Index to course material for Math 251:19-20, fall, 1996 (in GIF)

This material is also available in Postscript format.

Starting with the syllabus

The syllabus and textbook problem assignments presented here are versions of the common syllabus suggested by the course coordinator, Gene Speer. This syllabus was too fast. I didn't cover change of variables in multiple integrals (13.9), parametric surfaces (14.6), or Stokes' Theorem (14.8). Late in the course I talked about vector fields as fluid flows in some detail. I explained divergence as the "infinitesimal" source/sink rate of a flow, aiding in verification of the Divergence Theorem, and then showed how the Divergence Theorem could be used to get the Diffusion/Heat equation with the standard mild assumptions on heat flow. This material is not in the textbook. I decided this was more useful than rushing through vector calculus with barely enough time to state results and compute examples without motivation.
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.

Exams and Review Sheets

Review sheets for the exams were prepared for all sections of Math 251 by the course coordinator, Gene Speer. The review sheet for the first exam included a problem on gradients (#17) which was not covered on my first exam. It was covered on the next exam. I also prepared solutions with some comments to the first exam for students.
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.


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.


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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