thread.

Returning students, even slightly older, do not seem to follow these gendered patterns at all.

Sounds about right. I noticed myself being less of an insufferable prick in class after I had left college for a couple of years, at least.

My parents were present for the interview, and afterward they kind of half-smiled and said, "In these situations, you just have to tell them what they want to hear."

You know, that's not really right, though, and those kinds of instructions (though natural) do kids a disservice. In those situations you have to demonstrate that you are a functional conversationalist who understands what they want to hear. I realize that you know this, but the difference between the two is actually significant, I think.

60: Maybe that wasn't the exact formulation, but it was something along the lines of, "Please give us some indication you care about other human beings, can be at least moderately sociable, etc."

long tails of the IQ distribution are going to be almost completely irrelevant in the typical classroom

No doubt true, although it depends on how large the classroom is. When you get up to 100ish people, you're dealing with 1st and 2nd percentile at the top of the class, which is 2 SD out in a normal distribution. It's certainly far enough out in the tails where you start to see notable differences between two populations with identical means and very slightly difference standard deviations.

But the bigger point is that there's lots of cultural factors that also screw with subject-based achievement, so leaping to immutable laws of nature as the first choice is a sign of pure dickitude from many of those who pull it as a rhetorical gambit. Poor souls like RHC who actually want to suss out how it accords with classroom experience get caught in the crossfire.

LB, what are the more promising articles you've found reference to? You're on a forum with a crapload of professors and grad students, so I'm pretty sure one of us could pull them.

being strong and authentic and tangibly awesome is coded male

But it's already been established that we aren't talking about erections.

A friend of mine who taught classes at the men's college attached to my women's college said, in response to my theory that women often take on the stereotypical brilliant-boy roles in a single-sex environment, that he was surprised at how emotionally expressive the men were in the single-sex classroom. He said the boys would get tearful sometimes in response to literature and be really emotionally supportive of one another, in ways they usually aren't in co-ed classes. It reminded me of Keats's "On First Looking into Chapman's Homer," which my co-ed students think is, like, a pretty gay poem, man.

65: Footnote 16 in the linked Wikipedia article looks promising. But what I'd really like is a meta-analysis of what a bunch of studies have found in terms of differing variance between the genders, and I haven't even found references to something like that.

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Healthcare rant.

I misplaced part of a birth control pill pack, so I got a refill and started at the same place I was in the pack. I just tried to refill my prescription so that I could get back on track, and I was told that I can't because it's too soon.

Now, it's true, that I can go to the beginning of the pack that I do have, but this seems colossally stupid to me. It doesn't work if you don't take it every day, you know.

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Since BG has conveniently posted something irrelevant, I feel less bad about saying that whenever someone mentions that Keats poem online I automatically link this.

Can't you just rick-roll people?

I've always liked long thick tales, myself

Alright, who's been telling stories about tale?

70: Aw! That was before I knew you! What a nice Shelleyized-Popized-Homerish-Keatsian thing to do!

68: Alright, I'm trying to look for it. I also took a look at the WSJ article linked in 17 or 18, since it was referenced in an interesting point on the Wikipedia article. The underlying article looked like some interesting research in Science that found the gender gap in average performance on standardized math exams had finally disappeared between men and women, but then explored the variance issue.

WSJ headline: "Boys' Math Scores Hit Highs and Lows"
Actual journal article title: "Gender Similarities Characterize Math Performance"

Thanks. You shouldn't put too much effort into it if you're not fascinated yourself -- I have to admit that my reaction to the 'greater variance' claim is, "What, a difference in mean IQ could be socialized, but a difference in variance is definitely innate? How does that work?"

The research, of course, is that girls are more verbal than boys from the beginning--earlier readers and talkers. PK is a supremely verbal kid by any standards, but he didn't read early.

But boy culture, especially in more conservative (southern, Christian) places is yes, very anti-verbal. Boys especially aren't taught to talk about feelings, so getting them to discuss things they're enthusiastic about is tough.

My brother went from being a chatty enough child to a virtually monosyllabic teenager. He eventually started talking again and now he's a 40 year old insurance executive who hardly ever shuts up.

Who knows.

I think it's also that boys tend to worry more about "the right answer" than girls in interview type situations, which girls see more as a conversation. Again, there's some good research on different negotiating strategies and verbal styles around this stuff.

76: Mine, as usual, are out of step. Newt read significantly earlier than Sally (but I attribute that to being around while she was learning.) They're both very verbal; Sally was unsettling as a toddler in the clarity and complexity of her utterances (it's largely a mercy that two-year-olds don't talk well; you really don't often want to know exactly what they're thinking. Sally, at that age, to a little girl who had affronted her in the sandbox: "I am going to cut you up into pieces and eat them." I'm sure other toddlers think that sort of thing, but mostly don't have the sentence structure to say it with.) Newt was less startling, but again, second kid, we may simply have been paying less attention. But he's now alarmingly talkative.

I can go to the beginning of the pack that I do have

This seems like the obvious answer. Also, there's nothing that says you have to stick to a 28-day cycle. Finish up the pack you have, get your new one and just delay your period a couple weeks (or skip this one altogether).

75: Well, it is pretty interesting stuff, and it comes up often enough that it would be nice to know the actual data underlying all the hot air.

The Science article only has an online version available, but with a pretty interesting pdf supplement that specifically addresses the variance findings (basically, they're around, but they're not particularly relevant until you go a couple standard deviations out, and most interestingly, they're not universal because the ethnic Asians in the sample group actually showed more girls in the highest achieving percentiles on the math testing than guy). I'll email it through to the Unfogged address once I put in a good try at finding the Intelligence article.

"I am going to cut you up into pieces and eat them."

I might steal this line.

My experience as a shy, introverted HS senior in these situations was that my reaction depended primarily on how confident I was about ultimately getting into the college in question.

My experience of recently returning to finish my BA after a long absence suggests to me that it is just about brute socialization for most young adults. The vast majority of young (white, straight, middle-class, Gentile) men get the sports-video games-fantasy violence thing pounded into them so hard they can barely think, let alone converse. I wasn't even trying to engage the real dullards at Big State, and I wasn't taking a lot of gut courses, so I think I ran across a fair number of above-average intelligent young men, and they really are emotionally, intellectually and socially stunted at that age. Working at the campus paper, I had the opportunity to watch several of them grow up into fairly decent guys (not so much with the politics, you understand) but that 15-20 y.o. range is pretty bleak.

Oh, one other thing, although it probably wouldn't apply to heebie's specific case: When I took college classes while still in HS, the overwhelming majority of the other students who took advantage of that possibility were female. It seemed like an 80-20 ratio, but I'm sure it was probably more like 65-35. Still, if that was the case elsewhere, it would tend to speak to more reasonable expectations about what an academic interviewer wants to hear.

I might steal this line.

Heh. I said something like this to someone on the subway once, and the next day decided to make a visit to the St. Vincent's psychiatric emergency room. (Bad birth control reaction--murderous outbursts stopped the day I went off the pill.)

82: "I will kill you and eat you" was a staple of my childhood and adolescent threat arsenal, but I never did eat anybody I killed. Kids are all talk.

84 - I have never heard of anyone having that reaction to BC. Humans are complicated in all sorts of weird and interesting ways.

When I was about 10 and my sister was about 6 some kid threatened to kill me and I pointed out that if he did he'd go to prison. My sister chimed in "And you'll have to clean up the mess." Very practical, my sister.

86: My mother said the BC pill made her crazy, and she took herself off it after a few weeks because my father and I were both in very real danger of being murdered in our sleep.

I have a similar reaction to hallucinogenic recreational drugs--sudden rage and eerily calm death-threats. Some people just have a really touchy nervous system w/r/t drugs, and I'm one of them. I barely take vitamins and Advil without constantly checking my behavior for signs of weird reactions.

The pill made me a complete lunatic. Not rage-filled, but deranged: alternating panicky and distraught with dissociated and creepy.

Also, there's nothing that says you have to stick to a 28-day cycle. Finish up the pack you have, get your new one and just delay your period a couple weeks (or skip this one altogether).

Some folks might insist on seeking the advice of a professional gynecologist, but apostropher proves that a dedicated amateur can become quite knowledgeable with years of study.

One of the approximately six birth control pill formulations made me depressive and unbalanced-feeling (with gigantic, sore boobs). Then I went through a couple of formulations where I had my period like twice a month. Now I'm on a very nice pill that appears to have ceased my period altogether. I'm okay with that.

Ah, hormones! It's pretty clear we have no fucking clue how they work in individuals.

made me depressive and unbalanced-feeling (with gigantic, sore boobs)

Given the parenthetical, I can't tell if this mean emotionally-unbalanced or physically-unbalaced.

Both!

Two points of clarification on 45:

1. I didn't mean to imply anything sexist, because I specifically doubt that IQ tests measure anything important, and certainly not anything innate.

2. I know the difference between tales and tails.

You should have the email in your Unfogged address, LB.

Yesterday my 3 year old threatened to shove me up his butt and then poop me till I was dead.

)) >> ((
Romantic!

Oh, hell. HTML stole my brackets!

98: Three-year-olds either have exceptional pain tolerance or an extremely poor grasp of relative sizes. Either way, probably something parents-to-be should know.

But boy culture, especially in more conservative (southern, Christian) places is yes, very anti-verbal. Boys especially aren't taught to talk about feelings, so getting them to discuss things they're enthusiastic about is tough.

I have direct experience with the same culture/population that Heebie is talking about and that Bitch mentioned above. I would attest that the average, young Texan male is a particular species with distinct, learned traits that lean toward fumbling idiocy in academic situations.

Note that Heebs and I both are dealing with the ones that are expressly not going to the more highly-ranked local institution up the road but rather to the regional, cheaper, less rigorous ones. I feel strongly that most of the phenomena we're talking about is a product of local culture, but I'm a sociologist so I think that about most things. In any event, I have met and tried to teach a fairly large number of smart Texan males who think it uncool or uppity to seem smart or interested in school. I realize this isn't a trait specific to Texans in and of itself, but the way it gets played out here is quite specific.

The behavior of the six year old is more disturbing. She's taken to reciting the patriotic claptrap she learns at school while she's at home. She'll say the Pledge of Allegiance or mangle some patriotic songs. We've tried to talk with her about this, but I think all the gets out of it is conflicting messages from authority figures that she doesn't have the ability to process.

In any case, today, while she was saying the Pledge of Allegiance, the three year old walked up and hit her, prompting her to yell "What's the matter with you Joey, do you HATE AMERICA!"

Oh, hell. HTML stole my brackets!

And so soon after contraceptives stole your periods! Soon you'll be wholly unpunctuated.

In my classes, there is not much difference between the genders in terms of in-class participation. If there is a difference, it's in favor of women speaking up rather than men.

104: For those keeping score at home, it was Jackmormon that lost her period, and White Bear that lost her brackets.

Eh, white women all look alike.

105: This is a bit of a hobbyhorse after an unsettling teenage experience (follow the link in 59 for an account), but have you ever checked that in any quantitative manner, or are you just eyeballing it?

Here is the "Gender Similarities Characterize Math Performance" article:

Even at the 99th percentile, the gender ratio favoring males is small for whites and is reversed for Asian Americans.

"small" meaning a ratio of 2 to 1.

The ratio for Asian americans is weak sauce. Apparently, two asian girls and a single asian boy scored at the 99th percentile level for the whole state of Minnesota. Are you even allowed to do statistics on 3 people?

The breakdown of Asian groups in Minnesota is also different from the US as a whole:

http://www.lmic.state.mn.us/datanetweb/maplib/demogs/race/asamgrow.htm

[insert unwarranted "grand torino"- based speculations here]

Here is SAT score curves from 1992 for boys and girls:
http://www.nsf.gov/statistics/seind93/chap1/gif/01-0793.gif

No doubt true, although it depends on how large the classroom is. When you get up to 100ish people, you're dealing with 1st and 2nd percentile at the top of the class, which is 2 SD out in a normal distribution. It's certainly far enough out in the tails where you start to see notable differences between two populations with identical means and very slightly difference standard deviations.

In a fit of extreme procrastination (and because I can't do what I really meant to do today until some code finishes compiling), I did a simple check of what these statistics would mean for a classroom-sized set of students. I generated 1000 samples of 100 IQ scores ("classrooms"). IQ scores are pulled from a half-Gaussian (requiring them to be above 100), to model some sort of "college admissions". In each set of 100, there are 50 from a distribution of variance 18 ("men") and 50 from a distribution of variance 15 ("women"). (I don't know what the actual claimed difference of variances is.) It turns out that in about 3/4 of these classrooms, the highest IQ in the class is male, reflecting the difference in variance. The mean difference between the highest "male" IQ in the class and the highest "female" IQ in the class is about 7, so it's a little more than twice the assumed difference in variances.

Maybe I'm wrong, but I doubt that an average IQ difference of 7 points translates into anything particularly noticeable to a teacher or other observer of the classroom.

Closing parenthesis should be after "college admissions", not "above 100".

And "variance" should be "standard deviation".

109: I'm glad to see they plainly state "Gender differences in math performance, even among high scorers, are insufficient to explain lopsided gender patterns in participation in some STEM fields." And yet people keep citing things like this to try to explain precisely those patterns.

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I stupidly did not check to make sure that the cream that has been sitting in my fridge for the last, oh, I don't know, 2 weeks, was still good in order to make a fennel gratin. It is not. It is raining. I just got back from the markets. I do not want to leave the house again. Wah. I need a delivery service.

(Essear runs statistics when he's procrastinating, I cook).

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109

The ratio for Asian americans is weak sauce. Apparently, two asian girls and a single asian boy scored at the 99th percentile level for the whole state of Minnesota. Are you even allowed to do statistics on 3 people?

The paper is confusing but I think the n values are the number in the top 5%. So it is more than 3 people for Asians in the top 1%.

The Asian variance for boys is 1.06 the variance for girls which is less than the 1.12 or so typical of the other groups but still greater than 1.

110, 114: You people are so productive when you procrastinate. I prefer to avoid doing things by not doing anything.

108:

I'll tally next time, but my participators tend to be a very small subset of a very large class and are therefore a memorable group. They are also not usually the top people in the class (probably because the top people don't bother coming to class).

The paper is confusing but I think the n values are the number in the top 5%. So it is more than 3 people for Asians in the top 1%.

That makes more sense now.

113

I'm glad to see they plainly state "Gender differences in math performance, even among high scorers, are insufficient to explain lopsided gender patterns in participation in some STEM fields." And yet people keep citing things like this to try to explain precisely those patterns.

It would be more accurate to say "explain completely" as opposed to "explain" as it does account for a large part of the difference. And that is granting that math ability in the top 1% is sufficient.

116: I just realized that our procrastination behavior is of course highly gendered.....

Cooking sounds like a much better way to procrastinate.

120: But being genuinely useless is completely genderless! You should try it.

Oh, don't worry, rob, I'm pretty good at being genuinely useless...just ask my adviser.

(But I'm rewarding myself for having been observed in lecture today by taking most of the afternoon off to try cooking two things I've never made before. We'll see if the results are edible).

119: And that is granting that math ability in the top 1% is sufficient.

Part of what seems so weird about using statistics about high-school math performance to talk about later academic success (note, I'm not directly disagreeing here, I'm raising a different issue) is that any math you'd plausibly test a highschooler on is not at all tightly related to serious academic math. I'm (for example) comfortably in the top 1% of the population for math aptitude on any test that you'd plausibly give a large population of high school kids. On the other hand, I reacted to anything more advanced than Diff. Eq. with confusion and distaste; I've got a knack (at this point sadly rusty and unused) for math at the calculus and below level, but not so much for math-major level math.

And on the third hand, the kind of math aptitude I have is way plenty for most technical fields other than literal mathematics or theoretical physics.

Trying to explain the gender differences in technical fields by looking at the distribution of results among a population of high school kids seems really off-base; at some level, the SAT (for example) is testing speed and accuracy of calculation, which really hasn't got a lot to do with high-level math aptitude.

I just did a little gendered chart of about 250 recent students and what I found doesn't really match up with what I said above. If I generalize from what I found, I get this:

1. Boys are far more likely than girls to get very bad failing grades (like from plagiarism or not handing in assignments).

2. Girls are more likely to get high Fs, Ds, and C-s. (Zero boys fell into this category.)

3. Girls and boys are equally likely to get Cs, C+s, and B-s.

4. Boys are far more likely to get Bs and B+s.

5. Girls are far more likely to get A-s and As.

6. Girls and boys are equally likely to get A+s (boys have a slight edge).

I don't know what this means, but it's more complex than I thought.

And on the third hand, the kind of math aptitude I have is way plenty for most technical fields other than literal mathematics or theoretical physics.

You would be surprised how little of theoretical physics involves anything more advanced than Diff. Eq.... But yes, I agree. Whatever it is that leads to so few women working in my field, I'm pretty sure it has nothing to do with mathematical aptitude and everything to do with culture. (Both the broader culture of the U.S., and the culture of theoretical physics in particular.)

125 is fascinating.

125 is very similar to my own grade break-down, though I think girls actually have the edge in my courses. (This could have something to do with the gender break down at my institution).

Yeah, I should also add something I didn't realize until I did the numbers, but the ratio of girls to boys in those classes (about 10 classes' worth) was 5:2.

Oh, though - I think boys and girls are equally likely to get B and B+ but boys are more likely to get B-, in my class.

On the other hand, I reacted to anything more advanced than Diff. Eq. with confusion and distaste; I've got a knack (at this point sadly rusty and unused) for math at the calculus and below level, but not so much for math-major level math.

This is sort of me, except I flamed before Diff. Eq. (couldn't get it at all). I did well on math SAT and GRE, they were not advanced enough to show my actual math lousiness. My experience with math has convinced me that something genetic is involved, since it would have helped me to be better at math, I tried to be better, but I just could not fucking get it. It was like a wall. I think if I had a great teacher who taught me advanced stuff when I was young (say early high school) it might have helped, in the sense that I might have been drilled in some good recipes or tricks, but when I tried to learn it was too late for that stage of education.

My sense is that I have a good genetic verbal ability that allowed me to sort of rephrase math techniques verbally and do that that way, but then the ability to do that runs out at a certain level of complexity. I can still *talk about* math in a way that gives some impression of facility to the uninitiated, or verbally describe what an advanced math technique is *supposed to do*. But I can't fully grasp it.

And on the third hand, the kind of math aptitude I have is way plenty for most technical fields other than literal mathematics or theoretical physics.

that's probably not true. Except perhaps in the sense that many technical fields have cookbook recipes you can do by rote using computers without fully understanding them.

And, overall, girls have a much better chance of getting an A (of any kind) than boys do. It's funny, because in each of my classes I think of "the boys" as the loud chatty ones, when really that's maybe two or three boys, and the rest of them make themselves nearly invisible, and come out with a B or an F.

I have stereotypical female pattern problems with math. I'm not especially bad at math in the overall college population, and I'm not math-phobic, but I have trouble caring or getting interested. I tend to think concretely, and to ask myself "When will I ever be filling a storage tank of a given volume and dimensions with a pipe of one diameter, while simultaneously draining it with a pipe of a different diameter? Never, that's when."

124

... the SAT (for example) is testing speed and accuracy of calculation, which really hasn't got a lot to do with high-level math aptitude.

I don't think this is accurate. The problem with using the SAT for identifying extreme math talent is that it is (especially since it was renormed) too easy not that it is measuring the wrong thing. It is measuring something (call it problem solving ability) that is definitely a component (but not the sole component) of high level math ability.

110: Certainly the 7 points wouldn't be a big enough difference on their own. But look at a cutoff in the 130-135 range (basically the top few people in each class assuming that 18 and 15 are your standard deviations rather than variances), and check the gender balance of people above it. I bet you'd find that group of "top students" skews male to a fairly significant degree. Of course, there are plenty of other reasons why the guy students would be louder without necessarily being smarter.

Now this would only occur in an unbounded right-tailed distribution, so it would be noticeable mostly at elite schools. The result would also say nothing about whether the difference in variability comes from natural or societal factors. But I bet these statistical differences can explain a top group of students that seems gender-unbalanced toward guys (though nowhere near to the extent that professorships or executive positions are).

I'm way behind on this thread, but I can chip in with some anecdotal evidence. All of the teenage male members of my family are completely monosyllabic. Getting a conversation out of them is like pulling teeth.

With teaching, I have no idea.* Most of the students I've taught have been hyper-confident, articulate kids who've been trained and coached by their schools to be hyper-confident, and articulate. I've not really noticed much of a gender breakdown.

Anecdotally, the students who totally thought they were God's gift, but didn't have the chops to really bring it, were more likely to be male. The students who thought their opinions were valuable, because they were, like, their opinions, and opinions are precious flowers, were more likely to be female. Perhaps I'd say that the less confident male students tended to clam up, the less confident female students tended to be publicly self-deprecating. But, on the whole, all of the various stereotypical student 'types' seemed fairly evenly distributed by gender.

* I haven't taught at all for nearly 2 years, so my memory may not be perfect.

The problem with using the SAT for identifying extreme math talent is that it is (especially since it was renormed) too easy not that it is measuring the wrong thing.

This is not at all the case for me. I didn't do that great on the math SAT, (for unfogged standards I practically failed it with a 1420 or something, PRE-realignment, of course.) But I never flamed out taking higher and higher math courses, even though the level of abstraction seems to keep ratcheting up indefinitely.

A considerable proportion, maybe 10%, of the Math II SAT aptitude takers get 800 scores. It isn't designed to find geniuses.

It's probably true in a limited way that someone who works faster and more accurately than someone else at a medium level of math, and who figures out tricky questions better, does so because he or she is smarter. But it's a very crude screen.

126

You would be surprised how little of theoretical physics involves anything more advanced than Diff. Eq....

General relativity? String theory?

139: Yes. The strings are really, really little.

I've gotten perfect scores on every standardized test of mathematical or logical reasoning, from high school to grad school, and have always sucked (comparatively) on the verbal sections.

In my college coursework, however, I struggled like hell with my second semester of calculus, failed out of a few sciencey/mathy courses, and did not come anywhere near statistics. I majored in English and Spanish literature, minored in PoliSci, and did pretty well in them.

Standardized tests have told me nothing that seems relevant to my performance in coursework.

I just did a brief review of my classes and there is no difference in the propensity of women and men to get either A+s or As in general. And I mean dead even 50-50.

General relativity? String theory?

The snarky answer is that Einstein's equation is just some particular nonlinear partial differential equation, and string theory is like a whole bunch of those coupled together.

What I really meant, though, is that the day-to-day work of a theoretical physicist is a lot of conceptual stuff, maybe some computer programming, and the typical calculation involves differential equations and linear algebra. Yeah, there's more underlying machinery that leads to the differential equations you're trying to solve, which might involve differential geometry (as in GR) or complex analysis or whatnot, but even then you're rarely using more than what a mathematician would consider the most basic definitions from those fields. There's very little that a math major would recognize from their coursework. Almost no physicist is going to ever compute a spectral sequence, for instance.

There's this thing which is often said by textbooks and math course syllabi: that what a student needs as prerequisites includes some set of standard things (calculus, linear algebra) and "mathematical maturity". I would doubt that any standardized test is measuring that thing called "mathematical maturity", and that very many people in technical fields outside mathematics actually require it (though it may help).

If you want to make a test hard when the math is not that hard, you're testing something other than abstract comprehension of math concepts and into something like "spot the trick". Not universally, of course, but in practice I believe this is how it shakes out.

The same thing happens with weeder courses, like Organic Chem. (I loved that course though. Little puzzles!)

I'm living proof that organic chem is a great weeder course, having been weeded by it.

143: The really damn annoying thing is just how many real-world applications of math involve differential equations. Sure, they're useful, but it's not the type of math I was particularly good at because it just felt way too messy to me. But now I'm in a field where messy differential equations are really important. Boooo!!!

143
... but even then you're rarely using more than what a mathematician would consider the most basic definitions from those fields. There's very little that a math major would recognize from their coursework.

I don't think this is accurate. I have a PhD in math and the math involved in high level Physics is still beyond me. It is not undergraduate math major stuff. Or do you really mean "would"?

If you want to make a test hard when the math is not that hard, you're testing something other than abstract comprehension of math concepts and into something like "spot the trick".

Yessss. I am great at spotting tricks, not so great at actual math.

Plus, when I have to do all my own calculations, I sometimes go to Crazytown. When it's multiple choice, you can't end up saying, "Oh, x must equal 6.7945123!" because the answers on offer are 7, -7, 0, and 100.

Or! Those "which is greater" questions! I love those.

A. A is greater.
B. B is greater.
C. A and B are equivalent.
D. Not enough information.

Hooray for no hard thinking!

Which is greater: oreo ice cream or purring cats?

137
(for unfogged standards I practically failed it with a 1420 or something, PRE-realignment, of course.)

That doesn't sound bad at all to me, although maybe I'm just admitting to being even dumber than most people here. I got a 1350, the sum of the best of two tries. I don't know what realignment you're talking about, but I'm 26, so you do the math.

I've gotten perfect scores on every standardized test of mathematical or logical reasoning, from high school to grad school, and have always sucked (comparatively) on the verbal sections.

I'm the exact opposite. I suck, suck, suck at standardized tests in math (I believe I scored in the 34th percentile on the GRE).

In college, I struggled through my chemistry courses, succeeding in organic chemistry only with so much work that I realized that this was not my path (hardest fucking A- ever), and, scared of calculus, downsized my science major to a minor. I feel like I have the classically "female" problem of being unable to model things in my head. (Similarly, I can't tell right from left 95% of the time).

152: B!

153: Oh yeah, I meant to give my math portion, which was a circa 720. I suspect there's plenty of people here who got near an 800.

I really hope I didn't come off as obnoxiously flaunting my score while being falsely modest about it - I generally think of myself as a solid test-taker, but not outstanding.

I don't think I was very clear about how my 126, 143 and 144 relate to some of the underlying issues. So returning to LB's 124:

I'm (for example) comfortably in the top 1% of the population for math aptitude on any test that you'd plausibly give a large population of high school kids. On the other hand, I reacted to anything more advanced than Diff. Eq. with confusion and distaste; I've got a knack (at this point sadly rusty and unused) for math at the calculus and below level, but not so much for math-major level math. And on the third hand, the kind of math aptitude I have is way plenty for most technical fields other than literal mathematics or theoretical physics.

I think anyone with that level of math aptitude has enough technical chops to do just about any sort of science. The other skills required are pretty different from the ones that mathematicians need.

Anyone who's been reading here for a while knows that LB has damned good logical reasoning skills; can put together arguments, take them apart, and explain them clearly; and can home in on things that are important versus things that are peripheral. Someone with those skills, and enough mathematical aptitude to get through a class in differential equations, has plenty of potential to be a better physicist than someone farther on the high end of the mathematical aptitude distribution but less strong in the logical/rhetorical/insight aspects. And yet the gender ratio in my field is depressing, and people still cite variance in mathematical aptitude to try to explain it.

for unfogged standards I practically failed it

Oh, crap, I didn't realize there was a procedure for submitting our SAT scores before we could comment. Who do I have the College Board send them to?

158: I'm the gatekeeper.

My sense is that I have a good genetic verbal ability that allowed me to sort of rephrase math techniques verbally and do that that way, but then the ability to do that runs out at a certain level of complexity. I can still *talk about* math in a way that gives some impression of facility to the uninitiated, or verbally describe what an advanced math technique is *supposed to do*. But I can't fully grasp it.

PGD, that is a pretty exact description of my math abilities. A good memory and words got me through differential equations, but I always knew full well that I didn't have the intuition some of my classmates did. If we're making crude guesses about genetics, then I suppose we point to our mutual eastern European Jewish heritage. (Watered down in my case, although it is pretty obvious that I got my package of skills from that side of the family.)

159: I'll get right on it.

(And, for my two cents, you didn't come off as being boastful. Then again, two people in my senior class scored a perfect 1600, so I'm already used to feeling inadequate about my scores).

148 I have a PhD in math and the math involved in high level Physics is still beyond me. It is not undergraduate math major stuff. Or do you really mean "would"?

Are you sure it's not the physics that you don't have the background in? (Unless you're just saying that you don't know as much math as Ed/ward Wit/ten, in which case, welcome to the club.)

re: 150

I once marked an undergrad admissions paper, where the economics prof had set the students a little model with a certain honey-loving bear, and his donkey friend. The students were asked to calculate the optimal number of hives per acre [given some starting information]. Every student except one gave an integer number. The professor on the other hand, listed his answer as '3.2456' or something equally idiotic.

re: 159

What do you do with Scottish Highers? I expect ajay and myself are the only ones here who have 'em. Do we get extra credit for being exotic?

My mathematical ability, as measured as ability to solve proofs is much, much better than my mathematical ability measured as ability to solve equations.

165: Mine too. Geometry was the last math I was good at. Proofs felt like something I could handle because words were involved.

This verbal-symbolic divide is stereotypical, but keeps returning. I was always on the symbolic end because I could actually visualize what went where (okay, move the three from the numerator on the left to the denominator on the right). This skill was great for OChem too (rotate the molecules in my head), but geometry-style (and number theory and real analysis) proofs were much harder.

143: This exactly matches my experience of theoretical physics, and for that matter of theoretical statistics as well (though that substitutes combinatorics for diff. eq.).

And of course the SAT and GRE (general) questions are too easy, and too dependent on "guess the trick" and speed, rather than on properly mathematical thinking.

Now, on to the sexy stuff: by coincidence I just so happen to have recently read this paper, which reanalyzes data sets from 1932 and 1946, when the government of Scotland gave an IQ test to basically every 11 year old child in the country. The sample sizes are huge, 87k and 71k respectively, and the representativeness of the samples is great. (State power is the health of statistics.) The empirical distributions are not Gaussian at all, though compatible with a mixture of two Gaussians, one for normal people and one "reflecting ... variation in effects of genetic and environmental conditions involving mental retardation". (Figure 1, which makes this point, is helpfully reproduced here.) The size of this second Gaussian component was substantially larger for males than for females, giving the former a higher variance --- but not telling you about how many high-scoring boys there were, because the actual distribution isn't Gaussian. They do find slightly more boys than girls in the high-scoring upper tail, but nowhere near enough to account for observed differences in the professions.

(To be really pedantic, what they do here does not seem to me to be the best way to fit a mixture of two Gaussians to their data, so I wouldn't give them the highest score on an exam, but they have so much data that it almost doesn't matter.)

I think 131 is a bit off-track. I think rather than there being a genetic component that lets you avoid the wall, instead high-level math selects for people who are willing to keep going after hitting a wall. How long would you be ok being stuck on a problem before figuring out what to do?

I ran across some research on this point recently, but can't find the link. Basically most people give up on a problem that they don't know how to answer after 5 or 10 seconds. Higher level math even for the very smartest is going to involve questions you're stuck on for at least an hour. For research it's worse, I just found a two line answer to a minor question three of us had been stuck on for at least a month.

159:I'm the gatekeeper.

Hey, how about that? I'm the keymaster. Heard anything on the rectification of the Vuldrini? See ya' around.

168 is fascinating.

Shouldn't they just have done a fit to the sum of two unspecified normal distributions to see if the two they chose would fall out naturally? They have plenty of data for it.

I suspect I've sat the same test [or a successor to it]. We all got some mysterious test the year we left primary school to start high school.

162

Are you sure it's not the physics that you don't have the background in? (Unless you're just saying that you don't know as much math as Ed/ward Wit/ten, in which case, welcome to the club.)

I don't know tensor calculus which I believe is a prerequisite for general relativity.

BTW, Figure 2 in that paper provides pretty strong support for the "wide tail" theory. It graphs the proportion of all subject receiving a certain score that are male. It looks like a smile with a minimum at IQ = 100, at which 45% are male, and maxima at the ends approaching 60-70% male.

168

... The empirical distributions are not Gaussian at all, ...

I was under the impression that this is meaningless, that you can change the shape of the empirical curve by changing the mix of questions. So what is real is the percentile rank which for IQ is used to generate an IQ score assuming a mean of 100 and standard deviation of 15 (or sometimes 16). So IQ scores are normal more or less by definition. Whether this is sensible thing to do is a complicated question.

Almost 200 comments and nobody has mentioned the veldt?

I don't know tensor calculus

And they let you get a PhD in math? Huh.

On the veldt, nobody cared about your SAT score.

||
This is amusing the hell out of me.
|>

169: Commenter! Snarkout and I were just saying this morning that we missed seeing you around.

Aw, man, and I didn't get to discuss my experience hitting a wall in math. (It was a one-two punch of real analysis and complex analysis, for me; complex analysis is the only math class I got a C in in my entire life and convinced me that I didn't want to go to grad school.)

"These students are not high-achieving by Unfogged standards"
?!!!

177: He could easily be on the pure maths side of things. Had I gone on to do my extra year and the PhD, I certainly never would have learned tensor calculus. I didn't take any calculus-based course past methods and dynamics (our annoying-as-hell physics requirement) nor did I take any analysis past complex analysis (unless you count topology, which I did a bit of).

182: No, really. This school is not very competitive. Much easier to get into than a big state school. We do have some strong students, and these are them, but really.

180: Aw, thanks. I'm graduating this year, so between applying for jobs and trying to finish papers so that I'll have a thesis, my Unfogged reading time has suffered a bit.

183: Sorry, I was mostly just giving in to the "mock Shearer" urge. But I know that at the U of C, differential geometry was part of the first-year grad curriculum, so I was sort of assuming that was pretty standard. (I'm pretty sure it was covered in an undergrad class or two also, but maybe that was specialized to differential forms.)

260: ah, a fellow sufferer. Frustrating, isn't it? I think the very high verbal/low math combo is stereotypically Jewish, but plenty of Jews are, like, really good at math.

I think 131 is a bit off-track. I think rather than there being a genetic component that lets you avoid the wall, instead high-level math selects for people who are willing to keep going after hitting a wall.

I've often contemplated this possibility as well. I noticed some of my classmates had what to me were superhuman focus abilities. The ability to focus on one difficult math problem for hours straight could still have a genetic component, though. I can't imagine being able to do such a thing. Also, there's probably some relationship between a subject being "easier" in some sense, or at least more transparent, and the ability to focus on it for a long time without going insane.

Anyone who's been reading here for a while knows that LB has damned good logical reasoning skills....Someone with those skills, and enough mathematical aptitude to get through a class in differential equations, has plenty of potential to be a better physicist than someone farther on the high end of the mathematical aptitude distribution but less strong in the logical/rhetorical/insight aspects.

I think people who are good at math underestimate the importance of it. Also, I wonder if LB really was able to master DiffEq (no offense, LB). The whole thing of DiffEq -- mixing differential equations, linear algebra, and matrix algebra, truly being able to manipulate equation systems easily -- was to me a real sticking point. I don't doubt that if you were *truly* fluent in that you could go far, but I wonder if a lot of people don't stagger through DiffEq without really fully getting it. I also found advanced algebra (the kind where proofs last for pages and you're moving things into and out of integral signs all the time) to be a real struggle. It seems like both these things are pretty basic to people who are really good at math.

whoops, 260 s/b 160.

Not suffering over it anymore; my life very rarely requires more than an understanding of calculus (not even doing calculus, just understanding it). The lack of visualizing hurt more in chemistry than math anyway.

(Dude. You're calling 'only differential equations' "low math" ability, which suggests that you run with an impressive crowd.)

At my next physical, I'm going to ask my doctor whether I'd be a suitable candidate for a copper IUD. What I really want is not available in the US. It's a mostly copper IUD (on beads that you have to sew into the uterus/ sadly no t-framed version) which puts out a tiny amount of levonogestrel--around 5 mcg just to help with cramping and excessive bleeding.

Also, in Europe, they have smaller ones for smaller and nulliparous women. Is there a way to push the FDA to approve more contraceptive options?

The U of C math program is very prestigious. Ahmed Chalabi got his PhD there. However, Math majors are very impractical, and George Bush outsmarted him at every turn.

Even though I'm a girl, I can be tongue-tied in a lot of situations. One of my college interview questions was sone by an alumnus who was a college professor, and it was clear that he likedintellectuals, so I felt totally comfortable.

But I have a much harder time around high academic achievers who are not at all intellectual and am not sure how to figure out what the answers they want are, since I don't quite feel safe being myself exactly.

175: So IQ scores are normal more or less by definition.

The way it works with modern IQ tests is to administer the test to a reference sample, and then transform their actual scores (which are usually very skewed) so that they follow a Gaussian distribution; then other people get their scores by seeing where the fall in the reference distribution, and what the equivalent Gaussian would be. (This is part of how the Flynn effect was missed for so long: the tests are renormed every so often, and people were just comparing the normed scores, not the raw performances.) This does not, however, guarantee that later scores are Gaussian. The paper I linked to shows that they are not.

190: BG, I haven't followed the thread much at all, but this sounds like an ouch situation which may or may not bear a relation to engaging in higher maths in one's daily life.

This is why applied physics is so great. We get to call ourselves physicists, and we only rarely have to worry about existence theorems.

I can't remember what the name of the math course I seized up in was. All I remember are triangles on the board, with a capital letter at each corner and arrows for sides. Some of the arrows were double-ended. I have a faint memory that this was linear algebra; the lack of explicit matrices would explain why I was so surprised and pleased to meet yer more straightforward, squish-the-rectangle linear algebra when I caromed off applied mathematics on my way into soil science.

Gods arose from the soil science of the veldt.

167: O, lord, the rotating molecules. I am not certain if I'm actually mentally deficient in this area, or what, but I couldn't visualize them to save my life. It's the part of my brain that is the stereotypical veldt female waiting around for the men to spear the objects rotating in space. (I also was never sure why it was necessary; sure, there are left-handed and right-handed molecules with different effects, but we can't see the little fuckers anyway, so why does it matter if I can draw a diagram?) So, so frustrating.

Snarkout, have you seen this?

196: sounds like Axler's Linear Algebra Done Right (for a pure mathematician's value of "done right"). For me, the course where I hit the wall was undergrad real analysis.

198: !!!

198: No! My God. I'm blushing.

Especially when people like Cosma are floating around. Hi, Cosma! I don't think I've ever mentioned on Unfogged how much I enjoyed (and found inspirational, back when I was making a serious effort at blogging) your Notebook.

202: Oh, those old things? Mumble, stammer.

undergrad real analysis

Agreed. The hardest class I've ever taken. Harder than all my grad classes.

The last course I ever participated in was in analysis, I think maybe. The first course of the first sequence after calc at chicago. I was auditing for a few weeks and pulling Bs on the homeworks when I decided that it was taking up too much time for something I wasn't actually invested in. I understand that not long after I stopped attending they got their first nontrivial proof.

Math textbook-wise, I'm pretty excited to buy this little number.

193

... This does not, however, guarantee that later scores are Gaussian. The paper I linked to shows that they are not.

Well obviously they don't have to be if the reference sample is not representative of the overall population. But if the reference sample is a random sample of the entire distribution doesn't that force the overall distribution to be Gaussian (within the limits of statistical sampling error).

Also, catching up, I am currently blundering ahead in a field despite being fairly sure I lack the required mathematical chops. It's fun to be the dumb one!

199: The sample chapter 6 of Axler has a sketch of a R² vector as an example. That's way more concrete than anything I remember, but very possibly I was hearing nothing but white noise by the equivalent of chapter 6... I don't think we had a text, just a mad purist at the chalkboard.

It was probably heaven for the people who were to be algebraists, though.

While I have had classmates/colleagues who showed no sign of hitting a wall, certainly lots of us did, so there was selection for willingness to keep slogging for progressively tinier and more abstract insights. Actually, this links to my only hypothesis about different male and female responses to math education, which is my answer to the much later peak (IIRC, and a report ten years old) for female math productivity. Doing a whole lot of math makes you an outsider for a most social circumstances, so I hypothesized that younger men and older women lose less by being mathematicians in a patriarchal society: they're scrap, they don't have much to win *in* society. Of course, there isn't much of a re-entry path for pure math, although there's more than there used to be (partly because social structures are changing), so late bloomers were likely to bloom unseen (And very few to love).

The young men, see, they're inarticulate, so they might as well sit around and do math. And yet, that's not quite what Heebie is reporting.

I happen to know that in the future I will not have the slightest use for algebra, and I speak from experience.

You people remember your SAT scores? I've got a rough sense of range (i.e., nowhere fucking near Heebie), but don't have a damn clue what the number was. I do recall that I was badly hungover, but that's about it.

I've recently been trying to teach myself Real Analysis from textbooks. It doesn't seem as bad as I'd feared, based on its reputation, which I suspect means I'm completely missing most of the nuance.

206: That fucking book is beyond the fuck of this fucking blog, but interested students should look at other fucked books.

211: I went back and did that after learning more math-at-the-physicists-level, and it was immensely easier. (Plus, I think, it was a lot easier for me to concentrate.)

212: I don't even know if I'm fucking interested in the fucking subject, but I think the fucking concept is pretty fucking fucked.

Doing a whole lot of math makes you an outsider for a most social circumstances, so I hypothesized that younger men and older women lose less by being mathematicians in a patriarchal society: they're scrap, they don't have much to win *in* society.

Indeed, the tradition of primogeniture is responsible for many of the greatest mathematical discoveries of Europe.

207: At this point reasonable people ask themselves "exactly why again was it so important that this variable be Gaussian?"

216: all variables must be Gaussian! If the world is not organized such that its conjugate priors are easy to work with, then the world is wrong!

Real analysis stinks. Complex analysis is much awesomer.

206: It would be even better if the swear-words were used with technical meanings (so that a "fucked complex root" wasn't the same as an un-fucked complex root).

217: I think Mr. Jaynes needs his pills now, nurse.

Complex analysis was relatively a piece of cake, which never made much sense to me.

Although the thread has moved on a bit, I'd like to throw in a bit more anecdata from a current student (though in another country). In the literary-style papers, in Spanish, my experience does not at all match with what people have said about participation. Far and away the majority of class participation is from women, particularly the unsolicited contributions. In mathematics and CS papers they're disproportionately active as well, particularly the latter.

I don't know if that's a cultural difference, or particular to the institution I'm at. The comments above have been strikingly unfamiliar.

221: Holomorphy, dude. It's all about teh holomorphy.

Analysis stinks altogether. Stupid Riemann zeta function! You don't make no goddamned sense.

How can it be concrete and abstract?

216

At this point reasonable people ask themselves "exactly why again was it so important that this variable be Gaussian?"

Well as I said the issue of whether it is sensible to force IQ to be Gaussian is complicated but it is my understanding that it generally is effectively forced to be Gaussian.

225. Like Agassiz?

||

Finally got the "TeX Text" extension for Inkscape running properly on my Mac. Am spending more time making TeX-labeled illustrations of research results than generating the results themselves.

|>

Are you sure analysis stinks? Maybe you haven't given it enough time. Me, I say 15 years is about right.

Completely off-topic, but am I understanding the non-competition clause in this EULA correctly? As far as I can tell, the non-competition requirement expires one year after first agreeing to the EULA, regardless of whether you've terminated the agreement. Or do you have to formally terminate the agreement in order to get out of the non-competition clause (after its expires, of course). I'm wondering if someone who signs up for a free trial account, doesn't use it, but doesn't formally cancel it, will be subject to the clause in 2 or 3 or more years because the account still exists.

I don't plan on creating a competing product within the next year, mind you. But who knows what the further future could bring?

ANALYZE IT? FUCK THAT SHIT! I WANT TO FUCK IT!

229: That's funny, because I happen to have Stefan Banach right here.

Even more off-topic than 230, but since when does one have to sign up for the Guest Rewards program in order to get free wireless at a hotel? (The only alternative is paying $10/day.) Then they have the nerve to demand everything short of your firstborn child as demographic data for the rewards program. Bah.

I'm really fuming about this. Pay $240 a night for a hotel room* and THEN pay for wireless on top of it? What's next, a la carte pricing for electric lights?

*Yes, I know it's outlandish, but I'm at a conference and it made logistical sense.

233: It seems like every cheap, no-frills hotel I've stayed in in the last five years has free wireless, and any hotel with pretensions of eliteness charges for it.

Even more off-topic than 230

Hey! There's only so much room for off-topicness.

233: Lie out your nose.

Dammit Cosma. Your linked blog is pretty damn interesting, but I've already got more than enough nasty equations and concepts to slog through at the moment.

It's a bit scary trying to go back and learn some of these areas outside of class. Without the guidance and curatorial role of a professor, I always feel a little lost digging through the original literature. There's a major fear in the back of my mind that, after spending several nights puzzling through a paper dealing with unfamiliar literature, unfamiliar methods, and interesting results, that I might later find the paper's results were later disputed and feel most of my effort was wasted.

I really will have to go back to statistics though, dig into the rift between and thought behind Bayesians and Frequentists. I have a pretty decent intuition for probability (combinatorics and graph theory was certainly among the areas I was best at in school), and I thought I could grasp the use of statistics in most of finance, but your blog's left me feeling that I must be missing something huge that I haven't noticed these two different ways of viewing probability. More realistically, I think surprisingly few of the practitioners/professors actually have decent insight into the underpinnings of the statistical analyses they're doing (not being statisticians since I rarely deal with econometrics), but that I should get to know this stuff well if I ever want to produce proper criticisms.

Learning is way too hard.

It looks like Tom Friedman will be doing his own laundry and brown-bagging lunch pretty soon. He may have to moonlight cranking out extra metaphors freelance, just to get by.

If this is the off-topic thread, I have to say I'm in love with my maple glazed short ribs. Apparently procrastination really does have excellent benefits.

235: Not so! The set of all off-topic comments in Unfogged is dense. I leave the proof of this simple theorem to the reader.

236: Lie out your nose.

Continuously.

re 230: Maybe I should just cancel my account and not worry about the EULA beyond that. I'm really kind of annoyed that the news service that recommended the site (which tracks versions of webpages) didn't mention it wasn't free. I suppose I should have paid more attention before clicking through.

Lie out your nose.

Actually, the last time I had to register when I didn't want to, I said I was born in 1900 and it retorted, "You are too old to register." !!

I had to change it to 1920.

234: Now you tell me. Rats.

235: Very funny.

240: Plus it's a big blog. Very big blog.

I dunno, Stormcrow...it'd take one hell of a meetup.

234: Now you tell me. Rats.

Well, yes, there may be those in the cheaper hotel, but you decide which is more important: free wireless or freedom from vermin?

Are the vermin gratis?

I remember math through multivariable calc being fairly easy, finding linear algebra simultaneously easy and incomprehensible (problems easy - take set of basic rules, work from there; understanding what the hell we're talking about - incomprehensible). I slammed into a complete wall with Abstract Algebra. I also had serious issues with my intro freshman honors physics class - the math was fine, simple calc at most, but the long proofs - ugh. I wasn't alone, some two thirds of us got an F on our first midterm. My poor parents - the math dad, physics mom, so disappointed. Not that they had much in the way of illusions by that point. As for making lower math hard - my IB Maths Higher math teacher had to give all his students a full point plus for college transcript purposes - In 13th grade I think one person got a 5.5 (B) 3 of us got 5's (B-) and down it went from there. (The teacher had the attitude that if anyone got a 7 or 6.5 on a test, he'd clearly fucked up, but also didn't want to penalize us kids applying to US schools.) The girl with the 5.5 couldn't pass a standardized multiple choice math test to save her life, btw. She was absolutely brilliant at math, but didn't have whatever skill it took for your SAT type of thing.

I found math up until calculus painful and unfriendly, math calculus through linear algebra fun and easy, differential equations fine, although so. much. writing. and vector calculus mostly fun. But that's as far as I've gone.

The introductory statistics class I took online through a respected university's continuing ed/extension program was easily the worst math class I've ever taken. Also, the easiest, mostly because it was all about choosing equations and tables and plugging in numbers and not at all about why or how those equations and tables came to be used. I can't believe it's the equivalent in units of a regular college class, but they say it is. But now I can wave a piece of paper to satisfy a prerequisite (which I probably won't even need, since I've changed plans a bit) so that counts for something.

229: That's funny, because I happen to have Stefan Banach right here.

You know what's bizarre about that scene? McLuhan says to the bloviating dude, "you're saying my whole fallacy is wrong!". What?

Our vector calc prof was hilarious. He would jump up on the desk and cry out "I'm the normal vector!"

This was made more amusing and precarious by his advanced age, fierce comb-over, and googly eyes.

I doubt that noncompete clause is worth the pixels it's printed on. Just some dipshit lawyer trying to justify his existence. You might write them a nasty letter saying that you are telling everyone you know not to use their product, that you hope they do sue you just for the awesomely negative effect it will have on their sales, and that you hope that they didn't spend too much money on the lawyer that drafted their EULA.

IANAL and all, but seriously. If you're planning to make billions selling software that does the same thing you might want to talk to a real lawyer, otherwise fuck 'em.

230: Given that noncompete clauses in employment contracts are pretty much worthless, I'm with water moccasin.

254-255: Thanks. Stuff having to do with archiving of web pages is one of the few areas of web-related work I could see myself getting involved in in the future. But not at the level of billions in profit, anyway.

The introductory statistics class I took online through a respected university's continuing ed/extension program was easily the worst math class I've ever taken

Nearly all introductory statistics courses are worthless, mainly because they are usually taught by mathematicians who think that statistics is just applied probability (not the single best method to confuse oneself, that's believing that engineering mathematics is going to be useful in econometrics, but nonetheless a sure-fire path to confusion).

In general, I have never, ever, found any branch of mathematics that I needed to use, that couldn't be learned in a couple of weeks if you were prepared to go through the pain of doing it. The doing of loads of problem sets is the only reliable way to pick something up if you're not in the habit of doing maths all the time (remember; how did you learn long division?).

219: I have attempted to drag game theory in this direction with my eponymous extension of the Folk Theorem. So far, game theory has proved resistant.

255 -- The worthlessness of a particular clause is pretty fact dependent, the most important fact being geographical.

255 -- The worthlessness of a particular clause is pretty fact dependent, the most important fact being geographical.

131: My sense is that I have a good genetic verbal ability that allowed me to sort of rephrase math techniques verbally and do that that way, but then the ability to do that runs out at a certain level of complexity. I can still *talk about* math in a way that gives some impression of facility to the uninitiated, or verbally describe what an advanced math technique is *supposed to do*. But I can't fully grasp it.

Hah. This is pretty familiar.

137: This is not at all the case for me. I didn't do that great on the math SAT, (for unfogged standards I practically failed it with a 1420 or something, PRE-realignment, of course.) But I never flamed out taking higher and higher math courses, even though the level of abstraction seems to keep ratcheting up indefinitely.

This is precisely what I was thinking of. In terms of serious math aptitude, you've (Heebie, that is) clearly got a whole lot of stuff I don't. In terms of accurately doing easy problems at speed (SAT, Math Achievement), I can (or could as a teenager) kick your ass. The skills are probably related somehow, but they're really not the same thing.

I don't know tensor calculus which I believe is a prerequisite for general relativity.

And which, by coincidence, is the topic that convinced me that being a physics major was a poor idea.

Also, I wonder if LB really was able to master DiffEq (no offense, LB). The whole thing of DiffEq -- mixing differential equations, linear algebra, and matrix algebra, truly being able to manipulate equation systems easily -- was to me a real sticking point.

Probably not, depending on what "really master" means. All I was saying was that an initial DiffEq class wasn't remarkably difficult or traumatizing. But I'm sure there was much more to master than I ever knew.

197: O, lord, the rotating molecules. I am not certain if I'm actually mentally deficient in this area, or what, but I couldn't visualize them to save my life.

I've never really understood what this difference is supposed to refer to. I test well on anything that's supposed to test spacial ability, but I can't actually see anything usefully in my imagination -- I'm brute-force figuring stuff out: "This side is next to that side, which means this bit pops out on the left..." I'm not clear if anyone is actually just watching the movie in their heads to pull out the answers.

'm not clear if anyone is actually just watching the movie in their heads to pull out the answers.

Yes. I thought that was how everyone did it. There are loads of basic cognitive psychology experiments -- the sorts of things that they give undergraduates to do -- where they measure the speed of people's answers to those sorts of rotational problems. The results suggests that [many] people really are just rotating little images in their heads. It takes longer to answer problems that require a 270 degree rotation than it does to answer problems that require a 180 degree rotation, etc.

This is one of those tasks beloved of gender essentialists, since, afaik, men seem slightly better at using the 'just rotate the little picture in your head' model.

FWIW, I basically hit the effort/benefit wall with maths around first/second year university level. With high school maths I got really high scores and was able to work really quickly and with minimal effort. At university level it required effort and I lost interest.

The times I've had to use 'mathematical' techniques for my academic work -- some stats, some probability stuff, some 'game theoretic' things, some geometric stuff when I had to write a paper on optics, etc. -- I've never had any problem teaching myself how to do it, though. I don't lack the basic ability, I just lack the motivation to really stick at it unless I genuinely need it.

Whereas with other skills, I'll happily pursue them just for the hell of it.

261: I've seen those studies (on additional time to rotate through a greater angle). What's puzzled me is that talking to people who say they can usefully see models in their heads, they don't, IME, seem to be able to easily answer the type of question that would be obvious if you had a real model to look at.

The specific question that led me to wonder about this was figuring out, in Samoa, the relationship between the phase of the moon and the time of moonrise and moonset, and then talking to other people about it. (It's one of those perfectly obvious things that I'd never thought about living in a city where the moon doesn't matter; living someplace where it was dark at night, the pattern was obvious and then once I thought about it the geometry made sense.) For someone who doesn't know the relationship offhand, figuring it out should be pretty effortless if you've got a 'model' to look at. No one I've talked to who didn't know the answer beforehand, though, seems to find it easy in the way I think it would be if they were really getting useful information out of visualizing the situation.

I'm definitely a "see the model in my head" person. Trying to predict the phases of the moon is a perfect example of this.

they don't, IME, seem to be able to easily answer the type of question that would be obvious if you had a real model to look at.

I certainly make mistakes, but I'm definitely using primarily visual muscles in my brain.

re: 263

I'm not sure if that's a good example. In fact, it seems sort of a crazy counter-example. You're talking about the relationship between three bodies -- the earth, the sun, and the moon. Not the properties of a single shape that's easy to hold in one's 'mind's eye'. It requires some astronomical knowledge to know how to form the model in the first place.

With three dimensional solids of the type used in the classic experiments I really do rotate it in my head. The phenomenological experience is one of rotating it.

I feel like the following can't possibly be true, that it must just be power of suggestion from some psych class, but I swear it feels like the visual image resides in my visual cortex. Like if I had to assign a concrete location to the image, it would obviously be in my forebrain. But it seems bizarre that your brain would have its own (proprioception? is that the word?) and that you could feel where in your brain a process is occuring.

re: 266

Your visual cortex is at the back, though.

ttaM and I do not vote the same way on the Visualizing The Planets referendum. We might have to have a run-off.

267: A-HA! Oh.

re: 2688

Oh, I do visualize it that way. But only once I'd learned the astronomy. It's perfectly possible that people i) really do use visualization in solving particular geometric problems, but ii) don't use it in particular cases because they don't have enough factual information to form a visual model.

Rotating simple 3-D shapes that you have been shown [as in various IQ tests, etc.] isn't like that. It doesn't really require any additional information.

What astronomical information do you need beyond "The Earth revolves around the Sun every year and rotates daily, and the Moon revolves around the Earth every month"? I suppose it's possible not to know that, but I don't think I've every talked about this with anyone who wouldn't have.

I don't know. Perhaps I just didn't intuitively connect those basic facts up to form the appropriate model until i'd already seen the model represented somewhere [when I learned the basic astronomy].

I guess what I'm really wondering is whether people who have the phenomenological experience of rotating shapes before their minds' eye are strongly superior in objective measures of getting the right answer on 'spatial problems' to people who don't. I don't have that subjective experience at all, but I do pretty well on answering the sorts of questions for which it would be useful.

273: I was totally shocked the first time I taught undergrad Linear Algebra to discover that everyone wasn't just picturing everything. I think that Linear Algebra is a really hard subject to do verbally, and a really easy subject to do visually.

272: Well, that's kind of my point. The quoted sentence has all the information the necessary model contains -- that is, if you had an Acme Adjustable Home Orrery Kit (work with me here) you wouldn't need to know more than that to set it up to accurately represent the Earth-Moon-Sun system. And with a physical model in front of you, the relationship is beyond obvious; but it doesn't seem to be obvious from a mental image of the relationship.

The sort of image visualizers build in their heads seems to be significantly less useful, or more precisely to contain significantly less information, than a physical model would. This isn't important, it's just always interested me.

The Earth revolves around the Sun every year and rotates daily, and the Moon revolves around the Earth every month"?

Well, fwiw, I've been trying to picture this problem since you first mentioned it in 263, but I didn't know this. I thought the moon revolved around the earth daily. I've never really thought much about lunar phases, I guess. I thought they were related to the relative positions of the sun and the moon.

274: Sort of related: the first real musical instrument I was trained on was the piano, and so I visualize scales and intervals and keys and the like on a keyboard.

Several years later, when I started playing the viola, it was hard for me to figure out how people without a piano background were thinking about these things, but many of them seemed to have a stronger and faster grasp on, for example, transposing things into new keys than I did.

Part of it may have been that I don't have much musical talent beyond being decently competent at it, but it felt like I had to go through more steps to figure these things out than non-piano people did.

274: Huh. And I had an easy time with Linear Algebra (assuming I'm remembering the course-names right. That's matrices, right?), but I can't imagine what there is that you could picture.

I thought they were related to the relative positions of the sun and the moon.

An important first step.

274: my college instructor tried to teach the whole Linear Algebra class visually, and I'm not sure I ever understood what the hell he was talking about. (I think I sort-of almost grasped it at the time, but it's totally lost now.) Whereas the class seemed to me relatively easy to understand verbally (as systems of equations).

278: I have to run to teach, but if the conversation's still alive later I'll say more.

Also I haven't heard back from the colleague about his experiences interviewing the seniors yet.

278, 280: This is probably how we both ended up in law school.

For someone who doesn't know the relationship offhand, figuring it out should be pretty effortless if you've got a 'model' to look at. No one I've talked to who didn't know the answer beforehand, though, seems to find it easy in the way I think it would be if they were really getting useful information out of visualizing the situation.

I just thought it through, by visualizing the earth-moon-sun system, but it took a while. I'm not sure if it would have been faster actively trying to avoid visualizing it. Somehow composing a mental image and holding it in my head for long enough to manipulate it and figure out the answer is not something I can do quickly. (And there's extraneous information that it takes a few moments to figure out you don't need; like, the only important thing about the sun is that it's a light source hitting the earth and moon from one particular direction.)

I don't think I usually visualize things in detail when solving problems. There are some images that accompany the process but not a whole visual model that I hold in my head all at once.

It's a very useful sort of image if you're talking about what a molecule looks like from behind. Interestingly, though, I think that's a different skill from visualizing the earth-moon-sun system. You're not asking someone to imagine what the view looks like from behind the moon. (Well, spheres are easy.)

This is the sort of data gender essentialists love, because it's supposed to prove that men are better at fixing cars or something like that, but from my own experience, it's a very, very circumscribed ability that hasn't had much of a effect on my life besides a) meaning I got a B on that o-chem exam and b) ensuring that there is a class of video games that is very, very hard for me.

So, wow, the moon rises and sets each night due to the daily rotation of the earth around its axis, not because it's revolving around the earth every day? That actually makes a lot more sense.

re: 277

Yeah, I'm a guitar player, but had a brief foray into saxophone when I went through a 'jazz' phase in my late teens. It was an eye-opener. A whole new way of visualizing music.

re: 275

Yeah, but I'm just not sure that there's a one-size fits all answer. I think I'm a bit of a jack of all trades, master of none, intellectually speaking. Sometimes I use visual methods, sometimes I think verbally, sometimes quantitatively. What's easiest for one thing might not be easiest for another, etc.

For certain types of geometric problems I just 'see' them in my head, and really do do the mental rotation, etc. For some others, I immediately reach for formulas or equations or some verbal representation, etc. Furthermore, even in cases where I reach for non-visual tools, I often intuitively known why the tool works because of some visual model that I have.

I presume we all sort of work that way -- mixing things up -- with the exception of a few people who are extremely visual, or extremely verbal.

278: Every matrix can be broken down into a translation, a rotation, and a scaling. Those are all the facts you need to build an entirely adequate mental model to visualise all of linear algebra. (Err... maybe... but I do find linear algebra very visual.)

278, 274: Understanding that a set of eigenvectors is like a coordinate system for the matrix helped me a little bit in terms of understanding what an eigenvector was or why anyone might care about one, but from then on in, I found it much easier to just deal with it symbolically ("verbally"?). Similarly with stochastic calculus, where drawing little diagrams of blobs and lattices helped a bit to begin with, but I ended up chucking them away.

I'm sure Heebie will be back later with some professional guidance, but here's a (very) basic short introduction to linear algebra that highlightes some (very) basic geometric interpretations.

Every matrix can be broken down into a translation, a rotation, and a scaling.

A translation?

I don't visualize linear algebra; the pictures don't generalize so well once you want to do similar things where the entries are complex numbers, or in relativity where you're not in Euclidean signature, or if your "eigenvectors" are really eigenfunctions of some operator, or.... (All the math does, but the pictures don't. I suppose if the pictures serve as a mnemonic for the math for you, you could still make it work in some way.)

Similarly with stochastic calculus, where drawing little diagrams of blobs and lattices helped a bit to begin with, but I ended up chucking them away.

Huh, I'm afraid I'm not familiar with this style of teaching stochastic calculus (though I've mostly had to self-teach bits). What do they use the blobs and lattices to represent?

Oh yeah, and I also wanted to thank you for the recommendation of The Politics of Large Numbers. I've only gotten to flip through it a bit, but it looks quite good. It's currently sitting on my coffee table waiting for this damnable term to end.

All the math does, but the pictures don't.

essear only buys linear algebra for the articles.

essear only buys linear algebra for the articles.

Really, I have no idea how that box in the top of my closet got there.

What do they use the blobs and lattices to represent?

Basically if you approach it the easy way by discrete stochastic calculus (or if you're learning it for option pricing, the Cox-Ingersoll-Ross model), you end up drawing a lot of decision trees and writing out discrete probability measures on them. "Financial Calculus" by Baxter & Rennie is the textbook I used.

"Flipping through" that book is probably the best thing to do with it - I wasn't joking when I said that it was written in dull, pedantic French and then expertly translated into dull, pedantic English.

I suppose if the pictures serve as a mnemonic for the math for you, you could still make it work in some way.
This is the way it works for me. Being able to visualize what "invertible" or "ill-conditioned" means in two or three real dimensions allows me to feel like I have more of an intuitive grasp of more complicated problems.
Regarding learning math outside of a classroom, I've only had success when there was a very specific problem I wanted to solve. This would serve to focus my attention and provide motivation for an otherwise arbitrary-seeming set of rules. Otherwise, I get hopelessly frustrated and bored. It seems to me that too many math textbooks and papers present math completely differently than I learn

Oh, and I can't imagine a way to solve Lizardbreath's moon problem other than visualization.

Apparently some people call the universal constants Feigenbaum found investigating chaos "Feigenvalues." Which is totally cute.

I wish I had a tiny, beautiful desktop orrery. I am not bad at visualizing rotations of 3-D objects, or doing those problems where you have to figure out what a flat object will fold up into, but every time I want to remind myself of the relationship between the time of moonrise and moonset and the phase of the moon, I have to re-enact the scenario, usually by recruiting two friends. Then we all spin around (with great dignity, of course) and walk around each other and it makes sense again. There's something about visualizing that one that is the mental equivalent of patting my head and rubbing my stomach, somehow.

Acme Adjustable Home Orrery Kit

wantwantwant

— I often have some sort of mental image connected with a math problem, but I don't use it, if that makes sense. That needs either algebra or words, or a physical diagram I can manipulate and re-draw. (But when I'm doing algebra in my head I often find myself visualizing what the equations would look like, written out.)

— Re "blobs and lattices", I imagine d² is talking about something like the binomial model for evaluating functions of price changes (at each time, the price can go up a tick or down a tick, so you draw your lattice of possible paths and sum over paths).

Oops, crossed post with 294.

296: The way it works for me is talking through cases: "So, when there's a full moon, where does it have to be in relation to the earth and the sun to look 'full' from the surface of the earth. Okay, got that. Now a half moon. Okay. Now that I know where the full moon is, when is it going to rise and set? Okay, now a half moon. Oh, I see how it works, let me check by thinking about a new moon."

293: On the one hand, we know from Hilbert that geometry is really all about tables and chairs. On the other hand, this. Constructing the actual joke is left as an exercise for someone who doesn't have to run teach.

(And once I know the answer, I can sort of visualize it. I just can't learn anything from the visualization I didn't know without it.)

301: Using LB's exercise, I have just convinced myself that there should be one lunar eclipse per month. Astronomy: I'm Doing It Wrong.

No, you're doing it exactly right -- it's just that the spot for a lunar eclipse is tiny and everything's wobbly enough (technical term) that it usually misses. But yes, if all the orbits were perfectly in the same plane, you'd get an eclipse every month when the moon was precisely full.

271

What astronomical information do you need beyond "The Earth revolves around the Sun every year and rotates daily, and the Moon revolves around the Earth every month"? I suppose it's possible not to know that, but I don't think I've every talked about this with anyone who wouldn't have.

You also need to know that the moon is much closer than the sun, that the axes of rotation of the earth, the earth around the sun and the moon around the earth are roughly parallel and that all the rotations are in the same direction.

Are you visualizing those individual cases? And just not letting the model spin through the various configurations? If so, I guess that makes sense: using arguments to fill in what I would think of as the blanks.

the moon is much closer than the sun, that the axes of rotation of the earth, the earth around the sun and the moon around the earth are roughly parallel and that all the rotations are in the same direction.

All true, but all the sort of thing that most people thinking about the problem would know offhand.

re: 306

Yes, quite. I had that sort of thing in mind, and then forgot to bring it up.

307: Subjectively, I 'figure it out' and then visualize (to the extent I do at all) after I already know the answer.

309: Maybe I was a more astronomy-interested kid than most (well, I know that was true), but wouldn't anyone who'd ever seen a diagram of the solar system know all that?

the other day i saw almost horizontal moon serpent, never mentioned that before, it was like always kinda vertical when i looked at it

The sort of image visualizers build in their heads seems to be significantly less useful, or more precisely to contain significantly less information, than a physical model would.

Well, of course. It takes some time to set up the mental model, just like it does to set up the physical model. And given the time it would take to scrounge up materials for the physical model, the mental model is usually faster.

For instance, it's not immediately clear from a) the earth rotates, b) the moon revolves around the earth, and c) the earth revolves around the sun how those things all fit together. That's where the model construction comes in. Does the earth rotate in the same direction as it revolves or the opposite? Does the moon rotate in the same direction as the earth rotates or not? I do remember the realization when I had set up the model correctly that the moon also orbits the sun in an orbit that is nearly the same as the earth's, just with a little wobble for the revolution around the earth.

And given the time it would take to scrounge up materials for the physical model

Which is why the Acme Adjustable Home Orrery Kit would be awesome!

311: What's a moon serpent? Google isn't particularly helpful, although it does turn up this suggestive, but inaccessible article.

BTW, I use basically the same procedure as LB to set up the mental model (i.e. limiting cases, which are also good for equation graphing), and then animate to fill in the gaps.

||
The movers have come to cart our belongings over to the East. Classic big guy-little guy team. I am afraid it may end in a whacking, it is so cinematic. Lars, the big guy, has a fully shaved head.

|>

I initially read that a Home Ornery Kit, which it seems that many of us have in our domiciles.

This looks nice..

serpent
i thought it's called that and i saw a reversed to this and narrower moon

318: Oh man does it ever!

319: Oh. That makes sense.

Oh, and I can't imagine a way to solve Lizardbreath's moon problem other than visualization.

I do it purely verbally: "full moon rises at sunset, new moon at sunrise," and derive what information I want by interpolation: "that means first quarter at noon, last quarter at midnight," etc. I can visualize some parts of the orbital relationship, but can't solve problems with said visualization.

Not only nice, but affordable! All I found was this, which is nicer, but price-r.

215: One of my historically-minded profs claimed that through the 18th and 19th centuries math talent was transmitted from father to son-in-law. Not primogeniture so much as a vicar-and-curacy.

I think it's because of the moon serpent/dragon that "solar eclipse" in Japanese, going by kanji, is "sun-eat" (and same with lunar, mutatis mutandis).

re: 323

I found that that site stocks astrolabes the other day, and it was hard not to buy.

The types of things that I would ideally want my undergrads to visualize in a Linear Algebra course are things like: The span of two (lin indep.) vectors is a plane. If three vectors are linearly dependent, then they've got a linear relationship between them, and visually, what does this redundancy look like? Why will it result in the span of those vectors having a lower dimension than 3?

If they're verbal thinkers, than I'm sure they can still fully understand the material verbally, but it surprised me that not everyone reverted to a picture for these ideas.

Huh. I'm far enough away from this stuff that I don't quite understand what you said there (that is, I think you're saying that three linearly dependent vectors are co-planar, but I can't remember quite what linearly dependent means.) But as your sample non-visualizer, I'd be able to answer questions like "what does this look like" without much difficulty; it's just that figuring that out would be an endpoint, rather than a natural process I'd go through, if you see what I mean.

Okay, I just looked up 'linear independence', and now that I'm reminded, yes, I wouldn't spontaneously think of that spacially. I'd spontaneously think of it in terms of whether the third vector is redundant of the first two, or adds new information (this is garbled by the fact that while I dimly recall the subject, it's pretty dim.) The spacial representation makes sense, but I wouldn't go there on my own.

Exactly - I assumed that students would use those visualizations as a guide for their intuitions, and many of them do not. It's fine with me if they understand the material through any framework that works, but I hadn't realized that not everyone reaches for the visual.

(Linearly dependent means that there are non-trivial constants c1, c2, and c3 such that c1v1 + c2v2 + c3v3 = 0. Visually, right - this means that they are coplanar or colinear.)

294, 299: Oh, so the lattices refers to the binomial tree discretization? Alright, not sure why that didn't come to mind. I think I've got a firm separator in my mind between the tree methods and the calculus methods (especially since the latter involves so many damn multi-factor problems which my spreadsheet doesn't have enough dimensions to contain). I guess lattices could also refer to finite-difference methods, which I really need to get sharper on sometime in the next... 5 days.

323.2: I heard that line too, but with a specifically German application. As in, who does the aging Herr Prof. Dr. recommend take over his chair, of the many privatdozenten available who'd like it? Why, his son-in-law, of course.

Doesn't anybody want to talk about swimming?

Something here recently (in relation to the basketball stats) made me look up nullspaces, and I happened upon this set of lectures by Gilbert Strang from a Linear Algebra class at MIT. I watched the one (to my son's incredulous horror, "You're using YouTube to watch a lecture on the most boring subject in the world?") and found it quite engaging for Linear Algebra. If you are someone who learns better from lectures than a book (and are also willing to do some problem sets, per d² above) you could do worse.

Man, I find it so frustrating thinking about old math classes; it's maddening having known something that I no longer know -- like having a missing tooth. Anything past calculus is pretty much gone, and I'm sure calc is shaky as well.

Not that I have occasion to use math more complex than doing present value calculations for settlements. But I hate having forgotten things. (Oddly, no longer being able to read Latin at all doesn't bother me.)

Is Heebie one of those Gaussian mathematicians who collapsed the international financial system? I say stone her.

The Gaussian orientation of the Anglo-Dutch consortium is well known. The Knights of Malta, whom the Consortium controls, has a secret training school for Gaussians. Someone should ask Heebie about this.

Is Heebie one of those Gaussian mathematicians who collapsed the international financial system? I say stone her.

I'm not supposed to get stoned for another seven weeks.

I'm not supposed to get stoned for another seven weeks.
What happened the last time?

336: Hmmm, seven weeks is right around your due date. You've got one of those hippie doulas, don't you?

336: Sucks to be you, man.

That reminds me, when is the Unfogged Lottery Day?

Nothing about Gauss was Normal. Go figure.

I HAVE A VERY STRONG DESIRE NOT TO FEEL SO ALL ALONE!!!!!!

335: Emerson, haven't you been told to lay off reading Executive Intelligence Review? (Did the LaRouchies change the name of 21st Century Science and Technology when the century changed?)

For the record, the problem using Gaussian copulas to model correlations isn't the Gaussian bit, it's the copulas.

What do bell towers have to do with anything?

344: They're very cinematic.

344: These copulas. Of course since it's Wired, all the quotes might be made up to be provocative (but Salmon is usually more reliable than that).

The secret Gaussian initiation requires copulation with one of the senior Gaussians Knights of Malta, some of whom are approaching 100 years old.

347: Does the 1.96^th degree of initiation feature zombie Pierre-Simon Laplace and his vast and considerable ... intellect?

Man, I find it so frustrating thinking about old math classes; it's maddening having known something that I no longer know -- like having a missing tooth.

Tell me about it! So much of my undergraduate classes is completely gone now. All that remains of analysis are names: I know there was this thing called the Arzela-Ascoli theorem, but I couldn't begin to tell you what it says. There's a bit more algebra, and still quite a lot of geometry and topology, that I haven't completely lost yet, but still, that was a lot of time spent on things I don't use and can't remember. Sigh. At least I was having fun at the time, I guess.

No seriously, swimming is a fascinating sport.

Sigh. At least I was having fun at the time, I guess.

AHA! So you were looking at the pictures!

350: with many intriguing applications of partial differential equations!

I used to invert matrices in my head whilst doing laps.

and his vast and considerable ... intellect?

I remember watching Young Frankenstein as a kid one time on network television and for the line when Inga says "He will have an enormous schwanstuker!" they dubbed in "personality".

354: Actually I think I meant "schwanz-stuke".

Ah hell I don't know how it's spelled. Damn German.

It must be impressive if it's a Schwan Stuka

357: The terrifying high pitched whistling takes some getting used to, I can tell you that much.

I AM BECOME SWIMMING COMMENTS, THE DESTROYER OF THREADS.

Man, I find it so frustrating thinking about old math classes; it's maddening having known something that I no longer know -- like having a missing tooth.

I think of old maths classes more like ex-girlfriends - they were perhaps the right thing for me at the time, but we were really going in different directions and they have nothing to do with my present life, probably for the better. And I am sure I could go back to them at any time I wished, with no complications and my wife wouldn't object. So not really like ex-girlfriends at all, when you come to think of it.

Who knows, your wife might object to a reawakened interest in obscure math. So, not so different from old girlfriends.

Thinking about it, this would only really be an analogy if my missus had taken all my old maths classes, and got better grades from them than I did.

362: Wait, are you saying that your wife had relationships with all your ex-girlfriends, and they were better than your relationships with them?

I can see we're going to have to start calling you "Sherlock", Cosma.

That's a fascinating anecdote in terms of gender differences in math aptitudes, certainly.

364: "calling you Sherlock" s/b "getting you to write for Standpipe's blog".

366: I can barely write for my own. (As for "Sherlock": one of my exes became convinced that I really had sub-clinical autistic spectrum disorder, and there are definitely times when I wonder if she wasn't on to something...)

The male colleague who I e-mailed does not notice any gender gap. He was in a coed group of interviewers, with one undergrad male and one female.

368: Yay, data! Though it doesn't help us get to the "teenage boys are uncomfortable being judged by women" versus "teenage boys are uncomfortable being in a room with so many breasts" distinction.

One could perhaps provide an equal number of non-judgmental breasts.

sub-clinical autistic spectrum disorder

I thought we'd all already learned that another word for this is, 'male'.

371: On the veldt, males would compete in proving elaborate but impractical theorems, the better to demonstrate their fitness to potential mates.

the better to demonstrate their fitness to potential mates.

Who would apparently be doing just fine amusing themselves without male assistance.

But the handicap-arms-race can't take off unless the females can judge male displays. So, on the veldt, females evolved into refined critics of mathematics, letting them sort actual theorems from gobbledygook.

No, pretty much I think you've proved that the human race became extinct in the Pleistocene.

On the veldt, males would compete in proving elaborate but impractical theorems

… and so you HAVE to sleep with me, it's all here in black and white!

one of my exes became convinced that I really had sub-clinical autistic spectrum disorder

One day, we men will invent a medicalised term for the average female personality type, and then armed with this weapon we will rule the world, monopolise the best jobs, earn 23% more than women, not do much housework, etc.

One could perhaps provide an equal number of non-judgmental breasts

If I could provide any number of non-judgemental breasts, I'd never get out of the house.

375: so there are some idealizations in the model. Didn't Uncle Milton teach us this was for the best?

377: I was thinking of plastic anatomical models, actually, but when I went to look for suitably disturbing pictures to link to, I see that the plastic ones have gone out of fashion in favor of digital images. Bah!

377: Please. You're being hysterical.

375, 378: In the long run we have always already been dead.

always already dead

Hot dense homogeneous plasmas become cold dilute homogeneous plasmas; the rest is commentary.

DON'T FORGET THE STAMP COLLECTING!

Then the cold dilute homogeneous plasma fluctuates into Boltzmann brains posting on blogs?

Very cold spread-out distributed Boltzmann brains, which think they are posting on blogs. "Had we but worlds enough and time, our threads might grow vaster than empires, and more slow."

Brains? I amuse myself. (I think.)

There is always the classic Borges story "Dream-Bloggers".

I totslly suck at visualizing. And when I manage to do it, I only see a very vague sort of half-there picture. It makes organizing stuff in physical space a trial, because I have a hard time figuring out how the bottles will fit in the closet until I've already done it.

Someone should have caught this when my brain was more plastic and helped me to develop it. I did well enough in math using verbal skills.

Talking out of my hat, here, Bostoniangirl, but I believe someone demonstrated a lasting improvement in spatial visualization after not-very-long spent playing video games that depended on it. Maybe you just need to play some Tetris.

(It's a useful skill to have. My mother once built a very large shed/small horse barn on uneven ground by laying all the pieces out on top of each other, plywood and posts and diagonal trusses and all, and making four cuts around the edges with a circular saw. --No, I lie, she had to flip some of the offcuts onto the stack, line them up, and make a second cut.

The trusses weren't necessarily 45deg, but they all fit flush to the beams. She can't explain how she did it, other than that those were the pieces that needed to fit together.)

It makes organizing stuff in physical space a trial

I cannot for the life of me tell you how a couch/picture/etc would look against a different wall without moving it. Figuring out how to rearrange a room is a long and arduous task that I always have to delegate to somebody else. I just don't think in spatial terms that way.

One of my exes became convinced that I really had sub-clinical autistic spectrum disorder, and there are definitely times when I wonder if she wasn't on to something.

Edgar Schneider, "Living the Good Life with Autism".

From a review:

Schneider wholly believes that his perspective is one of clarity, not disordered. Rather than misinterpreting the world as it is through the eyes of Asperger's, the rest of the world misinterprets truth through socially belabored ritual and miscomprehension.

Anand (who sees a map with locations for all the numbers and whom I am very sure is watching math movies in his head) and I were going over some graphs the other day. I appalled him by writing in words what the display was saying. "Huge flood" next to one data point. "Over here is expensive AND bad. No one likes this quadrant."

Clew, I am in awe of your Mom.

No one likes this quadrant

Mouseover text.

Well, she'd be in awe of you, Megan!

Not if she saw me trying to assemble a shed. I could, however, carry beams for her. She'd have to point to where they should go.

389: The experimental evidence is that playing first-person shooter video games improves this sort of spatial ability. I'd be surprised if Tetris did, because I went through a phase of playing it obsessively, but still suck at this sort of thing.

The only thing my mother likes better than someone who will carry heavy beams is someone who will haul rocks. Srsly, if you're ever on the Olympic Peninsula and want pie, get in touch with me, I'll set you up. Heavy labor not required, but of course it helps you pack down more pie.

The spatial task that FPSs improved doesn't seem to involve any spatial rotation, just memory (of a kind useful in a FPS, but less impressive than the memory game in Kim). I'd like to see if an actual spatial rotation task got better -- there are ways of understanding, say, knitting that are very spatial-rotation-exercising, and no evidence that women are incompetent at it.

I think I have mentioned my ambivalence about my skills here before.

YAY!!! There is something I can do to earn pie in the Post-Collapse future!

Boo!!! It is carrying heavy things.

I would love to visit your Mom some time.

I'm pretty sure you'll be able to earn pie by 'splaining to other people how to reroute water. Washington has certainly had many problems with this of late.

I read somewhere that the only major improvement in the average HS student's test scores 1970-present was in spatial imagination or whatever they call it. I'm sure that video games are responsible. I wasn't that bad on the "which of these shapes is not like the others" tests, but I can see that if I'd played a lot of the right kind of game, I would have been a whiz.

I am now Google's major authority on "How/why did Iceland go bankrupt?"

Google-proofing Howwhy's name. Very clever.

399, 400: My father's family were the quasi-feudal landlords of some villages in Afghanistan (mainly, one called Shaliz...), and according to family tradition, they bargained their way into that as the people who could make the irrigation run. (I believe one of my grand-uncles was the last of us to actually get his hands dirty inside the kareezes.) So I foresee a bright, dynastic Post-Collapse future for Megan.

401: According to Flynn, kids have also been getting better at analogy problems.

This is a difficult thread to read only part of and still follow, but it seems on topic to say that I found it much easier to do non-geometry - I mean that in an informal sense, based on high school course titles - math when I had a sense of what the concepts looked like geometrically. Things like knowing what a derivative represents in space, or what an integral represents (I don't remember precisely any of this anymore, but you should be able to know what I mean here if you do remember this stuff) were initially explained with pictures, but it was all equations after that as far as the work went. But I spent some time trying to make sure I understood the geometrical correspondences. Or whatever the correct terms are.

I am quite good at rotating 3-D objects in my head. You definitely want me to be the person who packs your car when you move. And I'm decent at Tetris. But I have to interact with maps in a very specific way. For instance, if I'm plotting out a route that goes from north to south, using a map with the typical north-at-the-top layout, I MUST have the map upside down. That is, so that the path I'm plotting goes forward, away from me.

405: What would The Ogged (pbuh) say about that?

On the other hand, I absolutely sucked at the shells/washers stuff for integrals. You have no idea how glad I was to get to multiple integrals and no longer have to decide whether to add or subtract from something or other.

the people who could make the irrigation run

I should start working on ritual sacrifices that must be made to the priestesses before the gods will bless the fields with water.

Pi/e.

Also, curly-haired young men who will rise again.

the people who could make the irrigation run

s/b "make the drains run on time"

Those were exactly my thoughts. Want to be a priestess? I bet you'd make a good dowser, if youthe gods found the sacrifices satisfactory.

We'd gang aft agley.

Then the sacrifices weren't good enough!

[I thought this thread was dead.]

368: Yay, data! Though it doesn't help us get to the "teenage boys are uncomfortable being judged by women" versus "teenage boys are uncomfortable being in a room with so many breasts" distinction.

Wha? Kristina won this thread way back there (searching) at... 102:

Note that Heebs and I both are dealing with the ones that are expressly not going to the more highly-ranked local institution up the road but rather to the regional, cheaper, less rigorous ones. I feel strongly that most of the phenomena we're talking about is a product of local culture, but I'm a sociologist so I think that about most things. In any event, I have met and tried to teach a fairly large number of smart Texan males who think it uncool or uppity to seem smart or interested in school. I realize this isn't a trait specific to Texans in and of itself, but the way it gets played out here is quite specific.

The point is, is you're not supposed to be uppity, period. Further, you're are supposed to behave respecfully to a) your elders b) women. So these guys are walking into a roomful of women who are older, professional and (going by credentials), smarter, and as a consequence, they're shutting down, that is, shutting up. (This sort of thing is how you know George Bush is a pretend Texan. Way too egomaniacal. He can sorta cover it up, but not for long.)

The correct antidote (that I would use, if I were heebie) would be to turn on the female drill sargeant routine.

('Every young man that comes in here does the same thing and refuses to talk. So, son, lemme make it clear to you that we are here solely to hear what you have to say, we want to hear what you have to say, and will be satisfied with nothing less. If we didn't want to hear what you had to say, you wouldn't be here, and instead we'd be outside talking to trees; they don't say much, but they're pleasant to be around. So I expect you to give us full and complete answers and not waste our time with hemming and hawing. These are difficult questions; you may not be expert on much but surely after 18 years, you're an expert on yourself. So as long as you cooperate, I'm sure you'll do fine. Is all that clear?' Which should net a 'Yes, ma'am!' And that's a good point to add, 'Now sit up and take your hat off.' Injections of 'Very good!' with a big smile and 'You can do better than that' should keep the thing moving along nicely.)

407: For instance, if I'm plotting out a route that goes from north to south, using a map with the typical north-at-the-top layout, I MUST have the map upside down. That is, so that the path I'm plotting goes forward, away from me.

And for the far end of the curve here, there's me. Once I study a map, I don't really have to look at it much anymore, except to check some part I didn't study closely. I just visualize it in my head. I start out visualizing everything in my head, and only then do I dump it into the translator, so I can emit this chicken scratch stuff.

For example, the question was, 'what's that bright star?' To which I said, 'I dunno, but it's probably a planet. What time is it?' Given the time, I could relate the sun to my position and relate the bright star position to the sun, and then I could just click through the current planet positions to come up with, 'Ah-ha. That's Jupiter. Can't be anything else. See? That's Mars right overhead.'

Everything will go so much better when they get around to letting me plug into a machine directly.

max
['See? Graph!']

Once I study a map, I don't really have to look at it much anymore, except to check some part I didn't study closely.

Don't have to look at it anymore, like, ever? Or just not for a while?

I have decent mental maps of a lot of cities, but if I go to, say, Boston for the first time in a year or two, I have to look at the subway map to remember which branch of the green line goes where.

Boston is an odd, special case. It is impossible to learn the geography of Boston systematically; you must have it all in your head at once.

I have decent mental maps of a lot of cities, but if I go to, say, Boston for the first time in a year or two, I have to look at the subway map to remember which branch of the green line goes where.

[Long epic of memory checking deleted.] My memory tends to fade over time, like everyone elses, but if I am in a place I have been in before, I can find my way around and relate it to the map and vice versa. Ex-person would always get mad at me because I would 'take a different route every time' because I always knew where I was, and I was always trying to find a shorter route. During my deleted part there, I was again reminded that I can remember exactly the map I was looking at, and also remember exactly where I was, but can still get confused after a number of years because the map is wrong, and so that means I did something different than the map says and I remember that too! But not strongly enough after two and a half decades to immediately differentiate. Er, the map gets memorized, and the adjustments I have to make (because of errors on the map) also gets memorized and written onto the map on my head, and then the map in my head no longer quite matches the map I'm checking. Which can be annoying if I look at the map again many years later and it doesn't quite match the map in my head, which then makes me think I'm wrong, but I'm not!

max
['GRR.']