Thank you for your reflections and for the excellent contributions you made at different places in the course. I particularly liked the discussion of fold-outs and slices and duality that came together in your final presentation, incorporating some of your best work from the earlier parts of the semester. I also found your response to chapter 9 very captivating since I recall going through a number of those same thoughts myself back when I was in college. At one stage I was strongly tempted to switch over into English, and I mean that literally in that one of the best professors at the institution was trying to make me an offer he didn't think I could possibly refuse--I could graduate in three years and have a guaranteed place in Stanford's Ph. D. creative writing program. I thanked him but said, "No thanks". I was not all that comfortable with the subjectivity of his subject, and every time I got a criticism on a literature or philosophy paper, I was ready to fight for my own interpretation no matter what the instructor thought. It was hard for me to be convinced that I was wrong, in humanities, that is. In mathematics, I was all too conscious of was it was like to think I had a right argument and then seeing it destroyed by a counterexample. Gradually I found that I could check my own arguments in mathematics and have confidence that they would stand up to a rigorous test, but I never felt that way about English.
That isn't precisely the question you were wrestling with in the Week 12 response, and in fact I think that Abbott would be a good person to reread in that context. In "Flatland" the rationalist polygons were only comfortable with concepts that were quantifiable and measurable--they relegated such unwieldy concepts as "love" and "loyalty" to the women, with their alternate language with its different notions of precision. But Abbott's point is that we have to deal with that tension and not try to separate it away. Mathematics is pretty wonderful, I'll be the first to agree, but it isn't wonderful enough to shut out all the rest of the world. I do like the fact that Math 8 is a place where such ideas can come up.
Back to the Reflections comments:
I'm not sure how some of this internet technology would have worked in Math 35. Some homeworks are just too symbolically complicated to be suitable for web treatment, at least with the present condition of html typography. That might change. The other limitation is in the graphic communication, although you more than anyone else were able to overcome that with your scanning efforts. In any case, there might be some place for a webpage in calculus when we are dealing with those "thought problems", or with some of the group exercises that Dr. Reynolds suggests. I think we will try some of that in Math 9 next fall, and certainly I intend to do some mixture of paperless and paper assignments in Math 141. (What courses are you taking next semester anyway, since I am going to be your sophomore advisor?)
I agree that it will be good when we can incorporate some interactive graphical software like Fnord or Java, and I am hoping that once we get some facility with Java we will be able to duplicate much of the Fnord functionality on platforms less sophisticated than the UNIX workstations. I look forward to a standardization of the html software too. It will be nice for example when it has a spell checker available, even though that would not help with words like "Schlegel" that you will have to look up in another reference. I found myself making spelling corrections in the responses of several people in the class (two in your room) so I wouldn't be distracted by errors when I was trying to respond to the ideas. I do put the ideas at a higher priority, but in the days when I used to write on people's papers with red pencil, I succumbed to early training as a copy editor on the high school newspaper and corrected as I went along. (Maybe you experienced some of that in Math 35?) I don't know exactly how to deal with that in the electronic response forum--just one more question raised by our experiment.
Enough for now--and by the way: Course Grade, Satisfactory.
Comments on the Polyobjects Group
Your investigations were particularly successful in giving access to new tools for interacting with familiar geometric objects in three- and four-dimensional space. The Java applets will doubtless add quite a bit to future versions of this course, especially when they are extended to deal with filled-in objects in a reasonable amount of time. There should be algorithms for slicing general objects from arbitrary directions, of course, and some demonstrations showing how to move from an object to its dual. The ideas of truncating polyhedra or polytopes to produce semi-regular objects is also ripe for interactive illustrations. All in all your work has advanced the level of geometric interaction considerably, not just in the final project but throughout the semester, and it was effective to see how some of your earlier work was incorporated in your presentation.
The links to other sites are also very good to have, and it will be interesting in the future to explore the relationship between VRML, Java applets, and highly rendered individual objects. It would also be useful to link to some of the other projects, in particular those that deal with the axiomatic approach or the literary and artistic topics that use polytopes, duality, et cetera.
There is so much more to be done, taking off from the platform you have set up here. Good work in getting this effort off to such a strong start.