Final Reflections

David Akers

0) Have you submitted responses for weeks 5 through 13? (This is the minimum requirement for satisfactory completion of the course. Folders should be complete by Sunday May 5 at midnight.)

I have, and I believe they are all there and intact.

1) How has your view of yourself in relationship to mathematics changed over the course of the semester?

I think that I've really had the chance to explore a lot of areas that have interested me before, but that I hadn't found the time to explore prior to this course. (In particular, I am referring to the study of higher-dimensional geometry.) Before I took this course, geometry was always my "weak" subject.. something that I just didn't "get" for one reason or another. Reading the textbook helped make it something more understandable and familiar. Perhaps the main difference in my attitude is this: Before, I saw geometry as something to wonder at, but now I think of it as something that can also be analyzed by mathematical techniques. For example, I really liked the use of principles of combinatorics to analyze the slicing of the hypercube. The discussions of intersecting curves (and the related topic of topology) gave me new ways to look at things.

I guess what I'm saying is that the class helped me to make a lot of connections between different "topics" in mathematics. I've found that the more I learn about math, the less it seems a collection of disjoint principles, but rather more of a unified theory.. When I see a particular example (for example, a hypercube), I now have many different ways to look at it, rather than the one I had already known.

2) For you, what are the most positive and the most negative aspects of the course? Would you suggest any major changes in structure or emphasis?

The most positive for me was the final project. It was fun and exciting, both on an individual level and on a group level. I loved creating all those applets.. and learning JAVA in the process. But I also liked working in my group, and I think that the end project really was more than the sum of its parts. I could never have created something so complex on my own.. But I suppose this is getting into my answers for #7, so I will stop here. The presentations at the end were interesting, and the web pages all looked pretty well constructed and thought out.

I guess the most negative aspect of the course was the use of the www server for weekly assignments. I think there were probably only about ten people in the class who actually took the responses seriously, and put some real effort into them. Some wrote up meaningless little paragraphs which resembled "book-reports," as one student said. Others never quite figured out how to get their responses onto the web server, despite our offers of assistance. There were a lot of people who were only "sort of" involved in the course -- who showed up (or didn't), but didn't put any more effort into it than the bare minimum, if that. It was a little frustrating, because I thought that the class could have been a lot more interesting with only a little more participation.

3) Comment at length on the concept of the paperless course. What are the advantages or disadvantages of this approach? In what ways could such an approach work in other courses?

The main advantage of a paperless course, it seems to me, is that students get to read each others responses. This is still possible within a paper-course, but it is not nearly so easy. Being able to see the responses of others on the web was a real treat, despite the fact that the volume of responses was limited. (see response to #2).

Then again, the paperless course does present its difficulties, primarily technical ones. It is easy to hand in a sheet of paper for an assignment, and every student knows how to do this. But placing a document on the world-wide-web is a considerably more complicated task. It took about three or four weeks just for most to adjust to this, and some never did manage to adjust. I think that a lot of people who would normally have contributed greatly to the class might have been scared away during the first week.

Perhaps a more serious problem is that I'm not sure people really read each other's work. I know that I always tried to read everyone's papers, but I don't think that there was enough interactivity. An effort was made at one point to get everyone to respond to three people's comments, but this seemed to fall through. Somehow there needed to be an additional incentive to read the work of others...

4) Comment on your experiences with the technology used in the course. What can be done to make things easier in the future?

I think that the whole system would have worked, except that it took too long for things to develop in the beginning. There was a lot of confusion about the direction the server/web page would take, how it would be set up, how students would be able to enter their responses, etc. So I think it would have made more sense to have prepared the technological side of the course beforehand. In fact, I think that there should definitely have been a "technology TA" who was in charge of keeping things going. This way the entire task wouldn't get stuck with students taking the course, who often have conflicts with other things. At times I found that a lot of my time was being taken up with helping people.. While I didn't mind this, it sometimes got to the point where it was interfering with my other work, and this became somewhat of a problem.

This technology TA would then have time to devote to information sessions, handouts, helping individuals, etc. I think the course would have run a lot smoother in this case.

5) Describe your experience with the weekly assignments and the "response from Prof. B." feature. Comment on the public nature of these interchanges, and the possibility of linkings among student work and communication with the other class members. To what extent did you read the submissions of other students (and/or the professor's responses)?

Generally, I liked the weekly assignments. The "response from Prof. B." feature was nice -- it was really great to know that any question that I could possibly have about a chapter would be answered. When writing my own responses, I don't think I ever really thought about the public nature of the interchanges. I think I treated it as any other assignment, and didn't really change my responses based on the fact that others would be reading them. It was nice to be able to read the professor's responses to all of the papers online. It would be even better to be able to somehow link in the responses of other students.

As I think I mentioned before, I always made an effort to read everyone's responses, at least cursorily. After some time, I'll admit that I focused on reading the responses of certain people whose writing I found more interesting from week to week. I sometimes found myself skimming the student's writing and going directly to Prof. B's comments, which usually focused in on particularly interesting parts of the response. If I was still curious, I would then go back and read the student's paper again.

6) Describe in some detail your activities as part of your final project team.

My main task in the group was to create all of the JAVA applets for the page. I took Sun Microsystems' source code for a three-dimensional animation applet, and modified it to include slicing. I created a duality applet which handled general cases for finding and projecting the dual images of regular polyhedra. I also made a hypercube applet, which involved extending the mathematics of projection up a dimension.

As part of the group, I also attended several organizing meetings and brainstorming sessions which helped decide the layout of the page. During the last week, I was in charge of putting the whole thing together, assembling all of the images, pictures, and text which the other members of my group had created. I worked with Jeremy to design the final layout of the page, including an internally linked table of contents. I helped to edit a lot of the text, looking for syntactical errors and ways to improve the writing style.

At times it felt like I was being dragged along by the group.. and at other times, when others were tired or sick, I felt like I was the one dragging it along. It varied from meeting to meeting, and from week to week.

7) In the old days, the final project was mostly an individual effort, on the order of a ten-page paper. How would you characterize the experience of working on a team, and how did that affect your effort in the final project?

I think I've already mentioned this, but I think that working in a group was a great idea. Our final result was much more intricate than it could possibly have been if we worked individually. When I found myself looking for a good way to project slices of hypercubes, I was able to talk to another student in my group.. and together we worked all of the math out much faster than I would have independently. To a certain extent, I think that the "paperless" class REQUIRES group final projects. There were just enough people in the class who knew how to write good HTML code to split up among the groups. I don't think it would have been possible to have required everyone to create their own web page for their final project.

The whole idea behind working in groups is that the individuals within the groups will complement each other in their talents. From my own perspective, this worked pretty well.

Prof. Banchoff's response.

David Akers