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From: Richard Fateman on 19 Jan 2010 05:16 Do you have any evidence that, taken collectively, the students know more calculus? Can you show that they do better on the final exam than students who haven't used computer systems? Typically the calc teachers I've encountered want to know "what to leave out to make room" for computer stuff. I tell them to leave out Logarithmic Derivatives. Some students like computers because they are neat, and may be enthusiastic about this aspect of the course (though not all...). Maybe it is unimportant that they learn calculus at all, and they should just learn about computers. This would be an important but divisive claim: i.e. calculus is unimportant; we should require that students learn computer skills (Mathematica??) instead. Maybe David Park's point is really somewhere along that spectrum, and we should hold students who learn Mathematica to a lower standard regarding the traditional curriculum. To be clear, I don't object to teaching students about computer algebra systems. I do so when I get a chance (in computer science courses). I just am unaware of evidence that it makes them better calculus students. I don't doubt that a teacher using a computer to do graphics can enliven a calculus class. And even students doing graphics on their own (e.g. TI graphing?) can have fun. But can you show they learn more calculus if they have Mathematica at hand? RJF Helen Read wrote: > > My students come into university level Calculus I or II with no > Mathematica experience, and learn to use it in my calculus class while > learning calculus. ...
From: Murray Eisenberg on 20 Jan 2010 06:46 It's been some years since I've seen studies comparing performance of students taking a traditional calculus course, on the one hand, and those taking a Mathematica-using course, on the other hand. As I recall, the studies showed the latter group performing at least as well as the former, and sometimes better, at traditional pen-and-paper skills. In addition, the latter could solve more complex or more realistic problems than the former. You might want to take a look, e.g., at the link to "A guide to the studies done on the Mathematica-based courses" at matheverywhere.com. This concerns the "Calculus& Mathematica" project created by Jerry Uhl, Horacio Porta, and Bill Davis, at University of Illinois and Ohio State University. There have been a number of Mathematica-permeated calculus and post-calculus courses taught over the years. Others you might wish to look at are: Calculus, Keith Stroyan, University of Iowa Interactive Multivariable Calculus, Stroyan also Differential Equations, Selwyn Hollis, Armstrong Atlantic State University Abstract Algebra, Al Hibbard and Ken Levasseur, Central College and University of Massachusetts/Lowell There have been presentations about some of these projects at various Mathematica conferences. I've based several courses myself upon Mathematica use. Those of us who do so often just grow tired of trying to justify what we're doing to those who are dubious or skeptical or just plain ignorant of what's possible and how. My own experience is that those who don't want to "believe" simply will not believe. And often they don't want to make the effort to reconceptualize what it is they're teaching (and why) and how. On 1/19/2010 5:14 AM, Richard Fateman wrote: > Do you have any evidence that, taken collectively, the students know > more calculus? Can you show that they do better on the final exam than > students who haven't used computer systems? > > Typically the calc teachers I've encountered want to know "what to > leave out to make room" for computer stuff. I tell them to leave out > Logarithmic Derivatives. > > Some students like computers because they are neat, and may be > enthusiastic about this aspect of the course (though not all...). > > Maybe it is unimportant that they learn calculus at all, and they should > just learn about computers. This would be an important but divisive > claim: i.e. calculus is unimportant; we should require that students > learn computer skills (Mathematica??) instead. Maybe David Park's point > is really somewhere along that spectrum, and we should hold students who > learn Mathematica to a lower standard regarding the traditional > curriculum. > > To be clear, I don't object to teaching students about computer algebra > systems. I do so when I get a chance (in computer science courses). > I just am unaware of evidence that it makes them better calculus > students. I don't doubt that a teacher using a computer to do graphics > can enliven a calculus class. And even students doing graphics on their > own (e.g. TI graphing?) can have fun. But can you show they learn more > calculus if they have Mathematica at hand? > > RJF > > > Helen Read wrote: > >> >> My students come into university level Calculus I or II with no >> Mathematica experience, and learn to use it in my calculus class while >> learning calculus. ... > -- Murray Eisenberg murray(a)math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305
From: Richard Fateman on 21 Jan 2010 04:50 Murray Eisenberg wrote: > It's been some years since I've seen studies comparing performance of > students taking a traditional calculus course, on the one hand, and > those taking a Mathematica-using course, on the other hand. As I > recall, the studies showed the latter group performing at least as well > as the former, and sometimes better, at traditional pen-and-paper > skills. Rather than looking at all the material you suggest, I will take your word for it, since it generally agrees with what I recall. What I recall is that, essentially, the group of students using computers did, on the whole, substantially the same as the students not doing so. In addition, the latter could solve more complex or more > realistic problems than the former. This would be an interesting consequence (did they use the computer to solve more complex problems?) I can see it either way -- a student who has more confidence and has seen problems solved where the answer did not magically drop out and become some remarkably simple expression -- may be more willing to attempt to attack a more ambitious realistic problem. Often calculus students (indeed, I recall a graduate course in applied math where this was true) could guess that any time the solution of a homework problem required a large expression or a number with more than 3 digits, that he/she had made a blunder somewhere. The right answer in the back of the book was always small. > > You might want to take a look, e.g., at the link to "A guide to the > studies done on the Mathematica-based courses" at matheverywhere.com. > This concerns the "Calculus& Mathematica" project created by Jerry Uhl, > Horacio Porta, and Bill Davis, at University of Illinois and Ohio State > University. The success of such courses obviously depend on the enthusiasm, energy, and charisma of the teachers. To what extent does it depend on the computing aspect? In the case of this particular link, this leads to a business, where the professors are apparently selling courseware. I'm not saying this is a bad thing. Just that I would not expect statements on that web site to present nuanced opinions on pro/con teaching math with computers :) > ... snip.. > I've based several courses myself upon Mathematica use. Those of us who > do so often just grow tired of trying to justify what we're doing to > those who are dubious or skeptical or just plain ignorant of what's > possible and how. I haven't been in the business of teaching calculus for a long time. (I taught math at MIT before I taught computer science at Berkeley). My experience is that math teachers are, on the whole, quite conservative. My own calculus lab (this was in 1971-2) was dropped when I was not available to teach it. I used a computer algebra system, obviously not Mathematica. In the absence of a controlled experiment, it is hard to convince skeptics. Even the experiments that might be tried would probably be flawed -- e.g. two sections of the same course -- may be defective if the better students self-select to come to the "experimental computer-based" course. What we would like, I think, is some mechanical, automatic, guaranteed-to-win, technique (computer based or not!) that would take all students, including below average, and get them to have a superior understanding and appreciation of mathematics. If someone could convincingly show that using Mathematica (or anything else) was a sure-fire mechanism, then people in charge of schools who are in a position to hire/fire teachers, might take notice. There have been period efforts to improve education. e.g. I participated in conferences on "Calculus for the New Century". (uh, I think that was the previous one!).. Using graphing calculators -- still a big sell -- was one innovation. I personally don't doubt the efficacy of using computers to actually DO calculus problems. I also suspect that by far the most lucrative sales of programs like Mathematica are to schools to make them available for courses like calculus, for those semesters in which someone uses them (perhaps for an additional calculus lab.). But do students learn more? Note that most calculus students just want to pass the course so they can graduate and never use calculus for anything -- so they do not really want to learn "more" than necessary :) . Calculus is not the only course, though if you are selling educational software, it is probably the biggie (or maybe pre-calc, but that's not so much fun and often remedial). I would like to believe that my colleagues who teach engineering of various sorts would see computer algebra systems as more or less essential tools in their fields, and would feel an obligation to make room in the curriculum for the teaching of such software. While I do not have the 50 years of experience of AES, I do have 35+ years of experience in a similar situation. Maybe an occasional guest lecture is the best I could do. I suspect the engineering (undergraduate) curriculum is even less prone to experimentation than math. One article I have read on using computers to teach math (modern algebra, not calc.) included a study on whether students learned more (with or without computers). Observations: students stopped learning the computer aspects when they realized the final exam was going to be held in a room without computers. On the final exam, the students with/without computers did about the same. The authors conclude (with a straight face..) that more money should be spent on developing the computer programs so that they would be more successful. Good luck. RJF > My own experience is that those who don't want to > "believe" simply will not believe. And often they don't want to make > the effort to reconceptualize what it is they're teaching (and why) and how. > > On 1/19/2010 5:14 AM, Richard Fateman wrote: >> Do you have any evidence that, taken collectively, the students know >> more calculus? Can you show that they do better on the final exam than >> students who haven't used computer systems? >> >> Typically the calc teachers I've encountered want to know "what to >> leave out to make room" for computer stuff. I tell them to leave out >> Logarithmic Derivatives. >> >> Some students like computers because they are neat, and may be >> enthusiastic about this aspect of the course (though not all...). >> >> Maybe it is unimportant that they learn calculus at all, and they should >> just learn about computers. This would be an important but divisive >> claim: i.e. calculus is unimportant; we should require that students >> learn computer skills (Mathematica??) instead. Maybe David Park's point >> is really somewhere along that spectrum, and we should hold students who >> learn Mathematica to a lower standard regarding the traditional >> curriculum. >> >> To be clear, I don't object to teaching students about computer algebra >> systems. I do so when I get a chance (in computer science courses). >> I just am unaware of evidence that it makes them better calculus >> students. I don't doubt that a teacher using a computer to do graphics >> can enliven a calculus class. And even students doing graphics on their >> own (e.g. TI graphing?) can have fun. But can you show they learn more >> calculus if they have Mathematica at hand? >> >> RJF >> >> >> Helen Read wrote: >> >>> My students come into university level Calculus I or II with no >>> Mathematica experience, and learn to use it in my calculus class while >>> learning calculus. ... >
From: AES on 21 Jan 2010 04:56 In article <hj40q4$sgk$1(a)smc.vnet.net>, Richard Fateman <fateman(a)cs.berkeley.edu> wrote: > But can you show they learn more > calculus if they have Mathematica at hand? Speaking only for myself (and noting that my calculus-learning days are far in the past, and that I'm not at all sure what the operational meaning of "learn more calculus" might be), I can only say that having Mathematica at hand whenever I'm doing any kind of "maths" �� whether it's learning more about some familiar or new mathematical topic, or trying to solve some real problem using math �� certainly enables me to gain immensely more insight and/or intuition into what the symbols on the paper mean, or how the mathematically described system of interest will actually behave. Mathematica can really be "insanely great" at helping do that, and I'm grateful for it. But it's Mathematica that's the "tool" for producing results here, and the conventional mathematical symbols as conventionally written on paper and the real physical systems that are the important realities �� the things that most of us want to concentrate on �� not the arcane and sometimes inconsistent or even bizarre innards of Mathematica. Which is why it's so egregious and some of us so unsympathetic when attempts to apply Mathematica to some conventional mathematical input in what would seem a sensible and consistent fashion instead trigger some arcane Mathematica "gotcha"; and Mathematica acolytes then try to convince us that, hey, that's the way Mathematica works, and we must therefore accept it as near divinely inspired, and focus unlimited energies on learning the arcane (and often very ill-documented) details of what Mathematica does, not the tasks we want to accomplish with it. Mathematica is a _commercial tool_, not a divinely endowed accomplishment of human creativity before which we must all bow down (and that remains true not withstanding the large amount of great human creativity that has obviously gone into developing it).
From: Murray Eisenberg on 22 Jan 2010 05:37
Comments inserted below. On 1/21/2010 4:50 AM, Richard Fateman wrote: > Murray Eisenberg wrote: >> It's been some years since I've seen studies comparing performance of >> students taking a traditional calculus course, on the one hand, and >> those taking a Mathematica-using course, on the other hand. As I >> recall, the studies showed the latter group performing at least as well >> as the former, and sometimes better, at traditional pen-and-paper >> skills. > > Rather than looking at all the material you suggest, I will take your > word for it, since it generally agrees with what I recall. What I > recall is that, essentially, the group of students using computers did, > on the whole, substantially the same as the students not doing so. In the case of Calculus & Mathematica, the "treated" (Mathematica-using) group distinctly performing better than students in the control (standard course). Before even imagining the possibility of bias on the part of those who conducted the study, one might begin by looking at what the study found and how it was conducted. For example, see: https://cm.math.uiuc.edu/?q=node/30 (on the University of Illinois Mathematics Department's "Calculus & Mathematica") web site). One thing you'll learn about was the concomitant change for the treatment group from lecture to hands-on, lab-based approach. So yes, it was not simply that computing was used per se. Rather, the use of a computing-base course made the hands-on approach possible. > >> >> ... You might want to take a look, e.g., at the link to "A guide to the >> studies done on the Mathematica-based courses" at matheverywhere.com. >> This concerns the "Calculus& Mathematica" project created by Jerry Uhl, >> Horacio Porta, and Bill Davis, at University of Illinois and Ohio State >> University. > > The success of such courses obviously depend on the enthusiasm, energy, > and charisma of the teachers. To what extent does it depend on the > computing aspect? > In the case of this particular link, this leads to a business, where the > professors are apparently selling courseware. I'm not saying this is a > bad thing. Just that I would not expect statements on that web site to > present nuanced opinions on pro/con teaching math with computers :) I don't recall for sure whether the Calculus & Mathematica project was one of quite a number of pedagogical innovations backed by NSF funding. But one of the NSF program's objectives was dissemination of successful projects, and this included commercial dissemination as one route. -- Murray Eisenberg murray(a)math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305 |