I have yet to find an undergraduate course advanced enough that it doesn't have students who have never heard of citing page numbers for quotations. While I've had plenty of freshmen who arrive at college saying their high school teachers had taught them that you have to cite or you get an F, I've also had senior-level lit majors taking their last electives say they've never heard of this. This cannot possibly be true! And yet I end up having to teach it, over and over again, with the pretense that I guess I can't ever take for granted that this is something you could be held responsible for at any time in your education.
Sure, I could just fail half the class, but I don't think that would give them what they need. I make them rewrite the assignment until they get it right. Hopefully that impresses the necessity on them? Maybe? Or will they take someone else's class next semester and go, "Whoa, parenthetical citations? What are those? None of my other profs has ever mentioned that." I guess I get irritated on behalf of the probably half-dozen or more teachers they're insisting didn't think citation was important.
My fabulous high school math teacher used to assign roles to people. For a full year, Manish was the Derivative of a Constant guy. It was his job to either shout "Zero!" or "Plus C" every single time. Every time. If Manish didn't pipe up, we'd all turn to him.
Manish is now a successful lawyer in LA with a beautiful wife and two kids, so I think the system worked.
(1/n) * n = 1 -> n^-1 * n^1 = n^(-1 + 1) = n^0
sqrt(n^2) = n -> (n^2)^(1/2) = n^(2*1/2) = n^1
Come on, Heebie, you're cute! Your explanations should be ten times as effective as normal people's!
Alternately, most people are mopes about math.
What's the practice when you want to cite something you've already cited, but you want to cite the whole damn thing, not a particular page number? Do you write passim? It seemed strange to just write the author, short title without a page number but I did that recently because I didn't have time to look it up properly.
Why did the Derivative of a Constant guy have to do Indefinite Integral guy's job?
Is it too facile to blame No Child Left Behind for this? Because I feel like maybe it needs the fuck blamed.
This sounds like a combination of
1) People being used to accessing a specific bit of information in their brain under specific circumstances (e.g., 1 specific class) and having a difficult time accessing it under any other circumstance.
and
2) Students having learned that, when in doubt, it's safer to just pretend to know nothing.
I'm not sure how to address this. (2) actually seems like the more difficult of the two to change.
I'll take the fact that I couldn't possibly muster even a half-assed definition of "derivative" as used in the post and in math generally to be proof positive that I would be, like, heebie's favoritest student EVAR.
A derivative is a function f' whose values are systematically related to another function f, but which isn't as good as f.
So you're saying that a derivative is a sequel?
Sequels aren't necessarily related systematically to their predecessors. (Except insofar as between any original and its sequel you could gin up some system that relates them.)
Oh man, I always make the mistake in the OP. Advice: do not attempt to teach math to morons.
4,8: I was recently told that we shouldn't use ibid, passim, etc. anymore. Something about them being out of style? I never really got in the habit of doing anything besides short cites (Chicago Manual) so I confess to being woefully ignorant.
Sounds like the heebster had a bad class today.
I just found out today that my next door is a math professor at the local liberal arts university. (I didn't have time to ask her speciality.) I could print out this question and go knock on her door, if you'd like.
You should just tell them to look at a slide rule, and it will all make sense.
Speaking of which, what the fuck sort of university doesn't put faculty bios online? I see her name listed on the department's page, but that's it--a name. That's bullshit.
This sounds like my "savants". I have a subset of students who do really well on exams, but when I ask them a direct simple question in person they either stare at me blankly or start throwing out random words. Somehow they have learned how to take the test without learning any of the underlying concepts. It's amazing.
When do they start learning this stuff? Here, the ones who are good at maths do it for GCSE (so they first learn it at about 14 or 15). Your whole system is so different to ours though - people who are as crap as some of your students at maths are spared having to carry on studying it into adulthood in this country.
19: Lots of them. I have a friend in his second year of teaching at a major university who has yet to get his NAME up on the website (which appears to have last been updated in 1999). And it is a tenure-track position, not lecturer.
Your whole system is so different to ours though - people who are as crap as some of your students at maths are spared having to carry on studying it into adulthood in this country.
Our students are forced to study everything in college, meaning that most college classes contain mostly people who have no interest in the subject and learn nothing but need to get the same grades as people who are learning things. This is called producing well-rounded adults.
I didn't really learn all the math I "learned" in school until I was trying to write audio signal-processing software and had to really figure the shit out. You need to find out what your students are actually interested in and give them concrete examples to work on. Something about the volumes of hand-rolled marijuana joints of varying sizes, perhaps.
re: 21
I often feel that way reading education threads. That the stories of people in their late teens or early 20s sound like stories appropriate to 14 year olds. But then again, I have some of the same stories myself; indeed, of 'elite' students who've had the most expensive education money can buy.
21, 23: This is dangerous to admit on Unfogged, but I never went beyond pre-calculus. It's not like EVERYONE takes calculus.
mostly people who have no interest in the subject and learn nothing but need to get the same grades as people who are learning things.
This is another thing I don't understand. What role is 'need' playing in that sentence?
[I'm moderately Btock style at this moment, but the harsh Calvinist side of me wants to rewrite it as:
"mostly people who have no interesting in the subject and learn nothing and want the same grades as people who are learning things but who are shit out luck if they think they are going to get them"]
26: oh, you're so failing the quiz.
26 -- me too! On the other hand, I've had a nagging sense of shame my whole life that I don't know calculus. I even bought a teach yourself calculus book once, but realized that I'd also forgotten most of my pre-calc, an so I went back to watching TV. Maybe in the retirement home I'll go and audit some lectures or something.
re: 26
We do calculus at about age 15. Pretty much every Scottish student who goes to university will have done calculus. They might not remember it, or have been any good at it, but Higher Maths* would be pretty standard among almost everyone entering university. Even people studying humanities subjects.
* 'Higher' doesn't express any particularly advanced skill level. Highers are the standard exams everyone sits at about 16/17. When I did it, you'd standardly do 5, and Maths and English would be near compulsory. Higher Maths would involve quite a bit of calculus.
It's not like EVERYONE takes calculus.
Very true. My junior year in HS I took pre-calc. Then my senior year, I wanted to do Statistics, so the HS made a special arrangement where I got out early and took a Stat class at VCU. Then, when I got to U.Va. they accepted the stat class for credit and told me I was all done with math. Never took calculus.
29: there used to be a really great pre-calculus and calculus teacher at Santa Monica Community College. Hopefully he's moved on, but you never know.
28, 29: I know, it's an occasion for shame.
30: This is interesting to me. I did my study abroad at St. Andrews and it seemed like a number of Scottish students I knew were taking calculus and struggling miserably. Probably just the subset of people I knew.
27.last is what bothers me about students in lit courses claiming they need to get A's. A's are not impossible to get in a lit elective, but they tend to be earned by those who really care about the material, show an investment in research and writing, make interesting arguments well, etc.
The students who "need" A's are often trying to raise their GPAs because they haven't done as well in their own majors and think they can argue that not wanting to go to grad school in lit should make the requirements different. I'm like, oh, yes, and that's why you see so many struggling English majors taking advanced courses in Physics.
I've had a nagging sense of shame my whole life that I don't know calculus
This is as it should be.
I took, enjoyed, and did well in calculus. I remember none of it.
26: In my (UMC keener NJ) hs, one only took calc prior to college if one was on the AP track. At some point in junior high, the chosen kids skipped over "pre-algebra" and just took algebra, putting them, once they arrived in high school, a year further in math than most of their age cohort. Plenty of kids didn't skip that year* -- kids who would be off to Harvard, Princeton, Yale, etc.
*I skipped the year, but refused to take honors math in high school. This put me in "advanced" math with the people a year ahead of me, and completely borked my schedule. It's not even clear what I would have done senior year (taken the easier calc AP?), but I went to college after junior year.
Also, 21 compared with 30 confuses me. Just that the Scottish system is different than the English? Cause really, I don't need to know calculus and I'm not sure how I'd have been better off if I'd taken it at 15.
36: Same here. I have no idea if, were I to pick it up for some reason, it would come back to me comparatively quickly. I sense ... not. It was really too long ago.
37 actually sounds like what I did as well; I was in a class with sophomores and juniors when I was a freshman. Sadly, that was the last year I understood the math I was being taught.
re: 33
Quite possible. I struggled with 1st year university calculus. There's a gap between Higher Maths and 1st year university maths. It's usually bridged by doing what's called 6th year studies maths, but some students won't have done that,* or will [rarely now, I think, but much commoner in the past] gone straight to university from their 5th year. For those students [and I was one] you have to work really quite hard at the maths in the first month or two of first year not to get irretrievably lost. It's not impossible, but you need to focus as a lot of your peers will have done SYS [6th year studies] or A-level maths and be a year ahead. Or at least that was my experience [late 80s/early 90s].**
* not that common in people studying science subjects, I think, but not unknown.
* I got irretrievably lost ...
38: The idea is that you learn calculus at 15-16-17, then if you are doing maths or sciences at uni they can assume from the moment you walk in you have a working knowledge of calculus.
Struggling with calc may also indicate a lot harder calc course than what you'd be getting as equivalent in the States.
re: 38
Very different systems. Scottish kids are about a year ahead, usually, because of the way the system is structured, and, at least in the past, would study more subjects in high school, earlier, but cover them in slightly less depth than English kids would at A-level. The structure of the Scottish system made for a certain type of precocious shallow generalism. It's changed now, I think, as English kids do more GCSEs and A-levels than in the past, and it's more common for Scottish kids to do the optional 6th year, so there's more convergence between the two schemes, I think.
It's still, afaik, harder in the Scottish system to do the kind of absurdly narrow thing that is possible in English A-levels. I taught people for A-level philosophy [when I was teaching revision classes] whose subjects were things like Philosophy, English Literature, and Theatre Studies. In Scotland you'd pretty much have to have Maths and at least one science combined with those.
41: Ah! You're bringing back memories, as I remember asking someone to explain to me the system. Thanks for the additional explanation.
42.2: Since it was my American friends tutoring them, I didn't get the sense that that was the case. (Also, I was terrified that I'd be horribly undereducated compared to the Brits at the university. This turned out to be very much not the case, which, well, it's not Oxford so perhaps that's why (not that I was at an elite school in the states).)
(I'm totally over-commenting here to keep myself from provoking an argument about taxes on Facebook. I should just hide everyone I would ever have a potential disagreement with.)
it would come back to me comparatively quickly. I sense ... not.
I don't remember calc being so bad, except for a block I have with transcendentals and trig. Learned pre-calc in hs, umm, linear equations?, and a start on integration and differentiation. Maybe I had a good hs advanced math teacher, because freshman calc wasn't at all bad, and I feel like I could pick it right up.
Never could write worth a damn.
44.2 entirely possible. A lot of it would depend which courses people were doing when.
(Also St. Andrews' is pretty elite in terms of the class origins of students there, even compared to Oxbridge.)
45: Ahahah. I just posted this in order to provoke an argument over taxes on FB.
Which isn't what you meant by elite, of course, but.
35 was sort of half-joking, but truthfully (revealing my biases): a basic knowledge of calculus is one of the comparatively few things I think ought to be considered completely non-negotiable for a well-educated adult. The subject is one of the most important intellectual achievements in human history.
I don't think people need to know how to solve particular problems (even simple ones), but they do need a basic theoretical understanding of what derivatives are, what integrals are, how they're related and why they're important.
I'm open to the possibility that this is irrational. But honestly, unless you want to argue that there's nothing in particular that a well-educated person should be expected to know (which probably isn't a bad argument), I don't know how the basic concepts of calculus could be left off the list.
47: Although that William fellow who went there? I seem to recall his uncle explaining (at William's mother's funeral, no less) that William's paternal family were just a bunch of Bosch pretenders.
re: 44
It's worth remembering that the Scottish students were probably a year or two younger than their US 'peers' because the way the system is/was structured, too. Although, again, I think there's more convergence now than there was 15 or 20 years ago. I was 16 when I started university, which was young, even then, but I wasn't an accelerated student. I hadn't skipped a year, I'd just gone at the earliest opportunity that the sytem, as then constituted, allowed.
One third of the students at non-elite universities in the US take one math course, and it is remedial math that you all had when you were twelve: solving quadratic equations, intersecting lines to find a solution of a system of linear equations, etc.
Another third takes a Math Is Fun! course, which I actually think is completely appropriate and fine, which covers a hodge-podge of logical thinking math, baby stats, maybe a topic like voting systems or baby probability.
The last third starts at precalculus or calculus or higher, and of these students half have complete teflon-brain of all material in the first course above.
a basic knowledge of calculus is one of the comparatively few things I think ought to be considered completely non-negotiable for a well-educated adult.
I disagree. I think it should be statistics.
51: I'm told geography were a bit unfortunate about how that worked out, actually, because art history got the shiny stuff and then he ended up doing geography, but that's all third-hand rumour.
48: I considered stealing it, in fact, after I saw it reposted by another. But then I decided I would just ramble incoherently here, instead.
47,49: I was friends with maths/computer sciences people. God knows how that happened, given my own interests. And I know St. Andrews is an elite school, in both senses of the word. I was worried that it would be very obvious very soon that I was woefully unprepared (and had I tried to take calculus, I suppose it would have been), as I'd heard lots of discussions about how much better the various British systems were. But I was pleasantly surprised to find that I was not at a disadvantage in any of my classes.
50: The only time in my life that I have felt hampered by my lack of calculus is on Unfogged. If I had decided to do more than a biology minor, then yes, I would have needed to learn it. But it turns out you don't need to do calculus to be good at the types of sciences that interest me.
I don't disagree with anything in 50.
However,even in retrospect, not taking calculus was clearly the right instrumental/ strategic choice for me,in both HS and college, since it allowed me to take classes where I'd do better, instead. It probably would have been "better" for me to have been at schools with stricter mandatory requirements.
What else goes on the bare minimum knowledge list? I'd add "learn to speak or read at least one foreign language passably well" to the itemization of things that any educated person should have done, but real foreign language learning in school is rare even among very well educated Americans.
52: By the time I got there (in 2002) this gap seemed to have largely disappeared. And/or, I was oblivious to it because I'm a year younger than my school cohort too.
I disagree. I think it should be statistics.
And this is why Heebie is always right. (Not in the least because I did actually take statistics in college.)
re: 59
Yeah, I think grade inflation and other things eroded that before 2002. When I started university the first time round [1988 aged 16] the points score that I had racked up by age 16 [derived from 'Higher' exam results] was enough to get me into pretty much every university course in the country except, I think, medicine at one or perhaps two elite institutions [Oxbridge, maybe Imperial or Kings or somewhere like that]. By 2000 it'd have been impossible to get into most of those courses without completing an extra year's worth of courses, as the entry level in terms of grades had been raised. So the age difference largely got erased.
54: both!
56.3: I don't know that it necessarily has instrumental value for most people. (Although I think most people who don't know anything about it seriously underestimate its instrumental value, even for people who don't professionally utilize mathematics.)
58: I don't have a comprehensive list. A foreign language is probably a good item to include on the list. (I had four years of Spanish, but wish I knew it a lot better than I do--I never use it, and doubt I could even speak it functionally today, much less proficiently.)
solving quadratic equations, intersecting lines to find a solution of a system of linear equations, etc
This was my junior HS year.
I said "Linear equations" above, but I meant " the area under the curve", umm, summations(?), infinite series...all this my senior year in hs.
I think it should be statistics.
Probability and statistics.
And somehow I feel all the above were somehow taught as related. I obviously had a good HS teacher or course, to remember and still feel a little confident after 40+ years.
To expand on my response to 54, I actually think a basic knowledge of statistics is something that ought to be considered non-negotiable for an adult with even a basic education (i.e., high school). I don't think calculus is something we ought to require of all high school students--when I said "well-educated", I meant college.
61.2: I sort of get this, I suppose, since I think knowing another language certainly makes me think about English differently.
However, I don't think I would really understand the concepts of calculus in a deep, life-altering way.* Unless heebie was teaching me.
*I base this on previous experiences with math that led to me memorizing a whole bunch of crap in order to get by on the test (and even get an A) without actually understanding what was going on.
56: Eh, to be honest people overestimate the importance of systems to individuals. I would be surprised if a bright student would have much difficulty at undergrad level in any country having gone through pretty much any decent secondary system.
(Outside some particularly specialised things obviously.)
Glancing thru Wikipedia articles, I think I will go ahead and say that probability should be required knowledge but not necessarily statistics.
Unless heebie was teaching me.
You really would. There is absolutely no one that I can't teach calculus.
I wonder if I'd need to take that remedial math course if I wanted to take a few math classes at a community college like Santa Monica College. I can remember quadratic equations well enough, but I'm having a hard time remembering exactly how to do even equations with intersecting lines, etc.
I've basically done no math in about 20 years, except for the kinds of arithmetic and simplified probability estimates that are necessary for work. I guess I've reviewed a bunch of expert stuff that's mathematical, but that's all been me trying to figure out how to best put their concepts into words, rather than actually doing math myself.
I'm not really aiming my comments at you Halford; don't go back to college just to avoid my irrational prejudice.
Actually, thinking about it, a remedial math class could be a great place to meet chicks.
Your sense of shame, if genuine, is enough for me.
I want Heebie to teach me calculus!!!!!!
I'm not really aiming my comments at you Halford;
So you're aiming them at me? Gee, thanks.
I want Heebie to teach me calculus!!!!!!
Seconded.
29 - I knew calculus once, and am very sure that I could never learn it from a book today. (I was not a textbook-reading type of student. I was a go-to-lecture type of student.) I do want to re-learn calculus, because I haven't used it in years. But I would have to go to an actual class with problem sets to do that. On the other hand, a math prof friend of mine, on hearing that I want to re-learn calculus, recommended a set of tv shows on it. She thought they illustrated the concepts beautifully.
Since I what I want is problem sets on engineering paper, with the equals signs lined up neatly down the middle, that look like Real Math, a tv show wouldn't quite do it. But a TV show could get you to the level Brock is talking about, where you understand how speed and acceleration and jerk are related.
[Cut to montage set to the theme song from Rocky, showing Halford staying up nights doing math, being in class, pumping iron, getting frustrated and snapping a pencil, running up a set of stairs, etc., and returning to Unfogged six months from now to impress Brock by dropping som serious maths on us all.]
74, 75:
That would be fun. I could teach you the basic idea of why and how the derivative formula works in an hour.
Stanley, there's a town named after you in Idaho.
There's a town named grasshopper?!
[Cut to montage set to the theme song from Rocky, showing Halford staying up nights doing math, being in class, pumping iron, getting frustrated and snapping a pencil, running up a set of stairs, etc., and returning to Unfogged six months from now to impress Brock by dropping som serious maths on us all.]
Like my brother and sister-in-law after I innocently beat them in Boggle! Boy did they show me.
I would love to see a series on TV that does intro-college type lessons on various things, taught by talented college instructors. There could be a whole channel! God knows there are enough shows giving advice about stuff you're not already insecure about. IME, lots and lots of people feel insecure about wanting to go back to school but they lack (or have forgotten) the basic knowledge they'd need to take a college course.
I do enjoy the In Our Time podcast for this reason, though they only cover humanities topics.
I always thought teaching calculus was relatively easy if the students came into the class with the proper pre-calculus foundation, and more or less impossible otherwise. And students are split down the middle on this; hence the bi-modal distribution of grades in my classes.
Pop quiz! what is the volume of the solid created by rotating the curve y = x2, 0 ≤ x ≤ 2 around the y axis?
79 -- sure, but could you do it on the Internet in a comments section? 'Cause that would be awesome.
Next, Stanley teaches me how to drum in the comments section.
I do realize that a plausuble interpretation of 85 is: some students are going to be able to learn the material well no matter what the dope in the front of the classroom is muttering on about; other students need a competent instructor. I was not a competent instructor, hence the bi-modal distribution of grades in my classes.
Goddamnit. I just wrote out a big explainy answer to 86 and it got eaten. I think the answer was 8pi.
I always thought teaching calculus was relatively easy if the students came into the class with the proper pre-calculus foundation, and more or less impossible otherwise.
Honestly, the kids without the background understand the calculus concepts just fine. They just can't execute the problems because they all require lots of algebra.
||
Since that other thread is dead: a relative of mine who grew up in Appalachian Tennessee, and was in exactly the same year-cohort, says "not a chance" to that fellow's assertion. She says any male who flunked college in 1969, there or anywhere, would be at instant jeopardy of being drafted, and it was a constant topic of conversation. In addition, the college in question is near Knoxville, which at the time had 3 network TV stations, which anyone at that college would have watched.
|>
So now I just went to the Wikipedia page for "calculus.". WTF -- could they make that any more confusing and incomprehensible?
"mostly people who have no interesting in the subject and learn nothing and want the same grades as people who are learning things but who are shit out luck if they think they are going to get them"
Because education is so expensive, it really is unfair not to pass them out of the class as long as they don't actively destroy the learning experience of others. It's scary to even contemplate the situation of someone who's gone into sizable debt to pay tuition and then doesn't end up with a bachelor's degree, the minimum possible requirement for a career.
In Our Time is fun.
I'm now exploring these Open Courseware thingies, starting with a Yale course on the Civil War. Is there a good one that gets you into calculus a bit? MIT's options are too intimidating.
95: I've rarely seen a student fail just out of total inability; usually it's plagiarism that does that. But there's a huge range between C and A, and while most of my students are currently in the B to A- range right now, the B-s and Cs are sort of giving me the "wah this advanced elective is so hard and i'm just a [whatever] major can't i get an A if i promise not to go to grad school in lit?" line.
I'm now exploring these Open Courseware thingies, starting with a Yale course on the Civil War.
The one Yglesias is always talking up?
I think the answer was 8pi.
I think so too, so let's call that right.
I guess it especially bothers me because I'm getting this from students who are barely paying attention, or seem fully aware that they don't understand the material, while they have classmates who are talking on a level that is so sophisticated I've rarely seen it among my PhD-student peers. Can't they hear what A thinking sounds like?
98: Yeah, that's where I got the idea. I know very little about the Civil War and can't be bothered to read a book about it, but I'm always looking for good gym listening. I'm three lectures in and so far it's not bad.
Isn't 8pi the integral of pi*x^3, not pi*x^4?
yes, but I believe that what you want to integrate (or can integrate anyway) is πydy from 0 to 4.
On the grounds that at each point on the y axis between zero and four one can create a radius to sqrt(y), so that the area of the circle with that radius is πy.
50. I remember being told that for a well-rounded education I should take quantum mechanics.
I guess it especially bothers me because I'm getting this from students who are barely paying attention, or seem fully aware that they don't understand the material, while they have classmates who are talking on a level that is so sophisticated I've rarely seen it among my PhD-student peers. Can't they hear what A thinking sounds like?
I imagine they think they're working just as hard or harder than those sophisticated students, and it's not their fault that they're not as smart. Smart people will get all the advantages out in the real world, why should they get all the good grades too?
And anyway 8π would only be the integral of πx3 if it went (assuming it starts at zero) to the fourth root of 24.
I did it using the shell method, so your typical cylinder has formula 2 pi r h, where r = x and h = x^2. Integrate from 0 to 2.
Neb did it with the washer method, I believe. If he's integrating wrt y he is.
Oh, I rotated around the wrong axis. Sorry Professor Heebie! I'll try hard and make you real proud of me some day!
On the grounds that at each point on the y axis between zero and four one can create a radius to sqrt(y), so that the area of the circle with that radius is πy.
You described a region which is not adjacent to (0,0) except at the origin. So you have washers, not disks.
I don't think I can convey this degree of teflon-brain. Why can we spend an entire six weeks on logarithms and exponential functions, and then in the following semester, if you mention logarithms they'll say "Oooh! Never heard of them!" Yes you did. And you correctly answered test questions. And you conversed semi-fluently in them.
I wish I had time to bore you all with the details, but my working seat-of-the-pants theory on this phenomenon goes like this:
1. Everybody makes choices about which new knowledge gets filed carefully in long-term storage, which is jumbled in there higglety-pigglety, and which is vaporized as soon as short-term memory needs the space.
2. Some of those choices are conscious and some are unconscious.
3. In either case, students are making a series of rapid judgments about the value of the material, the likelihood they will need it in the future, the circumstances under which they will need it, the punishment they may face if they need it and don't have it, and the burden to them of bothering to store it properly.
4. IME, two of the biggest areas of knowledge-drop* are people thinking they will never need something again, and people resenting/resisting the idea that the knowledge has value.
The former is often a failure of imagination, which is often eradicated by life experience. (I.e., the fourth or fifth time you face a situation that draws on that knowledge, you'll start to think it has wider application that the context in which you learned it.) The latter can be well-nigh impossible to shift.
*This excludes people who never really understood it in the first place, such that the "knowledge" was limited to "when you see this sign, plug in this formula."
Where to go with this, I don't totally know, but being able to diagnose something as "Oh wow, I think when we discussed this thing last week, he didn't encode that in the general category 'Ways we can influence public policy,' he put it in some much smaller box" has cut down on my own frustration.
I mean, not adjacent to the y-axis except at the origin.
You described a region which is not adjacent to (0,0) except at the origin.
Yeah, but, uh, so?
I mean, I know I have washers. I couldn't remember the cylinder-surface-area-based method.
||
Grrr. Remember my passive-aggressive smart-ass editor from last fall? Yesterday I got an e-mail from him that said, basically, "There's an error on the fourth page, and the referee feels he can't go on until it is corrected."
Okay, fine, the error is real and stupid of me, and the result is used all the time in papers, so there's no doubt it's true, and yes, I need to correct it although I don't get why it halted all proceedings.
I e-mailed back today and said, "Oh yes. I'll take care of that immediately." (It's almost done already.)
He just wrote back and said "Great! I hope to see your next revision VERY shortly."
Oh fuck you.
|>
No, wait. Why have I not got disks here?
IME, two of the biggest areas of knowledge-drop* are people thinking they will never need something again, and people resenting/resisting the idea that the knowledge has value.
This rings completely true, but this:
The former is often a failure of imagination,
is not the culprit. It's standardized testing and bite-size, one skill at a time, no compound learning approach to K-12 math. They're actually completely correct in their belief that up to that point, a given skill would never be tested again. So they just purge, purge, purge, out of habit.
h = x^2
Wait, I meant the bullet-shaped solid created, not the solid with a bullet-shape removed from its center.
I think that both the idea of a function and the idea that most functions can be approximated by any of several series to arbitrary accuracy are both pretty subtle, and take some time to sink in. Not sure how to translate this into anything pedagogically-- the only ways I can think of require the student to be curious about some topic, and a roomful of kids will be all over the place.
Amusingly, it comes out the same either way.
No, wait. Why have I not got disks here?
I understand your region to be the area under y=x^2, for 0
If you're integrating wrt y, one horizontal Riemann rectangle will trace out a washer because of that missing core.
for "0 less than or equal to x less than or equal to 2". Stupid html.
Um, heebie, I wasn't actually talking about math. (Should I confess I never took calculus? I didn't!)
But yeah, you're completely right. Even in some subjects other than math.
Right, we understood the region in opposite ways.
Note that all I actually said was "created by rotating the curve".
True. I debated which region you were describing, and decided your bounds on x implied that the region was bounded by the lines x=0 and x=2.
It's bounded by those lines either way. It was simply ambiguous.
He just wrote back and said "Great! I hope to see your next revision VERY shortly."
Am I allowed to give a smarmy reply about how I'll get it done in between X, Y, and Z?
117: all-caps on "very"? What an ass.
It's bounded by those lines either way. It was simply ambiguous.
Actually, first I thought you meant the infinite region, bounded by x=0 and x=2, and letting y go to infinity. I started writing up improper integrals, etc. Then I decided I was overthinking it.
ha ha ha. This is why it's good I don't actually set math problems.
131: Yes!! I'm completely irritated.
125 -- it's OK, Witt. We're assembling a remedial class for Heebie, and you can join.
130: I think my response might be along the lines of "Thanks. I hope to get it back to you VERY shortly."
Obviously he never learned that basic human civility is a transferable skill set. Encoding FAIL.
Would it be appropriate if I e-mailed him back and was completely honest, and said:
"Attached is my corrected proof. This is not how I produce my best work; quite often after a week or two I will be able to catch almost all of my typos and small errors. But, as per your request, here is an immediate correction."
139: I don't know academic protocol well enough to undersatnd if it would be appropriate, but it would certainly help inoculate you from further obnoxiousness if he should happen to find any additional typos.
||
It's remarkable what a difference .2 inches can make. Would that I had known that earlier.
|>
I took calculus 15 years before I was born. It didn't help me in chronology class, I can tell you that.
I'm enjoying imagining that 141 was reported to us as soon as possible after the realization happened.
143: That's been noted before.
I do enjoy the In Our Time podcast for this reason, though they only cover humanities topics.
They did a Leibniz vs. Newton once. I still haven't listened to it, but I downloaded it. I assume it's more historical than mathematical.
See, I think it's perfectly plausible to live as a relatively well educated person, and more importantly, to be able to make choices in life, with math at the junior high level. In addition to junior high science in the major fields.
History is one field in which one needs a better than junior high education. I'm not as sure about lit.
147 is wrong if you count statistics as math.
98% of people I surveyed don't count statistics as math, Sifu.
What was your sampling methodology?
150: I asked myself the question "Hey! Stat math?" 100 times. Some of the times, I lied when responding.
And what about the margin of error? The standard deviation?
I don't think lit is that necessary for life, but college-level writing is pretty hard for a lot of people who have come out of shitty high schools, and even for some from good hs programs. About lit, I'll say that it's something that people tend to feel insecure about, to the point that it makes college really scary, not knowing how one reads analytically.
Some of my favorite courses to teach have been fiction analysis and poetry analysis courses taken mostly by non-majors, many of whom are immigrants whose first language isn't English. When I was learning Spanish, I didn't think what I'd learned had really stuck until I could pick up a novel or a poem and understand it.
I dunno. I don't want to say that English study is the key important thing, but it ranks as one of the major (possibly psychological) barriers to being confident and successful in college. If you can read and write well, you feel like there's not much in the humanities you couldn't learn. I don't know that history classes focus as much on literacy.
The chi-square? The coefficient of variation?
My study's position on p-values borrowed heavily from the lyrics of The Rentals. (I'm not sure this joke actually works.)
I'm not sure this joke actually works.
It certainly flew over my head. So there's that.
This jokery does make me think a really good class project for a STAT 101 class would be to design the worst survey ever. STAT 101 professor lurkers: you're free to steal that idea.
The implication being, being friends with P would be good and hence we welcomed more P.
(cross-posted at the requisite joke-explanation blog)
Some colleges have moved towards allowing a wider range of departments to teach the composition courses, rather than just having it done in English or Lit departments. So history departments sometimes do that course. There are also "writing in the major" courses sometimes.
I did composition in comp lit when the options were basically only lit or English departments. I really hated, and still hate, writing school papers. And I sort of hate writing in general. But I often like having written.
Some colleges have moved towards allowing a wider range of departments to teach the composition courses, rather than just having it done in English or Lit departments.
Cornell is big on this.
I sort of hate writing in general. But I often like having written.
Me too.
Wait, all those comments and never a mutual agreement that there were no "washers" involved in neb's calculation, just disks?
And since I've never taught calculus this is probably full of shit, but I feel like part of the problem is that people are taught to memorize nonsense about "the washer method" or "the disk method" or "the shell method" with particular formulas in each case instead of just thinking about what it is they're trying to add up and then adding it up. Somehow a lot of the actual thinking has been distilled away in most of the textbooks and replaced with a bunch of rote problem-classes which are presented as independent techniques.
"washer" and "shell" might not be so bad, really. As names go they do suggest what the actual calculation you're going to have to make will be. I dunno; I learned the techniques for solving that kind of problem by those names, and (even if I can't state problems clearly) I can still do them despite not having had to since learning them. Though I (a) couldn't remember the cylinder one and had to think, there was another method here, what was it, uh, probably cylinder surface areas? and (b) it took me a while to figure out exactly what to integrate and with what values even in the circle-based method I did remember. Which is fine. This comment is pointless, isn't it? I will say that I had a really good calculus teacher.
Because education is so expensive, it really is unfair not to pass them out of the class as long as they don't actively destroy the learning experience of others.
That's satire, yes?
Fuck washer and shell methods for calculus. No, I'm not still angry over how I used to mess up the sign, and how I didn't do the homework, and how this led to me getting a C on one exam involving washers and shells. Really, I just want to say that multivariable calculus rules for this stuff.
Funny, while I generally think I remember calc fairly well, I'm drawing a blank on washer/shell integration to find the volume of a solid -- I'd have to either look it up or spend some serious thought on remembering how to go about it. (although I'm with neb that those are good names, in terms of physically conveying what you're doing, and while I was drafting this comment I remembered, at least well enough for the problem neb stated).I liked multivariable calculus (really, I'm a sucker for anything that teaches exciting new notation. "Neat, now I can understand expressions with those weird triangle things in them."
Isn't it a generally accepted fact that US high schools just don't teach as much as other developed country secondary schools? We catch up in college, I think, but a US kid who graduated with good grades from a good high school is way behind her European/Antipodean counterparts.
but I feel like part of the problem is that people are taught to memorize nonsense about "the washer method" or "the disk method" or "the shell method" with particular formulas in each case instead of just thinking about what it is they're trying to add up and then adding it up.
Washer and shell methods problems are good because students can understand the whole problem from the ground up, so that there's no magic you're borrowing from upper-level courses. So I think this is wrong.
159: (cross-posted at the requisite joke-explanation blog)
Standpipe's Blog!
See, I think it's perfectly plausible to live as a relatively well educated person, and more importantly, to be able to make choices in life, with math at the junior high level. In addition to junior high science in the major fields.
History is one field in which one needs a better than junior high education. I'm not as sure about lit.
Well, if you're willing to throw out any knowledge of major works of literature in your definition of well-educated, then I'm certainly fine with you throwing out calculus as well. I would't claim that one is more important than the other. But I don't think many people would agree, or would regard someone with no more than a junior-high familiarity with literature and mathematics as "well-educated". "Able to make choices in life"? Sure. Is that supposed to be a comparable standard?
164: Not so much with specifically the washer or the shell method, but I had this gripe as a student with a lot of classes -- we'd get types of problems artificially broken down, each with their own little hyperspecific algorithm, and I'd have to ignore the breakdown and generalize what was going on so I could understand it. And this worked okay generally, but I ended up with a vocabulary disconnect, where I'd get lost because I hadn't internalized the specific problem-type, and would be staring dumbly at someone talking about a problem until I rederived the method they were talking about.
As I try to describe this, it sounds like a very, very minor problem, doesn't it.
and I'd have to ignore the breakdown and generalize what was going on so I could understand it
No, it's a real problem because most students aren't capable of this kind of independent learning and disregard for the teacher's momentum.
I confess I have no idea what the "shell method" is. My working hypothesis is that it's a shotgun shell, and you threaten to shoot Neb if he doesn't tell you the answer.
A better way to get the volume of Neb's solid: Dunk it in water. The volume of displaced water is the volume of the solid. Wrap it, ship it, done!
On the other hand, I think sometimes teachers are presenting material in a way that another teacher would say "Oh yes. Each small problem is being related back to the big overhead outline of the material" but it is completely lost on the students.
It's very hard to teach, simultaneously, the big picture and the tiny details, and often one or the other gets lost. And it's not consistent across students within a class which one they miss.
Isn't it a generally accepted fact that US high schools just don't teach as much as other developed country secondary schools? We catch up in college, I think, but a US kid who graduated with good grades from a good high school is way behind her European/Antipodean counterparts.
But `way behind' doesn't mean too much, because most European kids are way behind the people getting good grades.
(& also to be honest high school is a pretty inefficient way of teaching; a year at high school is a uni semester at best, and there's way more than a semester's variance in any uni's intake.
also to be honest high school is a pretty inefficient way of teaching; a year at high school is a uni semester at best, and there's way more than a semester's variance in any uni's intake.
I don't think this is a meaningful comparison, because of the self-selection of college students. Mandatory attendance complicates all comparisons.
Sorry, yeah. I mean, if you have a kid who would choose to be at uni, I think you can teach them a lot more at uni than at high school. Is that sensible sounding?
175: If you integrate neb's solid over y, which seems easier to me, you're adding up an infinite number of discs (which could be washers if there were a hole in the middle.) The radius of the disc is sqrt(y), so the volume (area x dy) of each disc is π(sqrt(y))2dy, integrate from 0 to 4, and bob's your uncle.
If instead you integrate over x, you're adding up the volume of nested cylinders. The surface area of each cylinder is (2πx)x2, times dx for volume. Integrate from 0 to 2, and bob is once again your uncle.
Oh, and each cylinder is what we're calling a shell.
Agreed on compulsory statistics courses. Despite having a pretty decent maths education at school (top stream until I did an early A-level, at which point I stopped), we didn't do anything more than really basic stats. I couldn't begin to tell you how to do a regression analysis, for instance, which is a bit embarrassing given that my reporting beat is stats heavy.
(which could be washers if there were a hole in the middle.)
Which is the case if you interpreted neb's problem per heebie's 123.
True, but it's easier the way neb meant it.
re: 183
I think you really need to be making regular use of it for it to stick in a really meaningful/useful way, though. Or at least that's my experience. I studied a bit of psychology as an undergrad, and did a compulsory stats course. Then when I started getting deeper into the philosophy of biology and the human sciences I decided I needed to do more stats, so I went through a fat volume of stats for econometricians, and then again a couple of years later, I did the same again with another stats textbook, for the same reason. I've also studied probability theory, and Bayes and all that stuff.
Despite that, I still end up taking statistics in science papers mostly on faith, as I just don't make regular use enough of stats to keep the methods fresh enough to assess other people's use of them. I can usually spot stupid misuses of statistics in the popular press, or by politicians, but my chances of spotting anything other than the most obvious error made by a working scientist is very very low.
I think it's easy to overestimate how much use 'stats 101' would be to the ordinary thinking person if they weren't making regular, and fairly deep use of it.
181, 184: Whoops, except that when I described the problems, I think I ended up with the volume of the interior 'bullet' for the disc method, but the negative-space 'bowl' for the shell method. Should have been (2πx)(4-x)2, integrate from 0 to 2, for the bullet.
I think it's easy to overestimate how much use 'stats 101' would be to the ordinary thinking person if they weren't making regular, and fairly deep use of it.
Actually, the stats course that I think everyone should have is more a how to lie with stats course, ie how to critically read journalism and develop a gut instinct for stats that are fishy. Enough math to justify your intuition, but really heavy on critical thinking and reading skills, and how articles tend to be distorted.
186: Well, I think most of what people need is "stupid misuses of statistics in the popular press, or by politicians." You're right, I think, that no one who isn't doing stats professionally is going to meaningfully check scholarly work (and probably people who do stats professionally aren't likely to meaningfully check scholarly work offhandedly). But if you can spot the really dumb errors, and get a sense of what kind of stats don't mean all that much regardless of whether they're formally valid (small 'n', badly chosen sample, whatever), you're still better off.
Yeah. I mean, I personally would benefit quite a lot from a full-on stats course, but I don't think everyone would. I was definitely thinking about a "how to lie with stats" type primer, as advocated by Ben Goldacre. Or they could just make students read "How To Lie With Statistics", I suppose.
re: 189/190
Yeah, that sort of thing could be meaningfully taught, I suppose.
re: 190
Years ago, the OU used to produce a little paperback stats primer for social scientists/psychologists/business studies types. Don't know if it still exists, though.
Oy. I wouldn't have done that writing on paper -- typing math is difficult.
See, I think it's perfectly plausible to live as a relatively well educated person, and more importantly, to be able to make choices in life, with math at the junior high level. In addition to junior high science in the major fields.
People are arguing for math, literature and history? Biology should be mandatory at a high school or greater level. I say greater because we had fewer biology classes than we had math classes in high school because apparently there are a lot of people who don't think that biology is as important as math (biology! the study of life!)
If you have a somewhat basic understanding of biology you know something about foods, medicine, animals (and study design but that may be wishful thinking). I had a roommate that didn't know there were no yeast living in her stomach (TCM quackery). Learning about statistics would be much more helpful if people could then use it to figure out what to believe about new studies.
Plus birds give you a pretty portable hobby.
Plus birds give you a pretty portable hobby.
This big bird around my neck is a huge limitation to my mobility.
If we're talking about basic knowledge, 194 is absolutely correct. Biology should be an absolute requirement, not just because it's important to the individual, but because it's important to all of us that our fellow citizens know about things like the development of antibiotic resistant bacteria due to overuse and improper use of medicines.
In my ideal high school kids would also be taught useful stuff in sex ed beyond what goes where and why with a brief mention of birth control and STDs. They ought to at least have the schoolyard nonsense demolished so they don't think that women who enjoy sex are all sluts, for example.
Not just any tits, great tits.
Also tit-mice which seem uncomfortable.
197: SOUNDS LIKE YOU'RE GOING TO THINK TWICE BEFORE SHOOTING ANOTHER ALBATROSS, SMART GUY.
... Martin Gardner's update of Silvanus P. Thompson's _Calculus Made Easy_ worked for me, Robert Halford.
But! Clearly there is far more that Everyone should know than we currently teach kids, and they should all be forced to wear drab clothes and kept at the grindstone until they've mastered it. Or they'll find themselves drab and grinding at forty, as I am.