tag:blogger.com,1999:blog-264226589944705290.post1359411827687607943..comments2024-09-25T08:51:01.854-07:00Comments on God Plays Dice: Solids of revolution?Michael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger33125tag:blogger.com,1999:blog-264226589944705290.post-41472015826162687052008-08-13T07:51:00.000-07:002008-08-13T07:51:00.000-07:00I teach non-AP calculus to students, and I agree w...I teach non-AP calculus to students, and I agree with a lot of what has been said... I agree with you though, I don't recall ever seeing it or using it in college. <BR/><BR/>But there are two things which I think it's easy to overlook. <BR/><BR/>First is what Calculus Dave said in the comments. The pedagogical reason.<BR/><BR/>Second is the fact that at least my students seemed to be more interested/excited about it than a lot of the other calculus topics -- they could actually "see" it work. And any math my kids are excited about naturally is something I want to continue. (My students aren't "mathy" students -- most are going off to college to be poets and the like.)<BR/><BR/>And even I remember thinking that it was insanely cool, when I took calculus. So yeah, maybe it's not the most used thing, but I think it has a purpose.<BR/><BR/>If we're talking about things that students will never see again, ever, let's do away with two column proofs in geometry. If we want to teach them proofs, I'm sure there is an infinitely better way out there that doesn't make you want to jab a compass in the back of your hand.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-76390692276017230072008-07-26T10:01:00.000-07:002008-07-26T10:01:00.000-07:00Some more references for Kurt:Calculus: A Genetic ...Some more references for Kurt:<BR/><BR/>Calculus: A Genetic Approach by Otto Toeplitz.<BR/>Basic Calculus: From Archimedes to Newton to its Role in Science by Alexander J. Hahn.<BR/>Analysis by Its History (Undergraduate Texts in Mathematics. Readings in Mathematics) by Gerhard Wanner and Ernst Hairer.<BR/>Free Calculus: A liberation from Concepts and Proofs, by Qun Lin. <BR/>The Feynman Lectures on Physics, by Richard Feynman.mishahttps://www.blogger.com/profile/01166708933155105921noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-37603448718881209162008-07-26T09:30:00.000-07:002008-07-26T09:30:00.000-07:00I think solids of revolution are just nice applica...I think solids of revolution are just nice application problems, nothing more. The washers, disks, and what's the third "method"? unhelpful.<BR/><BR/>Here we have something to visualize, not too tough, how are you going to slice your solid so that the dimensions of the slice are easy to calculate and add up? Which is going to be your "thin" direction? How far do you need to go (limits)? and can you take advantage of symmetry?<BR/><BR/>JonathanAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-50877158864579021172008-07-25T16:41:00.000-07:002008-07-25T16:41:00.000-07:00unnecessarily=necessarilyunnecessarily=necessarilyKurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-53526687209480009382008-07-25T13:51:00.000-07:002008-07-25T13:51:00.000-07:00"Teachers and curriculum designers have limited re..."Teachers and curriculum designers have limited resources. I'm not an expert in secondary mathematics education, and I'm not willing to second-guess the trade-offs made by those who are based on anecdotal evidence."<BR/><BR/>That's how I feel about the axioms. I'm not going to believe these things are useful or even true unless the instructor demonstrates why and how this is unnecessarily the case. So that I if forget the formula I can derive it from knowledge I already have. <BR/><BR/>P.S. this "Mathematics for the Million" seems like a very good starting place, thanks HypatiaKurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-79970285195451850702008-07-25T13:04:00.000-07:002008-07-25T13:04:00.000-07:00I'm not going to say that any teacher is great, an...I'm not going to say that any teacher is great, and you may well have had a bad one, but you're indicting the entire trigonometry curriculum because <I>you</I> had a bad time of it.<BR/><BR/>So we take out the axioms and put in a history lesson. What, now, happens to the students who would have learned better from formal rules? And what happens in either case to students who'd do better cutting and pasting physical scraps of paper?<BR/><BR/>Teachers and curriculum designers have limited resources. I'm not an expert in secondary mathematics education, and I'm not willing to second-guess the trade-offs made by those who are based on anecdotal evidence.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-19275892053915452522008-07-25T12:19:00.000-07:002008-07-25T12:19:00.000-07:00Unapologetic, I'm saying that I decided not to pay...Unapologetic, I'm saying that I decided not to pay attention in high school math because I saw no use for the math and intended to avoid any career in which math would be necessary. I quite easily remembered the definitions of the trig functions because they were useful for solving physics problems, which I enjoyed. However, the math itself was of no interest to me for its own sake.<BR/><BR/>Now I have taken the course again! And looking back as a person who is interested in the subject now, and very committed to learning, I can see exactly why I hated it in high school. As topological musing points out, its the axioms. When I was in high school I wouldn't have bothered, to think about all of this, I would have just tossed it into the "this is stupid" category and used my calculator. Its only now that I'm studying the subject intently that I can see that all the axioms are a serious problem. <BR/><BR/>When you question my "disinterest" as a young age you assume that I would automatically be interested in something just because, it is presented to me in school. As someone who later went on to study a bit of economics I look at things from a completely different perspective. My time is valuable, there are a limited number of things that can be read and studied, each (required) class that I take I approach from the perspective that it is a waste of my valuable time, until the instructor has demonstrated to me that their subject is worth my attention. History, English, physics, they are all constantly justifying the validity of their usefulness to you in the class. Even when when their validity is dubious (most of the humanities). I am not blaming the subject for my ignorance, I openly say i chose not to learn. But math must compete in the market place of ideas with other (lesser) subjects. And yet no one ever made a case to me for why the subject matter was important. This would simply be my problem if America were producing plenty of math students, but i feel that I'm in the majority here, people are proud of saying they're "bad at math" in this country. And, what I am trying to say is I think an alternate option in the the approach to teaching the subject would go a long way. <BR/><BR/>Somewhere, on this blog I think, there was a statement that many students become graduate students in math only to find out that they're not good at math, they're good at following instructions. I am one of those people who never follows instructions to assemble things, the type of person who just dumps everything on the floor tries to infer how it must logically fit together. I suppose that's the problem here. But, again, from an economic perspective, we are customers too! Someone somewhere should be presenting the material in way we find accessible!<BR/><BR/><BR/>P.S.<BR/><BR/>The unit circle is a beautiful eloquent logical extension. <BR/><BR/>There's nothing logical about memorizing sheets trig identities.Kurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-13122120045800995682008-07-25T10:04:00.000-07:002008-07-25T10:04:00.000-07:00I am now going back and trying to learn some math ...<EM>I am now going back and trying to learn some math because I've found it has some useful properties. I find reading about the properties and the reason they originally were discovered, by whom, when, what discoveries proceeded and what discoveries followed is quite helpful to my understanding of the nature of the concept being explored.</EM><BR/><BR/>I think Kurt is emphasizing the fact that a lot of times examining the history of a concept, say, that of a function, can go a long way in motivating one to study that concept with much greater "zeal". All of mathematics today is axiomatized, which is good for the subject, but from a pedagogical standpoint it tends not to elicit interest in the minds of beginners/novices. In high school, as well as in college, almost no emphasis is placed on the progression of ideas and how they interact with each other, leading to the discovery/creation of new concepts in mathematics. Of course, one could argue that it is the student that must work hard to keep his/her level of motivation high, but I also think that teachers can do a much better job in presenting the history of mathematical ideas in a way that can genuinely arouse interest in the minds of students about the subject. <BR/><BR/>Also, I think Kurt is saying that had mathematics been presented in high school the way he wanted them to be, then he would have had learned much more, and I strongly believe that there are a lot of students who think the same. In the end, it really boils down to good pedagogy.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-11692099876261715082008-07-25T07:41:00.000-07:002008-07-25T07:41:00.000-07:00Kurt,Assuming your are being serious, there is a b...Kurt,<BR/>Assuming your are being serious, there is a book you might want to get. It's an old one that I have that is probably out of print. But I'm certain you can still find copies available on the internet. It's called "Mathematics for the Millions" by Lancelot Hogben. I really enjoyed reading it.<BR/><BR/>Strangely enough I have seen mention of this book several times during the past year while looking at a wide variety of sites. David Mumford from Brown University mentions it in an article you can find on his website listed under 'articles of general interest'.<BR/><BR/>HypatiaAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-20770094966907077602008-07-25T06:52:00.000-07:002008-07-25T06:52:00.000-07:00I did a lot of this in several calc classes (with ...I did a lot of this in several calc classes (with the same prof for 1 thru 3). To try answering Isabel's question in the post, my view was always that these calc classes are often the branching point when some people decide to head off towards engineering, and some head off towards something like pure math. Aside from DiffEq, the engineers I know didn't really have the same math curriculum as the math students, save for these early classes in calc. So the calc class seems to have to serve those who need it to help with intuition on "physical" things, and as a starting point to understanding analysis for those who will study math as math. Sure, it's not a clear line by any means, but I think it just may be that early calc has always had to serve multiple kinds of interests, so things like "solids of revolution" are kept because it's a decent example that may either be relevant later on, or not.Anonymoushttps://www.blogger.com/profile/12387725027602840136noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-69215387832190335602008-07-25T03:34:00.000-07:002008-07-25T03:34:00.000-07:00So Kurt, just to be clear: you had already decided...So Kurt, just to be clear: you had already decided not to pay attention to math in school because you thought it was useless. As a result you didn't understand the definition of the trigonometric functions when they were presented, and you only remember the formulæ which followed (which are examples of reasoning about the functions from their geometric definitions rather than in terms of calculational rules). And now you want to blame the course instead of your own disinterest.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-43685983301026881992008-07-25T03:32:00.000-07:002008-07-25T03:32:00.000-07:00Long ago I used to have a match teacher eplaining ...Long ago I used to have a match teacher eplaining that trigonomertry and such were very useful to us Dutch, while roaming the seas, finding out where we were shooting stars. Generally speaking I feel the only use of any math is that it shapes the mind into logical/abstract thinking.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-32100849784195331702008-07-25T01:57:00.000-07:002008-07-25T01:57:00.000-07:00It's because "Centroids of Solid Revolution" would...It's because "Centroids of Solid Revolution" would be a great name for a rock band.JimBhttps://www.blogger.com/profile/00397889451057033724noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-42517207798435701472008-07-25T00:44:00.000-07:002008-07-25T00:44:00.000-07:00Kurt, the reason no one gives you the 'rule' is th...Kurt, the reason no one gives you the 'rule' is that the rule is completely unusable. The Wikipedia page gives a number of different ways of defining the sine function; most of them are the ones we've brought up. I personally like the unit circle definition. But if you want a formula, you can find it <A HREF="http://en.wikipedia.org/wiki/Sine#Series_definitions" REL="nofollow">here</A>. The problem is that it's an infinite series and almost impossible to work with effectively, and it will never give you an exact answer. But if you want it, there it is.Jadagulhttps://www.blogger.com/profile/06494463929384804840noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-21085139425355651742008-07-24T21:44:00.000-07:002008-07-24T21:44:00.000-07:00I'm not trying to "troll". I don't know what "stu...I'm not trying to "troll". I don't know what "students" understand. When I was in high school, i didn't bother to learn any math because I didn't think it was useful and I just did enough to pass, when I was in college I got A's in math but taking non-major courses. <BR/><BR/>I am now going back and trying to learn some math because I've found it has some useful properties. I find reading about the properties and the reason they originally were discovered, by whom, when, what discoveries proceeded and what discoveries followed is quite helpful to my understanding of the nature of the concept being explored. I'd rather read a 20 page essay on the function and do 10 problems using it than read 3 pages and do 30 problems. I feel like i learn so much more from reading 1 research paper than an entire chapter of a math book.<BR/><BR/>I want to understand what is happening not memorize 16 rules which are connected in someway which the instructor does not see fit to share with me, those 16 rules are already in my calculator, if when I get done and I still don't what the properties of the function are. what is that point, to save on batteries? <BR/><BR/>But as I said I am an odd bird, and I am verbal. Maybe it is precisely to avoid such explication that others study math.Kurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-57919851003873654812008-07-24T21:03:00.000-07:002008-07-24T21:03:00.000-07:00Isabel and unapologetic, thanks. I missed that. ...Isabel and unapologetic, thanks. I missed that. =)Unknownhttps://www.blogger.com/profile/06229416831400925212noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-21019694845094852002008-07-24T20:34:00.000-07:002008-07-24T20:34:00.000-07:00Kurt, why do I get the idea you're trolling here? ...Kurt, why do I get the idea you're trolling here? You say that students don't understand what the sine function is even when it's defined geometrically (meaning we can get our hands on it and reason about it without focusing on the calculations you're so obsessed with), and you propose replacing it with <I>infinite series</I>?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-80146267360523648552008-07-24T18:43:00.000-07:002008-07-24T18:43:00.000-07:00right exactly, your memorize the ratios of height ...right exactly, your memorize the ratios of height over hypotenuse for a few easy to remember angles and you can know if you see Pi/2 the sine is 1 but you don't actually know what Sine is doing in the "black box" if someone gives you an angle you didn't happen to memorize then you pile on more ad hoc formulae for Sine (Something I know + something I don't know) great of I can solve the problem. But how about they just give me 1 formula for every possible input to sine and I’ll just use that. In instead of memorizing half angles, double angles, power reductions. Why is this necessary? Sure it’s built into my calculator, but its still a black box. I'd like to see an actual equation with just F(x) and X's. <BR/><BR/>I see the Taylor series for Trig functions over there on Wikipedia hanging out, why can't they just use that?Kurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-12677313857559989922008-07-24T17:26:00.000-07:002008-07-24T17:26:00.000-07:00kurt: what are you talking about? Every trigonome...kurt: what are you talking about? Every trigonometry curriculum I've ever seen starts by defining the sine and cosine in terms of triangles (and the unit circle). That's why it's <I>trigon</I>ometry.<BR/><BR/>Now, if you didn't pay attention... but we can hardly fault the course for that, now, can we?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-90160467364400406122008-07-24T16:34:00.000-07:002008-07-24T16:34:00.000-07:00The whole concept of all these formulae in math is...The whole concept of all these formulae in math is completely ridiculous, I think (though I am abnormal and very verbal). I realize now this is why I hated math in loved physics in high school. In physics the formula naturally flowed from the physical properties we sought to describe. That's why when I first saw calculus, I just repeated over and over in my head "acceleration is the derivative of velocity". Now that I'm trying to go back and learn relearn all the math I never bothered to learn before, I can see it’s all just a bunch of adhoc special case formulae to memorize. You can sit through an entire semester of Trig and still not know what Sine is. Is just a black bock, you know when you put things into it what pops out, but as soon as you forget all the formula you memorized you're back to not knowing anything! (I would say you didn’t actually know anything to begin with, in retrospect) How many years do people go on knowing pi is~ 3.14 and having no idea it is the ratio between the circumference and the diameter? Ask a student who’s taken one semester of (school mandated) statistics, what a standard deviation is, I’m 95% confident if they give you an answer at all it will be a formula and if you ask the student “yeah but what is it designed to measure” they’ll just stare at you blankly. <BR/><BR/>The whole freakin system is out of order! I’d like a math book that read like a history book, starts with the most basic problems people we’re able to understand, tells the story of how they gained that knowledge and what properties they discovered. Then moves on the next advance from there, at least, I would have enjoyed this, I wish it were at least an option….though I guess it would take too long? <BR/><BR/>P.S. As for the normal distribution, it feels like and exponentiation of Cosine to me, I try to figure out how de moivre came up with it when I get bored sitting in my statistics class, so far I’ve got y=exp(-2*pi*x^2) i'm not sure how the root-mean-square got into the mix.....thoughKurt Osishttps://www.blogger.com/profile/08186216783828286206noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-87589683790399364352008-07-24T14:42:00.000-07:002008-07-24T14:42:00.000-07:00It turns out that the railroads have a requirement...It turns out that the railroads have a requirement that loads attached to flatbed railcars be loaded with their center of mass centered on the car. This is to prevent trains from fallin over, I guess. So I used calculus once to figure out the center of mass of a half cylindrical shell. That's the only use I can remember from my blue-collar career.CarlBrannenhttps://www.blogger.com/profile/17180079098492232258noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-33238293336716860632008-07-24T13:45:00.000-07:002008-07-24T13:45:00.000-07:00Isabel is right, Boris. "Converting to polar coor...Isabel is right, Boris. "Converting to polar coordinates" is <I>the exact same thing</I>. The whole point of solids of revolution is that they look really nice in cylindrical coordinates, so we can do two of the three integrals necessary to find a volume without using antiderivatives. Then we leave the last one as a single integral for the student to do as the work in the problem.<BR/><BR/>And I make a big deal of this when I teach calculus 3: these formulæ from calculus 2 are all just special cases of the techniques we just learned.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-62063044392370368462008-07-24T13:35:00.000-07:002008-07-24T13:35:00.000-07:00I agree with "D" that the point of covering volume...I agree with "D" that the point of covering volumes of revolution is the derivation, not the result. It would help if textbooks and instructors would emphasize that.<BR/><BR/>As for the paint bucket that can't hold enough paint to paint itself, this isn't as unrealistic as it sounds. You could actually create such a container. Because of surface tension, the end of the container would have no paint inside. Or if you want to consider an idealized liquid with no surface tension, you could point out that painting the outside implies a certain minimum thickness.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-10127572556119460222008-07-24T13:19:00.000-07:002008-07-24T13:19:00.000-07:00Boris: you're considering the integral of exp(-(x^...Boris: you're considering the integral of exp(-(x^2+y^2)) over the xy-plane. The usual way to do this is to change to polar coordinates. But the volume between the graph of exp(-(x^2+y^2)) and the xy-plane is in fact a solid of revolution, namely the solid obtained by revolving the area between the x-axis and the graph of exp(-x^2) around the y-axis.<BR/><BR/>Calculus Dave: you make a good point about doing this with various other cross-sections (and in the course I've taught, this comes up). But the same complaint occurs to me -- when are you going to have to compute the volumes of these things? I suspect that the value here is really in getting students to think geometrically and to conceptualize the integral as an infinite sum.Michael Lugohttps://www.blogger.com/profile/15671307315028242949noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-21601669851632398772008-07-24T13:09:00.000-07:002008-07-24T13:09:00.000-07:00unapologetic,I usually see that by squaring the ne...unapologetic,<BR/><BR/>I usually see that by squaring the necessary integral and changing to polar coordinates. How can I use a solid of revolution?Unknownhttps://www.blogger.com/profile/06229416831400925212noreply@blogger.com