RE: Re: Re: Mathematica and Education

*To*: mathgroup at smc.vnet.net*Subject*: [mg65101] RE: [mg65075] Re: [mg64957] Re: [mg64934] Mathematica and Education*From*: "David Park" <djmp at earthlink.net>*Date*: Wed, 15 Mar 2006 06:28:13 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

Peter, I don't think that anyone would argue that students and researchers should never use pencil and paper. What I am arguing is that one can use a Mathematica notebook just like pencil and paper, and it's advantages (which I listed in a previous posting) are so great that eventually one will find oneself doing more and more in the notebook and less and less on the side. And if one learns how to write, derive and calculate clearly in a Mathematica notebook, this might actually carry over to pencil and paper work. It isn't something I thought up myself but was built into the notebook concept from the beginning by Theodore Gray. And there are plenty of examples of good style from Paul Abbott, Stan Wagon and many others. But, despite its being right there in front of us, many users fail to take advantage of it. Some users don't use any Text cells or Sections at all. Some users do most everything in word processing mode, typing expressions in Text cells, and barely use the kernel. Some users want to design a whole new GUI without learning how to use the notebook GUI. Some users want to morph Mathematica into their favorite programming language. The standard Mathematica usage and notebook interface are very good and if students and researchers would learn it they would get up to speed much faster. Let's agree that we are talking about people who are seriously interested in a technical career. Teaching rudimentary math to the masses is an entirely different matter. I don't even know what the objectives and issues are for mass math education. A CAS would almost certainly be too great a hurdle for most of them so let's leave that out of the discussion. We're talking about 'the happy few'. Many of your objections relate to calculators and not to Mathematica. I'm arguing that one should not think of Mathematica as a 'calculator' (although it clearly can act as one) or as a 'programming language' (although it clearly is a type of programming language) but as 'pencil and paper' for writing text, making definitions, doing calculations, derivations and transformations, and drawing diagrams. The paradigm and mind set with which one approaches Mathematica can make a hugh difference in getting effective use from it. It may not be strictly true that one can do every technical derivation in Mathematica or that it would be the most efficient way, or lead to the deepest understanding. But, for all practical purposes, for almost all students and most researchers, it is the best way. There are far fewer errors than with hand calculation. There is an easier possibility and time for alternative approaches. Answers can be more easily checked against specific cases. Definitions and methods have to be explicitly given and these are usually the nuts and bolts of deeper understanding. Learning how to communicate substantive technical ideas clearly is far more important than being good at mental arithmetic. Mathematica notebooks follow the standard, time-honored style of technical communication. It is not at all counter to what students should be using or with standard pencil and paper work. It would be nice if there was one CAS that was very good, inexpensive and everybody used it. That might come about some day when CASs will be a similar as slide rules were. But in the meantime we have to live with the world the way it is and let market forces fight it out. It is definitely a problem. Students and researchers should be able to think in terms of their subjects and not have to worry about the 'intracacies' of the CAS they are using. In this sense I think there are many gaps in Mathematica relating to convenience of use. These can be filled by additional packages geared to broad fields. I think good packages will generally follow the Mathematica paradigm and principally add new convenience commands. (The Combinatorica package would be a good example.) Also, good packages for students should be developed with lots of actual usage on existing textbooks. In fact, the best method of development is to work through a good textbook and try to add general routines that fill the gaps and remove the 'intricacies' that might relate only to programming methods. My interest in Mathematica is in trying to learn some modern math and physics, specifically general relativity and differential goemetry. I'm currently trying to write a differential forms package to go with Tensorial. I'm trying to work through a textbook 'Advanced Calculus: A Differential Forms Approach' by Harold M. Edwards. (Edwards is a very good mathematician and teacher, and this is a very good book.) I haven't gotten very far at all without spinning off a number of routines that make it much easier to work with the material. An exercise I was working on yesterday involved deriving, using Cartesian coordinates!, a 2-form for spherical flow from a source at the origin. Edwards gives only a plausible derivation where he used 'analogy' and an 'enlightened guess'. With Mathematica I was able to derive an answer explicitly from the basic principles. I used GramSchmidt to calculate two orthoginal vectors to the radial vector {x,y,z}. These expressions were fairly complicated. I could not have done it by hand. I could then write a general expression for the 2-form that represents the flow, and equations for the flow through a surface perpendicular to the radial direction and the other two orthogonal surfaces. Mathematica could easily solve these equations to get the coefficients in the 2-form. But then I got an extra Sign[x] factor in my solution that Edwards did not have. (Suprise! Error of method?) I then realized that GramSchmidt will have a singular line (x == 0) in the way that I set it up, and the orientation of the vectors might change when we crossed that line. I wasn't certain of that so I made an animation as we ran around the equator, and sure enough, there was a discontinuous flip in one of the vectors as we crossed the x == 0 line. So I could throw out the Sign[x] factor to keep the orientation the same. Then I realized that the derivation was no good for x == 0. But I've read about atlases and charts and realized that I could set up GramSchmidt so that the singularity was at y == 0, or z == 0, instead - and always get the same final result. So I learned or reinforced a mathematical method and principle. I wouldn't have learned any of that if I had tried to 'analogize' it through with pencil and paper. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: King, Peter R [mailto:peter.king at imperial.ac.uk] To: mathgroup at smc.vnet.net David, (and all the othes who responded), I have now had the time to read all the responses to my initial response and I can't really argue with the main points, in fact I don't think I ever stated that Mathematica should not be used in the teaching of mathematics (with the caveat below). Yes it does enable you to do all the things that you and others have stated and can enormously increase a student's abilities to do things. This wasn't the thrust. My concern was about students who claimed never to have used pen and paper and only to have used Mathematica. I think this is dangerous. Why? 1) suppose there is a bug (shock horror they do exist) or the student has mistyped things, how do they check the results if they can't do some kind of manual check themselves? Can the student do a rough estimate of what they expect the answer to look like? Do they understand the answer and what it means? Sure they could plot it out (but then why not just write a program to solve the problem numerically in the first place). This doesn't mean that students shouldn't use MAthematica but it does mean they should also be able to do calculations by pen and paper when they are comfortable with that they can move on and use the tools that enable them to do "more advanced" and "more interesting" things. 2) Related to this, actually I am very concerned about the current generation that has been brought up on calculators. it HAS generated people who cannot do simple calculations without one. When a student asks me how to divide 1 by 2/3 because he hasn't got a calculator I get worried. When I see exam scripts where people give the answer E (for error) when they take the square root of a negative number I get worried. More importantly students (not all but a significant minority) don't actually understand what numbers mean. I see lengths quoted to 10 significant figures (implying a measurement accuracy on the sub atomic scale). This has happened over a period of probably 20 years and reflects poor education policies towards mathematics and is probably beyond the scope of this thread (or indeed this list) but it has happened because people have taken the attitude why bother to learn to do multiplication when a calculator can do it quicker and more accurately than you can. I would be worried to go down the same track with more advanced mathematics. I strongly believe that the basic skills should be learnt first on pen and paper and then reinforced using tools like mathematica. I do also believe that Mathematica can be used as part of the learning and reinforcing of the basic skills - just not as a sole substitute. This isn't just an issue of preserving old skills. After all we bother to teach people to read. Why? technology can give us spoken text. I think there are some skills (and this includes mathematics) that are so basic that if we cannot perform them we are missing something. Also often we are forced to operate without the use of these tools. Such as in the field, in meetings without access to computers, in companies that can't afford or don't want to pay for software licenses (I spent many years working for a large multinational that I had to convince very hard to buy a single licence for MAthematica because they couldn't see how it would affect their business performance - this is not uncommon). 3) Why Mathematica (this is the caveat I referred to above). Now this is probably heresy or blasphemy to this list but there are other computer tools for doing mathematics. All these tools have there pros and cons. They all have their quirks some of which distract from the underlying mathematics (some of which may enhance). There is a danger that students get caught up with the intricacies of how to do a particular operation in that particular package rather than the underlying mathematics. You could argue that the mathematics is the basic "truth" and the implementation package is something different (a bit like Plato's shadow worlds). However, this is an interesting philosophical question that I don't really want to go into here (pen and paper, is if you like, another package and how much is mathematics limited by our ability to write things down and solve analytically by hand and how much is it enhanced by using the power of computers, expecially for visualising complex data or phenomena). I haven't seen this with mathematical packages but for other commercial software I have seen students held back by learning the idosincracies of packages and claiming something can't be done simply because the software can't do it. In other owrds it can limit the student's abilities to do things because of the limitations of the package. Again this is not a reason for not using Mathematica in education but it is a reason not to rely on it solely and to teach students there are other ways of doing things (including by hand or with other packages). Finally I would like to say that the response on this list has been almost overwhelmingly in favour of using Mathematica in education and I would support that wholeheartedly. But that support is tempered by the requirement that the students are actually learning how to do the mathematics properly, when required they can think on their own feet and not rely any particular package and that they are learning not just how to use a tool but how to use the underlying subject. I would also point out that that the support for Mathematica on this list is not entirely unbiased (it is after all made up of people who are Mathematica users and experts). If I went to the other packages forums (which must exist, I have never checked) I expect i would see them strongly advocate the use of their own particular package and if I were to go to the general group of educators I expect i would see a very different response. It is easy to dismiss them as being behind the times or out of touch, but they do represent a very big experience bank. Peter King