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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

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

From: Paul Abbott [mailto:paul at]
To: mathgroup at

Maybe, maybe not. I think that Bruno Buchberger has got it exactly right

Here is what Buchberger says:

>  The White-Box / Black-Box Principle for Using Symbolic Computation
>  in Math Education:
> Although math software systems, in particular those based on advance
> 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
> 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
> teaching the mathematical techniques covered by theses systems is not any
> more necessary and , rather we should confine ourselves to teach how to
> of these systems.
> For bridging this disagreement I introduced, in 1989, the "White-Box /
> Black-Box Principle" for the didactics of using symbolic computation
> in math teaching: I am advocating that, in the "white-box" phase of
> a particular mathematical topic (i.e. the phase in which the topic is new
> 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
> new topic), it is essential for modern teaching of math to use these
> 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
> "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
> 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.

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