# End-of-Semester Reflections

0) All of my responses have been submitted.

1) I feel quite good about my progress this semester in my mathematical thinking. I was hoping that this course would give me a good amount of time (and inspiration) to contemplate the mathematics that I have studied while at Brown, and I am happy to report that it has done just that. Each week I was reminded of some topic in math and this course tied them all together by showing how they all were connected through geometry. Additionally, I cannot say enough about how my thoughts on dimensions have matured--I really understand the wide range of things we are talking about when we discuss dimensions now; when this semester began I felt I had only a tenuous grasp even on that. The time spent throughout the semester on imagining what four-dimensional objects might 'look' like has also gradually taken effect--I recall vividly my first drawing of a hypercube and how I not only could not represent it on paper, but also did not really understand how it should appear. Today, after months of staring at a couple of models on my desk, it is hard to imagine not knowing the hypercube. Similarly, the contrast between seeing the film of the rotating hypercube that first time and the second time is amazing. (I would really enjoy seeing that a million more times.) My visualizations of rotations in four-dimensional space has come a long way. This course has done wonders for my personal synthesis of many areas of mathematics. It has particularly given me some perspective on the work we did last semester in Differential Geometry, since all of these four dimensional things are so much more familiar. I am psyched that I can now proceed with the rest of my life having begun thinking more seriously about higher dimensions--my daydreams are much improved.

2) One of the great strengths of this course is the time spent on the ideas presented. In most college courses, so much material must be covered that there is no time to discuss any one thing more than once or twice. This is not usually sufficient to put the idea into long-term storage. In Math 8, however, every idea is presented, discussed and reviewed from a variety of approaches--mathematical, literary, artistic, etc., resulting in a much firmer grasp of the material. It is such a conceptual class that this is definitely necessary. This difference is a matter of quality versus quantity, a situation that always calls for some sort of compromise--one that was resolved in the right direction in this course.

3) The Paperless Course.Advantages: It was interesting to have access to everyone's thoughts on the chapters in our book. I think that this benefits everyone for at least two reasons. First of all, you can read others' ideas and get different perspectives from people who are also just being introduced to the material for possibly the first time. Also, it places a gentle pressure on students to write somewhat more carefully. It seems like just having to hand in papers to the professor should do that effectively, but sometimes peer exposure can have more pull. Since this class was rather experimental at first, there was not as much direct correspondence between students as there could have been, but I think that this would be a great benefit once everyone was comfortable with the workings of the computer. Students have different levels of understanding of some of the material, and having other students explain things and comment can be a very useful learning tool.

Another advantage is that the end result looks really cool. I think our web page is very aesthetically pleasing with all of those things to click on and read, especially now that the final projects are on line, too. It makes the whole class seem much more complete to have not only one's own work completed in a notebook somewhere, but to be able to see that a whole classful of students progressed through the semester. This is a nice element of closure.

Using the computer to do all of my work has helped me feel more comfortable with the computer. I know a lot more about how to use it--'html', 'links' and 'scanning' were all completely foreign concepts to me before--this is clearly a valuable intellectual gain in today's computer-reliant world.

Disadvantages: The coursework required extra motivation to get to the CIT (for those of us computerless students). I believe that I may have done more work if I could have done it while sitting in my own room--I tended to schedule trips to the CIT when I had large blocks of time, which is not as often as I would normally work on a class. Working in a paperless course forces one to use the computer, even if one is not sure of how to do that. This could be seen as a disadvantage to some, but I think that the benefits of this outweigh the detriments, for even the unwilling student becomes more competent at using the computer. (See Advantages.)

This approach would be difficult to implement in a typical math class because there are so many equations and symbols that must be used in assignments. It could be modified to be a useful tool, however, for the discussion rooms and student correspondence would provide another community environment (outside of the classroom) in which students could ask questions about the course material or even go off onto their own tangents that would take up too much time in a lecture. In non-math classes, this kind of set up would probably work very well for all of the same reasons that Math 8 worked quite well. I am not sure as a professor, however, how much time I would like to spend reading papers on the computer screen--perhaps Professor Banchoff will give us his feedback on that.

4) I love the computer generated images and films that were used in this course. I think they make geometry so much more approachable and exciting--they will continue to work their magic, I am sure. I do think the interactive stuff is even more wonderful (like the mouse-controlled rotations we did in MA106), and more of that in Math 8 would also make concepts even more accessible.

5) I did not mind the public nature of the weekly assignments and the 'response from Prof. B.' feature. Perhaps in a more competitive environment this would be something that I would prefer not to suffer through--it is difficult to say since I felt no such pressure in Math 8. I did like being able to see what Prof. B. thought of others' responses, because they would often touch on topics that I would leave out and I would therefore benefit from Prof. B.'s insight on even more subjects. It is also nice to know that any student could easily be accessed by e-mail regarding their work, although I cannot claim to have responded to others more than a few times. That is another thing I think I would have done more of if I had been casually reading responses in my own room rather than on a time-limited basis in the CIT.

6) I contributed the history section as well as the section on parallels to our group project. Keith actually gives me too much credit when he gives me co-authorship of his section, for I really was only conversationally involved. (Although discussing these concepts with Keith was probably the most mathematically productive time that I spent on the project, so I would not want to belittle that portion of time spent.)

7) I enjoyed the group aspect of the final project. It gave me a good opportunity to talk more in depth with some people in the class, which is nearly always a productive way to learn and to develop ideas. By the end of the work I felt that the project was truly a team effort with a nicely interwoven feel to it. My personal effort was perhaps increased due to the responsibility that comes from working as part of a team. It does seem like the group thing works to students' advantage because it would be impossible for one person to cover so many diverse areas in one project, and this way everyone focuses on one area, putting a lot of effort into it, and still benefits from the information and investigation of all members of the group (and of the other groups as well--another pro for the paperless nature of the course is that the presentation of the project is only an introduction, for students can then actually read other groups' work). These type of projects seem like an appropriate end note to this modern community of Math 8.

One final question: How long will all of this be on-line, anyway?