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Re: Recommended learning exercises for beginners?

  • To: mathgroup at
  • Subject: [mg64103] Re: Recommended learning exercises for beginners?
  • From: Paul Abbott <paul at>
  • Date: Wed, 1 Feb 2006 04:34:47 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <dqad5a$1io$>
  • Sender: owner-wri-mathgroup at

In article <dqad5a$1io$1 at>,
 "David Park" <djmp at> wrote:

> I'm absolutely convinced that the use of Mathematica in technical courses in
> universities can revolutionize the teaching and learning of technical
> material. 


> Nevertheless, I think there are some real roadblocks to
> implementing this. It is almost impossible to obtain anything near the full
> potential with the present approach. 

Which "present approach"?

> However, the problems can be solved.
> I'm writing basically as a student using Mathematica. I don't have
> experience as a teacher. But I've had a fair amount of communication with
> students and teachers and believe I have a reasonable view of the problem.

I do have considerable experience as a teacher, using Mathematica in my 
courses since 1992.

> I'm interested in hearing as much comment as possible, pro or con, on what I
> have to say.
> There are three principal problems. 1) Students do not have enough
> preparation in Mathematica before being asked to use it in technical
> courses. 

At my University, we have provided an introductory computer laboratory 
course using Mathematica, consisting of ~10 x 2 hour lab sessions, since 
1992, first to our 3rd year undergrads and, for the last 3 years to our 
2nd year undergrads. See

Note, however, that this course _uses_ Mathematica, it is not a course 
_on_ Mathematica. The learning of Mathematica is implicit, not explicit. 
But, I would argue, effective nonetheless.

> 2) There is a gap between Mathematica and what is needed in
> specific fields. 

In my experience the gap is small. It is more a matter of your following 
point ...

> 3) Mathematica is often approached with the wrong paradigm.

Strongly agree.

> It is not fair to ask students to learn Mathematica and difficult technical
> material at the same time. 

I disagree. Indeed, if material is designed well, the parallel learning 
can be particularly effective. I do agree that it is important to keep 
in mind the distinction between Mathematica and mathematics.

> Nor is it fair to ask professors and instructors
> to teach Mathematica in the introduction to their own courses. 

Why? If they are going to use or expect the use of Mathematica, then 
they should teach it. On the other hand, at our School, many lectures 
use Mathematica for examples and exercises, building on the foundation 
course I teach.

> Students
> couldn't obtain a working knowledge of Mathematica in the first class, or
> maybe not even in the first two weeks of a course. 

I find that two 2 hour introductory sessions can cover sufficient basic 

> Universities should offer
> first term Freshman courses in Mathematica and they might even make them
> required for all students entering technical fields. 

Maybe it would be nice -- but there is little room in most curricula ...

> Or professors could
> make completion of such a course a requirement for their particular course.
> Universities just have to bite the bullet on this, otherwise students will
> simply resist Mathematica as too time consuming or fumble with it. It would
> be best if students were to learn Mathematica in high school.

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 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.

You wrote:

> Mathematica by itself is not ideally suited to teaching various subjects.

Not convinced by your arguments here and below.

> 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.

> Students should be encouraged to write definitions, rules and routines but
> it is unreasonable to expect each student to write all the routines needed.

Agreed. The approach that I use is to give a number of skeleton routines 
that require modification. This is just that way that most of us learnt 

> This means there must be good packages for each field. There aren't really a
> lot of good packages around; most of them are far too special purpose. Good
> packages will be natural, follow the regular Mathematica paradigm and be
> 'broad and dense' in the sense that they provide the needed routines to
> manipulate the subject matter at every level. 


> 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.

> student should think he is 'doing mathematics', applying mathematical
> principles, axioms and propositions to his material. He should be thinking
> in terms of his subject matter and not about computer science and
> programming. 

The two are tied together very closely. In my experience, I do not 
really understand an algorithm or equation until I can implement it. 
During the implementation stage the implicit textbook assumptions need 
to be made explicit and the "coding" experience, often close to literal 
translation, assists with critical understanding of basic concepts.

> We paid good prices for all those hard working guys at WRI to
> do the computer science. The student and professor need to concentrate on
> the subject at hand.

A nice ideal -- but not really practical. 
> For teaching, Mathematica comes with a very good paradigm. Unfortunately,
> many users fail to appreciate it, or veer off from it. I don't think of
> Mathematica as a 'calculator' nor do I think of it as a 'programming
> language' ("I'm from computer science and I'm here to help you.") I think of
> Mathematica and the Mathematica notebook as 'pencil and paper'. It's a very
> magical piece of paper because it will remember what I have written, execute
> commands, and make diagrams come alive with animation. It is the paradigm of
> text-equations-diagrams (TED) but fantastically improved. It is the standard
> style of textbooks and research papers. It certainly goes back as far as
> Euclid. The TED style is pervasive and persistent because it is completely
> flexible and open. Theodore Gray has provided us with a nearly perfect GUI
> that follows this style. Most efforts to provide alternative GUI's are
> misguided and probably counterproductive.

There was a nice article in the TMJ 1(4): 37-40 entitled "Three Excerpts 
from Euler" by Porta Sullivan, and Uhl, recasting Euler into Notebook 
> Students need to learn literary and graphical skills as well as calculation.
> They need to learn how to organize the material, to write textual
> explanations (The Text cells are as important as the Input cells!) and to
> use diagrams to illustrate their material. 


> Rather than working a number of
> throw-away exercises, students could write notebooks on particular topics.
> When they have finished they will actually have something that they can keep
> and that might be useful to them in the future. It would be more interesting
> for everybody. In the time that they have, students could do MORE examples
> and MORE DIFFICULT examples. It would be a revolution.

There are many examples of such Notebooks. Our second and third-year 
students regularly submit quite elegant Notebook solutions to 
assignments and experiments. Many fourth year students produce their 
dissertation in this format. For example, see the compressed Notebooks 
by Harris, Klopper, and McCarthy in

Also see the MSc thesis by Falloon
> Mathematica and Mathematica notebooks are significant teaching tools - but
> it is a mistake to think that we have learned how to use them to anywhere
> near maximum effectiveness.

I agree, up to a point. It is more the point that most people do not use 
them effectively.

Gareth Russell [mailto:russell at] wrote:

> I am teaching a course (for the third time) using Mathematica to
> explore Theoretical Ecology. Students mostly have no prior experience
> with Mathematica, so in the first class we look mainly at Mathematica
> itself, and I give the students homework designed to get them using it
> and learning some of the basics. Do date, I have used a mish-mash of my
> own material, but it could certainly be more coherent. So I am
> wondering...
> Do any of you have recommendations for 'Introduction to Mathematica'
> notebooks that have worked well for you (or your students)? I'm
> thinking of practice exercises, as well material to study.

Apart from the above comments, I should also mention that the Wolfram 
Education Group courses

do provide an excellent and coherent introduction to Mathematica.


Paul Abbott                                      Phone:  61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)    

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