Office hours were a special problem with this class. Most of the students lived on the Douglass/Cook campus and took many of their courses on Busch. My Douglass office hours listed in the syllabus were not very accessible to many students. Therefore I decided to have "office hours" Thursday evenings at Douglass. This experiment was successful, since I usually had a half-dozen or more students asking me questions then. I also used these hours to conduct review sessions for exams. Attendance was especially high then.

Since students had classes on several campuses and I taught on two campuses (with my permanent office on a third!), the students and I used e-mail extensively. I probably received and responded to 15 to 20 e-mail messages from students in this course during a typical week. At the beginning of the course I collected, duplicated, and then gave out a list of student names, e-mail addresses, and local 'phone numbers in order to help mutual student communication.

Here's the review sheet for the second exam and my second exam. My students were told to ignore problem 13 on the review sheet since I had not discussed change of variables in double integrals. My second exam consisted entirely of verbatim textbook problems, previously assigned as homework, which had been discussed during the recitation/workshops. No special answer sheet was prepared or requested!

The final exam review problems were followed by my final exam. My students were told to ignore problems 19c and 20 on this review sheet since I hadn't discussed Stokes' Theorem in class. Please note that each lecturer prepared separate in-class exams and final exams since there was no uniform final exam time for the whole course.

The mean and median for the first exam were 61.67 and 61, respectively. For the second exam, these scores were 79.92 and 81, and for the final, they were 116.74 and 110.

I gave only one quiz in workshop/recitation, presented here with answers. Since I collected two textbook homework problems almost every week, I didn't think there was much point using recitation/workshop time for quizzes testing essentially the same material in the same manner.

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.

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

A few other minor handouts (e.g., an intricate Lagrange multiplier problem) were prepared and discussed in class.

The first

The second

The third

The last

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 .

Back to the Math 251:19-20 home page Back to Stephen Greenfield's home page