WebMath: mathematical exposition

[Alan Cooper]

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
=======================================================================


-----------------------------------------------------------------
WebMath at mail.math.ca - WebMath Mailing List
To unsubscribe:
via Web:     http://camel.math.ca/cgi-bin/wcms/webmath.pl
via e-mail:  send message a to majordomo at mail.math.ca with
"unsubscribe webmath" in the BODY of message
List Archives: http://camel.math.ca/mail/webmath/
-----------------------------------------------------------------