As you say, it doesn't resonate that much -- I'm only vibrating, rather than shaking -- because I teach in the social sciences. Still, curves would, in my courses, disadvantage a great number of students who actually master quite a bit of the material. So I rely on a healthy mix of grade inflation, giving more Bs than I probably should, and protecting the A. I then tell myself that the exceptionally bright kids, those who also get really invested in the course, will feel wonderful about being one of a very few to get an A. And the Bs, who swim in the vast sea of mediocrity, won't complain. Everyone wins.
Since you won't link the post, I'll ask here: what sort of curve do you mean? The sort where the distribution of grades is determined in advance, the sort where the top grade is taken as the reference A+, the sort where you kind of eyeball it so things come out "right", or what? I took two classes where (I'm pretty sure) grades were either entirely determined or heavily influenced by what quartile you fell into, and that didn't seem unfair.
(Once, one of the TAs was writing up the cutoff points for the quartiles, and put up the highest score, the quartiles, and the lowest score. The professor went over and erased the lowest score, but it was too late.)
I actually always thought that having tests average around 50% makes a lot more sense than having them average around 80% (or whatever that standard curve that you're hoping for centers around). It means that the exam actually has challenging questions that engage your ability to use the material, not just silly exercises that check your ability to perform with few errors under time pressure.
Did you know Curves is owned by wingnuts? It's true.
I think the problem is that the idea of what a given grade "means" is completely arbitrary. I mean, C is supposed to be "average," but what "average" means depends: average for this class? For this course? For students I have taught over several years? What if you need a C- to pass? Because "passing" does not mean "average."
It's all fucked up. Which is why grades are (should be) meaningless.
6: I had a HS teach who said that if anyone got 100% on his tests it meant that he had written a bad test because it failed to identify the limits of people's knowledge.
Seemed reasonable to me, but it seems like a hard sell for most people.
My most infuriating curve experience was my Engineering Ethics course and was the opposite of the usual problem. I thought I'd done really well in the class and got a 95 on the final but then opened my report card to find I had a B-minus in the class. I went to the professor and asked him to explain my grade and he said that everyone in the class had scored over 90% on the final and he'd applied a bell curve, making my 95 a C. This would have been ridiculous enough on a quantitative test but the final was all subjective short essays asking what you would do in certain situations ("Is it ethical to contribute to an open source project without the knowledge of your employer? Explain your reasoning.", etc.). It was the only time in my college career that protested something up to the department chair. I told him that if he was going to make us all take statistics as a requirement to get our degree, he had to hire adjuncts versed in it as well. I won the appeal and every person's grade in the class was recalculated.
Also, 7 is true. So the wingnuts get you whether you're eating junk food or exercising.
11: are you sure it isn't a conspiracy?
Does that mean a lot of people who got 90% or above failed the test? That sounds less like a test and more like a game of musical chairs.
The real issue here is that grades for undergraduates don't actually represent an accurate measure of anything at all.
C is not supposed to be "average" it is supposed to be "satisfactory." Reminding yourself of this will free you of the image that your grade distribution should look like a bell curve with the high point over a 75.
I've never used an explicit curve, and try not to do much of anything curve-like. I try to design the tests so that a 75 represents a satisfactory knowledge of the material. If too many people fail, I change my pedagogy, but not the test. I hope that doesn't sound preachy. I just think it is a part of "Best Practices" for teaching.
Becks, your engineering ethics course story is astonishingly fucked up. I imagine the teacher was an engineer, not a philosopher.
Christ, don't even get me started on undergrad science programs at big schools. Hey motherfuckers, put the med students in a separate cage like the preening rats that they are. Some of us are actually interested in the material and when you make us run your gauntlet to weed people out I get a serious urge to beat the professor with a pipe.
Undergrad bio and chem/biochem are much better taken at smaller schools and junior colleges.
Thing is, the curve will come in at one point or another. If no one masters more than 60% of the material, and that means the average grade in the physics department for a solid student is a B, then everyone's expectations just adjust downward.
7: Curves is run by wingnuts with weird ideas about the woman body, but they got my mother exercising in an environment with no showyoff girls and no mirrors, so I can only hate them a little bit.
6: I had a HS teach who said that if anyone got 100% on his tests it meant that he had written a bad test because it failed to identify the limits of people's knowledge.
Totally unreasonable. Tests shouldn't be written to the knowledge of the students but the material taught in the class. If some people kick ass, then maybe they should have enrolled in a more advanced class, but the class they're actually enrolled in is the one they're actually enrolled in. (Maybe I should say that one of the reason the quartile system I mentioned above seemed fair was that there was actually a fairly wide spread in scores.)
Which is why grades are (should be) meaningless.
Maybe they should be, but they aren't. And if they should be, they shouldn't exist.
One more thing: the prevalence of curves in the hard sciences probably has a lot to do with people not prioritizing teaching. The main reason people use curves is to make the outcomes of the course look right on paper, without having to go through the trouble of carefully designing a curriculum. People in the hard sciences are more able to get away with this, because their research actually brings money into the university, and is generally acknowledged to be important. Those of us in the humanities are under far more pressure to actually teach occasionally. I also imagine a fetish for nice clean bell curves drives some engineering instructors.
the idea of what a given grade "means" is completely arbitrary
C is not supposed to be "average" it is supposed to be "satisfactory."
See?
Why is average unsatisfactory, anyhow? IQ-ists.
16 - Actually, he was an engineer! And he wasn't even a professor! They just found some dude from industry who was willing to come in and try to brainwash us into being good employees for 3 hours a week. Other urgent "ethics" topics: class-long lectures on not using office property (computers, copy machines, etc.) for personal use and requiring that we give him a copy of all of our resumes so he could try to poach graduates before other companies had a chance. None of the profs in the department cared about that class -- they just had to make us all take something called "engineering ethics" to keep their certification.
23: that's unfortunate. Engineering ethics is quite vitally important. See e.g. Henry Petroski.
if they should be, they shouldn't exist
I would agree with this entirely; I'd be happy to do away with grades and move to written evaluations if it were possible.
Given that it's not, I wish people who traffic in grades were better at remembering, as you say, that a grade is a mark of one's mastery of some of the material in a given course at a given time, subject to factors like sometimes taking courses that are ill-suited to one's particular abilities, having a personal life that may well be incompatible with mastering course material according to the calendar, knowing a lot of things that aren't covered by conventional course work, and so on. I think it fucking sucks that we don't foreground that stuff rather than simply saying "grades matter."
10 is certainly a valid complaint. But as every assignment of grades is on one curve or another, I'm not sure what you're saying you want. Are you just saying that professors shouldn't make their exams so easy (or so hard) that one end is cut off making every attempt to force it into a bell-shape will be a disaster? (To be more explicit you want something like lambda being large in this picture.)
universities have a "teaching people stuff" function; they also have a "sorting people out" function--curving probably has more to do with the second function than the first...
Thing is, there are a lot of interesting topics you can teach in engineering ethics. Any engineering problem involves balancing time, money and safety. That alone means you have to deal all sorts of interesting problems. Can cost benefit analysis handle this issue? How do you quantify the value of human life or health? If not CBA, then some precautionary principle approach? Can the precautionary principle be phrased in some substantial way that doesn't reduce to CBA?
Also, any big engineering project has major impacts on society, meaning you need to think about social justice issues. I'll love to teach an engineering ethics course which uses the interstate highway system as its major case study.
24 - I agree. An engineering ethics class, especially one that talked about the tradeoffs between time, money, and safety could have been very good. Especially in New Orleans.
i remember getting 21% correct on an o-chem test, when that was the second highest score in the class (the first was not much higher).
i was pretty proud of that.
29: oh my. Yeah. Might have been a bit depressing, though.
29: Shit, that's a course proposal right there: Engineering Ethics and the New Orleans Levees. Teach the course four times and you have a textbook written that is a guaranteed sell.
An engineering ethics class, especially one that talked about the tradeoffs between time, money, and safety could have been very good.
I propose visual aids like that scene in The Machinist where a dude gets his arm ripped off.
30: See, if your second highest grade is a 29, you have to ask yourself "why do I expect students to know this much material by test time."
Most undergraduate teachers in the hard sciences are in love with the screening function of their courses. As far as they are concerned, they belong to an elite club, and supervising the hazing rituals is one of the advantages of membership.
The result is, of course, that the bulk of the population hates science and scientists, and is completely illiterate on any public policy issue that has a scientific component, which is, frankly, all of them.
32: and yet, still no way to justify rebuilding the city as it was. That's a puzzler.
34: given the job oppotunities available in the sciences, is that bad?
Sifu in 35: Issues in rebuilding the city is the second half of the textbook.
in 36: see the last paragraph of 34.
I'm going to defer to Becks here because a couple of comment threads that I wasn't following closely seemed to get pretty tense lately, and maybe that's partially her motivation, but assuming she's thinking of the same post about grading curves which I am I deem her non-linkage odd. Tortuously odd, even.
I had an economics class in which the professor decreed that there would be a maximum of two As and two Bs given out. Up to that point my anti-grading philosophy was through and detailed but not angry.
Afterwards -- given that the class had a half-dozen returning-to-college adult students who had to forfeit their scholarship money if they scored below a B -- it might best have been called volcanic.
The intervening years haven't done much to calm me down. Come to think of it, I'm edging my way towards JE's opinions about economists.
I read 1 as a joke about other times where people got bad at Becks for linking to McMegan, not about where she read the referenced post. The post I'm thinking of wasn't Atlantic affiliated. But having now found Megan's, it is the one Megan linked to.
In one of my college calculus classes, then professor responded to the question of "are the exam grades curved" by stopping, thinking for a second, and then coming back with, "I guess, in a cosmic sense, everything is curved." And then he went on writing out the exam statistics on the board. It was probably my favorite thing that anyone ever said in any of my classes, ever.
how to develop an exam that accurately reflects achievement and has a good distribution of scores isn't taught to faculty
Yep, there you go. I have never had a hard-science class where the professor genuinely had any idea of what percentage of the material the average, or 75th-percentile, or 25th-percentile, or top-of-the-top student was actually learning. The principle seems to be "There's an infinite amount of information in this subject, as you can see by opening the textbook, so it was tough enough to reduce an infinite amount of material to the amount that will fit in 18 breakneck-paced lectures. Narrowing it down to the amount a student can actually learn would be nigh-impossible, since I don't know how much that is because as an actual scientist we don't learn in this way, so I'll just present as much material as I can and they can guess which parts I consider to be important enough to put on the exam."
This is what I would do too if I was a professor. Maybe before I get to that stage someone should train me how to do it better. But I doubt it.
I hate grading tests so, so much. Tests that people do well on are much easier to grade than tests that people do poorly on, because each student screws up a problem in a horribly unique way, (much like the happiness of those Russian familes.)
And 43 is why comprehensive exams are impossible to pass (in theory). Fitting to a rigid curve, well the Greeks had the tale of Procrustes.
had to forfeit their scholarship money if they scored below a B
See? Grades suck.
Weird test experience from today: we got our stats midterm back, and there were three questions that apparently caused some confusion during the exam, and as a result the professor dropped them from the grade calculation.
Then, during class, going over the exam, there was a fourth question that apparently a lot of people got wrong, and a bunch of them protested that they were confused in some way, and she agreed to drop that question, too. I was appalled, on general principle.
She did tell us, two days before handing back the exams, what the range of scores was but—interestingly for a stats class—not the median.
I thought McMegan owned Curves.
When I was an undergrad I graded for a couple of profs. One told me the desired median and desired approximate dispersion of scores and left the rest to me. The other, which was much more painful, gave absurdly complete directions. The first was for a physics for physics majors course, the second for a calc based physics course (largely for premeds).
They knew what they were doing, because those premeds were dirty grade grubbers and needed to know how to answer EXACTLY that sort of question for their MCATs, while the physics majors weren't being weeded out and would have their grades curved reasonably.
In grad school, all of the grading has been more like the first, except the students at this (much more glorious) school are less likely to grub for grades, and also much more likely to get nearly perfect scores.
In math, in my experience, people who make noises about "challenging the students, etc" often mean "separate the students with natural ability from those who merely understand the material." (This is not usually intentional, of course)
For example, I was recently forced (since I am a lowly TA) to give a one hour exam with four questions. Two of the questions were difficult, and the other two each had two parts and were not quick. As a result, the scores were basically uniformly distributed between 0 and 100.
I had a professor who would give us a straightforward hour-long exam plus a week-long challenging exam. That strikes me as the most reasonable way to do things (at least in math).
Oh yeah, and grades are bullshit.
Hey, this post is a great way to avoid my own grading!
I've taught at two schools that imposed a curve school-wide, and one that did not.
I agree that the best courses have an achievement-based grading system, so that anyone performing at a pre-determined level will receive a clear grade. This makes the classroom a positive sum, rather than zero sum, environment.
That said, I think it's true that this still involves an implicit curve, often built through significant experience with students. As I've moved between schools and levels of teaching (undergrad, grad, exec), it's often taken a couple iterations of a course to get a feel for what is a reasonable expectation.
Also, while curves may be lazy grading, they also help support lazy evaluation of teaching. I think schools impose curves for two reasons - employers prefer that grades convey some consistent information (I'm in professional schools where there's some effort to respond to what employers want) & if the school uses student evals to judge if teaching is acceptable then an imposed curve keeps faculty from grading too easily in hopes of better evals.
Also, while I think that good feedback is one of the most important parts of teaching, I really struggle with grading and still feel like I suck at it.
Good grief Charlie Brown.
No grades?
Good luck getting the bottom 90% of students to do a single damn thing. If it doesn't have points attached, it's purely optional for most.
While they have their limits, and on occasion someone falls between the cracks, the notion that it would be better (or more fair) without them is a particularly goofy academic fantasy.
A week-long exam, feldspar? How does that work?
Related, I don't think that any of the challenging classes I've ever been in have been graded without a curve. This being the result of professors producing tests and not even being able to hazard a guess as to what the median grade (e.g. 45%? 75%?) will be.
the notion that it would be better (or more fair) without them is a particularly goofy academic fantasy
A fantasy, maybe, but not particularly goofy (wait till I tell you some of my other ones).
As a student, I always felt a disconnect between learning and feedback and the performance aspect of grades. As a teacher, I'd much rather be coaching students who cared about something to do it better. But I find that hard with (moderately) large classes and the need (and desire, to be honest) to keep my research in the foreground. I think really great teachers achieve parts of this, even in big lectures. But I'm not that gifted.
A week-long exam, feldspar? How does that work?
It's called a take-home. Most of my upper level undergrad math classes were this way - when you're proving things, not many of us can cough it up quickly in a sit-down exam.
52: I don't think anyone here advocated getting rid of grades.
People who don't care about the material will really only remember it for the duration of the class (more or less). I think very few people are motivated enough by grades to master material that they otherwise find irrelevant.
The solution to this problem is not to grade harder.
I find that hard with (moderately) large classes and the need (and desire, to be honest) to keep my research in the foreground.
That's the problem. You can do away with grades if you can get the classes small enough. Of course, you need good teachers who are able to motivate and interest students in (dear god) the course material, or students who are motivated and interested enough on their own. But you know, it's not as if the only reason things ever get done in the world is because people are afraid of being punished if they don't.
54 could have been written by me. The only question now is which one of us is the clone...
good teachers who are able to motivate and interest students
This is the best part of teaching, isn't it.
it's not as if the only reason things ever get done in the world is because people are afraid of being punished
In fact, research suggests that offering more external reward for a task often dampens internal motivation.
I just talked with my daughter about this performance vs. feedback issue this evening. She was obsessively checking her answers for a supplemental math assignment she gets every week: the stretch problems for thinking more about fundamental issues. I reminded her that you fall down a lot when you learn something really physically hard, like skiing (she's a great skier). And that we learn more from falling down and figuring out why then from skiing cautiously and never risking anything. The performance side of school (which her current teacher does a great job of keeping in the background), can really discourage the risk-taking and failure that leads to learning.
'... I think it's far more important to reflect how well a student masters a subject than their relative knowledge compared to other members in the class. ..."
I think this depends on who grades are for. Thinking as an employer I would rather know relative position (assuming a reasonably large class). Even as a student I would rather know relative position since I usually already had a pretty good idea of how well I knew the material.
I don't think anyone here advocated getting rid of grades.
FTR, I didn't say it explicitly above, but I absolutely do.
You can do away with grades if you can get the classes small enough. Of course, you need good teachers who are able to motivate and interest students in (dear god) the course material, or students who are motivated and interested enough on their own.
There is at least one college that does more or less exactly this. It seems to work okay.
Even as a student I would rather know relative position since I usually already had a pretty good idea of how well I knew the material.
I denounce you as a traitor to the Grading Revolution!
Kidding aside, those are valid points.
I think, judging from the fact that you have children old enough to math, I must be the clone.
teo, which college are you thinking of? I may have found my home.
65 -- Bennington doesn't give grades. Unless you make them. UCSC doesn't either, iirc.
teo, which college are you thinking of? I may have found my home.
Hampshire?
I'm surprised to hear that about Bennington. I thought it was the poor man's Middlebury, that is, really dumb, really rich kids going through a totally packaged experience.
I'm surprised about Bennington as well. Wikipedia verified! Add Evergreen State to the list.
Ned, that's a complete misimpression of Bennington. Except that it's damned expensive (and I've got cancelled checks to prove it).
Blame Radar Magazine, Charley.
Although the "poor man's Middlebury" idea came from someone else.
Middlebury, that is, really dumb, really rich kids going through a totally packaged experience
Is that a common impression of Middlebury, or are you a leetle bit jaded, Ned? Not one I've ever gotten, but I'm pretty out of date on liberal arts colleges.
I should say also, Ned, that I think this is a serious misread of Middlebury as well. It may have been a lesser light when my parents went there, but nowadays the entering class is plenty smart.
The joke/derogatory thing I heard about Middlebury is that the summer language programs are better (teachers, students) than the regular college. This might be true, but not really a good comparison.
71: Their account of Cornell is a pretty good roundup of the usual stereotypes, but it's not particularly accurate. I don't know enough about any of the other schools to judge.
72, what I meant is that it differs from Middlebury in that Bennington has the dumb kids and Middlebury has the smart kids but they are otherwise similar.
This was the perception at my small-town Pennsylvania prep school where kids were obsessed by such things, yes.
btw, in the Washington Monthly rankings, Middlebury is a 32 and Bennington is a 194 (St. John's is 200).
Evergreen, Hampshire, and Reed also do poorly, which surprises me.
A two minute stroll across each campus would completely cure even the most oblivious of this notion. Really.
Bennington students can use the library at Williams, and there's a shuttle bus going back and forth the short distances between the campuses. I wonder if ever in the entire history of the world a Bennington student has been mistaken for a Williams student. Even without a speech act of any kind.
77: The criteria the Washington Monthly uses are a little odd, and tend to advantage larger schools, I think.
I think you're right. I'm just not used to 2400 being "large."
The nearly $1b endowment may also have an effect.
I'm just not used to 2400 being "large."
Clearly your paradigmatic liberal arts college is not St. John's.
82 -- In fairness, SJ isn't anyone else's either. It's a pretty unique kind of thing. NTTAWWT
The only college stereotype I learned in school was that U.Va. was the awesomest thing that ever awesomed. Like times twenty. Brah.
Are you people trying to take that away for me?
84 -- Yeah, that WM put W&M ahead of UVA. That has to be alarming. But it puts Cal ahead of Stanford, which makes it God's own truth.
In fairness, SJ isn't anyone else's either.
Well, it's mine. Which probably explains a lot.
I actually thought about applying to SJ, but decided against it since I wanted to learn math, physics, and psychology.
Not that they would have taken me, anyways.
The SJ program contains a surprising amount of math and physics, actually. Though probably not to the level you would have preferred.
That said, I'm glad I didn't go there. It's not for everyone.
The whole rankings gambit always seems sort of, well, rank. The ongoing competition for #1 public university (in US News & World Report) always seemed to get far too much play at U.Va.
"Dood! We got #2! I can't believe this!"
"Yeah, hmph. So, what'd you think of the reading about microloans in LDCs?"
"Huh?"
Of course, this may say more about U.Va. than the rankers themselves. I don't know how it plays out at the other schools.
This is a problem near and dear to my heart. I agree with the above in that a) no one should get 100% on my exam, since that would mean that I had not adequately tested the extent of what my students are capable of, and that b) a mean of 60 is more fair than a mean of 80, since with a mean of 80, small mistakes (which everyone makes from time to time) can have a big effect on your grade.
I had an experience in college with a physics class in which the exam was a 25 question multiple choice test. The mean was an 85 (about 21/25), meaning that if you screwed up one question or guessed correctly on a question you didn't know, your grade could actually change by a whole letter grade. What a horrible exam.
The SJ program contains a surprising amount of math and physics, actually
Yes, but doesn't it stop at the beginning of the 20th century? Not exactly good prep for continued study in science.
Also, I once gave an exam where the mean was a 35, but the top grade was a 96. What do you do there? Clearly, it's not the case that the exam was impossibly hard. I'm willing to stipulate that I made some mistake in exam writing in that circumstance, but I'm still not sure how I could have made it better.
The physics seems to top out at around 1930 (which is about where I was before my interest petered out), while the math certainly doesn't make it to 1800 (except for Dedekind) which is right when it starts to get interesting (for me).
But a lot of the other books are things that I now wished I had read. And I'm much slower reading things on my own.
Yes, but doesn't it stop at the beginning of the 20th century? Not exactly good prep for continued study in science.
Yeah, I remember a conversation I once had on an airplane with a retired scientist who was complaining about how St. John's was (apparently) advertising their program as good preparation for medical school or something. I said "well, they read Einstein and stuff," to which he replied "but Einstein was 100 years ago!" There wasn't much I could say to that; it really is a liberal arts person's idea of what a science curriculum should be.
The physics seems to top out at around 1930 (which is about where I was before my interest petered out), while the math certainly doesn't make it to 1800 (except for Dedekind) which is right when it starts to get interesting (for me).
Part of my problem in the conversation described in 96 is that I wasn't quite sure how far the science goes. 1930 sounds about right. I think the math goes a little further, though; when was Lobachevsky?
Yeah, a little further, but not much past 1900. Dedekind seems to be at the end of junior year.
Here's an overview of the program, btw. The link at the bottom leads to more detailed lists of the seminar readings at both campuses, which are slightly different.
I think in a normal math major you get that in yr 2?
Missed him. But it's non-Euclidean geometry, which sucks!
They do skip the genuinely interesting analytical issues discussed in the 19th century by Weierstrauss, Cauchy, Cantor, etc, and the exciting algebra (Galois!). Though props for Dedekind.
That's not math, that's history of math. Should be a separate major, like Harvard's History of Science major (er, concentration).
Fucking Weierstrauss and his goddamn M test.
Of course, they're not trying to prepare future mathematicians, so I forgive their oversights.
They do skip the genuinely interesting analytical issues discussed in the 19th century by Weierstrauss, Cauchy, Cantor, etc, and the exciting algebra (Galois!).
Most of this is due to time constraints, I'd think. There's only so much you can fit in.
That's not math, that's history of math. Should be a separate major, like Harvard's History of Science major (er, concentration).
St. John's doesn't have majors. Everyone takes the same things.
Hey TJ, Weierstrass gave the first example of a continuous nowhere differentiable function...bitch.
109: Yeah, that's why they can't go into more depth on the math, because everyone's also taking science, music, language (Greek and French) and seminar (mostly philosophy and literature) at the same time.
Clearly, it's not the case that the exam was impossibly hard.
No, but weighted towards questions requiring approaches that most students had not mastered.
My ideal exam would probably have 60% material that everyone should have understood if they just read and showed up in class. Next 20% would be understood by students who'd applied themselves in a basic fashion to the work of the course. Next 10% would be those uninspired but hard working souls who had retained and could apply most of the concepts of the course in a reasonable fashion. The final 10% would be for those who had really become adept with the ideas, could apply them in a creative way to a new setting or problem.
Not that I've yet written an exam that accomplished this well.
That mean I have to like it? No, no it does not! Lick my Koch curve.
(That is pretty sweet, actually)
It is a great idea. If you want a broad and deep classical liberal arts education, that is.
Aha! The (graph of the) Weierstrass function is a fractal. Comity!
Yeah, it's really good for what it's designed to do.
There sure seem to be a lot of mathematicians around here.
I just looked up the Weierstrass function and it is damn cool. But the Blancmange function is even cooler, and the Hofstadter-Conway $10,000 sequence is awesome.
I'm a physicist. I had a double major, half of which was math, and which I never completed. Still, I do love the stuff.
1, 11, 21, 1211, 111221, 312211
See the pattern?
You can do away with grades if you can get the classes small enough
Big if. One of my current classes at Remedial Undergrad U is like that, and while I appreciate the A and all, the borderline retard in love with the sound of his own voice is fucking collapsing the whole enterprise. What's the professor going to do, expel him? Dr. Hippy hasn't got the balls to do anything beyond directing subtle insults in Mr. Borderline's direction, with predictable results. Going back to school has awakened in me vigilante instincts I never knew I had.
Funny story about the Blancmange function. It's a special case of the Besicovitch-Ursell function, which I did some undergraduate research on after my sophmore year (looking at its fractal properties)
one
one one
two ones
Is that the one?
I'll amend: I hate those things.
There sure seem to be a lot of mathematicians around here.
I was a math and social sciences undergrad. So I sampled around math, some algebra, topology, analysis, and a few stats classes. Not much history of math, though, and nothing on Weierstrauss functions, though my probability instructor had worked on fractals and fractional dimensions.
Yeah.
I don't like them much either, but that one is pretty cute. (It's something of Conway, and has crazy properties)
121: Going back to school has reminded me of how amazingly, amazingly paranoid I am in academic settings. It's very nearly pathological how suspicious I am of both students and faculty. It's been quite an awakening to deal with, since my experience has been that almost everybody in my classes is there 'cause they want to be there. The jackass quotient is actually really low.
Saint John's math story: One of their quirks is that all the professors (tutors) are capable of teaching all the classes. To work there, you have to be able to teach ancient Greek, and French, and philosophy, and physics, and so on. Understandably, given our social norms, this means that the math teaching is what gets fudged.
If you end up with a tutor who isn't terribly math comfortable, the class turns into "Let us all explore this material from a position of complete naivete together" and is led by the more assertive students. Under such conditions, Dr. Oops once proved, to the satisfaction of the tutor and the other students, that a parabola approaches two vertical asymptotes. When she discovered her error at home, she then had to spend next week's class apologizing for having misled her classmates, and leading them through the proof that she was entirely wrong.
It's a great school, but I wouldn't go there for math. (Although if you want to use cycles and epicycles to predict planetary motion, a la Ptolemy, it's probably one of the few places that will teach you how.)
Although if you want to use cycles and epicycles to predict planetary motion, a la Ptolemy, it's probably one of the few places that will teach you how.
That's cool. For a history of science project I did some work for a few years back I had to learn how to operate use an astrolabe and I found the astronomical mathematics of the time absolutely fascinating. One day I'll be able to buy a decent astrolabe [they are amazing things].
Back on topic, the UK changed the grading procedure for A-levels (the exam taken at age 18 which essentially governs university admissions) in 1986; before then, it had been a forced curve, so x per cent of students were by definition in grade y. The new system graded them on the basis of...how many of the fucking questions they got right!
Due to this, EVERY FUCKING YEAR we have a major political row when the results come out and turn out to be marginally better than the year before (flynn effect, anyone?). Which of course means that Standards Are Slipping!! And EVERY FUCKING YEAR I have to explain that pre-86 results NEVER CHANGED AT ALL.
re: 131
Also, given the increasing emphasis on university education and on preparing kids for exams, it's not a surprise that they get better at passing exams. Passing exams is essentially what they are being coached for.
FWIW, I have taught revision classes for A-level philosophy examinees and have looked at the past papers going back to the 1970s, and I had no sense at all that the exams had gotten any easier. More recent exams, the questions were perhaps a little more direct and less ambiguously phrased, but other than that, the material covered and the standard required looked essentially the same.
1, 11, 21, 1211, 111221, 312211
See the pattern?
Someone showed me this once and I complained that by its own logic it should begin with 11, not 1.
by its own logic it should begin with 11, not 1
Usually, sequences have a rule for transforming the last number to create the next. The first number is somewhat arbitrary (as in the fibonacci sequence where the function is arbitrarily defined at 0 & 1), but it can be easier to see the rule if it starts at 1.
133: Huh. I'd want the first term to be 01 (that is, there are no ones in the preceding term, given that there are no preceding terms), which is what it is given at. I'm not seeing the rationale for 11 as a first term.
I'm not seeing the rationale for 11 as a first term.
(I looked up the answer, because I can't figure those things out to save my life.) By the pattern-making rules, each number is an answer to "how many of each digit are there?" in the form of "there are x Ys and z As." So "1" isn't a proper answer, but "11," which means "there is one one" is.
So "1" isn't a proper answer
That's why LB suggested "01" which would be "zero 1's". I like that much better than starting with 11.
But see, if you start with "01" and you take "0" to be just some number, rather than a Very Very Very Special concept, then it screws up everything.
01, 1011, 111021, 31101211, 132110111221, ...
So. That doesn't work.
That's why LB suggested "01" which would be "zero 1's".
Totally skimmed over the first part of LB's comment. Ignore me.
I looked it up, and still can't reliably apply the rule, as leblanc just did.
137: Yeah, and there also isn't any real justification for starting with 01 rather than 010203040506.....
Really, I think you just have to start somewhere. I'm just not getting an argument that 11 is a better starting point, rather than just another arbitrary starting point. There isn't a prior term with one 1.
One of the humorous things about it is that you can start with basically anything (except 22) and get the same qualitative result. 1 is an okay starting point, but I admit that it does not follow the internal logic of the sequence.
74
The joke/derogatory thing I heard about Middlebury is that the summer language programs are better (teachers, students) than the regular college. This might be true, but not really a good comparison.
I don't understand. How is that derogatory? I'm just curious because I live in Middlebury, and some friends of mine went there (not me, though).
From SJ's curriculum page: Essays by: Young, Taylor, Euler, D. Bernoulli, Orsted, Ampere, Faraday, Maxwell
Do they actually read the original papers? I hope not. Back when I was in grad school, I tried reading Maxwell's paper on Maxwell's equations. It took the better part of two weeks and I finished with less understanding than I started with.
Even in a young field, like computer architecture, where you can read papers that are only 20 years old with no background whatsoever, reading the original work takes 10x as long as reading the same material from a good textbook. I can't even imagine actually trying to learn math or physics from primary sources.
They do read the original papers, but they also have lab manuals that (theoretically) explain the concepts more clearly.
144, 145: I've found that while the original paper may not be the best source to learn a topic from, it is often useful to read the original to understand the context of the discovery. One example from digital logic design: Rent's Rule, which is an empirical relationship between the number of I/Os in a logic partition and the number of components (and average pins per component). If you read the original paper by Landman and Russo, you find a couple of crucial qualifications on their result (which the Wikipedia article does reflect accurately): the relationship assumes that the partitions were determined to (approximately) minimize the number of signals crossing between partitions, and the relationship breaks down at the upper end of the scale (when one is dealing with, e.g., an entire processor).
But there are a lot of papers out there that use the formula blindly without qualification, assuming, e.g., that it holds for an entire microprocessor, or for randomly-determined logic partitions. I believe that most of those authors picked up the formula from a secondary source that was not so careful about defining the conditions under which the formula was expected to hold. So I think there is value in going back to the original sources wherever possible.
