Workshops
Each student in the course saw me for three 80 minute periods each week. Twice a week I was "the lecturer" and one day a week, the group of students divided in half and met me as "the recitation instructor" or "workshop facilitator". One plus of this arrangement was there were no coordination or communication problems between "the lecturer" and "the recitation instructor". One defect was that the students were not presented with a contrasting viewpoint of the subject - instructors teach differently and people learn differently.The instructional staff of Math 251 wanted the workshop/recitation periods to consist of: 1) a brief review of the material presented during the preceding two lectures, followed by going over that week's textbook problems (the standard recitation); then 2) formation of small discussion groups and followed by work on non-routine problems, the "workshop problems". (Some lecturers also used this period to give short quizzes - instead I asked for several textbook problems to be handed in.) By the third or fourth week it was clear to me that our initial model could not be followed. The material was difficult. Going over textbook problems took more time, and dominated the period as the semester progressed. Below are the workshops together with some comments which I actually handed out. Also, three workshop meetings were devoted to reviewing for exams (using the review material discussed above). Some of the workshop problems given here were devised by Gene Speer and others were my own, written either for this course or for Math 291 several years ago.
Most of these students had experienced the
"new" Math 151-2 and were familiar with workshop problems and the
format desired for written solutions. For the others I gave out a sheet describing the
style desired.
Workshop 1
Problems about adding vectors. I asked students to hand in problem 2. I
didn't like this problem because it was not representative of the type of
work we would be doing in the rest of the course.
Workshop 2
Problems about lines and planes. I asked students to hand in problem
2.
Workshop 3
Problems about tangent lines to curves. I divided the workshop into two
groups alphabetically and asked those students in the first half of the
alphabet to work on problem 1 and those in the second half to work on
problem 2. Rewards were given to the group solving its problem
(correctly!) and presenting its solution on the board first. I did not
ask that any problem solutions be handed in.
Workshop 4
A qualitative problem about curvature and another problem asking for some
neat surface sketches. I asked students to hand in problem 1, about
curvature. Here and on exams I wondered what the "fair" and correct way
was to ask students about curvature. The computations are elaborate, and I
now do them only rarely without electronic help (Maple or
something similar). Yet knowledge of curvature and other calculus-derived
tools to analyze the geometry of curves and surfaces has become
increasingly important in such areas as protein structure and material
science, and has been quite important in computer graphics. My current
"solution" to the question of what to ask about curvature is here and on
the first exam.
Workshop 5
A problem using the equality of mixed partial derivatives and a problem
about linear approximation. I didn't ask students to hand in
anything.
Workshop 6 A mistake by Euler is commemorated, and another problem
discusses the best fitting straight line to the exponential function on the
interval [0,1]. I asked students (most of whom were majoring in
experimental sciences) to hand in the problem on least square fit. I
indicated that Maple could make the computations (not the
explanations!) almost trivial. Most students did, in fact, use Maple
to help with this problem. By this time in the course, though, the
classical "workshop" pedagogy (break up into small groups, discuss the
problems, etc.) had almost collapsed through neglect. There just wasn't
time: review of the ordinary textbook homework problems was taking up
almost all of the period.
Workshop 7 A cute problem on Lagrange multipliers. I didn't ask that
it be handed in: we were swamped trying to set up and work out "routine"
Lagrange multiplier problems. I don't think that I did any
L.m. problem with more than one constraint - we found that the usual
problems took more than enough time.
Workshop 8
I asked students to hand in problem 2 on centers of mass. Problem 1
was about the syntax of double integrals. The pumpkin icons were
caused by the date of the workshop (Hallowe'en!).
Workshop 9
One problem is about the syntax of triple integrals, and another is
about bats and slugs. There was no time to do either during the
recitation/workshop, and the bat/slug problem was discussed during the
first half of the next day's lecture. The textbook problems involving
vector calculus (chapter 14) took a great deal of time to discuss
satisfactorily.
Thus "only" five writeups of workshop problems were requested of students during this semester. Students usually handed in two textbook problems each week to be graded, and handed in Maple labs . They had enough to do.
Notes and other material
On Tuesday, October 8, I gave a lecture on the important topic of
gradient. I didn't do a good job, and took advantage of e-mail to broadcast
a "lecture": I created a file
describing what I should have said and sent it to every student in the
class. I got some nice feedback, but perhaps the most embarrassing comments
were those written by several students who remarked that they saw little
difference between the quality of the lecture I thought I messed up and the
ones I usually gave! The actual e-mail message is shown, demonstrating the
difficulties of writing math in ascii.
I prepared and gave out some problems on the
several variable chain rule on October 4. I devoted most of one lecture
to working through these problems with students - I first let them try the
problems and then did the problems on the board. It seemed to be an
effective and amusing way of dealing with a complicated topic.
Late in the course (November 8) I gave a lecture to show a complicated
classical application of multivariable calculus. I verified Newton's result
that, if gravitation is given by an inverse square law, then (from the
point of view of an external observer) the mass of a homogeneous sphere can
be considered to lie at the center of the sphere. I prepared notes on this material. The principal
technical computation in these notes is an intricate integration using
clever choices to integrate by parts. I realized later that Maple
could easily do this integral, and that I need never compute the guts of
this problem in front of a class again!
A few other minor handouts (e.g., an intricate Lagrange multiplier
problem) were prepared and discussed in class.
Maple
A number of Maple labs were given in the course. The handouts were
the same in all lectures. They stemmed from material written by Rick Falk
and were rewritten by Gene Speer, the course coordinator. We first handed
out some general information
about Maple together with a
preliminary lab which students would hand in to be critiqued without
having their grades recorded: a method to get students familiarized with
Maple and with how to create a "lab report" or worksheet. We had
students work with some of Maple's simple calculus
instructions.
The first Maple lab asked
students to compute with vectors, to do calculus manipulations of a vector
function of a scalar variable, and, finally, to compute the Frenet frame
and curvature and torsion of some curves. It ended with a computation of the
curvature of certain ellipses. This lab was probably too long for the first
"real" assignment.
The second Maple lab asked
students to sketch quadric surfaces and certain curves (primarily slices)
associated with these surfaces using Maple. This lab was intended
to replace consideration in class of a section of the text (11.6) and was
also to replace textbook homework in that section. Although the lab seemed
to be relatively successful, I am not convinced that it totally replaced
lecturing about these concepts.
The third Maple lab asked
students to investigate critical points and max and min in several (two)
variables both graphically and symbolically, and also mixed in some of the
numerical techniques available in Maple . It was a good lab, but by
that time I thought that the students were spending too much time on their
Maple labs and were neglecting more traditional aspects of the
course. I asked that my students do only about half of this lab.
The last Maple lab was devoted
to setting up a slightly complicated triple integral (a moment of inertia
of a solid) in several ways. Again, I asked that my students do only a
portion of this lab.
Many of the students in these sections did all that was requested of them in the Maple labs and almost all of the students handed in work on all of the labs. I believe that the vast majority of the work handed in represented the students' own work. I told them repeatedly that learning to using programs such as Maple was "the wave of the future" and extremely applicable to every technical field. I think they realized they'd be cheating themselves if they didn't learn how to use the program. I also urged them to apply Maple in other relevant situations in the course (such as workshops and textbook homework problems). Solutions to the Maple labs, written by the course coordinator, Gene Speer, are available .
Ending with my comments on the course
To me, multivariable calculus is the most enjoyable part of the
standard calculus sequence to teach. The interaction of geometry and
algebraic/analytic computation is lovely. We changed the pedagogy in
this course a great deal. I don't think we "prioritized" the various
parts of the course adequately for the students. Their energies were
too diffused. I should have attempted to cover fewer topics "better"
at the beginning. During the semester I wrote two memos addressed to
others interested in the course. I have included the memos, slightly
edited. The first memo was written
about six weeks after the semester began, after the first exam in the
course. The second memo was
prepared late in the semester for a meeting of the instructional staff
of Math 251 held after the end of classes, and had some elements of a
longer perspective: how has the course changed over time, and what are
we doing with it now? The language is sometimes intended to be
humorous and not literal but the intent is serious.
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