WebMath: using online math tools

[Alan Cooper]

As the author of an applet which does just what you describe
(see
http://www.langara.bc.ca/mathstats/resource/GraphExplorer/index.htm    )
I agree that there are many ways we can use such tools (either web-based
applets or CAS's) to enhance student understanding and overcome difficulties.

 'Custom animation' is an important addition to our capabilities in this
regard. It can help a student to develop a better intuition for the process of
rescaling - especially when the end views look dramatically different, as well
as allowing a more "hands on" sense of the process of curve fitting - either to
data, or eg as in the definition of e by tangency to 1+x at x=0. Although the
animation may not actually add to the content, it may provide visual clues that
help to trigger understanding in some students (though admittedly there's a
trade off against exercise of the imagination). In this connection I also think
that it is important for us to design interfaces to our tools that give the
user an opportunity to involve kinesthetic as well as visual sensation. The
various "grab and rotate" 3D graphers and VRML are of course good examples of
this.

With regard to your question about what skills we should test, there are a
number of alternatives to the traditional(?) sketch the graph of f(x)=...
question.
Some that we have found effective are:

Sketch the graph of a function with the following properties...
(eg f'=0 at.. ,f''<0 in..., etc)

The graph of ... produced by ..(aCAS) appears to have a local max.   Determine
the exact coordinates of that point.

A question type which can be used to test the student's skill in effectively
using a CAS or graphing applet would be to ask for a complete description of a
graph of a function whose behaviour cannot be shown in a single standard
picture eg where there is substantially different non-trivial behaviour on
dramatically different scales (such examples are not unrealistic - they occur
frequently in various areas of mathematical physics)

Alan


June Lester wrote:

> Given that we now have a number of nice tools for teaching online math,
> what can/should we now be doing differently?  Example: there is a web-based
> grapher that does zooming (or the MathView/LiveMath plug-in will).  This
> means, for example, that instead of doing the standard "secant-line -->
> tangent-line" motivation for derivatives, I can easily do a "local linear
> approximation" motivation: keep zooming in on a point, the curve gets
> closer to a line, find the slope of the line (equivalent, but a slightly
> different point of view).
>
> What other things can we do differently with existing online tools?  Our
> treatment of graphing should definitely be changed, for example - requiring
> students to use calculus to graph a function is pointless in these days of
> online graphers (at least it certainly seems to to the students) - but then
> how do we use the tools to get the students to understand the relations
> between the calculus and the graphs? Or is it important to understand them
> anyway?
>
> I'd like to bring some fresh ideas to the Calculus I course I'm designing,
> but I'm interested in all innovative ways to use interactivity to teach
> mathematics online.  If you have any concrete examples you'd like to share,
> so much the better.
>
> Thanks.
>
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