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Re: Re: Recommended learning exercises for beginners?
I don't see that Buchberger is saying anything different than: Prove a theorem, then use the theorem. Students should be able to derive a result, or see how something is done with detailed steps, but afterwards employ a routine that is more convenient and hides the steps and which might have much more detailed code that handles special cases, or does error checking and other matters. This is what I mean by saying that packages should have breath and depth. The student should be able to work at all levels. If Buchberger is saying that anything special is needed in the CAS software, it is simply that students be able to do the elementary operations that underlie their subject matter. There are plenty of gaps in Mathematica that make things difficult for students. Take the case of linear algebra through the manipulation of matrices. Mathematica has all kinds of nifty routines but also has significant gaps. I would like to easily create a matrix structure where I adjoined a given matrix to a unit matrix, add a column on the left, add a row on the top, perhaps add another column of labels and row of labels. I would like to be able to display this array with divider lines between the different portions. Perhaps the display should be possible in a separate window, and for big matrices I would like to be able to easily shift between displayable portions. (This is one place where an added GUI would be useful and natural.) Then I would like a number of elementary commands such as adding a multiple of one row to another row, divide out common factors from a row if it is an integer matrix, or normalize on a pivot element, do a pivot step on one element over a specified range of rows, do a diagonalization over a given set of rows and columns, dot the left hand column into the columns of the matrix to generate the upper row. These are the kinds of things that students should be able to do. Maybe some of them can be done with Part expressions but it would be better to have specific commands. It takes more than a little good code to implement all these things, especially the display. Maybe they should just use Excel? There are plenty of other examples. The folks at WRI can't do everything; they have to concentrate on the more important tasks. This creates opportunities. In the mean time things are far from rosy for students. Yes, students will improve their Mathematica skills as they use it. But still, far too many students are asked to undertake difficult subject matter without adequate Mathematica preparation and with software that is not complete enough for the purpose. David Park djmp at earthlink.net http://home.earthlink.net/~djmp/ From: Paul Abbott [mailto:paul at physics.uwa.edu.au] To: mathgroup at smc.vnet.net Maybe, maybe not. I think that Bruno Buchberger has got it exactly right at http://www.risc.uni-linz.ac.at/people/buchberg/white_box.html Here is what Buchberger says: > The White-Box / Black-Box Principle for Using Symbolic Computation Systems > in Math Education: > > Although math software systems, in particular those based on advance symbolic > computation techniques, are now heavily considered for improving and > supporting math teaching all over the world, there is still a lot of > confusion about their appropriate use in math teaching. There seems to exist > an unbridgeable disagreement between those who believe that these systems > must not be used in teaching in order not to "spoil the abilities of the > students" and those who believe that, with the availability of these systems, > teaching the mathematical techniques covered by theses systems is not any > more necessary and , rather we should confine ourselves to teach how to use > of these systems. > > For bridging this disagreement I introduced, in 1989, the "White-Box / > Black-Box Principle" for the didactics of using symbolic computation systems > in math teaching: I am advocating that, in the "white-box" phase of teaching > a particular mathematical topic (i.e. the phase in which the topic is new to > the students), the pertinent parts of the SC systems should not be used, > while in the "black-box" phase (in which the students completely master the > new topic), it is essential for modern teaching of math to use these systems. > The principle is recursive because, what was "white-box" in a particular > phase of teaching becomes "black-box" in a later stage and new topics become > "white-box" that use earlier "black boxes" as building blocks. > > Quite some authors in math didactics refer now to this principle and a > couple of didactics textbooks appeared that are based on this principle. > Also, in several Austrian high-schools, based on my advide didactical > experiments incorporating this principle were pursued. > The student should be able to > derive results, step by step, in whatever detail is required for > understanding, all using Mathematica and the associated packages. No. See Buchberger's White-Box / Black-Box Principle above. > Mathematica is actually a metatool for making the tools needed in any field. > It will almost always require additional routines to provide the necessary > convenience and flexibility and to fill the many annoying little gaps. Not that much (good) code is required.