Briefly, what I cover here are (1) the mechanics and (2) the philosophy behind the Maple worksheets I've used in lab. By ``mechanics'' I mean (a) how to construct the worksheets, (b) how I make them available, and (c) what happens when I'm in the lab with the students. For (a) I've written a short Maple Worksheets Lab which covers the basics. (B) and (c) are covered under mechanics below.
The philosophy heading is self-explanatory.
So far I've made and given four MAT272 and two MAT242 labs using my
philosophy (you'll find them under labs);
but I've come up with a list of ideas for a few more and
included that here under examples.
The ECA lab computers are set up so that
when students click on the link in
Netscape,
Maple springs to life with a copy of the worksheet loaded.
My worksheets usually contain a short discussion of the relevant
Maple commands, some ``live'' examples, and then some exercises.
The students are expected to work through the text and examples and then work on
the exercises. They turn in their results at the end of the hour.
They're encouraged to work in groups. I don't lecture or do
demonstrations; I circulate and answer questions.
When students arrive at the lab, I give each one a paper copy of that
day's Maple worksheet.
This is partly for insurance against network failure.
The idea is that they should work directly in their downloaded copy of
the worksheet, but if that fails they can read the paper and
type in the commands by hand.
I give out the Exercises section of the worksheet on separate paper so
students can write their solutions on that and hand it in. Having people
print out their solutions tends to result in a printer logjam, lost work, and
wasted paper.
I make a Maple worksheet with solutions to the lab available
(also by ftp from the course web site) afterwards.
I want my students thinking about the subject material in the lab, not
learning the intricacies of Maple, so I keep the Maple
commands to a minimum. (If it's tempting to do some fancy programming,
it's probably better to let the students do it - but as an
Honors project, not as a one-hour lab.)
A corollary of this is that I don't have to know too much Maple
myself.
I don't know how to introduce new mathematics in the lab, so I only do
a lab on a topic after we've covered it in lecture. But because
Maple gives such quick feedback, an hour in the lab can be
much better than an hour of me working examples on the board.
A corollary of this is that I don't have to type a lot of mathematics
into the worksheets, which is onerous.
If I knew more Maple, maybe I could do these too:
Joe Rody
Mechanics
I put the worksheets in a public ftp directory,
and then I create links on my course home pages which point to the
worksheets. Example:
Philosophy
Maple is great at drawing pictures, so I try to use the labs to
reinforce the connection between the mathematics and the pictures.
Specifically, I give them Maple plots of curves, surfaces,
vector fields, etc, and ask them to find a Maple command
that will produce that plot. The Maple involved is basic but
useful; most of the work goes into finding out how the information in
the picture can be used to recover the right parameterization,
equation, function, etc.
This is also a pretty efficient way of creating exercises: I just make
a bunch of plots and then erase the Maple code, leaving the
pictures. The solutions worksheet is just the plots without the
Maple code erased.
Sample Labs
Exercises: Find a function of two variables given only a plot of its
graph.
Exercises: Find parametric equations for a curve given only its graph.
Exercises: Find the formula for a vector field given only a plot of
it.
Exercises: Find parametric equations for a surface given only a plot of
it.
Further Examples
Here's a list of calculus labs which could easily be written according
to my rough philosophy.
Exercises: Find a function whose graph is the given plot.
Exercises: Given superposed plots of f and f', which is
which? Find f and f'. Find f given only a plot of
f'.
Exercises: Plot a tangent line given only a plot of the function.
Exercises: Plot a Taylor polynomial given only a plot of the
function.
Exercises: Plot the inverse of f given only a plot of f.
Exercises: Find a polar equation for a given polar plot. Find a
Cartesian equation for a given polar plot.
Exercises: Find and plot a curve through a given set of points.
Exercises: Find and plot a curve whose local maxima/minima are a given
set of points.
Exercises: Find a function given only its contour diagram.
Exercises: Given only the plot of a surface, plot certain
cross-sections of the surface. Given only a plot of some
cross-sections, find a surface with those cross-sections.
Exercises: Plot the tangent plane to a surface given only a plot of
the surface, or a plot of its cross-sections.
Exercises: Given a plot of a region or solid, specify limits of
integration which determine it as a region or solid of integration.
Exercises: Find and plot a function whose gradient field is a given plot.
Find and plot its gradient field given only a plot of a function.
Exercises: Find or plot a vector field given only plots or formulas
for its div, curl, and grad.
Exercises: Find a vector given only a plot of the vector.
Exercises: Given superposed plots of
u, v, and u x v, which is which?
Find and plot u and v such that u x v =
w, given only a plot of w.
Exercises: Given superposed plots of
u, v, and the projection of u onto v, which
is which? Find and plot u and v such that the projection of
u onto v is w, given only a plot of w.
Exercises: Choose and plot a change of variables given only a plot of
the ``bad'' domain of integration. Choose and plot a change of
variables which transforms one given region into another given region.
Links
...to the web pages of other Workshop participants:
Paul Vaz
Katie
Kolossa
John Jones
Tempe, Arizona
U.S.A.
85287-1804