March 2004

I had teams work on a visualization/graphing exercise today in class. The handout can be found here. Aside from their project, this is the hardest exercise I've given them to date. The subject is a curve called Viviani's curve. This is not usually discussed in Calculus III. Certainly, it's beyond the level of most Calc III textbooks. I was very pleasantly suprised by both of my classes. At least one team in each class was able to parameterize and sketch Vivian's curve in about 40-45 minutes. I allowed student "ambassadors" to visit and collaborate with other teams, so this knowledge diffused fairly rapidly through the class. Most teams were on the verge of getting the answer at that point, and just needed a little helping hand. I liked having that help come from the students rather than from me.

It was interesting watching and listening to the students during today's exercise. Yesterday in class, I had carefully explained how to find the curve defined by the intersection of a sphere and cylinder whose origins coincide. Yet, the students did not seem to relate that discussion to today's exercise. Once they figured it out for themselves, today, I heard some comments like "Oh, that's kind of like what we did in class yesterday."

I still have one problem group. This is a team of four students in my 9AM section. They've positioned themselves in the back of the room which in and of itself is not a bad thing, but in this instance I think they'd be better off centrally located. When we have discussions the four of them sit there working on their own, or staring into space. As I visit them and try and prompt a discussion, it just doesn't stick. My sense is that one of the students thinks he knows the material already, two of the students are weak, and the fourth I can't read. I don't know how to fix this group dynamic! The did ok on their project, but not great. Ideas?

I'm still working on the M&M project. I want to hand this out right after spring break. I'm struggling to write dialog!

Mini-Exercises
March 2, 2004

Today we had half lecture, half group work. The group exercise involved finding the quadric surface that looked like a nuclear power plant cooling tower, finding the cross sections of this surface, and using Maple to sketch an example surface that looking as much like a cooling tower as possible. Walking around listening to the groups I realized a little bit about the value of this exercise. In lecture, we had discussed other quadric surfaces. My guess is that few of the students in the class could tell you how the shape of an ellipsoid varies as you vary parameters in its equation. On the other hand, after today, I bet most could answer the question for a hyperboloid of one sheet. In an ideal world, I'd like each student to have explored each quadric surface on their own. I'd like them to have plotted them using Maple, played with parameter values, computed traces, etc. I imagine that is an unrealistic expectation? I think it is an improvement having them as a team explore at least one quadric surface. BTW - today's mini-exercise could easily be turned into a longer group project.

Tommorrow I'm planning a full class period group exercise. Today, there was one student missing from each section. (That's pretty good attendance!) I let them know today that tommorrow would be a graded group exercise and that they might want to contact their errant team members. Tommorrow will be a bit of an experiment. I'm asking them to examine the space curve defined by the intersection of a sphere and cylinder. This is a curve called Viviani's curve. It's non-trivial to construct and not usually discussed in Calc III. I want to test their limits a little bit. Also, I'm asking them to decided whether or not they think the curve is "smooth" and "curvy." I want to see if they can anticipate the notions of differentiability and curvature.

Returning Project Grades
March 1, 2004

The mood in class today was subdued. I began by handing back grades for each team on project #1. An example of the grade sheet I handed back can be found here. I also gave each student a slip of paper with the average rating they received from their team. On this slip of paper I also included distilled comments that I thought might be useful to the student. As you might expect there was a strong correlation between the teams who ranked each other as productive and useful and the grades on the first project. I think that some students were just waking up to the fact that the team part of the class is serious. The grade distribution on the project ranged from a 72 to a 92. I think some students were surprised to see grades that low. I think I was hard, but fair in my grading. The grading was not very subjective, I had landmarks in mind that easily let me score the projects. While the feeling in class was subdued, student participation during todays lecture was at a high. (With the exception of our talk about presentations last week.) Also, we covered a topic related to their first project. I think this was good. They had a good idea of what I was talking about from their own work and I think they gained some new insight into the topic.

I've been thinking about PBL in another context. This June I'll be serving as a mentor at a mathematics graduate student modeling workshop. A description of the program can be found here. The idea is very very problem based learning oriented. Each mentor presents a real-world industrial problem to the group. The mentor then helps a team of 4-6 graduate students work on constructing and analyzing a mathematical model of the problem. At the end, the team gives a presentation and prepares a report. The camp is only four days long, so it is quite intense. When I was a graduate student, I attended a similar camp. It was a great experience. I enjoyed meeting students from around the country (many of whom are now faculty around the country). It was also a great confidence builder. I got to see that I was just as good as students elsewhere. On the other hand, in thinking about whether or not I learned any mathematical modeling skills, or any new mathematical content, an honest answer would be "very little." Our mentor (who I liked very much), gave us a difficult problem and then led us through it. I did learn a new technique for approximating solutions to a particular type of equation. (That did come in handy later.) But, again, I don't know that I learned anything about mathematical modeling. Probably, because it was done for us, rather than us doing it... So, the question is, what should I do as a mentor this time around? Any suggestions?

February 2004 « » April 2004