WebMath: mathematical exposition

[Rejeb Hadiji]

Please do not send me this kind of message in this adress.
Thank you

On Wed, 1 Mar 2000, Alan Cooper wrote:

> I would like to add my vote to those citing the columns of Alex Bogomolny as
> examples of good on-line mathematical exposition. The first criterion for this is
> to be good mathematical exposition - which entails a mix (depending on the intended
> audience) of engagement, surprise, elegance, and rigour. And Alex meets this
> finely, but he also makes good use of the special capabilities of the medium, both
> for interactivity and for hypertext linking. My one quibble might be with his use
> of a highlighted word or phrase to link to a source citation. I have come to expect
> such links to lead to more detail or insight, and to avoid disappointment would
> prefer to see citation links attached to a separate object such as a traditional
> footnote or reference symbol.
> 
> Another nice piece of work is Jim White's 'Cardano' MathWright workbook on cubic
> equations which also in my opinion uses the technology to effectively support what
> is already fine exposition.
> 
> The use of interactive material in such documents can of course often be emulated
> in text but this typically requires more commitment from the reader.
> Even with a CAS available, the setting up of an interactive example from text
> instructions can be more trouble than it's worth, but with the web (or a Maple or
> Mathematica workbook)  it can be all set up and ready to go.
> 
> Perhaps the one truly distinctive feature of the Web is the possibility of
> hypertext linking - not just within the same document, as in a CAS or other
> electronic workbook, but to documents elsewhere. This is a feature of the medium
> that I think is underused even in many of the best examples of web-based math
> exposition (except when the document is just an index to other resources).
> 
> ...and now the plug!
> <PLUG>
> A good example of this type of exposition (which is not necessarily to say "an
> example of good exposition of this type") is I think provided in some parts of my
> topic-by-topic resource guide at
> http://www.langara.bc.ca/mathstats/resource/onWeb
> where I have experimented in some places with embedding the links to external
> material in explanatory material of my own.
> </PLUG>
> 
> Another aspect of hypertext is that it allows one to break with the tradition
> of identifying "idea/concept/train of thought", and to think of a document as
> presenting a network of interrelated concepts through which there may be many
> paths. Understanding then becomes an awareness of the global structure rather than
> experience of any one path. This breaks with the public misperception of
> mathematics as a "left-brained" activity, but I think most mathematicians
> understand that the role of a sequential proof is not to define the unique route
> from A to B, but rather to play the role of Theseus' thread in the maze, and
> establish what is connected to what.  Of course many of our students just want to
> follow the thread to escape as quickly as possible without any interest in mapping
> the rest of the labyrinth.  How can we convince them that there may be treasure
> down those dark passages rather than another hungry Minotaur? Or how can we provide
> the options of alternate paths without creating unproductive confusion and
> frustration?
> 
> June Lester wrote:
> 
> > What I'm trying to get a handle on is how we use the new technologies to
> > present/communicate mathematics effectively. ...
> 
> > HYPERTEXT. How do we structure hypertext to communicate mathematics
> > effectively?  For an expository mathematical paper in the "theorem-proof"
> > style, for example, one obvious option would be to have the main sequence
> > of ideas and theorem statements on the first page, with linked proofs to
> > the theorems on subsidiary pages and sublinks to proofs of lemmas, etc.
> > Another would be an introductory section on the main page and the remaining
> > sections as sequential links. Is either better than the other, and why?
> > Are there other ways of organizing expository mathematics hypertextually?
> 
> Many long papers, perhaps especially in applied math and mathematical physics, have
> the structure of "linearized hypertext" with a major result proved in the first
> section on the basis of lemmas proved at what initially look like random locations
> throughout the next many pages. Such papers might be much more readable if the
> proofs of the lemmas could be linked to directly. They could then also be more
> clearly separated and be seen as truly independent and not circularly related.
> 
> >
> > Or, for more educationally-oriented material, is there something more we
> > can do with hypertext beyond the "click here to step through the details of
> > the example" model?
> 
> A model I like is the hyperlinked hint.
> Or the link to related enrichment info (perhaps an application or even a historical
> vignette).
> 
> >
> >
> > MULTIMEDIA AND INTERACTIVITY. What makes a mathematical animation or use of
> > interactivity relevant and effective rather than gratuitous or ineffective?
> > I'm looking for criteria to determine things like "how well does that
> > particular animation communicate the mathematical idea it was intended to
> > communicate?"  And then "how does it do that?"  Or "I can drag the points
> > of that geometric figure around like so.  Does my doing so increase my
> > understanding of the ideas the accompanying text is trying to communicate?
> > and just how?" Or "how *should* this equation behave when I click on it to
> > communicate something useful about the mathematics it represents?"  In
> > other words, just how does the animation or interactivity communicate/
> > illustrate/illuminate the idea, and how well does it do it?
> 
> I tend to suspect that the different sense areas in the brain can carry different
> and possibly conflicting models of reality, and thus that the more senses are
> engaged in the apprehension of a concept, the more likely it is to be correctly
> retained. And also (though this may be just some kind of 'puritan work ethic'
> speaking) that up to a point, the more work done in apprehending the concept, the
> more firmly it will be retained.
> 
> An example I like to use is comparison of Jim Morrey's award winning animated proof
> of Pythagoras' theorem and the IES group's version of the same proof. The IES
> version just gives the viewer the picture (with draggable points to implement shear
> and rotation) and basically says "play with it". It's no discredit to Morrey that I
> prefer the latter, especially as I have seen intelligent people get frustrated with
> it. Fortunately we can have the best of both worlds (without rewriting the applet)
> by presenting the IES version with a link to Morrey's via the "hint" button.
> 
>             Alan
> --
> ======================================================================
> Alan Cooper (acooper at langara.bc.ca , http://www.langara.bc.ca/~acooper)
> Dep't of Mathematics and Statistics (http://www.langara.bc.ca/mathstats)
> Langara College        (http://www.langara.bc.ca )
> 100 W 49th Ave. Vancouver BC
> Canada    V5Y2Z6       Tel(604)323-5676,Fax(604)323-5555
> =======================================================================
> 
> 
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